**Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference**

Valeriy I. Sbitnev

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53471

### **1. Introduction**

One of the crises of contemporary mathematics belongs in part to the subject of the infinite and infinitesimals [1]. It originates from the barest necessity to develop a rigorous language for description of observable physical phenomena. It was a time when foundations of inte‐ gral and differential calculi were developed. A theoretical foundation for facilitation of un‐ derstanding of classical mechanics is provided by the concepts of absolute time and space originally formulated by Sir Isaac Newton [2]. Space is distinct from body. And time passes uniformly without regard to whether anything happens in the world. For this reason New‐ ton spoke of absolute space and absolute time as of a "container" for all possible objects and events. The space-time container is absolutely empty until prescribed metric and a reference frame are introduced. Infinitesimals are main tools of differential calculus [3, 4] within chos‐ en reference frames.

Infinitesimal increment being a cornerstone of theoretical physics has one receptee default belief, that increment *δV* tending to zero contains a lot of events to be under consideration. Probability of detection of a particle within this infinitesimal volume *ρ*(*r* <sup>→</sup> )*δV* is adopted as a smooth differentiable function with respect to its argument. From experience we know that for reproducing the probability one needs to accumulate enormous amount of events occur‐ ring within this volume. On the other hand we know, that as *δV* tends to zero we lose infor‐ mation about amount of the events. What is more, the information becomes uncertain. It means the infinitesimal increment being applied in physics faces with a conflict of depth of understanding physical processes on such minuscule scales. This trouble is avoided in quan‐ tum mechanics by proclamation that infinitesimal increments are operators, whereas ob‐ servables are averaged on an ensemble.

© 2013 Sbitnev; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sbitnev; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In light of classical views Newton maintained the theory that light was made up of tiny par‐ ticles, corpuscles. They spread through space in accordance with law of classical mechanics. Christian Huygens (a contemporary of Newton), believed that light was made up of waves vibrating up and down perpendicularly to direction of the light propagation. It comes into contradiction with Newtonian idea about corpuscular nature of light. Huygens was a pro‐ claimer of wave mechanics as opposite to classical mechanics [5].

Ones suppose that random fluctuations of electron-positron pairs take place always. What is more, these fluctuations are induced by other pairs and by particles traveling through this random conglomerate. Edward Nelson has described mathematical models [6, 7] represent‐ ing the above random fluctuations as Brownian motions of particles that are subjected by random impacts from particles populating aether (Nelson's title of a lower environment).

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

loaded by a Wiener term *w* with diffusion coefficient equal to *ћ*/2*m*, where *m* is a mass of the particle and *ћ* is the reduced Planck constant. In this perspective Nelson has considered two Markov processes complementary to each other. One is described by forward-difference op‐ erator in the time, here *b*(*x*(*t*),*t*) is a velocity calculated forward. And other equation is de‐

of transition to two new variables, real and imaginary, finally lead to emergence of the

Nelson' vision that aether fluctuations look as Brownian movements of subparticles with *ћ*/2*m* being the diffusion coefficient of the movements, correlates with Feynman's ideas about quan‐ tum fluctuations of virtual particles in vacuum [8]. The Feynman path integral is akin to the Einstein-Smoluchowski integral equation [9]. The latter computes transition probability densi‐ ty. We shall deal with the modified Feynman path integral loaded by a temperature multi‐ plied by the Boltzmann constant. At that, probability amplitude stays as a fundamental mathematical object at all stages of computations. As an example we shall consider emergence of interference patterns at scattering heavy particles on gratings. The particles are heavy in the sense that they adjoin to both realms, quantum and classical. They are nanoparticles. Such nanoparticles have masses about 100 amu and more, as, for example, fullerene molecules [10] shown in Fig. 1. It is remarkable that there are many experiments with such molecules show‐ ing interference patterns in the near field [11-18]. On the other hand these molecules are so

**Figure 1.** The fullerene molecule C60 consists of 60 carbon atoms. Its radius is about 700 pm. De Broglie wavelength of the molecule, λdB, is about 5 pm at a flight velocity *v* =100 m/s [18]. The molecules are prepared in a thermal emission

gun which has temperature about 1000 K. It means that carbon atoms accomplish thermal fluctuations.

scribed by backward-difference operator in the time with the velocity *b*†

large, that they behave themselves as classical particles at ordinary conditions.

*dx t b x t t dt dw t* ( ) ( ( ), ) ( ) = + (1)

(*x*(*t*),*t*) [6, 7]. The two complementary processes, by means

(*x*(*t*),*t*) calculated

http://dx.doi.org/10.5772/53471

185

The model is viewed as a Markov process

backward. In general *b*(*x*(*t*),*t*) ≠ *b*†

Schrödinger equation.

We abstain from allusion to physical vacuum but expand Huygensian idea to its logical completion. Let us imagine that all Newtonian absolute space is not empty but is populated everywhere densely by Huygensian vibrators. The vibrators are silent at absence of wave propagating through. But as soon as a wave front reaches some surface all vibrators on this surface begin to radiate at a frequency resonant with that of incident wave. From here it fol‐ lows, that in each point of the space there are vibrators with different frequencies ranging from zero frequency up to infinite. All are silent in absence of an external wave perturbance. Thus, the infinitesimal volume *δV* is populated by infinite amount of the vibrators with fre‐ quencies ranging from zero to infinity. They are virtual vibrators facilitating propagation of waves through space.

Let us return to our days. One believes that besides matter and physical fields there is noth‐ ing more in the universe. Elementary particles are only a building material of "eternal and indestructible" substance of the cosmos. However we should avow that all observed matter and physical fields, are not the basis of the world, but they are only a small part of the total quantum reality. Physical vacuum in this picture constitutes a basic part of this reality. In particular, modern conception of the physical vacuum covers Huygens's idea perfectly. All space is fully populated by virtual particle-antiparticle pairs situated on a ground level. Such a particle-antiparticle pair has zero mass, zero charge, and zero magnetic moment. Fa‐ mous Dirac' sea (Dirac postulated that the entire universe is entirely filled by particles with negative energy) is boundless space of electron-positron pairs populated everywhere dense‐ ly - each quantum state possessing a positive energy is accompanied by a corresponding state with negative energy. Electron has positive mass and positron has the same mass but negative; electron has negative charge and positron has the same positive charge; when elec‐ tron and positron dance in pair theirs magnetic moments have opposite orientations, so magnetic moment of the pair is zero.

Let electron and positron of a virtual pair rotate about mass center of this pair. Rotation of the pairs happens on a Bohr orbit. Energy level of the first Bohr orbit, for example, *mv* <sup>2</sup> /2, is about 14 eV. Here *m* is electron mass and *v* is its velocity (on the first Bohr orbit the velocity is about 2.188 106 m/s). Energy of the pair remains zero since positron has the same energy but with negative sign. Quantum fluctuations around this zero energy are as zero oscilla‐ tions of electromagnetic field. Observe that, energy releasing of electron and positron from vacuum occurs at *mc*<sup>2</sup> = 1.022 MeV. So we see that there are about 7.5 104 Bohr orbits lying below this energy. That is, there is a vast scope for occupation of different Bohr orbits by the virtual electron-positron pairs.

A short outline given above is a basis for understanding of interference effects to be descri‐ bed below.

Ones suppose that random fluctuations of electron-positron pairs take place always. What is more, these fluctuations are induced by other pairs and by particles traveling through this random conglomerate. Edward Nelson has described mathematical models [6, 7] represent‐ ing the above random fluctuations as Brownian motions of particles that are subjected by random impacts from particles populating aether (Nelson's title of a lower environment). The model is viewed as a Markov process

In light of classical views Newton maintained the theory that light was made up of tiny par‐ ticles, corpuscles. They spread through space in accordance with law of classical mechanics. Christian Huygens (a contemporary of Newton), believed that light was made up of waves vibrating up and down perpendicularly to direction of the light propagation. It comes into contradiction with Newtonian idea about corpuscular nature of light. Huygens was a pro‐

We abstain from allusion to physical vacuum but expand Huygensian idea to its logical completion. Let us imagine that all Newtonian absolute space is not empty but is populated everywhere densely by Huygensian vibrators. The vibrators are silent at absence of wave propagating through. But as soon as a wave front reaches some surface all vibrators on this surface begin to radiate at a frequency resonant with that of incident wave. From here it fol‐ lows, that in each point of the space there are vibrators with different frequencies ranging from zero frequency up to infinite. All are silent in absence of an external wave perturbance. Thus, the infinitesimal volume *δV* is populated by infinite amount of the vibrators with fre‐ quencies ranging from zero to infinity. They are virtual vibrators facilitating propagation of

Let us return to our days. One believes that besides matter and physical fields there is noth‐ ing more in the universe. Elementary particles are only a building material of "eternal and indestructible" substance of the cosmos. However we should avow that all observed matter and physical fields, are not the basis of the world, but they are only a small part of the total quantum reality. Physical vacuum in this picture constitutes a basic part of this reality. In particular, modern conception of the physical vacuum covers Huygens's idea perfectly. All space is fully populated by virtual particle-antiparticle pairs situated on a ground level. Such a particle-antiparticle pair has zero mass, zero charge, and zero magnetic moment. Fa‐ mous Dirac' sea (Dirac postulated that the entire universe is entirely filled by particles with negative energy) is boundless space of electron-positron pairs populated everywhere dense‐ ly - each quantum state possessing a positive energy is accompanied by a corresponding state with negative energy. Electron has positive mass and positron has the same mass but negative; electron has negative charge and positron has the same positive charge; when elec‐ tron and positron dance in pair theirs magnetic moments have opposite orientations, so

Let electron and positron of a virtual pair rotate about mass center of this pair. Rotation of the pairs happens on a Bohr orbit. Energy level of the first Bohr orbit, for example, *mv* <sup>2</sup>

about 14 eV. Here *m* is electron mass and *v* is its velocity (on the first Bohr orbit the velocity

but with negative sign. Quantum fluctuations around this zero energy are as zero oscilla‐ tions of electromagnetic field. Observe that, energy releasing of electron and positron from

below this energy. That is, there is a vast scope for occupation of different Bohr orbits by the

A short outline given above is a basis for understanding of interference effects to be descri‐

m/s). Energy of the pair remains zero since positron has the same energy

= 1.022 MeV. So we see that there are about 7.5 104 Bohr orbits lying

/2, is

claimer of wave mechanics as opposite to classical mechanics [5].

waves through space.

184 Advances in Quantum Mechanics

magnetic moment of the pair is zero.

is about 2.188 106

bed below.

vacuum occurs at *mc*<sup>2</sup>

virtual electron-positron pairs.

$$d\mathbf{x}(t) = b(\mathbf{x}(t), t)dt + dw(t) \tag{1}$$

loaded by a Wiener term *w* with diffusion coefficient equal to *ћ*/2*m*, where *m* is a mass of the particle and *ћ* is the reduced Planck constant. In this perspective Nelson has considered two Markov processes complementary to each other. One is described by forward-difference op‐ erator in the time, here *b*(*x*(*t*),*t*) is a velocity calculated forward. And other equation is de‐ scribed by backward-difference operator in the time with the velocity *b*† (*x*(*t*),*t*) calculated backward. In general *b*(*x*(*t*),*t*) ≠ *b*† (*x*(*t*),*t*) [6, 7]. The two complementary processes, by means of transition to two new variables, real and imaginary, finally lead to emergence of the Schrödinger equation.

Nelson' vision that aether fluctuations look as Brownian movements of subparticles with *ћ*/2*m* being the diffusion coefficient of the movements, correlates with Feynman's ideas about quan‐ tum fluctuations of virtual particles in vacuum [8]. The Feynman path integral is akin to the Einstein-Smoluchowski integral equation [9]. The latter computes transition probability densi‐ ty. We shall deal with the modified Feynman path integral loaded by a temperature multi‐ plied by the Boltzmann constant. At that, probability amplitude stays as a fundamental mathematical object at all stages of computations. As an example we shall consider emergence of interference patterns at scattering heavy particles on gratings. The particles are heavy in the sense that they adjoin to both realms, quantum and classical. They are nanoparticles. Such nanoparticles have masses about 100 amu and more, as, for example, fullerene molecules [10] shown in Fig. 1. It is remarkable that there are many experiments with such molecules show‐ ing interference patterns in the near field [11-18]. On the other hand these molecules are so large, that they behave themselves as classical particles at ordinary conditions.

**Figure 1.** The fullerene molecule C60 consists of 60 carbon atoms. Its radius is about 700 pm. De Broglie wavelength of the molecule, λdB, is about 5 pm at a flight velocity *v* =100 m/s [18]. The molecules are prepared in a thermal emission gun which has temperature about 1000 K. It means that carbon atoms accomplish thermal fluctuations.

The article consists of five sections. Sec. 2 introduces a general conception of the path inte‐ gral that describes transitions along paths both of classical and quantum particles. Here we fulfill expansions in the Taylor series of terms presented in the path integral. Depending on type of presented parameters we disclose either the Schrödinger equation or diffusion-drift equation containing extra term, osmotic diffusion. In the end of the section we compute passing nanoparticles through *N*-slit grating. Sec. 3 deals with interference patterns from the *N*-slit grating. Specifically, we study blurring of the Talbot carpet (an interference pattern emergent under special conditions imposed on the grating [19, 20]) arising under decoher‐ ence of incident on the grating nanoparticles. In Sec. 4 we find equations for computing the Bohmian trajectories. Also we compute variance of momenta along the trajectories. These computations lead to emergence of the uncertainty conditions. In concluding Sec. 5 we sum‐ marize results. For confirmation of existence of the Bohmian trajectories here we mention in‐ terference experiments with single silicon oil droplet [21].

( ) ( ) ( ) 2 1 <sup>2</sup> <sup>1</sup> *p r r t p r r t t p r r t dV* , ; += +

ò r r r r rr

*R*

<sup>→</sup> 1 through all intermediate positions *q*

*R*

 d

1

*x*

d

 ,; , , ; . dt

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

This equation describes a Markovian process without memory. That is, only previous state bears information for the next state. Integration here is fulfilled over all a working scene R, encompassing finite space volume. Infinitesimal volume *δV* in the integral tends to zero. It should be noted, however, that this volume should contain as many particles as possible for getting a satisfactory statistical pattern. One can see that this claim enters in conflict with the

Next we shall slightly modify approach to this problem. Essential difference from the classi‐ cal probability theory is that instead of the probabilities we shall deal with probability am‐ plitudes. The transition amplitudes can contain also imaginary terms. They bear information about phase shifts accumulated along paths. In that way, a transition from an initial state *q*

1 0 <sup>1</sup> <sup>0</sup> ( , ; ) ( , ; ,) ( , ;) . *x xx*

Here function *ψ*(...) is a probability amplitude. Probability density *p*(...), in turn, is represent‐

<sup>1</sup>. It is called propagator [8, 25, 26]. We suppose that the propagator has the following

d

r r& r r (4)

**i**h (5)

*<sup>x</sup>*;*t* + *δt*, *t*) represents the transition amplitude from an intermediate state *q*

<sup>1</sup> (,) ( , ; , ) exp , *x x*

<sup>B</sup> G= + 2 . *kT t* d

Factor 2 at the first term is conditioned by the fact that the kernel *K* relates to transitions of the probability amplitude *ψ*, not the probability *p*. Observe that a fullerene molecular beam

ì ü ï ï += -í ý ï ï <sup>G</sup> î þ

where denominator Γ under exponent is a complex-valued quantity, i.e., Γ = β+**i***ћ*. The both parameters, β and the reduced Planck constant *ћ*, have dimensionality of energy multiplied by time. From here it follows that β = 2*k*B*Tδt*. Here *k*<sup>B</sup> is Boltzmann constant and *T* is temper‐

*Lq q t Kq q t tt <sup>A</sup>*

 *q q t t Kq q t tt q q t q* += + d y

ò r r rr rr

ed by square of modulo of the probability amplitude, namely, *p*(...)=|*ψ*(...)|2

→

*x* given on a manifold R3

3

(2)

187

http://dx.doi.org/10.5772/53471

→ 0

(see Fig. 2) is

. Integral kernel

*<sup>x</sup>* to a final

→

*D* (3)

dt

assumption *δV* → 0.

to a final state *q*

*K*(*q* → <sup>1</sup>, *q* →

state *q* →

standard form

ature. So, we can write

represented by the following path integral

y

### **2. Generalized path integral**

Let many classical particles occupy a volume *V* and they move with different velocities in different directions. Let us imagine that there is a predominant orientation along which en‐ semble of the particles drifts. As a rule, one chooses a small volume *δV* in order to evaluate such a drift, Fig. 2. Learning of statistical mechanics begins with assumption that the volume should contain many particles.

**Figure 2.** Infinitesimal volume δV contains many particles moving with different velocities having predominant orien‐ tation along blue arrow. The infinitesimal volume δV, as a mental image, is shifted along the same orientation.

The problem is to find transition probabilities that describe transition of the particle ensemble from one statistical state to another. These transient probabilities can be found through solu‐ tion of the integral Einstein-Smoluchowski equation [9]. This equation in mathematical phys‐ ics is known as Chapman-Kolmogorov [22-24] equation.. This equation looks as follows

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference http://dx.doi.org/10.5772/53471 187

$$p\left(\vec{r}\_2, \vec{r}\_1; t + \delta\tau\right) = \int\_B p\left(\vec{r}\_2, \vec{r}; t + \delta\tau, t\right) p\left(\vec{r}, \vec{r}\_1; t\right) dV. \tag{2}$$

This equation describes a Markovian process without memory. That is, only previous state bears information for the next state. Integration here is fulfilled over all a working scene R, encompassing finite space volume. Infinitesimal volume *δV* in the integral tends to zero. It should be noted, however, that this volume should contain as many particles as possible for getting a satisfactory statistical pattern. One can see that this claim enters in conflict with the assumption *δV* → 0.

The article consists of five sections. Sec. 2 introduces a general conception of the path inte‐ gral that describes transitions along paths both of classical and quantum particles. Here we fulfill expansions in the Taylor series of terms presented in the path integral. Depending on type of presented parameters we disclose either the Schrödinger equation or diffusion-drift equation containing extra term, osmotic diffusion. In the end of the section we compute passing nanoparticles through *N*-slit grating. Sec. 3 deals with interference patterns from the *N*-slit grating. Specifically, we study blurring of the Talbot carpet (an interference pattern emergent under special conditions imposed on the grating [19, 20]) arising under decoher‐ ence of incident on the grating nanoparticles. In Sec. 4 we find equations for computing the Bohmian trajectories. Also we compute variance of momenta along the trajectories. These computations lead to emergence of the uncertainty conditions. In concluding Sec. 5 we sum‐ marize results. For confirmation of existence of the Bohmian trajectories here we mention in‐

Let many classical particles occupy a volume *V* and they move with different velocities in different directions. Let us imagine that there is a predominant orientation along which en‐ semble of the particles drifts. As a rule, one chooses a small volume *δV* in order to evaluate such a drift, Fig. 2. Learning of statistical mechanics begins with assumption that the volume

**Figure 2.** Infinitesimal volume δV contains many particles moving with different velocities having predominant orien‐ tation along blue arrow. The infinitesimal volume δV, as a mental image, is shifted along the same orientation.

The problem is to find transition probabilities that describe transition of the particle ensemble from one statistical state to another. These transient probabilities can be found through solu‐ tion of the integral Einstein-Smoluchowski equation [9]. This equation in mathematical phys‐ ics is known as Chapman-Kolmogorov [22-24] equation.. This equation looks as follows

terference experiments with single silicon oil droplet [21].

**2. Generalized path integral**

186 Advances in Quantum Mechanics

should contain many particles.

Next we shall slightly modify approach to this problem. Essential difference from the classi‐ cal probability theory is that instead of the probabilities we shall deal with probability am‐ plitudes. The transition amplitudes can contain also imaginary terms. They bear information about phase shifts accumulated along paths. In that way, a transition from an initial state *q* → 0 to a final state *q* <sup>→</sup> 1 through all intermediate positions *q* → *x* given on a manifold R3 (see Fig. 2) is represented by the following path integral

$$
\mu\left(\vec{q}\_{1\prime}, \vec{q}\_0; t + \delta t\right) = \int\_B K(\vec{q}\_{1\prime}, \vec{q}\_{\times}; t + \delta t, t) \,\mu\left(\vec{q}\_{\times}, \vec{q}\_0; t\right) \mathcal{D}^{\ 3} q\_{\times} \,\tag{3}
$$

Here function *ψ*(...) is a probability amplitude. Probability density *p*(...), in turn, is represent‐ ed by square of modulo of the probability amplitude, namely, *p*(...)=|*ψ*(...)|2 . Integral kernel *K*(*q* → <sup>1</sup>, *q* → *<sup>x</sup>*;*t* + *δt*, *t*) represents the transition amplitude from an intermediate state *q* → *<sup>x</sup>* to a final state *q* → <sup>1</sup>. It is called propagator [8, 25, 26]. We suppose that the propagator has the following standard form

$$K(\vec{q}\_{1'}\vec{q}\_{\chi};t+\delta t,t) = \frac{1}{A} \exp\left\{-\frac{L(\vec{q}\_{\chi'}\dot{\vec{q}}\_{\chi})\delta t}{\Gamma}\right\},\tag{4}$$

where denominator Γ under exponent is a complex-valued quantity, i.e., Γ = β+**i***ћ*. The both parameters, β and the reduced Planck constant *ћ*, have dimensionality of energy multiplied by time. From here it follows that β = 2*k*B*Tδt*. Here *k*<sup>B</sup> is Boltzmann constant and *T* is temper‐ ature. So, we can write

$$
\Gamma = \mathcal{Z}k\_{\rm B}T\delta t + \mathbf{\dot{i}}\hbar.\tag{5}
$$

Factor 2 at the first term is conditioned by the fact that the kernel *K* relates to transitions of the probability amplitude *ψ*, not the probability *p*. Observe that a fullerene molecular beam for interference experiment prepared in a Knudsen cell at T=1070 K [27] spreads further in a vacuum. That is, fullerene molecules keep this temperature. From here it follows, that ther‐ mal fluctuations of carbon atoms from equilibrium occur at that temperature as the fullerene molecules propagate further within the vacuum chamber. One can see that at *T*=1000 K the term 2*k*B*Tδt* may be about *ћ* if *δt* is about 10-14 s.

Next let us imagine that particles pass through a path length one by one. That is, they do not collide with each other along the path length. The particles are complex objects, however. They are nanoparticles. Fullerene molecule, for example, contains 60 carbon atoms, Fig. 1. Conditionally we can think that the atoms are connected with each other by springs simulat‐ ing elastic vibrations. In this view the Lagrangian *L* (*q* → *<sup>x</sup>*, *q* ˙ → *<sup>x</sup>*) can be written in the following form

$$L(\vec{q}\_{\mathbf{x}}, \dot{\vec{q}}\_{\mathbf{x}}) = \sum\_{k=1}^{N} \left[ \frac{m}{2} \frac{(\vec{q}\_{1,k} - \vec{q}\_{\mathbf{x},k})^2}{\delta t^2} - \mathcal{U}(\vec{q}\_{\mathbf{x},k}) \right]. \tag{6}$$

The path integral (3) contains functions depending only on coordinates *q*

different from zero as well.

define a small increment

Here *U* (*q* →

**2.1. Expansion of the path integral**

function written on the left is expanded up to the first term

x

x

d

*<sup>x</sup>*)is sum of all potentials *U* (*q*

lor series up to the second terms of the expansion

→ *<sup>x</sup>*)=*U* (*q* → <sup>1</sup> −*ξ*

yx

The under integral function *ψ*(*q*

The potential energy *U* (*q*

<sup>→</sup> 1), *ξ* → ).

small parameter *ξ*

<sup>→</sup> 1)−(∇*U* (*q*

*U* (*q*

The Lagrangian (7) is rewritten, in such a case, in the following view

 x

d

→ *<sup>x</sup>*, *q* →

1 0 10

 y

y

positions of the center of mass. Whereas the Lagrangian (6) gives description for behavior of each atom constituent the complex molecule. Here we shall suppose that oscillations of all atoms are noncoherent. And consequently they do not give contribution to interference ef‐ fect on output. We believe that these oscillations provide a thermal noise. And next we shall replace this oscillating background by a corresponding thermal term. For this reason, we be‐ lieve that along with the reduced Planck constant *ћ* the parameter 2*k*B*Tδt* in Eq. (5) can be

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

The next step is to expand terms, ingoing into the integral (3), into Taylor series. The wave

y

¶

 x

1, , 1, ,

( ) 2 2

<sup>→</sup> also. Here we restrict themselves by the first two terms of the expansion,

<sup>2</sup> r r r r rr (11)

 y x

<sup>→</sup> ) is subjected to expansion into the Taylor series by the

yx

rr rr r r <sup>r</sup> & *<sup>U</sup>* (10)

 dd

> d

*<sup>x</sup>*,*<sup>k</sup>* ) given in the center of mass. Further we shall deal

<sup>0</sup>;*t*)is subjected to expansion into the Tay‐

 d¶

rr rr (8)

= - Þ =- *qq q* <sup>r</sup> r r *D D* (9)

1 0 1 0 ( , ; ) ( , ;) . *qqt t qqt t <sup>t</sup>*

+» +

As for the terms under the integral, here we preliminarily make some transformations. We

2 2

with the path integral (3) where the kernel *K* contains the Lagrangian given from Eq. (10).

2 2 1 1

<sup>1</sup> ( )( ) (,) ( ) 2 2 *N N N k xk k xk x x N x k k*

 d = = - - =- + - å å

dd

*<sup>m</sup> <sup>m</sup> Lq q <sup>m</sup> <sup>q</sup> t t tN t*

→

<sup>0</sup>;*t*)=*ψ*(*q* → <sup>1</sup> −*ξ* <sup>→</sup> , *q* →

( , ;) ( , ;) , *q qt qqt* - » - Ñ +Ñ ×

Taking into account the expressions (8)-(11) and substituting theirs into Eq. (3) we get

3 3 <sup>1</sup> . *x x*

 dy → <sup>0</sup>, *q* → <sup>1</sup>, *q* →

http://dx.doi.org/10.5772/53471

*<sup>x</sup>*relating to

189

Here *N* is amount of atoms, constituent complex molecule, and *m* is mass of a single atom. By supposing that there are no quantum permutations between atoms we may expand the Lagrangian further

$$\begin{split} \mathcal{L}(\vec{\boldsymbol{q}}\_{\boldsymbol{x}},\dot{\vec{\boldsymbol{q}}}\_{\boldsymbol{x}}) &= \frac{m}{2} \sum\_{k=1}^{N} \frac{(\vec{\boldsymbol{q}}\_{1,k} - \vec{\boldsymbol{q}}\_{1} + \vec{\boldsymbol{q}}\_{1} - \vec{\boldsymbol{q}}\_{x,k} + \vec{\boldsymbol{q}}\_{x} - \vec{\boldsymbol{q}}\_{x})^{2}}{\delta t^{2}} - \sum\_{k=1}^{N} \mathcal{U}(\vec{\boldsymbol{q}}\_{x,k}) \\ &= \frac{m\_{N}}{2} \frac{(\vec{\boldsymbol{q}}\_{1} - \vec{\boldsymbol{q}}\_{x})^{2}}{\delta t^{2}} - m\_{N} \frac{(\vec{\boldsymbol{q}}\_{1} - \vec{\boldsymbol{q}}\_{x})}{\delta t} \frac{1}{N} \sum\_{k=1}^{N} \frac{(\vec{\boldsymbol{\delta}}\_{1,k} - \vec{\boldsymbol{\delta}}\_{x,k})}{\delta t} + \frac{m}{2} \sum\_{k=1}^{N} \frac{(\vec{\boldsymbol{\delta}}\_{1,k} - \vec{\boldsymbol{\delta}}\_{x,k})^{2}}{\delta t^{2}} - \sum\_{k=1}^{N} \mathcal{U}(\vec{\boldsymbol{q}}\_{x,k}) . \end{split} \tag{7}$$

Here we admit that *q* → *<sup>x</sup>* and *q* <sup>→</sup> 1 are coordinates of center of mass of that complex molecule in intermediate and final positions. And *mN* = *Nm* is a mass of the molecule referred to its cen‐ ter of mass. Small deviations *δ* → 1,*<sup>k</sup>* =(*q* → <sup>1</sup> −*q* → 1,*<sup>k</sup>* ) and *δ* → *<sup>x</sup>*,*<sup>k</sup>* =(*q* → *<sup>x</sup>* −*q* → *<sup>x</sup>*,*<sup>k</sup>* ) are due to oscillations of the *k*th atoms with respect to the center of mass.

Let us consider the second row in Eq. (7). First of all we note that the term (*q* → <sup>1</sup> −*q* → *<sup>x</sup>*) / *δt* repre‐ sents a velocity of movement of the center of mass. In such a case the first term is a kinetic energy of the center of mass. The second term represents product of the center mass velocity on an averaged velocity of partial oscillations of atoms constituting this molecule. The aver‐ aged velocity we believe vanishes because of conservation of total momentum. The third term represents a thermal kinetic energy of partial oscillating atoms constituting this molec‐ ular object. This energy is small enough. But it is sufficient to exhibit itself in the Casimir effect. The last term is a total potential energy *U* (*q* → *<sup>x</sup>*)in the point *q* → *x*.

The path integral (3) contains functions depending only on coordinates *q* → <sup>0</sup>, *q* → <sup>1</sup>, *q* → *<sup>x</sup>*relating to positions of the center of mass. Whereas the Lagrangian (6) gives description for behavior of each atom constituent the complex molecule. Here we shall suppose that oscillations of all atoms are noncoherent. And consequently they do not give contribution to interference ef‐ fect on output. We believe that these oscillations provide a thermal noise. And next we shall replace this oscillating background by a corresponding thermal term. For this reason, we be‐ lieve that along with the reduced Planck constant *ћ* the parameter 2*k*B*Tδt* in Eq. (5) can be different from zero as well.

#### **2.1. Expansion of the path integral**

for interference experiment prepared in a Knudsen cell at T=1070 K [27] spreads further in a vacuum. That is, fullerene molecules keep this temperature. From here it follows, that ther‐ mal fluctuations of carbon atoms from equilibrium occur at that temperature as the fullerene molecules propagate further within the vacuum chamber. One can see that at *T*=1000 K the

Next let us imagine that particles pass through a path length one by one. That is, they do not collide with each other along the path length. The particles are complex objects, however. They are nanoparticles. Fullerene molecule, for example, contains 60 carbon atoms, Fig. 1. Conditionally we can think that the atoms are connected with each other by springs simulat‐

> → *<sup>x</sup>*, *q* ˙ →

2

ë û

2 ,

r r r r <sup>r</sup> & (6)

1 11

<sup>→</sup> 1 are coordinates of center of mass of that complex molecule in

= ==

 dd

> d

*N NN*

å åå

*N x k k kk*

1, ,

d*t* é ù - <sup>=</sup> ê ú -

*k xk x x x k*

Here *N* is amount of atoms, constituent complex molecule, and *m* is mass of a single atom. By supposing that there are no quantum permutations between atoms we may expand the

> 2 2 1 1 1, , 1, ,

dd

 d

intermediate and final positions. And *mN* = *Nm* is a mass of the molecule referred to its cen‐

sents a velocity of movement of the center of mass. In such a case the first term is a kinetic energy of the center of mass. The second term represents product of the center mass velocity on an averaged velocity of partial oscillations of atoms constituting this molecule. The aver‐ aged velocity we believe vanishes because of conservation of total momentum. The third term represents a thermal kinetic energy of partial oscillating atoms constituting this molec‐ ular object. This energy is small enough. But it is sufficient to exhibit itself in the Casimir

→

*<sup>x</sup>*)in the point *q*

→ *x*.

1,*<sup>k</sup>* ) and *δ* → *<sup>x</sup>*,*<sup>k</sup>* =(*q* → *<sup>x</sup>* −*q* →

() () <sup>1</sup> ( )( ) ( ). 2 2

rr rr rr rr <sup>r</sup>

2 2 ,

*<sup>m</sup> U q t t tN t*

( )) (,) ( ). <sup>2</sup>

2

2 ,

*Nx x k xk k xk*

Let us consider the second row in Eq. (7). First of all we note that the term (*q*


*<sup>m</sup> q q Lq q U q*

1

=

*k*

*N*

å

*<sup>x</sup>*) can be written in the following

*<sup>x</sup>*,*<sup>k</sup>* ) are due to oscillations of the

→ <sup>1</sup> −*q* → *<sup>x</sup>*) / *δt* repre‐

(7)

term 2*k*B*Tδt* may be about *ћ* if *δt* is about 10-14 s.

form

Lagrangian further

188 Advances in Quantum Mechanics

Here we admit that *q*

ing elastic vibrations. In this view the Lagrangian *L* (*q*

1, 1 1 ,

d

ter of mass. Small deviations *δ*

→ *<sup>x</sup>* and *q*

*k*th atoms with respect to the center of mass.

effect. The last term is a total potential energy *U* (*q*

*<sup>m</sup> q qqq qq Lq q U q <sup>t</sup>*


r rrr rr r r <sup>r</sup> &

1 1

= =

→ 1,*<sup>k</sup>* =(*q* → <sup>1</sup> −*q* →

*mqq qq m*

d

*N N k xk x x x x x k k k*

å å

( ) (,) ( ) <sup>2</sup>

d

The next step is to expand terms, ingoing into the integral (3), into Taylor series. The wave function written on the left is expanded up to the first term

$$
\rho\left(\vec{q}\_{1'}\vec{q}\_0; t + \delta t\right) \approx \nu(\vec{q}\_{1'}\vec{q}\_0; t) + \frac{\partial \,\nu}{\partial t} \delta t. \tag{8}
$$

As for the terms under the integral, here we preliminarily make some transformations. We define a small increment

$$
\vec{\xi} = \vec{q}\_1 - \vec{q}\_\chi \Rightarrow \mathsf{D}^{-3} q\_\chi = -\mathsf{D}^{-3} \xi. \tag{9}
$$

The Lagrangian (7) is rewritten, in such a case, in the following view

$$\mathcal{L}(\vec{q}\_{x},\vec{\tilde{q}}\_{x}) = \frac{m\_{N}}{2}\frac{\xi^{2}}{\delta t^{2}} - m\_{N}\frac{\xi}{\delta t}\frac{1}{N}\sum\_{k=1}^{N}\frac{(\vec{\tilde{\mathcal{S}}}\_{1,k} - \vec{\tilde{\mathcal{S}}}\_{x,k})}{\delta t} + \frac{m}{2}\sum\_{k=1}^{N}\frac{(\vec{\mathcal{S}}\_{1,k} - \vec{\tilde{\mathcal{S}}}\_{x,k})^{2}}{\delta t^{2}} - \mathcal{U}\left(\vec{\tilde{q}}\_{x}\right) \tag{10}$$

Here *U* (*q* → *<sup>x</sup>*)is sum of all potentials *U* (*q* → *<sup>x</sup>*,*<sup>k</sup>* ) given in the center of mass. Further we shall deal with the path integral (3) where the kernel *K* contains the Lagrangian given from Eq. (10). The under integral function *ψ*(*q* → *<sup>x</sup>*, *q* <sup>→</sup> 0;*t*)=*ψ*(*q* → <sup>1</sup> −*ξ* <sup>→</sup> , *q* → <sup>0</sup>;*t*)is subjected to expansion into the Tay‐ lor series up to the second terms of the expansion

$$\left\{\boldsymbol{\eta}\prime(\vec{q}\_{1}-\vec{\xi}\prime,\vec{q}\_{0};t)\right\}\approx\;\boldsymbol{\psi}\prime(\vec{q}\_{1},\vec{q}\_{0};t)-\left(\nabla\,\boldsymbol{\psi}\prime\vec{\xi}\right)+\nabla^{2}\boldsymbol{\psi}\cdot\vec{\xi}^{2}\Big/\Delta\tag{11}$$

The potential energy *U* (*q* → *<sup>x</sup>*)=*U* (*q* → <sup>1</sup> −*ξ* <sup>→</sup> ) is subjected to expansion into the Taylor series by the small parameter *ξ* <sup>→</sup> also. Here we restrict themselves by the first two terms of the expansion, *U* (*q* → <sup>1</sup>)−(∇*U* (*q* <sup>→</sup> 1), *ξ* → ).

Taking into account the expressions (8)-(11) and substituting theirs into Eq. (3) we get

$$\begin{split} \left| \nu \left( \bar{q}\_{1}, \bar{q}\_{0}; t \right) + \frac{\partial \nu}{\partial t} \delta t = -\frac{1}{A} \int\_{\mathbb{R}^{3}} \exp \left[ -\frac{1}{\Gamma} \left( \frac{m\_{N}}{2} \frac{\xi^{2}}{\delta t} - \xi \underbrace{\frac{m\_{N}}{N} \sum\_{k=1}^{N} \frac{(\bar{\mathcal{S}}\_{1,k} - \bar{\mathcal{S}}\_{x,k})}{\delta t}}\_{\text{(a)}} + \underbrace{\left[ \frac{m}{2} \sum\_{k=1}^{N} \frac{(\bar{\mathcal{S}}\_{1,k} - \bar{\mathcal{S}}\_{x,k})^{2}}{\delta t^{2}} \right]}\_{\text{(b)}} \right] \delta t \\ - \left( \mathcal{U} \left( \bar{q}\_{1} \right) - \left( \nabla \mathcal{U} \left( \bar{q}\_{1} \right), \bar{\mathcal{E}} \right) \right) \delta t \end{split} \tag{12}$$

In the light of this observation let us now solve integral (13) accurate to terms containing *δt*

¶ G + = +Ñ+ -

<sup>→</sup> , ∇*ψ*) here is absent since we consider *v*

y d

r r r r <sup>r</sup> *<sup>U</sup>* (16)

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

1 0 1 1 0

r r rr r rr *<sup>U</sup>* (17)

10 1 10

1 0 1 0

*N*

*N*

r r <sup>r</sup> r r *<sup>U</sup>* r r (20)

 y

r r <sup>h</sup> rr r rr <sup>h</sup> *<sup>U</sup>* (19)

 y

1 0 1 0 1 <sup>1</sup> ( , ;) ( , ;) () . <sup>2</sup> *<sup>N</sup> N qqt t qqt t qT t t m*

¶ G

( ) 1 0 <sup>2</sup>

<sup>B</sup> 2 22 2 *NN N kT t mm m*

Let *k*B*Tδt* = 0. It means that Γ = i*ћ*. Also *TN* = 0. One can see that Eq. (17) is reduced to

y

( ) 1 0 <sup>1</sup>

( , ;) ( ) ( , ;) ( , ; ). <sup>2</sup>

*qqt q T D qqt qqt <sup>t</sup> m D*

=- D +

( , ;) ( , ; ) ( ) ( , ; ). <sup>2</sup> *<sup>N</sup> qqt qqt q qqt t m*

2

Let *k*B*Tδt* >> *ћ*. We can suppose that Γ = *k*B*Tδt*. Eq. (17) takes a form

y

¶ - =D +

1 0

¶

y

¶

y

¶

**i**

It is the Schrödinger equation.

*2.1.2. Temperature is not zero*

y

¶ <sup>G</sup> =Ñ + - ¶ G

G

( , ;) <sup>1</sup> ( , ; ) ( ) ( , ; ). <sup>2</sup> *<sup>N</sup>*

d

= + **<sup>i</sup>** <sup>h</sup>

is seen to be as a complex-valued diffusion coefficient consisting of real and imaginary

*qqt qqt q T qqt t m*

( ) <sup>2</sup>

<sup>→</sup> 1, *q* →

> y

y d

<sup>→</sup> =0, as was mentioned

http://dx.doi.org/10.5772/53471

191

(18)

<sup>0</sup>;*t*) we come to the following dif‐

not higher the first order:

y

We note that the term (*v*

y

ferential equation

The parameter

*2.1.1. Temperature is zero*

parts.

Here

y

d y

above. By reducing from the both sides the function *ψ*(*q*

*N*

First, we consider terms enveloped by braces (a) and (b): (a) here displacement (*δ* → 1,*<sup>k</sup>* −*δ* → *<sup>x</sup>*,*<sup>k</sup>* ) divided by *δt* is a velocity *v* → *<sup>k</sup>* of k*th* atom at its deviation from a steady position relative to the center of mass. Summation through all deviations of atoms divided by *N* gives averaged velocity, *v* <sup>→</sup> , of all atoms with respect to the center of mass. This averaged velocity, as we mentioned above, vanishes. The velocity can be nonzero only in a case when external forces push coherently all atoms. This case we do not consider here. (b) this term is a thermal kinet‐ ic energy, *TN*, of the atoms oscillating about the center of mass. Observe that energy of ther‐ mal fluctuations, *TN*, is proportional to *k*B*T*. Because of its presence in the propagator intensity of an interference pattern diminishes in general. Further we shall add this term in‐ to the potential energy as some constant component.

In the light of the above observation we rewrite Eq. (12) as follows

$$\begin{split} \mathcal{Y}\boldsymbol{\eta}(\vec{q}\_{1},\vec{q}\_{0};t) + \frac{\partial \boldsymbol{\nu}}{\partial t}\boldsymbol{\delta t} &= -\frac{1}{A} \int\_{\mathcal{R}^{3}} \exp\left\{-\frac{m\_{N}}{2\Gamma}\frac{\vec{\boldsymbol{\xi}}^{2}}{\delta t}\right\} \Bigg(1 + \underbrace{m\_{N}}\_{\begin{subarray}{c} \begin{subarray}{c} \boldsymbol{\eta} \\ \end{subarray}} \frac{\vec{\boldsymbol{\xi}}^{2}}{\Gamma} + \underbrace{T\_{N}}\_{\begin{subarray}{c} \begin{subarray}{c} \boldsymbol{\eta} \end{subarray}} \frac{\delta t}{\Gamma} + \left(\boldsymbol{\mathcal{U}}\left(\vec{q}\_{1}\right) - \left(\boldsymbol{\nabla}\boldsymbol{\mathcal{U}}\left(\vec{q}\_{1}\right),\vec{\boldsymbol{\xi}}\right)\right) \frac{\delta t}{\Gamma}\bigg}\right) \\ \times \Big(\boldsymbol{\eta}\left(\vec{q}\_{1},\vec{q}\_{0};t\right) - \left(\boldsymbol{\nabla}\boldsymbol{\mathcal{V}}\boldsymbol{\mathcal{V}},\vec{\boldsymbol{\xi}}\right) + \boldsymbol{\nabla}^{2}\boldsymbol{\eta} \cdot \boldsymbol{\xi}^{2}\Big/2\Big]\mathsf{D}^{-3}\boldsymbol{\xi}. \end{split} \tag{13}$$

Here we have expanded preliminarily exponents to the Taylor series up to the first term of the expansion. Exception relates to the term exp{ -*mN ξ* <sup>2</sup> /2Γ*δt*} which remains in its original form. This exponent integrated over all space R 3 results in

$$-\frac{1}{A} \int\_{\mathcal{R}} \exp\left\{-\frac{m\_N}{2\Gamma} \frac{\xi^2}{\delta t}\right\} \mathcal{D}^{\ 3} \xi = -\frac{1}{A} \left(\frac{2\pi \Gamma \delta t}{m\_N}\right)^{3/2} = 1 \qquad \Rightarrow \qquad A = -\left(\frac{2\pi \Gamma \delta t}{m\_N}\right)^{3/2}.\tag{14}$$

To derive outcomes of integration of terms containing factors (∇*ψ*, *ξ* <sup>→</sup> ) and ∇2*ψ*⋅*ξ* <sup>2</sup> / 2 we mention the following integrals [8]

$$\frac{1}{A} \int\_{\mathcal{R}^3} \exp\left\{-\frac{m\_N}{2\Gamma} \frac{\xi^2}{\delta t}\right\} \xi \mathsf{D}^{~3} \xi = 0 \qquad \text{and} \qquad \frac{1}{A} \int\_{\mathcal{R}^3} \exp\left\{-\frac{m\_N}{2\Gamma} \frac{\xi^2}{\delta t}\right\} \xi^2 \mathsf{D}^{~3} \tilde{\xi} = \frac{\Gamma}{m\_N} \delta t \tag{15}$$

In the light of this observation let us now solve integral (13) accurate to terms containing *δt* not higher the first order:

$$\mathcal{W}\left(\vec{q}\_{1},\vec{q}\_{0};t\right) + \frac{\partial\mathcal{W}}{\partial t}\delta t = \mathcal{W}\left(\vec{q}\_{1},\vec{q}\_{0};t\right) + \frac{\Gamma}{2m\_{N}}\nabla^{2}\mathcal{W}\,\delta t + \frac{1}{\Gamma}\left(\mathcal{U}\left(\vec{q}\_{1}\right) - T\_{N}\right)\mathcal{W}\,\delta t.\tag{16}$$

We note that the term (*v* <sup>→</sup> , ∇*ψ*) here is absent since we consider *v* <sup>→</sup> =0, as was mentioned above. By reducing from the both sides the function *ψ*(*q* <sup>→</sup> 1, *q* → <sup>0</sup>;*t*) we come to the following dif‐ ferential equation

$$\frac{\partial \,\nu(\vec{q}\_1, \vec{q}\_0; t)}{\partial t} = \frac{\Gamma}{2m\_N} \nabla^2 \nu(\vec{q}\_1, \vec{q}\_0; t) + \frac{1}{\Gamma} (\mathsf{U}\,(\vec{q}\_1) - T\_N) \nu(\vec{q}\_1, \vec{q}\_0; t). \tag{17}$$

The parameter

( ) ( ( ) )

the center of mass. Summation through all deviations of atoms divided by *N* gives averaged

mentioned above, vanishes. The velocity can be nonzero only in a case when external forces push coherently all atoms. This case we do not consider here. (b) this term is a thermal kinet‐ ic energy, *TN*, of the atoms oscillating about the center of mass. Observe that energy of ther‐ mal fluctuations, *TN*, is proportional to *k*B*T*. Because of its presence in the propagator intensity of an interference pattern diminishes in general. Further we shall add this term in‐

<sup>→</sup> , of all atoms with respect to the center of mass. This averaged velocity, as we

() () 22 3

*D*

results in

*N N*

*D* (14)

 x

*D D* (15)

 d

 x

*N N a b*

 xd

æ ö ¶ ì ü ï ïç ÷ + =- - + + + - Ñ í ý ¶ ï ï G GG <sup>G</sup> î þè ø

Here we have expanded preliminarily exponents to the Taylor series up to the first term of

*<sup>m</sup> t t <sup>A</sup>*

ì ü ï ï æ ö G G æ ö

 y

( ) ( ( ), ) ( , ; ) , 2

*U U D*

1 0 <sup>2</sup> 1 1

d

 x

1 1 ( )( ) ( , ;) exp 2 2

öü ÷ïï - -Ñ ÷ý - Ñ +Ñ × ÷ï÷ øïþ

r r r r r r

First, we consider terms enveloped by braces (a) and (b): (a) here displacement (*δ*

*m m <sup>m</sup> qqt t <sup>t</sup> t A tN t t*

x

<sup>ì</sup> <sup>æ</sup> <sup>ï</sup> <sup>ç</sup> - - é ù ¶ <sup>ï</sup> <sup>ç</sup> + =- - - <sup>í</sup> <sup>+</sup> ê ú <sup>ç</sup> ¶ G <sup>ï</sup> ê ú <sup>ç</sup> ë û <sup>ï</sup> <sup>ç</sup> <sup>î</sup> <sup>è</sup>

ò å å

1 1 1 0

*q q t qqt*

xd

2 2

yx

*N N N N k xk k xk k k*

= =

dd

 d

1, , 1, ,

rr rr

 y x

22 3

d

 dd

d

→ 1,*<sup>k</sup>* −*δ* → *<sup>x</sup>*,*<sup>k</sup>* )

d

<sup>→</sup> ) and ∇2*ψ*⋅*ξ* <sup>2</sup> / 2 we

*N*

*t*

 d (13)

x

/2Γ*δt*} which remains in its original

p d

xx

(12)

x.

( ) ( )

*<sup>k</sup>* of k*th* atom at its deviation from a steady position relative to

{ { ( )

*U U*

*<sup>a</sup> <sup>b</sup>*

144424443 144424443

3

→

to the potential energy as some constant component.

3

ò

*R*

form. This exponent integrated over all space R 3

1 0

y

*N*

x

ï ï

d

x x

mention the following integrals [8]

x

d

*qqt*

In the light of the above observation we rewrite Eq. (12) as follows

( ( ) )

´ - Ñ +Ñ ×

r r r

the expansion. Exception relates to the term exp{ -*mN ξ* <sup>2</sup>

x

3 3

*R R*

yx

*N*

2 1 0 1 1

x

d

( , ;) , 2 .

<sup>1</sup> ( , ;) exp <sup>1</sup> ( ) ( ( ), ) <sup>2</sup>

<sup>r</sup> <sup>r</sup> r r <sup>r</sup> r r

*<sup>m</sup> t t qqt t mv T q q tA t*

 y x

3 2 3 2 <sup>2</sup> 1 1 <sup>3</sup> 2 2 exp <sup>1</sup> . <sup>2</sup>

*A t Am m*

To derive outcomes of integration of terms containing factors (∇*ψ*, *ξ*

2 2 1 1 <sup>3</sup> 2 3 exp 0 and exp 2 2 *N N*

*A t A tm*

*m m*

ì ü ï ï ì ü <sup>G</sup> í ý -= - = í ý ï ï î þ G G ï ï î þ ò ò


p d

*R*

y

divided by *δt* is a velocity *v*

y

3

*R*

 d  d

y

velocity, *v*

y

r r

190 Advances in Quantum Mechanics

$$\frac{\Gamma}{2m\_N} = \frac{2k\_B T \delta t}{2m\_N} + \mathbf{i} \frac{\hbar}{2m\_N} \tag{18}$$

is seen to be as a complex-valued diffusion coefficient consisting of real and imaginary parts.

#### *2.1.1. Temperature is zero*

Let *k*B*Tδt* = 0. It means that Γ = i*ћ*. Also *TN* = 0. One can see that Eq. (17) is reduced to

$$\mathrm{i}\hbar \frac{\partial \,\nu(\vec{q}\_1, \vec{q}\_0; t)}{\partial t} = -\frac{\hbar^2}{2m\_N} \Delta \,\nu(\vec{q}\_1, \vec{q}\_0; t) + \mathcal{U}\left(\vec{q}\_1\right)\nu(\vec{q}\_1, \vec{q}\_0; t). \tag{19}$$

It is the Schrödinger equation.

#### *2.1.2. Temperature is not zero*

Let *k*B*Tδt* >> *ћ*. We can suppose that Γ = *k*B*Tδt*. Eq. (17) takes a form

$$\frac{\partial \,\nu(\vec{q}\_1, \vec{q}\_0; t)}{\partial t} = D \,\Delta \,\nu(\vec{q}\_1, \vec{q}\_0; t) + \frac{\left(\mathsf{U}\left(\vec{q}\_1\right) - T\_N\right)}{2m\_N D} \nu(\vec{q}\_1, \vec{q}\_0; t) \,. \tag{20}$$

Here

$$D = \frac{k\_B T \delta t}{m\_N} \tag{21}$$

Due to appearance of the term (*D* / 2)∇ln(*ρ*) in Eq. (22) the diffusion equation becomes nonlin‐ ear. It is interesting to note, running ahead, that this osmotic term reveals many common with the quantum potential, which is show further. Observe that the both expressions contain the

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

http://dx.doi.org/10.5772/53471

193

Reduction to PDEs, Eq. (19) and Eq. (22), was done with aim to show that the both quantum and classical realms adjoin with each other much more closely, than it could seem with the first glance. Further we shall return to the integral path paradigm [8] and calculate patterns arising after passing particles through gratings. We shall combine quantum and classical

Computation of a passing particle through a grating is based on the path integral technique [8]. We begin with writing the path integral that describes passing the particle through a slit made in an opaque screen that is situated perpendicularly to axis *z*, Fig. 3. For this reason we need to describe a movement of the particle from a source to the screen and its possible deflection at

**Figure 3.** Interferometry from one grating *G*0 situated transversely to a particle beam emitted from a distributed

We believe, that before the screen and after it, the particle (fullerene molecule, for example) moves as a free particle. Its Lagrangian, rewritten from Eq. (17), describes its deflection from

(,) . <sup>2</sup> *NN N*

The first term relates to movement of the center of mass of the molecule. So that *mN* is mass of the molecule and *x*˙ is its transversal velocity, i.e., the velocity lies in transversal direction to the axis *z*. The second term is conditioned by collective fluctuations of atoms constituent

*<sup>x</sup> L m m xv T* =- + & <sup>r</sup> & (25)

2

passing through the slit, see Fig. 4. At that we need to evaluate all possible deflections.

term ∇*ρ* / *ρ* =∇ln(*ρ*) relating to gradient of the quantum entropy *SQ* = −ln(*ρ*) / 2 [28].

realms by introducing the complex-valued parameter Γ = 2*k*B*Tδt* + i*ћ*.

**2.2. Paths through** *N***-slit grating**

a straight path in the following form

source.

is the diffusion coefficient. The coefficient has dimensionality of [length2 time-1]. It is a factor of proportionality representing amount of substance diffusing across a unit area in a unit time - concentration gradient in unit time.

We can see that Eq. (20) deals with the amplitude function *ψ*, not a concentration. However, a measurable function is *ρ*=|*ψ*|2 - concentration having dimensionality of [(amount of sub‐ stance) length-3]. Let us multiply Eq. (20) from the left by 2*ψ*. First we note that the combina‐ tion 2*ψ Δψ*=2*ψ* ∇*ψ*−<sup>1</sup> *ψ* ∇*ψ*=2(*ψ* ∇*ψ*−<sup>1</sup> )(*ψ* ∇*ψ*) + 2∇(*ψ* ∇*ψ*) results in −(1 / 2)(∇ln(*ρ*) ⋅∇*ρ*) + *Δρ*. As a result we come to a diffusion-drift equation describing dif‐ fusion in a space loaded by a potential field (*U* (*q* → <sup>1</sup>)−*TN* ):

$$\frac{\partial \rho}{\partial t} + \frac{D}{2} \left( \nabla \ln(\rho), \nabla \rho \right) = D \Delta \rho + \frac{\left( \mathcal{U} \left( \vec{q}\_1 \right) - T\_N \right)}{2m\_N D} \rho \tag{22}$$

Extra term (*D* / 2)∇ln(*ρ*) in this diffusion-drift equation is a velocity of outflow of the particles from volume populated by much more number of particles than in adjacent volume. The term −ln(*ρ*) is entropy of a particle ensemble. From here it follows that ∇ln(*ρ*) describes inflow of the particles to a region where the entropy is small. Observe that the velocity

$$
\vec{\mu} = D \frac{\nabla \rho}{\rho} = D \nabla \ln(\rho) \tag{23}
$$

is an osmotic velocity required of the particle to counteract osmotic effects [6]. Namely, imagine a suspension of many Brownian particles within a physical volume acted on by an external, virtual in general, force. This force is balanced by an osmotic pressure force of the suspension [6]:

$$
\vec{K} = k\_\text{B} T \frac{\nabla \rho}{\rho}. \tag{24}
$$

From here it is seen, that the osmotic pressure force arises always when density difference exists and especially when the density tends to zero. And vice-versa, the force disappears in extra-dense media with spatially homogeneous distribution of particles. As states the sec‐ ond law of thermodynamics spontaneous processes happen with increasing entropy. The osmosis evolves spontaneously because it leads to increase of disorder, i.e., with increase of entropy. When the entropy gradient becomes zero the system achieves equilibrium, osmotic forces vanish.

Due to appearance of the term (*D* / 2)∇ln(*ρ*) in Eq. (22) the diffusion equation becomes nonlin‐ ear. It is interesting to note, running ahead, that this osmotic term reveals many common with the quantum potential, which is show further. Observe that the both expressions contain the term ∇*ρ* / *ρ* =∇ln(*ρ*) relating to gradient of the quantum entropy *SQ* = −ln(*ρ*) / 2 [28].

Reduction to PDEs, Eq. (19) and Eq. (22), was done with aim to show that the both quantum and classical realms adjoin with each other much more closely, than it could seem with the first glance. Further we shall return to the integral path paradigm [8] and calculate patterns arising after passing particles through gratings. We shall combine quantum and classical realms by introducing the complex-valued parameter Γ = 2*k*B*Tδt* + i*ћ*.

#### **2.2. Paths through** *N***-slit grating**

B *N kT t <sup>D</sup> m* d

of proportionality representing amount of substance diffusing across a unit area in a unit

We can see that Eq. (20) deals with the amplitude function *ψ*, not a concentration. However,

stance) length-3]. Let us multiply Eq. (20) from the left by 2*ψ*. First we note that the combina‐

−(1 / 2)(∇ln(*ρ*) ⋅∇*ρ*) + *Δρ*. As a result we come to a diffusion-drift equation describing dif‐

( ) <sup>1</sup> (() ) ln( ), 2 2

 r

Extra term (*D* / 2)∇ln(*ρ*) in this diffusion-drift equation is a velocity of outflow of the particles from volume populated by much more number of particles than in adjacent volume. The term −ln(*ρ*) is entropy of a particle ensemble. From here it follows that ∇ln(*ρ*) describes inflow of

is an osmotic velocity required of the particle to counteract osmotic effects [6]. Namely, imagine a suspension of many Brownian particles within a physical volume acted on by an external, virtual in general, force. This force is balanced by an osmotic pressure force of the

*<sup>D</sup> q T <sup>D</sup> t m D*

> *uD D* ln( ) r

> > <sup>B</sup> *K kT* . r

<sup>Ñ</sup> <sup>=</sup>

r

From here it is seen, that the osmotic pressure force arises always when density difference exists and especially when the density tends to zero. And vice-versa, the force disappears in extra-dense media with spatially homogeneous distribution of particles. As states the sec‐ ond law of thermodynamics spontaneous processes happen with increasing entropy. The osmosis evolves spontaneously because it leads to increase of disorder, i.e., with increase of entropy. When the entropy gradient becomes zero the system achieves equilibrium, osmotic

r

<sup>→</sup> 1)−*TN* ):

<sup>r</sup> *<sup>U</sup>*

r

is the diffusion coefficient. The coefficient has dimensionality of [length2

*ψ* ∇*ψ*=2(*ψ* ∇*ψ*−<sup>1</sup>

rr

the particles to a region where the entropy is small. Observe that the velocity

¶ - + Ñ Ñ = D+

time - concentration gradient in unit time.

fusion in a space loaded by a potential field (*U* (*q*

r

¶

a measurable function is *ρ*=|*ψ*|2

tion 2*ψ Δψ*=2*ψ* ∇*ψ*−<sup>1</sup>

192 Advances in Quantum Mechanics

suspension [6]:

forces vanish.

= (21)


*N*

r

<sup>Ñ</sup> = = Ñ <sup>r</sup> (23)

<sup>r</sup> (24)

*N*

)(*ψ* ∇*ψ*) + 2∇(*ψ* ∇*ψ*) results in

time-1]. It is a factor

(22)

Computation of a passing particle through a grating is based on the path integral technique [8]. We begin with writing the path integral that describes passing the particle through a slit made in an opaque screen that is situated perpendicularly to axis *z*, Fig. 3. For this reason we need to describe a movement of the particle from a source to the screen and its possible deflection at passing through the slit, see Fig. 4. At that we need to evaluate all possible deflections.

**Figure 3.** Interferometry from one grating *G*0 situated transversely to a particle beam emitted from a distributed source.

We believe, that before the screen and after it, the particle (fullerene molecule, for example) moves as a free particle. Its Lagrangian, rewritten from Eq. (17), describes its deflection from a straight path in the following form

$$L = m\_N \frac{\dot{\mathbf{x}}^2}{2} - m\_N \langle \dot{\mathbf{x}}, \vec{v} \rangle + T\_N. \tag{25}$$

The first term relates to movement of the center of mass of the molecule. So that *mN* is mass of the molecule and *x*˙ is its transversal velocity, i.e., the velocity lies in transversal direction to the axis *z*. The second term is conditioned by collective fluctuations of atoms constituent this molecule. This term is nonzero when atoms have predominant fluctuations along axis *x*. For the sake of simplicity we admit that *v* <sup>→</sup> is constant. The third term is a constant and comes from Eq. (17). It can be introduced into the normalization factor. For that reason we shall ig‐ nore this term in the following computations. A longitudinal momentum, *p*z, is much greater than its transverse component [16, 17, 29] and we believe it is constant also. By translating a particle's position on a small distance *δx* = (*x*b - *x*a)<< 1 for a small time *δt* = (*t*b - *t*a)<< 1 we find that a weight factor of such a translation has the following form

in Eq. (12). We shall believe that the ratio *v*/*v (a,b)* is small enough. We can define a new renor‐

By substituting the kernel (28) into the integral (27) we obtain the following detailed form

1 0 0 1 10 0 { } { } ( ( )) { } { } (( ) ) (,,) exp exp . 2 22 2

The integral is computed within a finite interval [-*b*0,+*b*0]. Observe, that the integrating can be broadened from -∞ to +∞. But in this case we need to load the integral by the step func‐ tion equal to unit within the finite interval [-*b*0,+*b*0] and it vanishes outside of the interval. The step function, that simulate a single slit, can be approximated by the following a set of

*m mx x m m x x xxx <sup>d</sup>* x


t

1

densely the step function. We rewrite Eq. (29) with inserting this form-factor

*k K bK k GbK*

h p=

*K*

1 2 ( ( (2 1))) ( , , , ) exp . 2( )

x

ì ü ï ï - -- <sup>=</sup> í ý -

Here parameter *b* is a half-width of the slit, real *η >* 0 is a tuning parameter, and *K* takes inte‐ ger values. At *K* = 1 this form-factor degenerates to a single Gaussian function. And at *K* → ∞ this function tends to an infinite collection of the Kronecker deltas which fill everywhere

1 0 0 0 0

= -+ í ý ç ÷

We do not write parameters *η* and *K* in the Gaussian form-factor and for the sake of simplic‐ ity further we shall consider they equal to 1. That is, for simulating the slit we select a single

First we replace the flight times *τ*0 and *τ*<sup>1</sup> by flight distances (*z*0-*z*s) and (*z*1-*z*0), see Fig. 4. This

*s z z*

0 0 1 10

ìï = - <sup>í</sup> ï = - î

t

t

( ) ( )

*z zv zzv*

{ } { } ( ( )) (( ) ) ( , , ) ( , ) exp . <sup>2</sup> <sup>2</sup>

*m m xx x x xxx G b <sup>d</sup>*

1 0 1 0

*N N s*

t

<sup>G</sup> <sup>G</sup> ï ï î þ è ø <sup>ò</sup> (31)

1 2 1 2 2 2 1 00 0 0

æ ö ì üì ü ï ïï ï - + æ ö + - <sup>=</sup> ç ÷ í ýí ý - - ç ÷ G GG G è ø ï ïï ï î þî þ è ø <sup>ò</sup> (29)

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

 pt  x

t

2

2 2

 x

> t

x

(32)

ï ï î þ <sup>å</sup> (30)

2

1 00 00

ì ü ï ï æ ö -+ +-

x

*b*

h

x

http://dx.doi.org/10.5772/53471

195

*N N N Ns*

malized mass {*m*}*N* = *mN* (1 - *v*/*v(a,b)*) and further we shall deal with this mass.

*2.2.1. Passing of a particle through slit*

p t

0

*b*

*s b*

y

0

the Gaussian functions [30]

*s*

Gaussian function.

replacement reads

y

x h

 x

*2.2.2. Definition of new working parameters*

¥


p tt

**Figure 4.** Passage of a particle along path (*z*s,*x*s) → (*z*0,*x*0) → (*z*1,*x*1) through a screen containing one slit with a width equal to 2b0. Divergence angle of particles incident on the slit, α, tends to zero as the source is removed to infinity.

$$\exp\left\{-\frac{L\delta t}{\Gamma}\right\} = \exp\left\{-\frac{m\_N}{2\Gamma}\frac{\left(\mathbf{x}\_b - \mathbf{x}\_a\right)^2}{\left(t\_b - t\_a\right)} + \frac{m\_N v}{\Gamma} (\mathbf{x}\_b - \mathbf{x}\_a)\right\}.\tag{26}$$

The particles flying to the grating slit along a ray α, Fig. 4, pass through the slit within a range from *x*0-*b*0 to *x*0+*b*0 with high probability. The path integral in that case reads

$$\Psi(\mathbf{x}\_1, \mathbf{x}\_0, \mathbf{x}\_s) = \int\_{-b\_0}^{b\_0} \mathbf{K}(\mathbf{x}\_1, \tau\_0 + \tau\_1; \mathbf{x}\_0 + \tilde{\mathbf{y}}\_0, \tau\_0) \mathbf{K}(\mathbf{x}\_0 + \tilde{\mathbf{y}}\_0, \tau\_0; \mathbf{x}\_s, \mathbf{0}) d\tilde{\mathbf{y}}\_0. \tag{27}$$

Integral kernel (propagator) for the particle freely flying is as follows [8, 26]

$$\mathcal{K}(\mathbf{x}\_{b},t\_{b};\mathbf{x}\_{a},t\_{a}) = \left[\frac{m\_{\mathcal{N}}(1-2\,\mathrm{v}/\mathrm{v}\_{(a,b)})}{2\pi\,\Gamma(t\_{b}-t\_{a})}\right]^{1/2} \exp\left\{-\frac{m\_{\mathcal{N}}(\mathbf{x}\_{b}-\mathbf{x}\_{a})^{2}}{2\Gamma(t\_{b}-t\_{a})}\left(1-2\frac{\mathrm{v}}{\mathrm{v}\_{(a,b)}}\right)\right\} \tag{28}$$

Here *v(a,b)* = (*xb – xa*)/(*t<sup>b</sup> – tb*) is a velocity of the molecule on a segment from *xa* to *xb*. And *v* is an average velocity of collective deflection of atoms constituent the molecule. It was defined in Eq. (12). We shall believe that the ratio *v*/*v (a,b)* is small enough. We can define a new renor‐ malized mass {*m*}*N* = *mN* (1 - *v*/*v(a,b)*) and further we shall deal with this mass.

#### *2.2.1. Passing of a particle through slit*

this molecule. This term is nonzero when atoms have predominant fluctuations along axis *x*.

from Eq. (17). It can be introduced into the normalization factor. For that reason we shall ig‐ nore this term in the following computations. A longitudinal momentum, *p*z, is much greater than its transverse component [16, 17, 29] and we believe it is constant also. By translating a particle's position on a small distance *δx* = (*x*b - *x*a)<< 1 for a small time *δt* = (*t*b - *t*a)<< 1 we

**Figure 4.** Passage of a particle along path (*z*s,*x*s) → (*z*0,*x*0) → (*z*1,*x*1) through a screen containing one slit with a width equal to 2b0. Divergence angle of particles incident on the slit, α, tends to zero as the source is removed to infinity.

*Nb a N*

*b a*

The particles flying to the grating slit along a ray α, Fig. 4, pass through the slit within a

*t t*

*b a*

(26)

(28)

*x x*

 xt

1 2 <sup>2</sup>

*b a b a a b*

 *t t tt v* é ù - ì ü - æ ö ï ï = -- ê ú í ý ç ÷ G - G - ç ÷ ë û ï ï î þ è ø

= ++ + ò (27)

 x

(,)

<sup>2</sup> ( ) exp exp ( ). 2( )

ì ü ì ü ï ï - íý í -=- + - <sup>ý</sup> î þ G G- G ï ï î þ

*L t m x x mv*

range from *x*0-*b*0 to *x*0+*b*0 with high probability. The path integral in that case reads

1 0 10 1 0 00 0 00 0 ( , , ) ( , ; , ) ( , ; ,0) .

*x x x Kx x Kx x d*

t t

Integral kernel (propagator) for the particle freely flying is as follows [8, 26]

(,)

(1 2 ) ( ) ( ,; ,) exp 1 2

*m vv mx x <sup>v</sup> Kx t x t*

*s s*

2 ( ) 2( ) *N a b Nb a*

Here *v(a,b)* = (*xb – xa*)/(*t<sup>b</sup> – tb*) is a velocity of the molecule on a segment from *xa* to *xb*. And *v* is an average velocity of collective deflection of atoms constituent the molecule. It was defined

 xt

d

0

*b*

y

*bb aa*

0

p

*b*


find that a weight factor of such a translation has the following form

<sup>→</sup> is constant. The third term is a constant and comes

For the sake of simplicity we admit that *v*

194 Advances in Quantum Mechanics

By substituting the kernel (28) into the integral (27) we obtain the following detailed form

$$\Psi(\mathbf{x}\_{1},\mathbf{x}\_{0},\mathbf{x}\_{s}) = \int\_{-b\_{0}}^{b\_{0}} \left(\frac{[m]\_{\mathrm{N}}}{2\pi\,\Gamma\,\tau\_{1}}\right)^{\mathrm{J}/2} \exp\left\{-\frac{[m]\_{\mathrm{N}}(\mathbf{x}\_{1} - (\mathbf{x}\_{0} + \boldsymbol{\xi}\_{0}))^{2}}{2\Gamma\,\tau\_{1}}\right\} \left|\frac{[m]\_{\mathrm{N}}}{2\pi\,\Gamma\,\tau\_{0}}\right|^{\mathrm{J}/2} \exp\left\{-\frac{\left\{m\right\}\_{\mathrm{N}}(\left(\mathbf{x}\_{0} + \boldsymbol{\xi}\_{0}\right) - \mathbf{x}\_{s}\right\}^{2}}{2\Gamma\,\tau\_{0}}\right|d\boldsymbol{\xi}\_{0}.\tag{29}$$

The integral is computed within a finite interval [-*b*0,+*b*0]. Observe, that the integrating can be broadened from -∞ to +∞. But in this case we need to load the integral by the step func‐ tion equal to unit within the finite interval [-*b*0,+*b*0] and it vanishes outside of the interval. The step function, that simulate a single slit, can be approximated by the following a set of the Gaussian functions [30]

$$G(\xi, b, \eta, K) = \frac{1}{\eta} \sqrt{\frac{2}{\pi}} \sum\_{k=1}^{K} \exp\left\{-\frac{\left(K\xi - b(K - (2k - 1))\right)^2}{2(b\eta)^2}\right\}.\tag{30}$$

Here parameter *b* is a half-width of the slit, real *η >* 0 is a tuning parameter, and *K* takes inte‐ ger values. At *K* = 1 this form-factor degenerates to a single Gaussian function. And at *K* → ∞ this function tends to an infinite collection of the Kronecker deltas which fill everywhere densely the step function. We rewrite Eq. (29) with inserting this form-factor

$$\Psi(\mathbf{x}\_{1},\mathbf{x}\_{0},\mathbf{x}\_{s}) = \bigcap\_{-\pi}^{\pi} \mathcal{G}(\boldsymbol{\xi}\_{0},\boldsymbol{b}\_{0}) \frac{\{m\}\_{N}}{2\pi \Gamma \sqrt{\pi\_{1}\tau\_{0}}} \exp\left\{-\frac{\{m\}\_{N}}{2\Gamma} \left(\frac{(\mathbf{x}\_{1} - (\mathbf{x}\_{0} + \boldsymbol{\xi}\_{0}))^{2}}{\tau\_{1}} + \frac{\{(\mathbf{x}\_{0} + \boldsymbol{\xi}\_{0}) - \mathbf{x}\_{s}\}^{2}}{\tau\_{0}}\right)\right\} d\boldsymbol{\xi}\_{0}.\tag{31}$$

We do not write parameters *η* and *K* in the Gaussian form-factor and for the sake of simplic‐ ity further we shall consider they equal to 1. That is, for simulating the slit we select a single Gaussian function.

#### *2.2.2. Definition of new working parameters*

First we replace the flight times *τ*0 and *τ*<sup>1</sup> by flight distances (*z*0-*z*s) and (*z*1-*z*0), see Fig. 4. This replacement reads

$$\begin{cases} \tau\_0 = (z\_0 - z\_s) / v\_z \\ \tau\_1 = (z\_1 - z\_0) / v\_z \end{cases} \tag{32}$$

where *vz* is a particle velocity along the axis *z*.

There is, however, one more parameter of time that is represented in definition of the co‐ efficient Γ= 2*k*B*Tδt* + **i***ћ*. It is a small time increment *δt*. The parameter *δt* first appears in the path integral (3) as the time increment along a path. In accordance with the uncertain‐ ty principle *δt* should be more or equal to the ratio of *ћ*, Planck constant, to energy of oc‐ curring events. In a case of a flying particle through vacuum it can be minimal energy of vacuum fluctuations (it is about energy of the first Bohr orbit of electron-positron pair that is about 14 eV). Evaluation gives *δt* ~ 2.8 10-16 s. From here it follows, that 2*k*B*Tδt* is less than *ћ* on about one order at *T* = 1000 K (almost temperature of fullerene evapora‐ tion from the Knudsen cell [27]).

Emergence of the term 2*k*B*Tδt* can be induced by existence of quantum drag [31] owing to different conditions for quantum fluctuations both inside of the fullerene molecule and out‐ side what can induce weak Casimir forces. Because of the weak Casimir force the quantum drag does not lead to decoherence at least in the near zone. However further we shall see that a weak washing out of the Talbot interference pattern is due to existence of this term.

Let us divide the parameter Γ by {*m*}*<sup>N</sup> vz*

$$\frac{\Gamma}{\{m\}\_N \upsilon\_z} = \frac{2k\_\text{B}T}{\{m\}\_N \upsilon\_z} \delta t + \mathbf{i} \frac{\hbar}{\{m\}\_N \upsilon\_z}.\tag{33}$$

22 2 0 00

> p*b*

è ø è ø <sup>å</sup> (37)

*x x x x*

*s*

http://dx.doi.org/10.5772/53471

1/2 comes from (30). Parameters Ξ<sup>0</sup>

(35)

197

0 0 0 00

ì ü é ù - X- æ ö ï ï <sup>=</sup> í ý - -+ ê ú ç ÷ LS - L - S - ç ÷ ï ï ê ú î þ ë û è ø

*z z z z z z*

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

*s s*

<sup>2</sup> () ( ) (,, , ) exp <sup>1</sup> . ( ) () ( )

p

Here argument of ψ-function contains apart *x* also *z* in order to emphasize that we carry out

0 0 0

X= - S= + - - - (36)

0

0 0 ( , ) ( , , , , ) ( , , , , ). *s s pxz xzdx xzdx* =Y LY L (38)

*λ*dB. So at *λ*dB = 5 pm the distance is equal to 500 nm. Require‐

<sup>=</sup> (39)

*xz n dx*

*x xzz zz zz*

In order to simplify records here we omit subscript 1 at *x* and *z* - an observation point that is

Let us consider that an opaque screen contains *N*0 slits spaced through equal distance, *d*, from each other. Numeration of the slits is given as it is shown in Fig. 3, *n*<sup>0</sup> = 0,1,2, …, *N*0 – 1. Sum of all wave functions (35), each of which calculates outcome from an individual slit,


0 0 2 0 0 0 0 ( )( ) ( )( ) <sup>1</sup> and ( )( ) ( ) 2 *s s s s*

*z z xx z z*

0

*N*

1


<sup>1</sup> (,,, , ) ,, , . <sup>2</sup>

æ ö æ ö - Y L= ç ÷ ç ÷ -

*s x*

Calculation of the wave function (37) is fulfilled for the grating containing *N*<sup>0</sup> = 32 slits. Dis‐

ment *λ*dB << *d* and *N*0 tending to infinity together with a condition that the particle beam is paraxial, that is, *x*s = 0 and *z*s → -∞, provides emergence in the near-field of an interference pattern, named Talbot carpet [19,20]. Here a spacing along interference patterns is measured

2

dB 2 , *<sup>d</sup>*

which is a convenient natural length at representation of interference patterns. Since we re‐ strict themselves by finite *N*0 we have a defective carpet, which progressively collapses as a

l

T

*z*

0

=

*n <sup>N</sup> xzdx* y

0 0 0

Probability density in the vicinity of the observation point (*x,z*) reads

0

*xzx x*

y

and Σ0 read

situated after the slit.

tance between slits is *d* = 105

in the Talbot length

*s*

p

observation in the point (*z*,*x*), see Fig. 4. The factor (2/*π*)

gives a total effect in the point (*z*,*x*) where a detector is placed:

Here *pz* = {*m*}*<sup>N</sup> vz* is a particle momentum along axis *z*. We can define the de Broglie wave‐ length *λ*dB = *h*/*pz* where *h* = 2*πћ* is the Planck constant. Let us also define a length *δ<sup>T</sup>* = 4*πk*B*Tδt*/({*m*}*<sup>N</sup> vz*). In this view we can rewrite Eq. (33) as follows

$$\frac{\Gamma}{\{m\}\_N \upsilon\_z} = \frac{\delta\_\Gamma}{2\pi} + \mathbf{i}\frac{\lambda\_{\text{dB}}}{2\pi} = \frac{1}{2\pi}\Lambda. \tag{34}$$

The length *δ<sup>T</sup>* tends to zero as *T* → 0. At *T* = 1000 K and at adopted *δt* = 2.8 10-16 s we have *δ<sup>T</sup>* ≈ 0.4 pm. On the other hand, the de Broglie wavelength, *λ*dB, evaluated for the fullerene mol‐ ecule moving with the velocity *vz* =100 m/s is about 5 pm [18]. So, we can see that the length *δ<sup>T</sup>* is less of the de Broglie wavelength on about one order and smaller. A signification of the length *δT* is that it determines decoherence of a particle beam. Decoherence of flying parti‐ cles occurs the quickly, the larger *δT*. Observe that the length *δ<sup>T</sup>* has a close relation with the coherence width - a main parameter in the generalized Gaussian Schell-model [32, 33].

#### **3. Wave function behind the grating**

Wave function from one slit after integration over *ξ*0 from -∞ to +∞ has the following view [28]

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference http://dx.doi.org/10.5772/53471 197

$$\mathcal{W}\{\mathbf{x},\mathbf{z},\mathbf{x}\_{0},\mathbf{x}\_{s}\} = \sqrt{\frac{2}{\pi\Lambda\Sigma\_{0}(\mathbf{z}\_{0}-\mathbf{z}\_{s})}} \exp\left\{-\frac{\pi}{\Lambda} \left[\frac{(\mathbf{x}-\mathbf{x}\_{0})^{2}}{\mathbf{z}-\mathbf{z}\_{0}}\left(1-\frac{\Xi\_{0}^{2}}{\Sigma\_{0}}\right) + \frac{(\mathbf{x}\_{0}-\mathbf{x}\_{s})^{2}}{\mathbf{z}\_{0}-\mathbf{z}\_{s}}\right]\right\}.\tag{35}$$

Here argument of ψ-function contains apart *x* also *z* in order to emphasize that we carry out observation in the point (*z*,*x*), see Fig. 4. The factor (2/*π*) 1/2 comes from (30). Parameters Ξ<sup>0</sup> and Σ0 read

$$\boldsymbol{\Sigma}\_{0} = 1 - \frac{\{\mathbf{x}\_{0} - \mathbf{x}\_{s}\} \{\mathbf{z} - \mathbf{z}\_{0}\}}{\{\mathbf{z}\_{0} - \mathbf{z}\_{s}\} \{\mathbf{x} - \mathbf{x}\_{0}\}} \qquad \text{and} \qquad \boldsymbol{\Sigma}\_{0} = \frac{\{\mathbf{z} - \mathbf{z}\_{s}\}}{\{\mathbf{z}\_{0} - \mathbf{z}\_{s}\}} + \frac{\Lambda \{\mathbf{z} - \mathbf{z}\_{0}\}}{2\pi b\_{0}^{2}} \tag{36}$$

In order to simplify records here we omit subscript 1 at *x* and *z* - an observation point that is situated after the slit.

Let us consider that an opaque screen contains *N*0 slits spaced through equal distance, *d*, from each other. Numeration of the slits is given as it is shown in Fig. 3, *n*<sup>0</sup> = 0,1,2, …, *N*0 – 1. Sum of all wave functions (35), each of which calculates outcome from an individual slit, gives a total effect in the point (*z*,*x*) where a detector is placed:

$$\left\{\Psi\_{0}(\mathbf{x},\mathbf{z},d,\mathbf{x}\_{s},\Lambda)\right\} = \sum\_{n\_{0}=0}^{N\_{0}-1} \psi\left(\mathbf{x},\mathbf{z},\left(n\_{0}-\frac{N\_{0}-1}{2}\right)d,\mathbf{x}\_{\mathbf{x}}\right).\tag{37}$$

Probability density in the vicinity of the observation point (*x,z*) reads

where *vz* is a particle velocity along the axis *z*.

tion from the Knudsen cell [27]).

196 Advances in Quantum Mechanics

Let us divide the parameter Γ by {*m*}*<sup>N</sup> vz*

There is, however, one more parameter of time that is represented in definition of the co‐ efficient Γ= 2*k*B*Tδt* + **i***ћ*. It is a small time increment *δt*. The parameter *δt* first appears in the path integral (3) as the time increment along a path. In accordance with the uncertain‐ ty principle *δt* should be more or equal to the ratio of *ћ*, Planck constant, to energy of oc‐ curring events. In a case of a flying particle through vacuum it can be minimal energy of vacuum fluctuations (it is about energy of the first Bohr orbit of electron-positron pair that is about 14 eV). Evaluation gives *δt* ~ 2.8 10-16 s. From here it follows, that 2*k*B*Tδt* is less than *ћ* on about one order at *T* = 1000 K (almost temperature of fullerene evapora‐

Emergence of the term 2*k*B*Tδt* can be induced by existence of quantum drag [31] owing to different conditions for quantum fluctuations both inside of the fullerene molecule and out‐ side what can induce weak Casimir forces. Because of the weak Casimir force the quantum drag does not lead to decoherence at least in the near zone. However further we shall see that a weak washing out of the Talbot interference pattern is due to existence of this term.

> <sup>B</sup> <sup>2</sup> . {} {} {} *Nz Nz Nz k T*

> dB <sup>1</sup> . {} 2 2 2 *T*

d

<sup>G</sup>

4*πk*B*Tδt*/({*m*}*<sup>N</sup> vz*). In this view we can rewrite Eq. (33) as follows

*m v N z*

**3. Wave function behind the grating**

*t mv mv mv* d

Here *pz* = {*m*}*<sup>N</sup> vz* is a particle momentum along axis *z*. We can define the de Broglie wave‐ length *λ*dB = *h*/*pz* where *h* = 2*πћ* is the Planck constant. Let us also define a length *δ<sup>T</sup>* =

l

The length *δ<sup>T</sup>* tends to zero as *T* → 0. At *T* = 1000 K and at adopted *δt* = 2.8 10-16 s we have *δ<sup>T</sup>* ≈ 0.4 pm. On the other hand, the de Broglie wavelength, *λ*dB, evaluated for the fullerene mol‐ ecule moving with the velocity *vz* =100 m/s is about 5 pm [18]. So, we can see that the length *δ<sup>T</sup>* is less of the de Broglie wavelength on about one order and smaller. A signification of the length *δT* is that it determines decoherence of a particle beam. Decoherence of flying parti‐ cles occurs the quickly, the larger *δT*. Observe that the length *δ<sup>T</sup>* has a close relation with the coherence width - a main parameter in the generalized Gaussian Schell-model [32, 33].

Wave function from one slit after integration over *ξ*0 from -∞ to +∞ has the following view [28]

ppp

<sup>G</sup> = + =L **<sup>i</sup>** (34)

(33)

= + **<sup>i</sup>** <sup>h</sup>

$$p(\mathbf{x}, z) = \left\langle \Psi\_0(\mathbf{x}, z, d, \mathbf{x}\_{s'}, \Lambda) \middle| \Psi\_0(\mathbf{x}, z, d, \mathbf{x}\_{s'}, \Lambda) \right\rangle. \tag{38}$$

Calculation of the wave function (37) is fulfilled for the grating containing *N*<sup>0</sup> = 32 slits. Dis‐ tance between slits is *d* = 105 *λ*dB. So at *λ*dB = 5 pm the distance is equal to 500 nm. Require‐ ment *λ*dB << *d* and *N*0 tending to infinity together with a condition that the particle beam is paraxial, that is, *x*s = 0 and *z*s → -∞, provides emergence in the near-field of an interference pattern, named Talbot carpet [19,20]. Here a spacing along interference patterns is measured in the Talbot length

$$z\_{\rm T} = 2\frac{d^2}{\lambda\_{\rm dB}},\tag{39}$$

which is a convenient natural length at representation of interference patterns. Since we re‐ strict themselves by finite *N*0 we have a defective carpet, which progressively collapses as a spacing from the slit increases. Fig. 5 shows the Talbot carpet, being perfect in the vicinity of the grating slit; it is destroyed progressively with increasing *z*T. As for the Talbot carpet we have the following observation. We see that in a cross-section *z* = *z*T/2 image reproduces ra‐ diation of the slits but phase-shifted by half period between them. At *z* = *z*T radiation of the slits is reproduced again on the same positions where the slits are placed. And so forth.

Evaluation of sizes of the interference pattern is given by ratio of the Talbot length to a length of the slit grating. In our case the Talbot length is *z*T = 0.1 m. And length of the slit grating is about *N*0*d* = 1.6 10-5 m. From here we find that the ratio is 6250. It means that the interference pattern shown in Fig. 5 represents itself a very narrow strip.

**Figure 6.** Blurred interference pattern in the near field. It is shown only the central part of the grating containing *N*0 =

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

http://dx.doi.org/10.5772/53471

199

**Figure 7.** Blurred interference pattern in the near field. It is shown only the central part of the grating containing *N*0 =

**Figure 8.** Destroyed interference pattern because of large δT = 4 pm ~ λdB = 5 pm. *N*0 = 32 slits, *d* = 500 nm.

32 slits; *d* = 500 nm, δT = 0.04 pm << λdB = 5 pm.

32 slits; *d* = 500 nm, δT = 0.4 pm < λdB = 5 pm.

**Figure 5.** Interference pattern in the near field. It is shown only the central part of the grating containing *N*<sup>0</sup> = 32 slits, λdB = 5 pm, *d* = 500 nm, and δT = 0. In the upper part of the figure a set of the Bohmian trajectories, looking like on zigzag curves, drawn in dark blue color is shown.

Zigzag curves, drawn in the upper part of Fig. 5 by dark blue color, show Bohmian trajecto‐ ries that start from the slit No. 15. One can see that particles prefer to move between nodes having positive interference effect and avoid empty lacunas. However the above we noted, that the ratio of the Talbot length to the length of the grating is about 6250 >> 1. It means that really the Bohmian trajectories look almost as straight lines slightly divergent apart. Zigzaglike behavior of the trajectories is almost invisible. Such an almost feebly marked zigzag-like behavior may be induced by fluctuations of virtual particles escorting the real particle.

As soon as we add the term *k*B*Tδt* different from zero (*T* > 0 K) we observe blurring the inter‐ ference pattern. The blurring is the stronger, the larger *k*B*Tδt*. For comparison see Figs. 6 and 7. Here instead of *k*B*Tδt* we write a more convenient parameter, the coherence length *δ*T. This length characterizes a dispersed divergence from initially tuned the de Broglie wavelength. Such a disperse medium can be due to quantum drag on the vacuum fluctuations. Here the Bohmian trajectories are not shown, since because of the thermal term *k*B*Tδt* > 0 a Brownian like scattering of the trajectories arises. This scattering we shall discuss later on.

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference http://dx.doi.org/10.5772/53471 199

spacing from the slit increases. Fig. 5 shows the Talbot carpet, being perfect in the vicinity of the grating slit; it is destroyed progressively with increasing *z*T. As for the Talbot carpet we have the following observation. We see that in a cross-section *z* = *z*T/2 image reproduces ra‐ diation of the slits but phase-shifted by half period between them. At *z* = *z*T radiation of the slits is reproduced again on the same positions where the slits are placed. And so forth.

Evaluation of sizes of the interference pattern is given by ratio of the Talbot length to a length of the slit grating. In our case the Talbot length is *z*T = 0.1 m. And length of the slit grating is about *N*0*d* = 1.6 10-5 m. From here we find that the ratio is 6250. It means that the

**Figure 5.** Interference pattern in the near field. It is shown only the central part of the grating containing *N*<sup>0</sup> = 32 slits, λdB = 5 pm, *d* = 500 nm, and δT = 0. In the upper part of the figure a set of the Bohmian trajectories, looking like on

Zigzag curves, drawn in the upper part of Fig. 5 by dark blue color, show Bohmian trajecto‐ ries that start from the slit No. 15. One can see that particles prefer to move between nodes having positive interference effect and avoid empty lacunas. However the above we noted, that the ratio of the Talbot length to the length of the grating is about 6250 >> 1. It means that really the Bohmian trajectories look almost as straight lines slightly divergent apart. Zigzaglike behavior of the trajectories is almost invisible. Such an almost feebly marked zigzag-like behavior may be induced by fluctuations of virtual particles escorting the real particle.

As soon as we add the term *k*B*Tδt* different from zero (*T* > 0 K) we observe blurring the inter‐ ference pattern. The blurring is the stronger, the larger *k*B*Tδt*. For comparison see Figs. 6 and 7. Here instead of *k*B*Tδt* we write a more convenient parameter, the coherence length *δ*T. This length characterizes a dispersed divergence from initially tuned the de Broglie wavelength. Such a disperse medium can be due to quantum drag on the vacuum fluctuations. Here the Bohmian trajectories are not shown, since because of the thermal term *k*B*Tδt* > 0 a Brownian

like scattering of the trajectories arises. This scattering we shall discuss later on.

zigzag curves, drawn in dark blue color is shown.

198 Advances in Quantum Mechanics

interference pattern shown in Fig. 5 represents itself a very narrow strip.

**Figure 6.** Blurred interference pattern in the near field. It is shown only the central part of the grating containing *N*0 = 32 slits; *d* = 500 nm, δT = 0.04 pm << λdB = 5 pm.

**Figure 7.** Blurred interference pattern in the near field. It is shown only the central part of the grating containing *N*0 = 32 slits; *d* = 500 nm, δT = 0.4 pm < λdB = 5 pm.

**Figure 8.** Destroyed interference pattern because of large δT = 4 pm ~ λdB = 5 pm. *N*0 = 32 slits, *d* = 500 nm.

We may think that the technical vacuum can be not perfect. It causes additional scattering of particles on residual gases. Because of this additional scattering the interference pattern can be destroyed entirely, as shown in Fig. 8.

Disappearance of interference fringes is numerically evaluated by calculating a characteristic called visibility [14, 34]. The fringe visibility [27] is represented as a ratio of difference be‐

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

max min max min . *P P*

Evaluation of *P*max and *P*min is shown in Fig. 9(b). As follows from the figure, the evaluations are fulfilled in a central region of the grating. That is, edges of the grating have to be left far off from the measured zone. The visibility *V* as a function of the parameter *δ*<sup>T</sup> is shown in Fig. 10. One can see that crossover from almost perfect interference fringes, *V* = 1, up to their absence, *V* = 0, begins near *δ*T. ~ *λ*dB. Transition from almost coherent particle beam to inco‐

**Figure 10.** Visibility of interference fringes as a function of the parameter δT ranging from 0.1 to 40 pm. Wavelength

Here we repeat computations of David Bohm [35] which lead to the Hamiltoton-Jacobi equation loaded by the quantum potential and, as consequence, to finding Bohmian trajecto‐ ries. But instead of the Schrödinger equation we choose Eq. (17) that contains complex-val‐

*V*

<sup>h</sup> (41)

→ <sup>1</sup>)−*TN* .

 b

b

yy

**i**

**4. Bohmian trajectories and variance of momenta and positions along**

2 2 <sup>2</sup> ( ) () ( ) . 2 () *<sup>N</sup>*

Here *β* = 2*k*B*Tδt* (in particular, the diffusion coefficient reads *D* = *β*/2*mN*) and *V* =*U* (*q*

h h <sup>h</sup>

¶+ - - = Ñ+ ¶ + **<sup>i</sup> <sup>i</sup>**

*t m* yb

b


http://dx.doi.org/10.5772/53471

201

*P P*

tween maximal and minimal intensities of the fringes to their sum:

herent is a cause of such a crossover [30].

of a matter wave is λdB = 5 pm.

ued parameter Γ=β+i*ћ*:

**paths**

*V*

Now let us draw dependence of the probability density *p*(*x, z*) as a function of *x* at fixed *z*. In other words, we calculate interference fringes in a cross-section of the interference patterns at *z* = *z*T/2 for different values of the length *δ*T. Such a cross-section is chosen because a self image of the slit grating appears phase-shifted by half period of the grating. For that reason we should see the interference fringes spaced between the slit sources of radiation.

Fig. 9 shows three characteristic patterns of the interference fringes. In Fig. 9(a) almost ideal interference fringes are shown obtained at *δ*T < *λ*dB. Fig. 9(b) shows interference fringes ob‐ tained at *δ*T ~ *λ*dB. It is instructive to compare these interference fringes with those that have been measured in experiments [14, 34]. And Fig. 9(c) shows disappearance of interference fringes because of strong scattering of the particles on residual gases in vacuum, *δ*T > *λ*dB.

**Figure 9.** Interference fringes in cross-section of the density distribution pattern by the Talbot half-length, *z* = *z*T/2, (the fringes are drawn in red): (a) δT = 0.4 pm, almost coherent beam; (b) δT = 4 pm, weak coherence; (c) δT = 40 pm, entirely noncoherent beam. Cyan strips show luminosity of slits. The grating consists of *N*<sup>0</sup> = 9 slits. Collapse of the interference pattern on edges of the grating is due to its finite length. Therefore visibility of the interference fringes is evaluated only for 5 central slits.

Disappearance of interference fringes is numerically evaluated by calculating a characteristic called visibility [14, 34]. The fringe visibility [27] is represented as a ratio of difference be‐ tween maximal and minimal intensities of the fringes to their sum:

We may think that the technical vacuum can be not perfect. It causes additional scattering of particles on residual gases. Because of this additional scattering the interference pattern can

Now let us draw dependence of the probability density *p*(*x, z*) as a function of *x* at fixed *z*. In other words, we calculate interference fringes in a cross-section of the interference patterns at *z* = *z*T/2 for different values of the length *δ*T. Such a cross-section is chosen because a self image of the slit grating appears phase-shifted by half period of the grating. For that reason

Fig. 9 shows three characteristic patterns of the interference fringes. In Fig. 9(a) almost ideal interference fringes are shown obtained at *δ*T < *λ*dB. Fig. 9(b) shows interference fringes ob‐ tained at *δ*T ~ *λ*dB. It is instructive to compare these interference fringes with those that have been measured in experiments [14, 34]. And Fig. 9(c) shows disappearance of interference fringes because of strong scattering of the particles on residual gases in vacuum, *δ*T > *λ*dB.

**Figure 9.** Interference fringes in cross-section of the density distribution pattern by the Talbot half-length, *z* = *z*T/2, (the fringes are drawn in red): (a) δT = 0.4 pm, almost coherent beam; (b) δT = 4 pm, weak coherence; (c) δT = 40 pm, entirely noncoherent beam. Cyan strips show luminosity of slits. The grating consists of *N*<sup>0</sup> = 9 slits. Collapse of the interference pattern on edges of the grating is due to its finite length. Therefore visibility of the interference fringes is

we should see the interference fringes spaced between the slit sources of radiation.

be destroyed entirely, as shown in Fig. 8.

200 Advances in Quantum Mechanics

evaluated only for 5 central slits.

$$V = \frac{P\_{\text{max}} - P\_{\text{min}}}{P\_{\text{max}} + P\_{\text{min}}}.\tag{40}$$

Evaluation of *P*max and *P*min is shown in Fig. 9(b). As follows from the figure, the evaluations are fulfilled in a central region of the grating. That is, edges of the grating have to be left far off from the measured zone. The visibility *V* as a function of the parameter *δ*<sup>T</sup> is shown in Fig. 10. One can see that crossover from almost perfect interference fringes, *V* = 1, up to their absence, *V* = 0, begins near *δ*T. ~ *λ*dB. Transition from almost coherent particle beam to inco‐ herent is a cause of such a crossover [30].

**Figure 10.** Visibility of interference fringes as a function of the parameter δT ranging from 0.1 to 40 pm. Wavelength of a matter wave is λdB = 5 pm.

### **4. Bohmian trajectories and variance of momenta and positions along paths**

Here we repeat computations of David Bohm [35] which lead to the Hamiltoton-Jacobi equation loaded by the quantum potential and, as consequence, to finding Bohmian trajecto‐ ries. But instead of the Schrödinger equation we choose Eq. (17) that contains complex-val‐ ued parameter Γ=β+i*ћ*:

$$(\boldsymbol{\beta} - \mathbf{i}\hbar) \frac{\partial \boldsymbol{\psi}}{\partial t} = \frac{(\boldsymbol{\beta}^2 + \hbar^2)}{2m\_N} \nabla^2 \boldsymbol{\psi} + \frac{(\boldsymbol{\beta} - \mathbf{i}\hbar)}{(\boldsymbol{\beta} + \mathbf{i}\hbar)} V \boldsymbol{\psi} \,. \tag{41}$$

Here *β* = 2*k*B*Tδt* (in particular, the diffusion coefficient reads *D* = *β*/2*mN*) and *V* =*U* (*q* → <sup>1</sup>)−*TN* . Further we apply polar representation of the wave function, *ψ* = *R* exp{i*S*/*ћ*}. It leads to ob‐ taining two equations for real and imaginary parts that deal with real-valued functions *R* and *S*. The function *R* is the amplitude of the wave function and *S*/*ћ* is its phase. After series of computations, aim of which is to put together real and imaginary terms, we obtain the following equations

$$\frac{\partial \mathbf{\hat{S}}}{\partial t} + \beta \frac{1}{\mathbf{R}} \frac{\partial \mathbf{R}}{\partial t} = \underbrace{- (\boldsymbol{\beta}^2 + \boldsymbol{\hbar}^2)}\_{\text{(a)}} (\nabla \mathbf{S})^2 + \underbrace{(\boldsymbol{\beta}^2 + \boldsymbol{\hbar}^2)}\_{\text{(b)}} \frac{\nabla^2 \mathbf{R}}{R} + \underbrace{(\boldsymbol{\beta}^2 - \boldsymbol{\hbar}^2)}\_{\text{(b)}} V\_{,} \tag{42}$$

( ) <sup>1</sup> <sup>1</sup> Im

positions of the particle in each current time beginning from the grating' slits up to a detec‐

Here *t* is a current time that starts from *t*=0 on a slit source and *δt* is an arbitrarily small in‐ crement of time. Some calculated trajectories of particles, the Bohmian trajectories, are shown in upper part of Fig. 5. It should be noted that the Bohmian trajectories follow from exact solutions of Eqs. (44)-(45). These equations give a rule for finding geodesic trajectories and secants of equal phases, *S/ћ*, at given boundary conditions. The geodesic trajectories point to tendency of the particle migration along paths. And the secant surfaces describe a

*rt t rt vt t* ( ) () () . += +

y

 d

2 2

¶ <sup>h</sup> <sup>1442443</sup> <sup>14243</sup> (49)

¶ <sup>1442443</sup> (50)

¶ <sup>123</sup> <sup>14243</sup> (51)

∇*R* =2∇ln(*R*)=∇ln(*ρ*). And *β*/2*mN* =*D* is the diffu‐

 b

(b) (a) () , <sup>2</sup> <sup>2</sup> *<sup>N</sup> <sup>N</sup> <sup>R</sup> RS R t m m*

(c)

<sup>2</sup> . In the second equation we may replace the term ∇*S* by *mN <sup>v</sup>*

2

(a)

( ln( ), ) ( , ). *N N*

r r r 14444244443 (52)

2

l

p

<sup>1</sup> ( 2 ), <sup>2</sup> *<sup>N</sup> <sup>S</sup> S RS tm R*

Here we take into consideration that in the first equation we may replace

 y

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

r rr (48)

(47)

203

http://dx.doi.org/10.5772/53471

<sup>→</sup> as follows

*N N*

d

2

=- Ñ + Ñ

2

2

( ) 2 2

*<sup>S</sup> D S m v D S m uv*

r

(c)

¶ = Ñ+ Ñ =Ñ+

¶ =Ñ -

dB (b)

<sup>4</sup> , *<sup>R</sup> D R DR*

= Ñ + ÑÑ

b

b


*v S m m*

coherence of all the passing particles created on a single source.

In case of *β* >> *ћ* we have the following two equations

¶

¶

sion coefficient. Now we may rewrite Eqs. (49)-(50) as follows

*t*

(∇*S* / ℏ)

<sup>2</sup> =4*<sup>π</sup>* <sup>2</sup> / *<sup>λ</sup>*dB

from Eq. (46). We notice also, that 2*R* <sup>−</sup><sup>1</sup>

*t*

¶

tor is calculated by the following formula

$$\frac{\partial}{\partial t^2} \frac{\partial \mathbf{S}}{\partial t} - \frac{1}{R} \frac{\partial \mathbf{R}}{\partial t} = \frac{(\boldsymbol{\beta}^2 + \boldsymbol{\hbar}^2)}{2m\_N \hbar^2} (\boldsymbol{\nabla}^2 \mathbf{S} + 2 \frac{1}{R} \boldsymbol{\nabla} \mathbf{R} \boldsymbol{\nabla} \mathbf{S}) - \frac{2\beta}{(\boldsymbol{\beta}^2 + \boldsymbol{\hbar}^2)} V. \tag{43}$$

Firstly, we can see that at *β* = 0 Eq. (42) reduces to the modified Hamilton-Jacobi equation due to loaded the quantum potential that is enveloped here by brace (b). And Eq. (43) re‐ duces to the continuity equation. These equations read

$$\frac{\partial \mathbf{\tilde{S}}}{\partial t} = -\underbrace{\frac{1}{2m\_N} (\nabla \mathbf{S})^2}\_{\text{(a)}} + \underbrace{\frac{\hbar^2}{2m\_N} \frac{\nabla^2 R}{R}}\_{\text{(b)}} + V\_{\text{\textquotedblleft}} \tag{44}$$

$$-\frac{1}{R}\frac{\partial R}{\partial t} = \frac{1}{2m\_N} \underbrace{\left(\nabla^2 S + 2\frac{1}{R}\nabla R \nabla S\right)}\_{\text{(c)}}.\tag{45}$$

Terms enveloped by braces (a), (b), and (c) are the kinetic energy of the particle, the quan‐ tum potential, and the right part is a kernel of the continuity equation (45), respectively. In particular, the term 2∇*R* / *R* =∇ln(*R* <sup>2</sup> ) relates to the osmotic velocity, see Eq. (23). Eqs. (44) and (45) are the same equations obtained by Bohm [35]. From historical viewpoint it should be noted that the same equations were published by Madelung1 in 1926 [36].

Momentum of the particle reads

$$
\vec{p} = m\_N \vec{\upsilon} = \nabla \mathcal{S}\_\prime \tag{46}
$$

where *v* <sup>→</sup> is its current velocity. The de Broglie equation relates the momentum *p* to the wave‐ length *λ*dB = *h/p*, where *h =* 2*πћ* is the Planck constant. Now, as soon as we found the current velocity

<sup>1</sup> My attention to the Madelung' article was drawn by Prof. M. Berry.

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference http://dx.doi.org/10.5772/53471 203

$$\vec{\boldsymbol{v}} = \frac{1}{m\_N} \nabla \mathbf{S} = \frac{\hbar}{m\_N} \text{Im} \left( \left| \boldsymbol{\nu} \right\rangle^{-1} \nabla \left| \boldsymbol{\nu} \right\rangle \right) \tag{47}$$

positions of the particle in each current time beginning from the grating' slits up to a detec‐ tor is calculated by the following formula

$$
\vec{r}(t + \delta t) = \vec{r}(t) + \vec{v}(t)\delta t.\tag{48}
$$

Here *t* is a current time that starts from *t*=0 on a slit source and *δt* is an arbitrarily small in‐ crement of time. Some calculated trajectories of particles, the Bohmian trajectories, are shown in upper part of Fig. 5. It should be noted that the Bohmian trajectories follow from exact solutions of Eqs. (44)-(45). These equations give a rule for finding geodesic trajectories and secants of equal phases, *S/ћ*, at given boundary conditions. The geodesic trajectories point to tendency of the particle migration along paths. And the secant surfaces describe a coherence of all the passing particles created on a single source.

In case of *β* >> *ћ* we have the following two equations

Further we apply polar representation of the wave function, *ψ* = *R* exp{i*S*/*ћ*}. It leads to ob‐ taining two equations for real and imaginary parts that deal with real-valued functions *R* and *S*. The function *R* is the amplitude of the wave function and *S*/*ћ* is its phase. After series of computations, aim of which is to put together real and imaginary terms, we obtain the

(b) (a)

*t Rt m m R*

*t Rt m R*

 b

duces to the continuity equation. These equations read

¶

be noted that the same equations were published by Madelung1

1 My attention to the Madelung' article was drawn by Prof. M. Berry.

particular, the term 2∇*R* / *R* =∇ln(*R* <sup>2</sup>

Momentum of the particle reads

where *v*

velocity

¶¶ + +Ñ - + =- Ñ + + ¶ ¶ +

2 2 <sup>2</sup> 2 2 2 2

¶¶ + - = Ñ + ÑÑ - ¶ ¶ + h

b

b

b

1 () ()() ( ) , <sup>2</sup> <sup>2</sup> *<sup>N</sup>* ( ) *<sup>N</sup> S R <sup>R</sup> S V*

(c) 1() 1 2 (2 ) . <sup>2</sup> ( ) *<sup>N</sup> S R S RS <sup>V</sup>*

Firstly, we can see that at *β* = 0 Eq. (42) reduces to the modified Hamilton-Jacobi equation due to loaded the quantum potential that is enveloped here by brace (b). And Eq. (43) re‐

2 2 <sup>2</sup>

h

<sup>1</sup> ( ) , 2 2 *N N S R S V t m mR* ¶ Ñ =- Ñ + +

(a) (b)

2

Terms enveloped by braces (a), (b), and (c) are the kinetic energy of the particle, the quan‐ tum potential, and the right part is a kernel of the continuity equation (45), respectively. In

and (45) are the same equations obtained by Bohm [35]. From historical viewpoint it should

11 1 ( 2 ). <sup>2</sup> *<sup>N</sup> <sup>R</sup> S RS*

*Rt m R* ¶ - = Ñ + ÑÑ

(c)

<sup>→</sup> is its current velocity. The de Broglie equation relates the momentum *p* to the wave‐ length *λ*dB = *h/p*, where *h =* 2*πћ* is the Planck constant. Now, as soon as we found the current

2 2 2 22 2 2 <sup>2</sup> 2 2 2

h hh

bb

hh h <sup>1442443</sup> (43)

b

b

14243 14243 (44)

¶ <sup>1442443</sup> (45)

) relates to the osmotic velocity, see Eq. (23). Eqs. (44)

, *<sup>N</sup> p mv S* = = Ñ r r (46)

in 1926 [36].

b

h h <sup>1442443</sup> <sup>1442443</sup> (42)

following equations

202 Advances in Quantum Mechanics

$$\frac{\partial \mathcal{R}}{\partial t} = -\underbrace{\frac{\beta}{2m\_N \hbar^2} R (\nabla S)^2}\_{\text{(a)}} + \underbrace{\frac{\beta}{2m\_N} \nabla^2 R}\_{\text{(b)}}\tag{49}$$

$$\frac{\partial \mathbf{S}}{\partial t}\_{\text{(\%)}} = \frac{\beta}{2m\_N} \underbrace{\langle \nabla^2 \mathbf{S} + 2\frac{1}{R} \nabla R \nabla S \rangle}\_{\text{(\%)}} \tag{50}$$

Here we take into consideration that in the first equation we may replace (∇*S* / ℏ) <sup>2</sup> =4*<sup>π</sup>* <sup>2</sup> / *<sup>λ</sup>*dB <sup>2</sup> . In the second equation we may replace the term ∇*S* by *mN <sup>v</sup>* <sup>→</sup> as follows from Eq. (46). We notice also, that 2*R* <sup>−</sup><sup>1</sup> ∇*R* =2∇ln(*R*)=∇ln(*ρ*). And *β*/2*mN* =*D* is the diffu‐ sion coefficient. Now we may rewrite Eqs. (49)-(50) as follows

$$\frac{\partial \mathcal{R}}{\partial t} = \underbrace{D \nabla^2 \mathcal{R}}\_{\text{(b)}} - \underbrace{\frac{4\pi^2}{\lambda\_{\text{dB}}^2} DR}\_{\text{(a)}} \tag{51}$$

$$\frac{\partial \mathcal{S}}{\partial t} = \underbrace{D \Big(\nabla^2 \mathcal{S} + m\_N (\nabla \ln(\rho), \vec{v})\Big)}\_{\text{(c)}} = D\nabla^2 \mathcal{S} + m\_N \langle \vec{u}, \vec{v} \rangle. \tag{52}$$

Here *u* <sup>→</sup> =*D* ∇ln(*ρ*) is the osmotic velocity defined in Eq. (23). We got the two diffusion equa‐ tions coupled with each other through sources. Namely, this coupling is provided due to the de Broglie wavelength and the osmotic velocity which can change with time. These diffu‐ sion equations cardinally differ from Eqs. (44)-(45). Because of diffusive nature of these sup‐ plementary parts blurring of interference patterns occurs. It leads to degeneration of the Bohmian trajectories to Brownian ones.

#### **4.1. Dispersion of trajectories and the uncertainty principle**

As for the Bohmian trajectories there is a problem concerning their possible existence. As follows from Eqs. (46) and (48) in each moment of time there are definite values of the mo‐ mentum and the coordinate of a particle moving along the Bohmian trajectory. This state‐ ment enters in conflict with the uncertainty principle.

Here we try to retrace emergence of the uncertainty principle stemming from standard probability-theoretical computations of expectation value and variance of a particle momen‐ tum. We adopt a wave function in the polar representation

$$\left| \Psi \right> = \operatorname{Re} \exp \{ \mathbf{i} \, \mathbf{S} / \hbar \} \,\tag{53}$$

Let us now calculate variance of the velocity V g. This computation reads

spective to multiply Var(V g) by *m*/2

the following result

*m*

(45). We rewrite the quantum potential as follows

As for the second term in Eq. (57) we have *i*(ℏ / 2)∇*V*

the real part of this expression. It reads

1 1 Var( ) . *<sup>m</sup> m m <sup>m</sup>* æ ö = Y - Ñ- Y = Y Ñ + Ñ+ - DY ç ÷ Y Y è ø Y Y

<sup>2</sup> <sup>1</sup> Var( ) . <sup>2</sup> 2 2

Y Y

2 2 2 2

22 2 2

2 2

**<sup>i</sup>** <sup>h</sup> <sup>h</sup> <sup>r</sup> *V V g g* (57)

14243 14243 <sup>144424443</sup> (58)

<sup>Ñ</sup> =- =- Ñ + Ñ hh h (59)

Ñ Ñ = Ñ- Ñ - **<sup>i</sup>** <sup>h</sup> <sup>h</sup> *Vg* (60)

*<sup>g</sup>* <sup>=</sup>*i*(<sup>ℏ</sup> / <sup>2</sup>*m*)*Δ<sup>S</sup>* <sup>−</sup>(ℏ<sup>2</sup> / <sup>2</sup>*m*)*ΔSQ*. It is fol‐

→

**<sup>i</sup> i i** <sup>h</sup> <sup>r</sup> hh h r r

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

*<sup>2</sup> Vg Vg V VV g gg* (56)

Terms over bracket (d) kill each other as follows from Eq. (54). It is reasonable in the per‐

*m* =- Y DY + Ñ

(a) (b) (c)

*<sup>R</sup> S R S S*

So, this expression has a dimensionality of energy. The first term to be computed represents

<sup>1</sup> <sup>1</sup> (, ) ( ) 2 .

*m m mR m R* <sup>Ñ</sup> æ ö Ñ Ñ - Y DY = Ñ - - D + ç ÷ Y Y è ø **<sup>i</sup>** h h <sup>h</sup>

Here the term enveloped by bracket (a) is a kinetic energy of the particle, the term envel‐ oped by bracket (b) with negative sign added is the quantum potential *Q*, and the term en‐ veloped by bracket (c) comes from the continuity equation. See for comparison Eqs. (44) and

> 22 2 2 2 2 () . 22 2 *Q Q <sup>R</sup> Q SS mR m m*

lows from computation by Eq. (54). As a result, the expression (57) takes the following view

*m m mR*

One can see that the variance consists of real and imaginary parts. Observe that the right side is represented through square of gradient of the complexified action [28], namely (∇(*S* + iℏ*SQ*))<sup>2</sup> / 2*m*. We shall not consider here the imaginary part. Instead we shall consider

<sup>2</sup> <sup>1</sup> 2 2 (,) Var( ) ( ) ( ) . 2 22 *<sup>Q</sup> <sup>m</sup> S R S S*

2

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205

(d)

1442443

where *R* = (*ρ*)1/2 is the amplitude of the wave function (*ρ* =*R*<sup>2</sup> = <*Ψ*|*Ψ*> is the probability den‐ sity) and *S/ћ* is its phase. Momentum operator *p* ^ <sup>=</sup> <sup>−</sup>*i*ℏ∇and corresponding velocity opera‐ tor *v* ^ <sup>=</sup> <sup>−</sup>*i*(<sup>ℏ</sup> / *<sup>m</sup>*)∇are kinetic operators in quantum mechanics. Here *m* is mass of the particle. Expectation value of the velocity operator reads

$$
\hat{\mathbf{V}}\_{\mathcal{S}} = \frac{1}{\left\langle \Psi \left| \Psi \right\rangle} \left\langle \Psi \right| - \mathbf{i} \frac{\hbar}{m} \nabla \left| \Psi \right\rangle = \frac{1}{m} \left( \nabla S + \mathbf{i} \hbar \nabla S\_{\mathcal{Q}} \right). \tag{54}
$$

The velocity V g is seen to be complex-valued. Here *SQ* = −ln(*R*)= −ln(*ρ*) / 2 is the quantum entropy [28] and (*ћ/*2*m*) is the quantum diffusion coefficient [6, 7]. Therefore its imaginary part is a quantum osmotic velocity

$$
\vec{\mu}\_Q = - (\hbar \,/\, m) \nabla S\_Q = (\hbar \,/\, 2m) \nabla \ln(\rho). \tag{55}
$$

It is instructive to compare this velocity with the classical osmotic velocity given in Eq. (23). As can see the osmotic velocity stems from gradient of entropy that evaluates degree of or‐ der and disorder on a quantum level, likely of vacuum fluctuations.

Real part of Eq. (54) gives the current velocity *v* <sup>→</sup> defined by Eq. (47). It should be noted that because of existence of imaginary unit in definition of the momentum operator, real part of Eq. (54) is taken as the current velocity. Whereas imaginary unit is absent in Eq. (47). There‐ fore at computing the current velocity by Eq. (47) we take imaginary part.

Let us now calculate variance of the velocity V g. This computation reads

Here *u*

204 Advances in Quantum Mechanics

tor *v*

Bohmian trajectories to Brownian ones.

**4.1. Dispersion of trajectories and the uncertainty principle**

ment enters in conflict with the uncertainty principle.

tum. We adopt a wave function in the polar representation

where *R* = (*ρ*)1/2 is the amplitude of the wave function (*ρ* =*R*<sup>2</sup>

Y Y

sity) and *S/ћ* is its phase. Momentum operator *p*

Expectation value of the velocity operator reads

Real part of Eq. (54) gives the current velocity *v*

part is a quantum osmotic velocity

<sup>→</sup> =*D* ∇ln(*ρ*) is the osmotic velocity defined in Eq. (23). We got the two diffusion equa‐ tions coupled with each other through sources. Namely, this coupling is provided due to the de Broglie wavelength and the osmotic velocity which can change with time. These diffu‐ sion equations cardinally differ from Eqs. (44)-(45). Because of diffusive nature of these sup‐ plementary parts blurring of interference patterns occurs. It leads to degeneration of the

As for the Bohmian trajectories there is a problem concerning their possible existence. As follows from Eqs. (46) and (48) in each moment of time there are definite values of the mo‐ mentum and the coordinate of a particle moving along the Bohmian trajectory. This state‐

Here we try to retrace emergence of the uncertainty principle stemming from standard probability-theoretical computations of expectation value and variance of a particle momen‐

^ <sup>=</sup> <sup>−</sup>*i*(<sup>ℏ</sup> / *<sup>m</sup>*)∇are kinetic operators in quantum mechanics. Here *m* is mass of the particle.

( ) 1 1 *<sup>S</sup>* . *m m* = Y- ÑY = Ñ + Ñ

The velocity V g is seen to be complex-valued. Here *SQ* = −ln(*R*)= −ln(*ρ*) / 2 is the quantum entropy [28] and (*ћ/*2*m*) is the quantum diffusion coefficient [6, 7]. Therefore its imaginary

It is instructive to compare this velocity with the classical osmotic velocity given in Eq. (23). As can see the osmotic velocity stems from gradient of entropy that evaluates degree of or‐

because of existence of imaginary unit in definition of the momentum operator, real part of Eq. (54) is taken as the current velocity. Whereas imaginary unit is absent in Eq. (47). There‐

( / ) ( / 2 ) ln( ). *Q Q u mS m* =- Ñ = Ñ

der and disorder on a quantum level, likely of vacuum fluctuations.

fore at computing the current velocity by Eq. (47) we take imaginary part.

**i i** <sup>r</sup> <sup>h</sup> *<sup>V</sup>* <sup>h</sup> *g QS* (54)

r<sup>r</sup> h h (55)

Y = *R S* exp , {**i** h} (53)

= <*Ψ*|*Ψ*> is the probability den‐

^ <sup>=</sup> <sup>−</sup>*i*ℏ∇and corresponding velocity opera‐

<sup>→</sup> defined by Eq. (47). It should be noted that

$$\mathrm{Var}(\mathsf{V}\_{\mathcal{g}}) = \frac{1}{\left\langle \boldsymbol{\Psi} \middle| \boldsymbol{\Psi} \right\rangle} \left\langle \boldsymbol{\Psi} \middle| \left( -\mathrm{i}\frac{\hbar}{m}\boldsymbol{\nabla} - \bar{\mathsf{V}}\_{\mathcal{g}} \right)^{2} \middle| \boldsymbol{\Psi} \right\rangle = \frac{1}{\left\langle \boldsymbol{\Psi} \middle| \boldsymbol{\Psi} \right\rangle} \left\langle \boldsymbol{\Psi} \middle| \left| \mathrm{i}\frac{\hbar}{m}\boldsymbol{\nabla}\bar{\mathsf{V}}\_{\mathcal{g}} + \mathrm{i}\frac{\hbar}{m}\boldsymbol{\mathring{\mathsf{V}}\_{\mathcal{g}}\boldsymbol{\nabla} + \underline{\mathsf{V}\_{\mathcal{g}}^{2}}} - \frac{\hbar^{2}}{m^{2}}\boldsymbol{\Delta} \middle| \boldsymbol{\Psi} \right\rangle. \tag{56}$$

Terms over bracket (d) kill each other as follows from Eq. (54). It is reasonable in the per‐ spective to multiply Var(V g) by *m*/2

$$\frac{m}{2}\text{Var}(\mathbb{V}\_{\mathcal{g}}) = -\frac{1}{\left\langle \Psi \middle| \Psi \right\rangle} \left\langle \Psi \middle| \frac{\hbar^2}{2m} \Delta \middle| \Psi \right\rangle + \text{i} \frac{\hbar}{2} \nabla \bar{\mathbb{V}}\_{\mathcal{g}}.\tag{57}$$

So, this expression has a dimensionality of energy. The first term to be computed represents the following result

$$-\frac{1}{\left\langle \Psi \left| \Psi \right\rangle} \left\langle \Psi \left| \frac{\hbar^2}{2m} \Delta \right| \Psi \right\rangle = \underbrace{\frac{1}{2m} (\nabla S)^2}\_{\text{(a)}} - \underbrace{\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R}}\_{\text{(b)}} - \underbrace{\hbar}\_{2m} \underbrace{\left( \Delta S + 2 \frac{\{\nabla S, \nabla R\}}{R} \right)}\_{\text{(c)}}.\tag{58}$$

Here the term enveloped by bracket (a) is a kinetic energy of the particle, the term envel‐ oped by bracket (b) with negative sign added is the quantum potential *Q*, and the term en‐ veloped by bracket (c) comes from the continuity equation. See for comparison Eqs. (44) and (45). We rewrite the quantum potential as follows

$$Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R} = -\frac{\hbar^2}{2m} (\nabla S\_Q)^2 + \frac{\hbar^2}{2m} \nabla^2 S\_Q. \tag{59}$$

As for the second term in Eq. (57) we have *i*(ℏ / 2)∇*V* → *<sup>g</sup>* <sup>=</sup>*i*(<sup>ℏ</sup> / <sup>2</sup>*m*)*Δ<sup>S</sup>* <sup>−</sup>(ℏ<sup>2</sup> / <sup>2</sup>*m*)*ΔSQ*. It is fol‐ lows from computation by Eq. (54). As a result, the expression (57) takes the following view

$$\frac{m}{2}\text{Var}(\mathcal{V}\_{\mathcal{G}}) = \frac{1}{2m}(\nabla \mathcal{S})^2 - \frac{\hbar^2}{2m}(\nabla \mathcal{S}\_{\mathcal{Q}})^2 - \mathbf{i}\frac{\hbar}{m}\frac{(\nabla \mathcal{S}, \nabla \mathcal{R})}{R}.\tag{60}$$

One can see that the variance consists of real and imaginary parts. Observe that the right side is represented through square of gradient of the complexified action [28], namely (∇(*S* + iℏ*SQ*))<sup>2</sup> / 2*m*. We shall not consider here the imaginary part. Instead we shall consider the real part of this expression. It reads

$$\frac{m}{2}\text{Re}(\text{Var}(V\_{\mathcal{g}})) = \frac{1}{2m}(\nabla \cdot \mathbf{S})^2 - \hbar \left(\frac{\hbar}{2m}(\nabla \cdot \mathbf{S}\_{\mathcal{Q}})^2\right). \tag{61}$$

The first term in this expression represents kinetic energy, *E*, of the particle. The second term, stemming from the quantum potential, contains under braces a term having dimen‐ sionality of inverse time, that is, of frequency

$$
\omega\_{\mathbb{Q}} = \frac{\hbar}{2m} (\nabla \cdot \mathbb{S}\_{\mathbb{Q}})^2 \tag{62}
$$

*δ p* <sup>→</sup> *δr*

Each nanoparticle incident on a slit grating passes only through a single slit. Its path runs along a Bohmian trajectory which is represented as an optimal path for the nanoparticle mi‐ grating from a source to a detector. Unfortunately, the Bohmian trajectory can not be observ‐ able since a serious obstacle for the observation comes from the uncertainty principle. In other words, an attempt to measure any attribute of the nanoparticle, be it position or orien‐ tation, i.e., the particle momentum, leads to destroying information relating to future history of the nanoparticle. What is more, any collision of the nanoparticle with a foreign particle destroys the Bohmian trajectory which could give a real contribution to the interference pat‐ tern. It relates closely with quality of vacuum. In the case of a bad vacuum such collisions will occur frequently. They lead to destruction of the Bohmian trajectories. Actually, they

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

Excellent article [21] of Couder & Fort with droplets gives, however, a clear hint of what happens when the nanoparticle passes through a single grating slit. In the light of this hint we may admit that the particle "bouncing at moving through vacuum" generates a wave at each bounce. So, a holistic quantum mechanical object is the particle + wave. Here the wave to be generated by the particle plays a role of the pilot-wave first formulated by Lui de Bro‐ glie and later developed by Bohm [37]. It is interesting to note in this context, that the pilot-

A particle passing through vacuum generates waves with wavelength that is inversely pro‐ portional to its momentum (it follows from the de Broglie formula, *λ* = *h/p*, where *h* is Planck's constant). One can guess that a role of the vacuum in the experiment of Couder & Fort takes upon itself a silicon oil surface with subcritical Faraday ripples activated on it [21]. Observe that pattern of the ripples is changed in the vicinity of extraneous bodies im‐ mersed in the oil which simulate grating slits. Interference of the ripples with waves gener‐ ated by the bouncing droplets provides optimal paths for the droplets traveling through the slits and further. As a result we may observe an interference pattern emergent depending on

Now we may suppose that the subcritical Faraday ripples on the silicon oil surface simulate vacuum fluctuations. Consequently, the vacuum fluctuations change their own pattern near the slit grating depending on amount of slits and distance between them. We may imagine that the particle passing through vacuum (bouncing through, Fig. 11) initiates waves which interfere with the vacuum fluctuations. As a result of such an interference the particle moves along an optimal path - along the Bohmian trajectory. Mathematically the bounce is imitated by an exponential term exp{i*S/ћ*}, where the angle *S/ћ* parametrizes the group of rotation

<sup>→</sup> =*mδv* → .

Here we take into account *δ p*

**5. Concluding remarks**

degenerate to Brownian trajectories.

waves have many common with Huygens waves [5].

amount of slits in the grating and distance between them.

<sup>→</sup> ≥ ℏ (67)

http://dx.doi.org/10.5772/53471

207

This frequency multiplied by *ћ* represents an energy binding a particle with vacuum fluctu‐ ations. This energy, as follows from Eq. (61), is equal to the particle mass multiplied by squared the osmotic velocity (55) and divided by 2. It is an osmotic kinetic energy. In the light of the above said we rewrite Eq. (61) in the following way

$$\frac{1}{2}\frac{m}{2}\text{Re}\{\text{Var}(V\_{\mathcal{g}})\} = E - \hbar\omega\_{\mathcal{Q}} \ge 0. \tag{63}$$

Let we have two Bohmian trajectories. Along one trajectory we have *E*1 - *ћωQ,*1, and along other trajectory we have a perturbed value *E*2 - *ћωQ,*2. Subtracting one from other we have

$$
\delta E - \hbar \delta \omega\_Q \ge 0. \tag{64}
$$

One can suppose that emergence of the second trajectory was conditioned by a perturbation of the particle moving along the first trajectory. If it is so, then emergence of the second tra‐ jectory stems from an operation of measurement of some parameters of the particle. One can think that duration of the measurement is about *δt* = 1/*δωQ*. From here we find

$$
\delta E \delta t \geq \hbar \tag{65}
$$

Now let us return to Eq. (48) and rewrite it in the following view

$$
\bar{\delta r}(t) = \bar{\upsilon}\_1(t)\delta t \ge \bar{\upsilon}\_1(t)\hbar/\delta E. \tag{66}
$$

The initial Bohmian trajectory is marked here by subscript 1. Observe that *δE* =*m*(*v*<sup>2</sup> <sup>2</sup> <sup>−</sup>*v*<sup>1</sup> 2 ) / 2≈*mv* → <sup>1</sup>*δv* <sup>→</sup> . Here we have calculated *v*<sup>2</sup> <sup>2</sup> =(*v* → <sup>1</sup> + *δv* <sup>→</sup> )<sup>2</sup> ≈*v*<sup>1</sup> <sup>2</sup> + 2*v* → <sup>1</sup>*δv* <sup>→</sup> . Substituting computations of *δE* into Eq. (66) we obtain finally

$$\begin{array}{c|c}\hline\hline\delta\vec{p}\delta\vec{r} \ge \hbar\\\hline\hline\end{array}\tag{67}$$

Here we take into account *δ p* <sup>→</sup> =*mδv* → .

### **5. Concluding remarks**

*m*

206 Advances in Quantum Mechanics

sionality of inverse time, that is, of frequency

<sup>2</sup> Re(Var(*Vg*)) <sup>=</sup> <sup>1</sup>

light of the above said we rewrite Eq. (61) in the following way

*m*

<sup>2</sup>*<sup>m</sup>* (∇*S*)

*<sup>ω</sup><sup>Q</sup>* <sup>=</sup> <sup>ℏ</sup>

The first term in this expression represents kinetic energy, *E*, of the particle. The second term, stemming from the quantum potential, contains under braces a term having dimen‐

This frequency multiplied by *ћ* represents an energy binding a particle with vacuum fluctu‐ ations. This energy, as follows from Eq. (61), is equal to the particle mass multiplied by squared the osmotic velocity (55) and divided by 2. It is an osmotic kinetic energy. In the

Let we have two Bohmian trajectories. Along one trajectory we have *E*1 - *ћωQ,*1, and along other trajectory we have a perturbed value *E*2 - *ћωQ,*2. Subtracting one from other we have

One can suppose that emergence of the second trajectory was conditioned by a perturbation of the particle moving along the first trajectory. If it is so, then emergence of the second tra‐ jectory stems from an operation of measurement of some parameters of the particle. One can

think that duration of the measurement is about *δt* = 1/*δωQ*. From here we find

<sup>→</sup> 1(*t*)*δt* ≥ *v*

<sup>→</sup> . Here we have calculated *v*<sup>2</sup>

→

The initial Bohmian trajectory is marked here by subscript 1. Observe that

<sup>2</sup> =(*v* → <sup>1</sup> + *δv*

Now let us return to Eq. (48) and rewrite it in the following view

*δr* <sup>→</sup> (*t*)=*v*

computations of *δE* into Eq. (66) we obtain finally

*δE* =*m*(*v*<sup>2</sup>

<sup>2</sup> <sup>−</sup>*v*<sup>1</sup> 2 ) / 2≈*mv* → <sup>1</sup>*δv* <sup>2</sup> −ℏ( <sup>ℏ</sup>

<sup>2</sup>*<sup>m</sup>* (∇*SQ*)2). (61)

<sup>2</sup>*<sup>m</sup>* (∇*SQ*)<sup>2</sup> (62)

<sup>2</sup> Re(Var(*Vg*)) <sup>=</sup>*<sup>E</sup>* −ℏ*ω<sup>Q</sup>* <sup>≥</sup> 0. (63)

*δE* −ℏ*δω<sup>Q</sup>* ≥ 0. (64)

*δEδt* ≥ ℏ (65)

<sup>1</sup>(*t*)ℏ / *δE*. (66)

<sup>2</sup> + 2*v* → <sup>1</sup>*δv*

<sup>→</sup> . Substituting

<sup>→</sup> )<sup>2</sup> ≈*v*<sup>1</sup>

Each nanoparticle incident on a slit grating passes only through a single slit. Its path runs along a Bohmian trajectory which is represented as an optimal path for the nanoparticle mi‐ grating from a source to a detector. Unfortunately, the Bohmian trajectory can not be observ‐ able since a serious obstacle for the observation comes from the uncertainty principle. In other words, an attempt to measure any attribute of the nanoparticle, be it position or orien‐ tation, i.e., the particle momentum, leads to destroying information relating to future history of the nanoparticle. What is more, any collision of the nanoparticle with a foreign particle destroys the Bohmian trajectory which could give a real contribution to the interference pat‐ tern. It relates closely with quality of vacuum. In the case of a bad vacuum such collisions will occur frequently. They lead to destruction of the Bohmian trajectories. Actually, they degenerate to Brownian trajectories.

Excellent article [21] of Couder & Fort with droplets gives, however, a clear hint of what happens when the nanoparticle passes through a single grating slit. In the light of this hint we may admit that the particle "bouncing at moving through vacuum" generates a wave at each bounce. So, a holistic quantum mechanical object is the particle + wave. Here the wave to be generated by the particle plays a role of the pilot-wave first formulated by Lui de Bro‐ glie and later developed by Bohm [37]. It is interesting to note in this context, that the pilotwaves have many common with Huygens waves [5].

A particle passing through vacuum generates waves with wavelength that is inversely pro‐ portional to its momentum (it follows from the de Broglie formula, *λ* = *h/p*, where *h* is Planck's constant). One can guess that a role of the vacuum in the experiment of Couder & Fort takes upon itself a silicon oil surface with subcritical Faraday ripples activated on it [21]. Observe that pattern of the ripples is changed in the vicinity of extraneous bodies im‐ mersed in the oil which simulate grating slits. Interference of the ripples with waves gener‐ ated by the bouncing droplets provides optimal paths for the droplets traveling through the slits and further. As a result we may observe an interference pattern emergent depending on amount of slits in the grating and distance between them.

Now we may suppose that the subcritical Faraday ripples on the silicon oil surface simulate vacuum fluctuations. Consequently, the vacuum fluctuations change their own pattern near the slit grating depending on amount of slits and distance between them. We may imagine that the particle passing through vacuum (bouncing through, Fig. 11) initiates waves which interfere with the vacuum fluctuations. As a result of such an interference the particle moves along an optimal path - along the Bohmian trajectory. Mathematically the bounce is imitated by an exponential term exp{i*S/ћ*}, where the angle *S/ћ* parametrizes the group of rotation given on a circle of unit radius. So, the path along which the particle moves is scaled by this unitary group, U(1), due to the exponential mapping of the phase *S/ћ* on the circle.

St.-Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Russia

[1] Dauben JW. Abraham Robinson: The creation of nonstandard analysis. A personal and mathematical odyssey. Princeton, NJ: Princeton University Press; 1995.

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

http://dx.doi.org/10.5772/53471

209

[2] Stanford Encyclopeia of Philosophy: Newton's Views on Space, Time, and Motion.

[3] Henson CW. Foundations of nonstandard analysis. In: Arkeryd L.O., Cutland N. J., & Henson C. W., (eds.) Nonstandard Analysis, Theory and application. The Nether‐

[4] Nelson E. Radically Elementary Probability Theory. Princeton, New Jersey: Princeton

[5] Huygens C. Treatise on Light. Gutenberg eBook, No. 14725. http://www.guten‐

[6] Nelson E. Dynamical theories of Brownian motion. Princeton, New Jersey: Princeton

[7] Nelson E. Quantum fluctuations. Princeton Series in Physics. Princeon, New Jersey:

[8] Feynman RP., Hibbs A. Quantum mechanics and path integrals. N. Y.: McGraw Hill;

[9] Einstein A., von Smoluchowski M. Brownian motion. Moscow-Leningrad: ONTI (in

[10] Sidorov LN., Troyanov SI. At the dawn of a new chemistry of fullerenes. PRIRODA

[11] Arndt M., Hackermüller L., Reiger E. Interferometry with large molecules: Explora‐ tion of coherence, decoherence and novel beam methods. Brazilian Journal of Physics

[12] Brezger B., Arndt M., Zeilinger A. Concepts for near-field interferometers with large

[13] Brezger B., Hackermüller L., Uttenthaler S., Petschinka J., Arndt M., Zeilinger A. Matter-Wave Interferometer for Large Molecules. Phys. Rev. Lett. 2002; 88 100404. [14] Gerlich S., Eibenberger S., Tomandl M., Nimmrichter S., Hornberger K., Fagan P. J., Tuxen J., Mayor M., Arndt, M. Quantum interference of large organic molecules. Na‐ ture Communications 2011; (2) 263. http://www.nature.com/ncomms/journal/v2/n4/

molecules. J. Opt. B: Quantum Semiclass. Opt. 2003; 5(2) S82-S89.

full/ncomms1263.html (accessed 5 April 2011)

berg.org/files/14725/14725-h/14725-h.h (accessed 18 January 2005)

http://plato.stanford.edu/entries/newton-stm/ (accessed 12 August 2004).

lands: Kluwer Acad. Publ.; 1997,p. 1-51.

Univ. Press; 1987.

Univ. Press; 2001.

Russian); 1936.

2011; 1153(9) 22-30.

2005; 35(2A) 216-223.

1965.

Princeton Univ. Press; 2001.

**References**

**Figure 11.** Bouncing a nanoparticle at moving through vacuum. Vertical dotted sinusoidal curves depict exchange by energy Δ*E* with vacuum virtual particle-antiparticle pairs over period of about Δ*t*=ћ/Δ*E*.

In conclusion it would be like to remember remarkable reflection of Paul Dirac. In 1933 Paul Dirac drew attention to a special role of the action S in quantum mechanics [38] - it can ex‐ hibit itself in expressions through exp{iS/ћ}. In 1945 he emphasized once again, that the clas‐ sical and quantum mechanics have many general points of crossing [39]. In particular, he had written in this article: "We can use the formal probability to set up a quantum picture rather close to the classical picture in which the coordinates q of a dynamical system have definite values at any time. We take a number of times t1, t2, t3, … following closely one after another and set up the formal probability for the q's at each of these times lying within specified small ranges, this being permissible since the q's at any time all commutate. We then get a formal probability for the trajectory of the system in quantum mechanics lying within certain limits. This enables us to speak of some trajectories being improbable and oth‐ ers being likely."

#### **Acknowledgement**

Author thanks Miss Pipa (administrator of Quantum Portal) for preparing programs that permitted to calculate and prepare Figures 5 to 8. Author thanks also O. A. Bykovsky for taking my attention to a single-particle interference observed for macroscopic objects by Couder and Fort and V. Lozovskiy for some remarks relating to the article.

### **Author details**

Valeriy I. Sbitnev

St.-Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Russia

#### **References**

given on a circle of unit radius. So, the path along which the particle moves is scaled by this

**Figure 11.** Bouncing a nanoparticle at moving through vacuum. Vertical dotted sinusoidal curves depict exchange by

In conclusion it would be like to remember remarkable reflection of Paul Dirac. In 1933 Paul Dirac drew attention to a special role of the action S in quantum mechanics [38] - it can ex‐ hibit itself in expressions through exp{iS/ћ}. In 1945 he emphasized once again, that the clas‐ sical and quantum mechanics have many general points of crossing [39]. In particular, he had written in this article: "We can use the formal probability to set up a quantum picture rather close to the classical picture in which the coordinates q of a dynamical system have definite values at any time. We take a number of times t1, t2, t3, … following closely one after another and set up the formal probability for the q's at each of these times lying within specified small ranges, this being permissible since the q's at any time all commutate. We then get a formal probability for the trajectory of the system in quantum mechanics lying within certain limits. This enables us to speak of some trajectories being improbable and oth‐

Author thanks Miss Pipa (administrator of Quantum Portal) for preparing programs that permitted to calculate and prepare Figures 5 to 8. Author thanks also O. A. Bykovsky for taking my attention to a single-particle interference observed for macroscopic objects by

Couder and Fort and V. Lozovskiy for some remarks relating to the article.

energy Δ*E* with vacuum virtual particle-antiparticle pairs over period of about Δ*t*=ћ/Δ*E*.

ers being likely."

208 Advances in Quantum Mechanics

**Acknowledgement**

**Author details**

Valeriy I. Sbitnev

unitary group, U(1), due to the exponential mapping of the phase *S/ћ* on the circle.


[15] Hornberger K., Hackermüller L., Arndt M. Inuence of molecular temperature on the coherence of fullerenes in a near-field interferometer. Phys. Rev. A. 2005; 71, 023601.

[30] Sbitnev VI. Matter waves in the Talbot-Lau interferometry, http://arxiv.org/abs/

Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference

http://dx.doi.org/10.5772/53471

211

[32] Mandel L., Wolf E. Optical coherence and quantum optics. Cambridge: Cambridge

[33] McMorran B.,Cronin AD. Model for partial coherence and wavefront curvature in

[34] Gerlich S., Hackermüller L., Hornberger K., Stibor A., Ulbricht H., Gring M., Gold‐

[35] Bohm D. A suggested interpretation of the quantum theory in terms of "hidden vari‐

[36] Madelung E. Quantentheorie in Hydrodynamischer form. Zts. f. Phys. 1926; 40,

[37] Stanford Encyclopeia of Philosophy: Bohmian Mechanics. http://plato.stanford.edu/

[38] Dirac PAM. The Lagrangian in Quantum Mechanics. Physikalische Zeitschrift der

[39] Dirac PAM. On the analogy between classical and quantum mechanics. Rev. Mod.

ometer for highly polarizable molecules. Nature Physics. 2007; 3, 711-715.

farb F., Savas T., Müri M., Mayor M., Arndt, M. A Kapitza-Dirac-Talbot-Lau interfer‐

[31] Volokitin AI. Quantum drag and graphene. PRIRODA, 2011; 1153(9), 13-21.

grating interferometers. Phys. Rev. A. 2008; 78 (1), 013601(10).

ables", I & II. Physical Review, 1952; 85, 166-179 & 180-193.

entries/qm-bohm/ (accessed 26 October 2001).

Sowjetunion. 1933; 3, 64-72.

Phys. 1945; 17 (2 & 3), 195-199.

1005.0890 (accessed 17 September 2010).

University Press; 1995.

322-326.


[15] Hornberger K., Hackermüller L., Arndt M. Inuence of molecular temperature on the coherence of fullerenes in a near-field interferometer. Phys. Rev. A. 2005; 71, 023601.

[16] Hornberger K., Sipe JP., Arndt M. Theory of decoherence in a matter wave Talbot-

[17] Nimmrichter S., Hornberger K. Theory of near-fieldmatter wave interference beyond

[18] Juffmann,T., Truppe S., Geyer P., Major AG., Deachapunya S., Ulbricht H., Arndt, M. Wave and particle in molecular interference lithography. Phys. Rev. Lett. 2009; 103,

[19] Berry M., Klein S. Integer, fractional and fractal Talbot effects, Journal of Modern Op‐

[20] Berry M., Marzoli L., Schleich W. Quantum carpets, carpets of light. Physics World,

[21] Couder Y., Fort E. Single-Particle Diffraction and Interference at a Macroscopic Scale.

[22] Stratonovich RL. Topics in the theory of random noise. N.Y.: Gordon and Breach;

[24] Ventzel AD. The course of stochastic processes theory. Moskow: Nauka (in Russian);

[25] Grosche C. An introduction into the Feynman path integral. http://arXiv.org/abs/hep-

[26] MacKenzie R. Path integral methods and applications. http://arXiv.org/abs/quant-ph/

[27] Juffmann T., Milic A., Müllneritsch M., Asenbaum P., Tsukernik A., Tüxen J., Mayor M., Cheshnovsky O., Arndt, M. Real-time single-molecule imaging of quantum inter‐ ference. Nature Nanotechnology - Letter. 2012; 7, 297–300. http://www.nature.com/ nnano/journal/v7/n5/full/nnano.2012.34.html?WT.ec\_id=NNANO-201205 (accessed

[28] Sbitnev VI. Bohmian trajectories and the path integral paradigm - complexified La‐ grangian mechanics. In: Pahlavani MR. (ed.) Theoretical Concepts of Quantum Me‐ chanics. Rijeka: InTech; 2011 p313-340. http://www.intechopen.com/books/ theoretical-concepts-of-quantum-mechanics/the-path-integral-paradigm-and-boh‐ mian-trajectories-from-the-lagrangian-mechanics-to-complexified-on (accessed

[29] Nairz O., Arndt M., Zeilinger A. Quantum interference experiments with large mole‐

[23] Gnedenko BV. Theory of Probability. N.Y.: Gordon & Breach; 1997.

Lau interferometer. Phys. Rev. A. 2004; 70, 053608.

263601.

210 Advances in Quantum Mechanics

1963.

1975.

25 March 2012)

2004-2012)

2001; (6), 39-44.

tics, 1996; 43(10) 2139-2164.

Phys. Rev. Lett. 2006; 97 (6), 154101(4).

th/9302097 (accessed 20 February 1993).

cules. Am. J. Phys. 2003; 71 (4), 319-325.

0004090 (accessed 20 April 2000).

the eikonal approximation. Phys. Rev. A. 2008; 78, 023612.


**Chapter 10**

**Chapter XX** 

**Quantum Intentionality and Determination of Realities**

© 2012 Bulnes, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2013 Bulnes; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Bulnes; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**in the Space-Time Through Path Integrals and Their**

8 In the universe three fundamental realities exist inside our perception, which share 9 messages and quantum processes: the physical, energy and mental reality. These realities 10 happen at all times and they are around us like part of our existence spending one to other 11 one across *organised transformations* which realise a linking field - energy-matter across the 12 concept of conscience of a field on the interpretation of the matter and space to create a 13 reality non-temporal that only depends on the nature of the field, for example, the 14 gravitational field is a reality in the space - time that generates a curved space for the 15 presence of masses. At macroscopic level and according to the Einsteinian models the time 16 is a flexible band that acts in form parallel to the space. Nevertheless, studying the field at 17 microscopic level dominated by particles that produce gravity, the time is an intrinsic part of 18 the space (*there is no distinction between one and other*), since the particles contain a rotation 19 concept (*called spin*) that is intrinsic to the same particles that produce gravity from quantum 20 level [1].Then the gravitational field between such particles is an always present reality and 21 therefore non-temporal. The time at quantum level is the distance between cause and effect, 22 but the effect (*gravitational spin*) is contained in the proper particle that is their cause on 23 having been interrelated with other particles and vice versa the effect contains the cause

25 Then the action of any field that is wished transforms their surrounding reality which must 26 spill through the component particles of the space - time, their nature and to transmit it in 27 organised form, which is legal, because the field is invariant under movements of the proper

28 space, and in every particle there sublies a part of the field through their *spinor*.

<sup>1</sup>**Quantum Intentionality and Determination of** 

<sup>2</sup>**Realities in the Space-Time Through Path** 

<sup>3</sup>**Integrals and Their Integral Transforms** 

**Integral Transforms**

http://dx.doi.org/10.5772/53439

6 http://dx.doi.org/10.5772/53439

Additional information is available at the end of the chapter

5 Additional information is available at the end of the chapter

cited.

24 since the particle changed their direction [1].

Francisco Bulnes

4 Francisco Bulnes

7 **1. Introduction** 

### **Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals and Their Integral Transforms** <sup>1</sup>**Quantum Intentionality and Determination of**  <sup>2</sup>**Realities in the Space-Time Through Path**  <sup>3</sup>**Integrals and Their Integral Transforms**

Francisco Bulnes 4 Francisco Bulnes

Additional information is available at the end of the chapter 5 Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53439 6 http://dx.doi.org/10.5772/53439

### 7 **1. Introduction**

8 In the universe three fundamental realities exist inside our perception, which share 9 messages and quantum processes: the physical, energy and mental reality. These realities 10 happen at all times and they are around us like part of our existence spending one to other 11 one across *organised transformations* which realise a linking field - energy-matter across the 12 concept of conscience of a field on the interpretation of the matter and space to create a 13 reality non-temporal that only depends on the nature of the field, for example, the 14 gravitational field is a reality in the space - time that generates a curved space for the 15 presence of masses. At macroscopic level and according to the Einsteinian models the time 16 is a flexible band that acts in form parallel to the space. Nevertheless, studying the field at 17 microscopic level dominated by particles that produce gravity, the time is an intrinsic part of 18 the space (*there is no distinction between one and other*), since the particles contain a rotation 19 concept (*called spin*) that is intrinsic to the same particles that produce gravity from quantum 20 level [1].Then the gravitational field between such particles is an always present reality and 21 therefore non-temporal. The time at quantum level is the distance between cause and effect, 22 but the effect (*gravitational spin*) is contained in the proper particle that is their cause on 23 having been interrelated with other particles and vice versa the effect contains the cause 24 since the particle changed their direction [1].

25 Then the action of any field that is wished transforms their surrounding reality which must 26 spill through the component particles of the space - time, their nature and to transmit it in 27 organised form, which is legal, because the field is invariant under movements of the proper 28 space, and in every particle there sublies a part of the field through their *spinor*.

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Bulnes; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Bulnes; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Bulnes, licensee InTech. This is an open access chapter distributed under the terms of the

2

1 Three fundamental realities perceived by our anthropometric development of the universe; 2 field - energy- matter between three different but indistinguishable realities are realised at 3 macroscopic level: one is the *material reality* which is determined by their atomic linkage 4 between material particles (*atoms constituted by protons, neutrons and electrons*), an energy 5 reality, called also quantum reality, since the information in this reality area exchanges the 6 matter happen through sub-particles (*bosons, fermions, gluons, etc*) and finally a virtual reality 7 that sublies like *fundamental field* and that is an origin of the changes of spin of the sub-8 particles and their support doing that they transform these into others and that they 9 transform everything around him (*Higgs field*). The integration of these three realities will be 10 called by us a *hyper-reality* by us. The hyper-reality contains to the *quantum reality* and to the 11 reality perceived by our senses (material reality).

Consider R<sup>d</sup> 12 u I*t*, like the space - time where happens the transitions of energy states into 13 space - time, and let *u*, *v*, elements of this space, the integral of all the continuous possible paths to particle *x*(*s*), that transit from energy state in *u*, to an energy state in *v*, in R<sup>d</sup> 14 u I*t*, is

$$\mathfrak{u}\left(L,\mathfrak{x}\{t\}\right) = \int\_{\mathcal{C}^{\omega}\{0,t\}} \exp\left\{\frac{i}{\hbar}\mathfrak{Z}\{\mathbf{x}\}\right\}d\mathfrak{x},\tag{1}$$

Provisional chapter 3

I(*x*), 215

x

http://dx.doi.org/10.5772/53439

I

(*x*), will be

1 where the entire action (2) is an intentional action (*for the whole infinity of paths* J*t*, that defines

f

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

 I

 I(

ª º

¬ ¼

I

I

\*, and that are trajectories of the space - time :R<sup>3</sup> u I*t*), if and only if Oc(*x*, *x s*( )

 I

 I

12 space of the sub-particles (*boson space*). Likewise a photon of certain class

I

20 involves a *connection* of the tangent bundle of the space of trajectories :(\*).

27 intentional action (*or simply intention*) of the field *X*, if and only if:

\* f

 ½ ª º ½ ° ° ° ° ® ¾ « » ® <sup>¾</sup> « » ° ° ° ° ¯ ¿ ¬ ¼ ¯¿ ³ ³ <sup>c</sup> ³ ³ ƺ(ƥ) ƺ(ƥ) ( )) ( ( )) Ǎ () ( ( ))d ( ) Ǎ , *TOTAL O x d x E E xs xs <sup>x</sup> <sup>s</sup>*

4 (3)

where the energy factor *E*+ *E* 5 , represents the energy needed by the always present force to 6 realise the action and *Oc*, is the conscience operator which defines the value or record of the 7 field *X* (direction), on every particle of the space :(\*), which along their set of trajectories \*,

 *<sup>O</sup>* ³ *X x O <sup>x</sup> <sup>d</sup> <sup>x</sup>* <sup>c</sup> <sup>c</sup> ( ) ( ( )) ( ( )) ( ), M 9 (4)

10 where the operator *O*c, invests an energy quasi-infinite, encapsulated in a microscopic 11 region of the space (*quantum space* M), and with applications and influence in an unlimited

13 generated by the quantum field (*if it manages to change its field spin*) and will be moved for

\* :\*

16 Interesting applications of the formula (3) to nano-sciences will happen at the end of the 17 present chapter. Also it will be demonstrated that (3) is a quantum integral transform of 18 bundles or distortions of energy in the space - time if it involves a *special kernel*. The bundle 19 stops existing if there is applied certain intention (*path integral transform*). The operator *O*c,

21 The operator *O*c, include a connection of the tangent bundle of the space of trajectories :(\*). 22 The integral (5) will determine on certain hypotheses the interdependence between the

24 Def. **2. 1** (*intentional action of the X*). Let *X*, be a field acting on the particles *x*1(*s*), *x*2(*s*), }, M, 25 and let , be their action on the above mentioned particles under an operator who 26 recognizes the "target" in M, (*conscience operator*). We say that , is an conscientious

29 b. , recognizes well their target, it is known what the field *X* wants to do (their

28 a. , is the determination of the field *X*, to realise or execute, (their force F(*x*)),

15 (5)

*X*

« »

³ ³ <sup>I</sup> <sup>c</sup> <sup>c</sup> <sup>ƥ</sup> ( ( )) ( ( ( ))) ( )) ( ( )) Ǎ , *<sup>O</sup>*

*x O x d x*

(I

8 realizes the action of permanent field *O*c, it being fulfilled that

14 the intention on a trajectory \*, by the path integral

23 material, quantum and virtual realities in M.

30 direction she follows a configuration patron)

3 wI(*x*))*d*I

, where then

2 ) = *L*(

16 where *h,* is the constant of Max Planck, and the action , is the one realised by their 17 *Lagrangian L*.

18 Since we have mentioned, the action of a field is realised being a cause and effect, for which 19 it must be a cause and effect in each of the component particles, "waking up" the particle 20 conscience to particle being transmitted this way without any exception. This action must 21 infiltrate to the field itself that it sublies in the space and that it is shaped by the proper 22 particles that compose it creating a certain co-action that is major than their algebraic sum 23 [2].The configuration space C*n, m* = {J*t*:(\*) \_J*t*o\*o\*/J} [3], is the model created by the 24 due action to each corresponding trajectory to the different splits it. Is clear here we must 25 have in mind all the paths in the space-time M, that contribute to interference amplitude in 26 this space, remaining the path of major statistical weight. The intention takes implicitly a 27 space C*n, m*. Any transformation that wants to realise of a space, has as constant the same 28 energy that comes from the permanent field of the matter and which is determined by the 29 quantum field of the particles *x*(*s*), constituents of the space and matter. If we want to define 30 a conscience in the above mentioned field, that is to say, an action that involves an intention 31 is necessary to establish it inside the argument of the action. Likewise, if *x*(*s*):, and (*x*(*s*)), 32 there is their action due to a field of particles *X*, and there is spilled an intention defined by 33 (1) the length and breadth of the space M, such that satisfies the property of synergy [2], for 34 all the possible trajectories that they fill :, we have that

$$\mathfrak{T}\_{\text{TOTAL}} \ge \int\_{\text{E}^-}^{\text{E}^+} \left\{ \sum\_{j} \int\_{\chi\_i} \mathfrak{T}\_j(\mathbf{x}(s)) d\langle \mathbf{x}(s) \rangle \right\} d\mu,\tag{2}$$

1 where the entire action (2) is an intentional action (*for the whole infinity of paths* J*t*, that defines \*, and that are trajectories of the space - time :R<sup>3</sup> u I*t*), if and only if Oc(*x*, *x s*( ) x 2 ) = *L*(I(*x*), 3 wI(*x*))*d*I, where then

2

214 Advances in Quantum Mechanics

17 *Lagrangian L*.

1 Three fundamental realities perceived by our anthropometric development of the universe; 2 field - energy- matter between three different but indistinguishable realities are realised at 3 macroscopic level: one is the *material reality* which is determined by their atomic linkage 4 between material particles (*atoms constituted by protons, neutrons and electrons*), an energy 5 reality, called also quantum reality, since the information in this reality area exchanges the 6 matter happen through sub-particles (*bosons, fermions, gluons, etc*) and finally a virtual reality 7 that sublies like *fundamental field* and that is an origin of the changes of spin of the sub-8 particles and their support doing that they transform these into others and that they 9 transform everything around him (*Higgs field*). The integration of these three realities will be 10 called by us a *hyper-reality* by us. The hyper-reality contains to the *quantum reality* and to the

Consider R<sup>d</sup> 12 u I*t*, like the space - time where happens the transitions of energy states into 13 space - time, and let *u*, *v*, elements of this space, the integral of all the continuous possible paths to particle *x*(*s*), that transit from energy state in *u*, to an energy state in *v*, in R<sup>d</sup> 14 u I*t*, is

½

¯ ¿ ³ <sup>I</sup> C [0, ] ( , ( )) exp ( ) , *u,v t*

15 (1)

16 where *h,* is the constant of Max Planck, and the action , is the one realised by their

18 Since we have mentioned, the action of a field is realised being a cause and effect, for which 19 it must be a cause and effect in each of the component particles, "waking up" the particle 20 conscience to particle being transmitted this way without any exception. This action must 21 infiltrate to the field itself that it sublies in the space and that it is shaped by the proper 22 particles that compose it creating a certain co-action that is major than their algebraic sum 23 [2].The configuration space C*n, m* = {J*t*:(\*) \_J*t*o\*o\*/J} [3], is the model created by the 24 due action to each corresponding trajectory to the different splits it. Is clear here we must 25 have in mind all the paths in the space-time M, that contribute to interference amplitude in 26 this space, remaining the path of major statistical weight. The intention takes implicitly a 27 space C*n, m*. Any transformation that wants to realise of a space, has as constant the same 28 energy that comes from the permanent field of the matter and which is determined by the 29 quantum field of the particles *x*(*s*), constituents of the space and matter. If we want to define 30 a conscience in the above mentioned field, that is to say, an action that involves an intention 31 is necessary to establish it inside the argument of the action. Likewise, if *x*(*s*):, and (*x*(*s*)), 32 there is their action due to a field of particles *X*, and there is spilled an intention defined by 33 (1) the length and breadth of the space M, such that satisfies the property of synergy [2], for

E

TOTAL

 ½ ° ° ® ¾ ° ° ¯ ¿ ³ ¦ <sup>³</sup> DŽ

*t j*

*j* 35 *xs dxs d* (2)

( ( )) ( ( )) Ǎ,

 t

E

*L xt x dx*

® ¾

*i*

*h*

11 reality perceived by our senses (material reality).

34 all the possible trajectories that they fill :, we have that

$$\mathfrak{T}\_{\text{TOTAL}} = \left[ \left\{ \mathfrak{T} \left[ \left. \int O\_c(\phi(\mathbf{x})) d(\phi(\mathbf{x})) \right| \right] \right\} \mathfrak{u}\_x = \left( E^+ - E^- \right) \int \left. \int \mathfrak{T}(\mathbf{x}(\mathbf{s})) d\mathbf{x}(\mathbf{s}) \right| \mathfrak{u}\_{s'} \tag{3}$$

where the energy factor *E*+ *E* 5 , represents the energy needed by the always present force to 6 realise the action and *Oc*, is the conscience operator which defines the value or record of the 7 field *X* (direction), on every particle of the space :(\*), which along their set of trajectories \*, 8 realizes the action of permanent field *O*c, it being fulfilled that

$$\mathfrak{T}\_{O\_c}(\phi(\mathbf{x})) = \int\_{\mathbb{X}(\mathbf{M})} O\_c(\phi(\mathbf{x})) d\phi(\mathbf{x}),\tag{4}$$

10 where the operator *O*c, invests an energy quasi-infinite, encapsulated in a microscopic 11 region of the space (*quantum space* M), and with applications and influence in an unlimited 12 space of the sub-particles (*boson space*). Likewise a photon of certain class I(*x*), will be 13 generated by the quantum field (*if it manages to change its field spin*) and will be moved for 14 the intention on a trajectory \*, by the path integral

$$\mathfrak{u}(\mathfrak{T}\_{O\_c}(\phi(x))) = \int\_{\mathbb{T}} \left[ \int\_{\chi(\Omega(\Gamma))} O\_c(\phi(x)) d(\phi(x)) \right] \mu\_{\Gamma'} \tag{5}$$

16 Interesting applications of the formula (3) to nano-sciences will happen at the end of the 17 present chapter. Also it will be demonstrated that (3) is a quantum integral transform of 18 bundles or distortions of energy in the space - time if it involves a *special kernel*. The bundle 19 stops existing if there is applied certain intention (*path integral transform*). The operator *O*c, 20 involves a *connection* of the tangent bundle of the space of trajectories :(\*).

21 The operator *O*c, include a connection of the tangent bundle of the space of trajectories :(\*). 22 The integral (5) will determine on certain hypotheses the interdependence between the 23 material, quantum and virtual realities in M.

24 Def. **2. 1** (*intentional action of the X*). Let *X*, be a field acting on the particles *x*1(*s*), *x*2(*s*), }, M, 25 and let , be their action on the above mentioned particles under an operator who 26 recognizes the "target" in M, (*conscience operator*). We say that , is an conscientious 27 intentional action (*or simply intention*) of the field *X*, if and only if:

28 a. , is the determination of the field *X*, to realise or execute, (their force F(*x*)),

29 b. , recognizes well their target, it is known what the field *X* wants to do (their 30 direction she follows a configuration patron)

4

Consider a particle system *p*1, *p*2,} in a space - time M # R4. Let *x*(*t*):(\*) R3 1 u I*t*, be a trajectory which predetermines a position *x*R<sup>3</sup> 2 , for all time *t*I*t*. A field *X*, that infiltrates its 3 action to the whole space of points predetermined by all the trajectories *x*1(*t*), *x*2(*t*), *x*3(*t*), 4 },:(\*), is the field that predetermines the points I*<sup>i</sup>*(*xi*(*t*)), which are fields whose 5 determination is given by the action of the field *X*, and evaluated in the position of every 6 particle. Every point have a defined force by the action , of *X*, along the geodesic J*t*, and determined direction by their tangent bundle given for Tx1 7 (:(\*)), that is the cotangent space 8 T\*(:(\*)) [4], which give the images of the states under Lagrangian, that is to say, the field 9 provides of direction to every point I*i*, because their tangent bundle has a subjacent spinor

bundle S [5], where the field *X*, comes given as I I I w w ¦ i i (,) <sup>i</sup> , *<sup>i</sup> i x <sup>X</sup>* I1, I2, I3, }x1 10 , on

11 every particle *p*i = *x*i(*t*) (i = 1, 2, }). Then *to direct an intention* is the map or connection:

$$O\_c: \ T\Omega(\Gamma) \to T\mathsf{Ar}^1(\Omega(\Gamma)) \ (\in T^\*(\Omega(\Gamma)), \tag{6}$$

13 with rule of correspondence

$$(\mathbf{x}^i, \partial\_i \mathbf{x}^i) \mathbf{I} \to (\phi^i, \partial\_m \phi^i) \tag{7}$$

Provisional chapter 5

217

http://dx.doi.org/10.5772/53439

1 where :\*(M), is some dual complex ("*forms on configuration spaces*"), i.e. such that "Stokes

4 then the integrals given by (8) can be written (to *m*-border points and *n*-inner points (see

I

( ( d) d ) ,

6 (11)

7 This is an *infiltration* in the space-time by the direct action [2, 3], that happens in the space 8 :u*C*, to each component of the space :(\*), through the expressed Lagrangian in this case 9 by Z, de (10)In (11), the integration of the space realises with the infiltration of the time,

11 The design of some possible spintronic devices that show the functioning of this process of

14 **Figure 1.** In a) The configuration space C*n, m*, is the model created by the due action to each corresponding trajectory 15 to the different splits. b) Example of a doble fibration to explain the relation between two realities of a space of particles: the bundle of lines *L*, and the ordinary space R<sup>3</sup> 16 . c) Way in as a quantum field *X*, which acts on a space - time

1 m

*t t n*

( ( )) d d

u

: \* uu u

q 1 T ( ƺ(ƥ ) ƺ(ƥ )

 I

ƺ 1 (ƥ ) ƺ(ƥ ) ƺ(ƥ )

12 m

*tt t*

*x d*

³ ³

³³ ³

12 transformation in the space M, will be included in this chapter.

) I

10 integrating only energy state elements of the field.

17 to change its reality, that is to say, to spill their intention.

!, ³

ǚ , dǚ <sup>ƺ</sup> *<sup>C</sup>* 3 (10)

I

1

 I

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

*m m n*

1

 I

*m m n*

*n*

2 theorem" holds:

5 **figure 1a**))) as:

13

which produces one to us i*th*- state of field energy I*<sup>i</sup>*15 [6], where the action , of the field *X*, 16 infiltrates and transmits from particle to particle in the whole space :(\*), using a 17 configuration given by their Lagrangian *L* (*conscience operator*), along all the trajectories of 18 :(\*). Then of a sum of trajectories ³*D*F(*x*(*t*)), one has the sum ³*d*(I(*x*)), on all the possible field 19 configurations *Cn, m*. Extending these intentions to whole space :(\*) M, on all the elections 20 of possible paths whose statistical weight corresponds to the determined one by the 21 intention of the field, and realising the integration in paths for an infinity of particles - fields 22 in T:(\*), it is had that

$$\mathfrak{a}(\phi^{i}(\mathbf{x})) = \int\_{T\mathfrak{A}(\Gamma)} \mathfrak{a}(\phi(\mathbf{x})) = \lim\_{\substack{N \to \infty \\ \delta s \to 0}} \frac{1}{B} \int\_{-\sigma}^{+\sigma} \frac{d\phi^{i}}{B} \cdots \int\_{-\sigma}^{+\sigma} \frac{d\phi^{n}}{B} \cdots = \prod\_{i=1}^{\alpha} \int\_{-\sigma}^{+\alpha} \mathrm{i}^{\pi[\phi^{i}, \vec{x}\_{\rho}\phi^{i}]} d\phi^{i} \langle \mathbf{x}(\mathbf{s})\rangle\_{\mathcal{Y}} \tag{8}$$

where , 2S ª º « » ¬ ¼ ! 1/2 Dž *<sup>m</sup> <sup>B</sup> i s* 24 is the amplitude of their propagator and in the second integral of

25 (8), we have expressed the Feynman integral using the form of volume Z(I(*x*)), of the space 26 of all the paths that add in T:(\*), to obtain the real path of the particle (*where we have chosen*  27 *quantized trajectories,* that is to say, ³*d*(I(*x*))). Remember that the sum of all these paths is the 28 interference amplitude between paths that is established under an action whose Lagrangian 29 is Z(I(*x*))=[(*x*)dI(*x*), where, if :M, is a complex with M, the space-time, and *C*(M), is a 30 complex or configuration space on M, (*interfered paths in the experiment given by multiple split*  31 [7]), endowed with a pairing

$$\left\{ \colon \mathbb{C}(\mathbb{M}) \times \Omega^\* \left( \mathbb{M} \right) \to \mathbb{R}, \tag{9}$$

1 where :\*(M), is some dual complex ("*forms on configuration spaces*"), i.e. such that "Stokes 2 theorem" holds:

4

Consider a particle system *p*1, *p*2,} in a space - time M # R4. Let *x*(*t*):(\*) R3 1 u I*t*, be a trajectory which predetermines a position *x*R<sup>3</sup> 2 , for all time *t*I*t*. A field *X*, that infiltrates its 3 action to the whole space of points predetermined by all the trajectories *x*1(*t*), *x*2(*t*), *x*3(*t*),

5 determination is given by the action of the field *X*, and evaluated in the position of every 6 particle. Every point have a defined force by the action , of *X*, along the geodesic J*t*, and determined direction by their tangent bundle given for Tx1 7 (:(\*)), that is the cotangent space 8 T\*(:(\*)) [4], which give the images of the states under Lagrangian, that is to say, the field 9 provides of direction to every point I*i*, because their tangent bundle has a subjacent spinor

3, }x1 10 , on

<sup>1</sup> : x )) ( ) ( ( )) ( ( \* ( , <sup>12</sup>*OT T T <sup>c</sup>* :\* o :\* # :\* (6)

*<sup>i</sup>*15 [6], where the action , of the field *X*, 16 infiltrates and transmits from particle to particle in the whole space :(\*), using a 17 configuration given by their Lagrangian *L* (*conscience operator*), along all the trajectories of

19 configurations *Cn, m*. Extending these intentions to whole space :(\*) M, on all the elections 20 of possible paths whose statistical weight corresponds to the determined one by the 21 intention of the field, and realising the integration in paths for an infinity of particles - fields

I

<sup>N</sup> <sup>1</sup> <sup>Dž</sup>s 0 <sup>1</sup> ( ( )) <sup>ǚ</sup>( ( )) lim e ( ( )), *i i <sup>i</sup> <sup>i</sup>*

*d d x x d x s BB B* 23 (8)

of : \* <sup>o</sup> f f f

24 is the amplitude of their propagator and in the second integral of

25 (8), we have expressed the Feynman integral using the form of volume Z(I(*x*)), of the space 26 of all the paths that add in T:(\*), to obtain the real path of the particle (*where we have chosen* 

28 interference amplitude between paths that is established under an action whose Lagrangian

30 complex or configuration space on M, (*interfered paths in the experiment given by multiple split* 

<sup>32</sup>³: \* *<sup>C</sup>*( ) ( ) M MR u: o , (9)

I

³ ³ ³ ³ <sup>I</sup> 1 n

( ( , ) ) , *ii i i*

I I

I

 I

f f <sup>f</sup> f w

(*x*), where, if :M, is a complex with M, the space-time, and *C*(M), is a

*i*

11 every particle *p*i = *x*i(*t*) (i = 1, 2, }). Then *to direct an intention* is the map or connection:

I

I I

w w ¦ i i (,) <sup>i</sup> , *<sup>i</sup> i x <sup>X</sup>*

I

I

I I

(*x*))). Remember that the sum of all these paths is the

<sup>Ǎ</sup> i[ , ]

*<sup>i</sup>*(*xi*(*t*)), which are fields whose

I1, I2, I

(7)

I

13

(*x*)), on all the possible field

4 },:(\*), is the field that predetermines the points

bundle S [5], where the field *X*, comes given as

*t m* 14 *x xI* w ow

*T* ( )

1/2

 I

which produces one to us i*th*- state of field energy

18 :(\*). Then of a sum of trajectories ³*D*F(*x*(*t*)), one has the sum ³*d*(

13 with rule of correspondence

216 Advances in Quantum Mechanics

22 in T:(\*), it is had that

I

where , 2S

31 [7]), endowed with a pairing

29 is Z(

I(*x*))=[(*x*)dI

*<sup>m</sup> <sup>B</sup>*

ª º « » ¬ ¼ !

27 *quantized trajectories,* that is to say, ³*d*(

Dž

*i s*

$$\begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \text{op.} \end{array} \end{array} \end{array} \begin{array}{c} \begin{array}{c} \text{op.} \end{array} \end{array} \end{array} \right. $$

4 then the integrals given by (8) can be written (to *m*-border points and *n*-inner points (see 5 **figure 1a**))) as:

$$\begin{split} \int\_{\mathrm{T}\Omega(\Gamma)} \mathfrak{Z}(\mathsf{\boldsymbol{\phi}}(\mathsf{x})) d\boldsymbol{\phi} &= \int\_{\Omega(\Gamma\_{\mathrm{r}}) \times \dots \times \Omega(\Gamma\_{\mathrm{r}}) \times \dots} \mathfrak{Z}\_{\mathsf{q}} \mathsf{d}\,\phi\_{1}^{m\_{1}} \dots \mathsf{d}\,\phi\_{n}^{m\_{n}} \dots \\ &= \int\_{\Omega(\Gamma\_{\mathrm{r}})} (\int\_{\Omega(\Gamma\_{\mathrm{r}})} (\int\_{\Omega(\Gamma\_{\mathrm{r}})} \mathfrak{Z}\mathsf{d}\phi\_{1}^{m\_{n}}) \dots \mathsf{d}\,\phi\_{n}^{m\_{1}}) \dots \mathsf{d}\,\phi\_{n}^{m\_{n}} \end{split} \tag{11}$$

7 This is an *infiltration* in the space-time by the direct action [2, 3], that happens in the space 8 :u*C*, to each component of the space :(\*), through the expressed Lagrangian in this case 9 by Z, de (10)In (11), the integration of the space realises with the infiltration of the time, 10 integrating only energy state elements of the field.

11 The design of some possible spintronic devices that show the functioning of this process of 12 transformation in the space M, will be included in this chapter.

14 **Figure 1.** In a) The configuration space C*n, m*, is the model created by the due action to each corresponding trajectory 15 to the different splits. b) Example of a doble fibration to explain the relation between two realities of a space of particles: the bundle of lines *L*, and the ordinary space R<sup>3</sup> 16 . c) Way in as a quantum field *X*, which acts on a space - time 17 to change its reality, that is to say, to spill their intention.

6

### 1 **2. Conscience operators and configuration spaces**

We consider M # R3 2 u I*t*, the space-time of certain particles *x*(*s*), in movement, and let *L*, be an 3 operator that explains certain law of movement that governs the movement of the set of 4 particles in M, in such a way that the energy conservation law is applied for the total action 5 of each one of their particles. The movement of all the particles of the space M, is given 6 geometrically for their tangent vector bundle TM. Then the action due to *L*, on M, is defined 7 like [8]:

$$\mathfrak{T}\_L: \operatorname{TM} \to \mathfrak{n}\_\prime \tag{12}$$

Provisional chapter 7

219

http://dx.doi.org/10.5772/53439

, is translated

(19)

I

Ǎ

I) = *L*(I, wPI)*d*I

1 As we can see, T\*M, carries a canonical symplectic form, which we call Z. Using *O*c, we

, wPI*<sup>i</sup>* 4 ), to Z*L*, modeling the space-time M, through H-

I

Ǎ Ǎ

w w <sup>w</sup>

ǚ dd dd, *i i j j*

 II

> I

*L dL h s s h s ds*

 I I

L

Z

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

 I

Z*L c O*

I

w ww w ww 2 2

*L i i j j <sup>L</sup>* 6 (20)

I

*dt* <sup>ƥ</sup> 9 d() ( ( ), ( )) ( ) , (21)

§ · w w ¨ ¸ © ¹ w w <sup>³</sup>

10 where *h*(*s*): \*o*TM*, and is such that WM o *h* = \* and *h*(*x*1) = *h*(*x*2) = 0, to extreme points of 11 \**x*(*s*1) = *p* y *x*(*s*2) = *q*. The total differential (21) is the symplectic form Z*L*, that constructs the application of the field intention expanding 2*n*-coordinates in (20). The space x1 12 (:(\*)), is the 13 space of differentiable vector fields on :(\*), and :(\*), is the manifold of trajectories (space-14 time of curves) that satisfies the variation principle given by the Lagrange equation that expresses the force F(*x*(*s*)j 15 ), (j = 1, 2, }, *n*) generated by a field that generates one

20 **Figure 2.** a) The particles act in free form in the space - time without the action of a quantum field that spills a force

*<sup>j</sup>* 21 ), is spilled, generated by a field that generates a "conscience" of 22 order given by their Lagrangian. For it, there is not to forget the principle of conservation energy re-interpreted in the

> w w § · ¨ ¸ © ¹ *j j dT T <sup>j</sup> - = F (x(s)),*

*dt x x* 23 (also acquaintances as "*living forces*")

24 transmitting their momentum in every *ith*-particle of the space E, creating a infiltred region by path integrals of

II

7 Likewise, the variation of the action from the operator *O*c = *d*(

I

16 "conscience" of order given by their Lagrangian (to see the figure 2).

that generates an order conscience. b) A force F(*x*(*s*)

Lagrange equations and given for this force like w w

25 trajectories :(\*), where the actions have effect. Here T, is their kinetic energy.

2 obtain a closed two-form Z*L*, on TM, by setting

Considering the local coordinates (I*<sup>i</sup>*

5 spaces, we have that (19) is

8 in the differential

17

18 19

 \* , 3

9 with rule of correspondence

$$\mathfrak{T}(\mathbf{x}(\mathbf{s})) \, = F \mathsf{I} \mathsf{u} \mathsf{x} \mathsf{L}\left(\mathbf{x}(\mathbf{s})\right) \mathsf{x}(\mathbf{s}).\tag{13}$$

11 and whose energy due to the movement is

$$E\_{-}=\mathfrak{T}-L\_{\prime}\tag{14}$$

But this energy is due from their Lagrangian *L* C<sup>f</sup> 13 (*T*M, R), defined like [9]

$$L(\mathbf{x}(s), \dot{\mathbf{x}}(s), \mathbf{s}) \, = T(\mathbf{x}(s), \dot{\mathbf{x}}(s), \mathbf{s}) - V(\mathbf{x}(s), \dot{\mathbf{x}}(s), \mathbf{s}) \,. \tag{15}$$

15 If we want to calculate the action defined in (7) and (8), along a given path \*= *x*(*s*), we have 16 that the action is

$$\mathfrak{T}\_{\Gamma} = \int\_{\Gamma} L(\mathbf{x}(s), \mathbf{x}(s), s)ds,\tag{16}$$

18 If this action involves an intention (that is to say, it is an intentional action) then the action is 19 translated in all the possible field configurations, considering all the variations of the action 20 along the fiber derivative defined by the Lagrangian *L*. Of this way, the conscience operator 21 is the map

$$\mathbf{O}\_c \colon \mathbf{T} \mathsf{M} \to \mathsf{T}^{\bullet} \mathsf{M} \tag{17}$$

23 with corresponding rule

$$O\_c(\mathfrak{v})\mathfrak{w} = \frac{d}{dt}L(\mathfrak{v} + t\mathfrak{w})\Big|\_{t=0},\tag{18}$$

25 That is, *O*c(*v*)*w*, is the derivative of *L*, along the fiber in direction *w*. In the case of *v* = *x*'(*s*), and *q* = 26 *x*(*s*), *q*M, *L*(*q*, *v*) = *E* – *V* = ½<*v*, *v*> - V(*q*), we see that *O*c(*v*)*w* = <v, w>, so we recover the usual map **s**b : TM o T\*M, (with b Euclidean inR<sup>3</sup> 27 ) associated with the bilinear form < , >. Is here where 28 the spin structure subjacent appears in the momentum of the particle *x*(*s*).

1 As we can see, T\*M, carries a canonical symplectic form, which we call Z. Using *O*c, we 2 obtain a closed two-form Z*L*, on TM, by setting

6

7 like [8]:

16 that the action is

21 is the map

23 with corresponding rule

9 with rule of correspondence

218 Advances in Quantum Mechanics

11 and whose energy due to the movement is

1 **2. Conscience operators and configuration spaces** 

We consider M # R3 2 u I*t*, the space-time of certain particles *x*(*s*), in movement, and let *L*, be an 3 operator that explains certain law of movement that governs the movement of the set of 4 particles in M, in such a way that the energy conservation law is applied for the total action 5 of each one of their particles. The movement of all the particles of the space M, is given 6 geometrically for their tangent vector bundle TM. Then the action due to *L*, on M, is defined

: , *<sup>L</sup>* 8 o *T*M R (12)

10  *x s FluxL x s x s*    , (13)

12 *E L* , (14)

14 *Lx s s Tx s s x* ( ) , , *x x* ( ), *s ss* ( )  ( ), *Vxs s* ( ) , ( ), , (15)

15 If we want to calculate the action defined in (7) and (8), along a given path \*= *x*(*s*), we have

18 If this action involves an intention (that is to say, it is an intentional action) then the action is 19 translated in all the possible field configurations, considering all the variations of the action 20 along the fiber derivative defined by the Lagrangian *L*. Of this way, the conscience operator

: T T\* , 22 *Oc* M M o (17)

*<sup>d</sup> O t dt* 24 *v w v w* (18)

25 That is, *O*c(*v*)*w*, is the derivative of *L*, along the fiber in direction *w*. In the case of *v* = *x*'(*s*), and *q* = 26 *x*(*s*), *q*M, *L*(*q*, *v*) = *E* – *V* = ½<*v*, *v*> - V(*q*), we see that *O*c(*v*)*w* = <v, w>, so we recover the usual map **s**b : TM o T\*M, (with b Euclidean inR<sup>3</sup> 27 ) associated with the bilinear form < , >. Is here where

28 the spin structure subjacent appears in the momentum of the particle *x*(*s*).

() ( *<sup>L</sup>* <sup>c</sup> <sup>0</sup> ) ,

*t*

 <sup>ƥ</sup> ³ ƥ 17 *L x s x s s ds* ( ( ), ( ), ) , (16)

But this energy is due from their Lagrangian *L* C<sup>f</sup> 13 (*T*M, R), defined like [9]

$$o\rho\_L = \left(\mathbb{C}\_c\right)^\* o\rho\_\epsilon\tag{19}$$

Considering the local coordinates (I*<sup>i</sup>* , wPI*<sup>i</sup>* 4 ), to Z*L*, modeling the space-time M, through H-5 spaces, we have that (19) is

$$\mathbf{a}\_{L} = \frac{\hat{\mathbf{c}}^{2}\mathbf{L}}{\hat{\boldsymbol{\alpha}}\boldsymbol{\phi}^{i}\hat{\boldsymbol{\alpha}}\boldsymbol{\phi}\_{\mu}\boldsymbol{\phi}^{j}}\mathbf{d}\boldsymbol{\phi}^{i}\wedge\mathbf{d}\boldsymbol{\phi}^{j} + \frac{\hat{\boldsymbol{\alpha}}^{2}\mathbf{L}}{\hat{\boldsymbol{\alpha}}\boldsymbol{\phi}^{i}\hat{\boldsymbol{\alpha}}\boldsymbol{\phi}\_{\mu}\boldsymbol{\phi}^{j}}\mathbf{d}\boldsymbol{\phi}^{i}\wedge\mathbf{d}\boldsymbol{\hat{\alpha}}\_{\mu}\boldsymbol{\phi}^{j},\tag{20}$$

7 Likewise, the variation of the action from the operator *O*c = *d*(I) = *L*(I, wPI)*d*I, is translated 8 in the differential

$$\mathrm{d}\mathfrak{T}(\phi)h = \int\_{\Gamma} \left( \frac{\partial \mathcal{L}}{\partial \phi} - \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{\phi}} \right) (\phi(s), \dot{\phi}(s)) h(s) ds,\tag{21}$$

10 where *h*(*s*): \*o*TM*, and is such that WM o *h* = \* and *h*(*x*1) = *h*(*x*2) = 0, to extreme points of 11 \**x*(*s*1) = *p* y *x*(*s*2) = *q*. The total differential (21) is the symplectic form Z*L*, that constructs the application of the field intention expanding 2*n*-coordinates in (20). The space x1 12 (:(\*)), is the 13 space of differentiable vector fields on :(\*), and :(\*), is the manifold of trajectories (space-14 time of curves) that satisfies the variation principle given by the Lagrange equation that expresses the force F(*x*(*s*)j 15 ), (j = 1, 2, }, *n*) generated by a field that generates one 16 "conscience" of order given by their Lagrangian (to see the figure 2).

17

18 19

20 **Figure 2.** a) The particles act in free form in the space - time without the action of a quantum field that spills a force that generates an order conscience. b) A force F(*x*(*s*) *<sup>j</sup>* 21 ), is spilled, generated by a field that generates a "conscience" of 22 order given by their Lagrangian. For it, there is not to forget the principle of conservation energy re-interpreted in the

Lagrange equations and given for this force like w w w w § · ¨ ¸ © ¹ *j j dT T <sup>j</sup> - = F (x(s)), dt x x* 23 (also acquaintances as "*living forces*")

24 transmitting their momentum in every *ith*-particle of the space E, creating a infiltred region by path integrals of 25 trajectories :(\*), where the actions have effect. Here T, is their kinetic energy.

8

1 How does it influence the above mentioned intention in the space - time? what is the handling of the force F*<sup>j</sup>* 2 (*x*(*s*))? What is the quantum mechanism that makes possible the 3 transformation of a body or space dictated by this intention?

4 It is necessary to have two aspects clear: the influence grade on the space, and a property 5 that the field itself "wakes up" in the space or body to be transformed though the quantum 6 information I(x), their particles. Consider the integral (8) and their Green function for *n*, 7 states I(*xj*) (*j* = 1, 2, }. *n*):

$$\mathbf{G}^{(n)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = \frac{\prod\_{i=1}^n \int\_{-\infty}^{\infty} \mathbf{e}^{i\Im[\phi^i, \hat{\boldsymbol{\beta}}\_p \phi^i]} d\phi^i(\mathbf{x}(\mathbf{s}))}{\int\_{-\infty}^{\infty} \mathbf{e}^{i\Im[\phi^i, \hat{\boldsymbol{\beta}}\_p \phi^i]} d\phi^i(\mathbf{x}(\mathbf{s}))},\tag{22}$$

Provisional chapter 9

221

http://dx.doi.org/10.5772/53439

,

I

I

derivative of their Lagrangian density L(0) 1 , that is a consequence of the differential (21),

I

4 But the equation (26) is the quantum wave equation (bearer of the information (*configuration*  5 *and momentum* of the intention)) due to *O*c, to the time *s*. Then the generating functional 6 takes the form (23), considering the property of the operator *O*c, given through the operator

> I

*W xs x x x x xs x x x s d x*

<sup>i</sup> [F ( ( ))] N exp ( ) ( ) ( ) i F ( ( )) ( )F ( ( )) ( ( )) <sup>2</sup>

½ ° ° ® ¾ ° ° ¯ ¿

)*h*, given by (21) (using the energy (*amplitude*) that their propagator contributes

*j j*


*j j j j*

O O

*j j j j*

*x x x x xs x x x s d x*

*j j j*

ƺ(ƥ) ƺ(ƥ)

8 (27)

½ ° ° ® ¾ ° ° ¯ ¿

10 The intention infiltrated by the conscience given for *O*c, establishes that the differential of the

12 *DF*) can be visualised inside the configuration space through their boarder points ("targets" 13 of the intention of the field *X*, and that happen in wM [12]),being also the interior points of 14 the space M, *int*M, are *the proper sources* of the field (particles of the space M, that generate the

( ), *T i* <sup>w</sup>M M *nt* 16 (28)

17 where this is a composition of the actions *int*M, and wM. These actions have codimensions 18 strata *k*, and *n* – *k*, respectively, when we want to form the space M, using path integrals [].

19 To extract the intrinsic properties of integrals over configurating spaces, we will follow the 20 proof of the formality theorem [13], and record the relevant facts in our homological-21 physical interpretation: admissible graphs are "cobordisms" Z(J) o [*m*], when Un is thought 22 as a state-sum model [14]. The graphs are also interpreted as "extensions" J o \*o J', when 23 considering the associated Hopf algebra structure. The implementations of these tools were 24 done in the [3]. Remember that using the Stokes theorem (10) a Lagrangian on the class G, of Feynman graphs is a *k*-linear map Z : *H* o : 25 (M), associating to any Feynman graph \*, a

1 The operator O(*x* - *xj*) = (*<sup>x</sup>*+ *m*2 – *i*H)G*n*(*x* - *x*j), and such that to their inverse O1(*x* - *xj*), the functional

<sup>i</sup> ' N exp () ( )( ) i F ( ( ))( )F ( ( ))( )) <sup>2</sup>

O O

'' -1

Ǎ Ǎ

I

³³ ³

 I

ƺ(ƥ) ƺ(ƥ)

I

I(*x*)] = [*d*Ic(*x*)].

15 field *X*). Then the intention of the field *X*, is the total action

³³ ³

f

f

f

f

I

9 Where we have used [*d*

w w   L L (0) (0)

 I

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

Dž Dž 0, Dž Dž *x x* 3 (26)

2 (whereas, by the application of their conscience operator *O*c) know

O(*x* - *xj*)1 7 :

11 action *d*(

ƺ(ƥ)

property is had:

 ³ *n n jj j xx x <sup>y</sup> dx x <sup>y</sup>* <sup>1</sup>

O O ( ) ( - ) Dž ( ),

9 These Green's functions can most straightforwardly be evaluated by use of generating functional where we are using an external force F*<sup>j</sup>* 10 (*x*(*s*)), given by the intention

$$\mathrm{W[F^{j}(x(s))]} = \mathrm{N} \int\_{-\infty}^{+\infty} \exp\left\{i\Im(\phi^{j}, \partial\_{\mu}\phi^{j}) + \mathrm{i} \int\_{\Delta(\Gamma)} \mathrm{F}^{j}(\mathbf{x}(s))\phi^{j}(\mathbf{x})\right\} d(\mathbf{x}(s)),\tag{23}$$

12 This operator is the operator of execution exe(I), which establishes in general form (5) that 13 has been studied and applied in other developed research (see [2, 10, 11] as an example).

14 Then the influence realised on the space :(\*) M, that there bears the functional one (23) 15 that involves the force of the intention given by the field (observe that the second addend of 16 the argument of *exp*, is the action which is realised from the exterior on the space :(\*)) can 17 go according to the functional derivative:

$$\mathbf{G}^{(n)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = \text{(-i)}^n \frac{1}{\text{W}[0]} \frac{\mathbf{\delta}^n}{\delta \mathbf{F}^j(\mathbf{x}\_1) \cdots \delta \mathbf{F}^j(\mathbf{x}\_n)} \text{W}[\mathbf{F}^j(\mathbf{x}(\mathbf{s}))] \Big|\_{\mathbf{F}^j = \mathbf{0}} \tag{24}$$

where <sup>1</sup> Dž DžF( ) DžF( ) *n j j <sup>n</sup> x x* 19 , describe the functional differentiation of nth-order, defined by

20 the formula [6]

$$\frac{\delta^n(\mathbf{F}^j(\mathbf{x}))}{\delta \mathbf{F}^j(\mathbf{x}\_1) \cdots \delta \mathbf{F}^j(\mathbf{x}\_n)} = \frac{\delta^n}{\delta \mathbf{F}^j(\mathbf{x}\_1) \cdots \delta \mathbf{F}^j(\mathbf{x}\_n)} \int \quad \delta \quad (\mathbf{x} \cdot \mathbf{x}\_1) \mathbf{F}^j(\mathbf{x}\_1) \delta(\mathbf{x} \cdot \mathbf{x}\_2) \mathbf{F}^j(\mathbf{x}\_2) \cdots$$
 
$$\cdots \delta(\mathbf{x} \cdot \mathbf{x}\_n) \mathbf{F}^j(\mathbf{x}\_n) d\mathbf{x}\_1 \cdots d\mathbf{x}\_n = \delta(\mathbf{x} \cdot \mathbf{x}\_1) \cdots \delta(\mathbf{x} \cdot \mathbf{x}\_n),$$

22 where these derivatives express impulses (force) of every particle placed in the positions *x*1, 23 }, *xn*. In case of receiving the influence of the field *X*, these impulses will be directed by the derivative of their Lagrangian density L(0) 1 , that is a consequence of the differential (21), 2 (whereas, by the application of their conscience operator *O*c) know

$$
\partial\_{\mu} \frac{\delta \mathbf{u}^{(0)}}{\delta \partial\_{\mu} \phi(\mathbf{x})} - \frac{\delta \mathbf{u}^{(0)}}{\delta \phi(\mathbf{x})} = 0,\tag{26}
$$

4 But the equation (26) is the quantum wave equation (bearer of the information (*configuration*  5 *and momentum* of the intention)) due to *O*c, to the time *s*. Then the generating functional 6 takes the form (23), considering the property of the operator *O*c, given through the operator O(*x* - *xj*)1 7 :

$$\begin{split} \mathcal{W}[\mathbb{E}^{j}\{\mathbf{x}(s)\}] &= \mathcal{N} \int\_{-\alpha}^{+\alpha} \exp\left\{-\frac{1}{2} \int\_{\varDelta\{\mathbb{D}\}} \phi(\mathbf{x}) \mathbf{c}(\mathbf{x}-\mathbf{x}\_{j}) \phi(\mathbf{x}\_{j}) - \mathrm{i} \int\_{\varDelta\{\mathbb{D}\}} \mathcal{F}^{j}(\mathbf{x}(s)) \mathbf{c}^{\ast}(\mathbf{x}-\mathbf{x}\_{j}) \mathcal{F}^{j}(\mathbf{x}\_{j}(s)) \right\} d(\phi(\mathbf{x})) \\ &= \mathcal{N} \int\_{-\alpha}^{+\alpha} \exp\left\{-\frac{1}{2} \int\_{\varDelta\{\mathbb{D}\}} \dot{\phi}^{\prime}(\mathbf{x}) \mathbf{c}(\mathbf{x}-\mathbf{x}\_{j}) \dot{\phi}^{\prime}(\mathbf{x}\_{j}) - \mathrm{i} \int\_{\varDelta\{\mathbb{D}\}} \mathcal{F}^{j}(\mathbf{x}(s)) \mathbf{c}^{\ast}(\mathbf{x}-\mathbf{x}\_{j}) \mathcal{F}^{j}(\mathbf{x}\_{j}(s)) \right\} d(\phi^{\prime}(\mathbf{x})), \end{split} \tag{27}$$

9 Where we have used [*d*I(*x*)] = [*d*Ic(*x*)].

8

6 information

I

7 states

I

220 Advances in Quantum Mechanics

(*xj*) (*j* = 1, 2, }. *n*):

17 go according to the functional derivative:

( )

<sup>1</sup> Dž DžF( ) DžF( ) *n j j <sup>n</sup> x x*

where

20 the formula [6]

1 How does it influence the above mentioned intention in the space - time? what is the handling of the force F*<sup>j</sup>* 2 (*x*(*s*))? What is the quantum mechanism that makes possible the

4 It is necessary to have two aspects clear: the influence grade on the space, and a property 5 that the field itself "wakes up" in the space or body to be transformed though the quantum

(x), their particles. Consider the integral (8) and their Green function for *n*,

I I

*<sup>n</sup> <sup>i</sup>*

i[ , ]

Ǎ

i[ , ] i

*i i*

½

° ° ¯ ¿

ƺ(ƥ)

Ǎ

*i i*

e ( ( ))

I

*d xs*

 I

*n*

³

11 22

*j j*

I

*d xs*

e ( ( ))

I I

 f f w

³

G(, ,, ) ,

8 (22)

9 These Green's functions can most straightforwardly be evaluated by use of generating

II

11 (23)

12 This operator is the operator of execution exe(I), which establishes in general form (5) that 13 has been studied and applied in other developed research (see [2, 10, 11] as an example).

14 Then the influence realised on the space :(\*) M, that there bears the functional one (23) 15 that involves the force of the intention given by the field (observe that the second addend of 16 the argument of *exp*, is the action which is realised from the exterior on the space :(\*)) can

> *<sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>j</sup> n j j*

18 (24)

19 , describe the functional differentiation of nth-order, defined by

1

22 where these derivatives express impulses (force) of every particle placed in the positions *x*1, 23 }, *xn*. In case of receiving the influence of the field *X*, these impulses will be directed by the

21 (25)

<sup>Dž</sup> (F ( )) <sup>Dž</sup> <sup>Dž</sup> ( - )F <sup>Dž</sup>( - )F

*xx x x s*

° ° ® ¾ <sup>w</sup>

1 2 F 0 1 <sup>1</sup> <sup>Dž</sup> G ( , , , ) ( i) W[F ( ( ))] , W[0] <sup>Dž</sup>F( ) <sup>Dž</sup>F( ) *<sup>j</sup>*

E

*n <sup>x</sup> xx x xx x*

*x x*

 

³ ³ *j j <sup>j</sup> <sup>j</sup> x s xs x dxs* <sup>j</sup> Ǎ

W[F ( ( ))] N exp i ( , ) i F ( ( )) ( ) ( ( )),

functional where we are using an external force F*<sup>j</sup>* 10 (*x*(*s*)), given by the intention

f

f

 

1 1 <sup>ƺ</sup>(ƥ)

Dž( - )F d d Dž( - ) Dž( - ),

*xx x x x xx xx*

*n n*

*n n1 n n*

*jj jj*

*j*

*n j n*

DžF( ) DžF( ) DžF( ) DžF( )

*xx xx*

( ) 1 1 2

*n i n*

*xx x*

f

³

f w

3 transformation of a body or space dictated by this intention?

10 The intention infiltrated by the conscience given for *O*c, establishes that the differential of the 11 action *d*(I)*h*, given by (21) (using the energy (*amplitude*) that their propagator contributes 12 *DF*) can be visualised inside the configuration space through their boarder points ("targets" 13 of the intention of the field *X*, and that happen in wM [12]),being also the interior points of 14 the space M, *int*M, are *the proper sources* of the field (particles of the space M, that generate the 15 field *X*). Then the intention of the field *X*, is the total action

$$\mathfrak{J}^{\prime} = \mathfrak{J}^{\prime \mu \prime} (\mathfrak{J}^{\prime \mu \prime \nu \prime}), \tag{28}$$

17 where this is a composition of the actions *int*M, and wM. These actions have codimensions 18 strata *k*, and *n* – *k*, respectively, when we want to form the space M, using path integrals [].

19 To extract the intrinsic properties of integrals over configurating spaces, we will follow the 20 proof of the formality theorem [13], and record the relevant facts in our homological-21 physical interpretation: admissible graphs are "cobordisms" Z(J) o [*m*], when Un is thought 22 as a state-sum model [14]. The graphs are also interpreted as "extensions" J o \*o J', when 23 considering the associated Hopf algebra structure. The implementations of these tools were 24 done in the [3]. Remember that using the Stokes theorem (10) a Lagrangian on the class G, of Feynman graphs is a *k*-linear map Z : *H* o : 25 (M), associating to any Feynman graph \*, a

$$\int\_{\mathfrak{Q}(\Gamma)} \mathsf{o}^{-1}(\mathsf{x} - \mathsf{x}\_{\mathsf{\jmath}}) \mathsf{o}(\mathsf{x}\_{\mathsf{\jmath}} - \mathsf{y}) d^n x\_{\mathsf{\jmath}} = \mathsf{S}^n(\mathsf{x} - \mathsf{y}),$$

<sup>1</sup> The operator O(*x* - *xj*) = (*<sup>x</sup>*+ *m*2 – *i*H)G*n*(*x* - *x*j), and such that to their inverse O1(*x* - *xj*), the functional property is had:

10

1 closed volume form on (\*), vanishing on the boundaries, i.e. for any subgraph J o \* 2 (viewed as a sub-object) meeting the boundary of \* : [*s*] o [*t*] (viewed as a cobordism), Z(J) 3 = 0. Then an action given by *int*M, is defined through their interior as:

4 Def. **2. 1.** An action on G ("int"), is a character W : *H* o R, which is a cocycle in the 5 associated DG-coalgebra (T(*H*\*), *D*), where G, is a class of Feynman graph.

6 Let C*<sup>n</sup>*,*<sup>m</sup>*, be the configuration space of *n*, interior points and *m*, boundary points in the 7 manifold M, with boundary wM (that is to say. [13], upper half-plane H). Its elements will be 8 thought as (geometric) "representations of cobordisms" (enabling degrees of freedom with 9 constraints). Then the action in (28) takes the form

$$\left( \{ \alpha(\chi) = 0 \} \xrightarrow{\{\varpi\}} \{m\} \right) \xrightarrow[\chi(\emptyset)]{} \left( \{ \alpha(\Gamma) = 0 \} \xrightarrow{\{\varpi\}} \partial \mathcal{M} \right) \,. \tag{29}$$

11 Let *H*, be the Hopf algebra (*associative algebra used to the quantised action in the space-time*), of a 12 class of Feynman graphs G [12]. If \*, is such a graph, then configurations are attached to 13 their vertices, while momentum are attached to edges in the two dual representations 14 (Feynman rules in position and momentum spaces). This duality is represented by a pairing 15 between a "configuration functor" (typically C\*, (configuration space of subgraphs and 16 strings [15], and a "*Lagrangian*" (e.g. Z, determined by its value on an edge, i.e. by a 17 propagator *D*F). Together with the pairing (typically integration) representing the action, 18 they are thought as part of the Feynman model of the state space of a quantum system. The 19 differential (21) considering the DG-structure [12], in the class of Feynman graphs G, can be 20 defined as one graph homology differential:

$$d\Gamma = \sum\_{\mathbf{e}:\mathbf{e}\to\mathbf{e}\_{\Gamma}} \pm \Gamma \;/\; \mathbf{y}\_{\mathbf{e}'} \tag{30}$$

Provisional chapter 11

223

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I(*x x*')

1 Points of phase space are called states of the particle system acting in the cotangent space of 2 M. Thus, to give the state of a system, one must specify their *configuration and momentum*.

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

**Figure 3.** a) In every plane there is a particle configuration for a given time. b) The evolution of the particles along everything t, happens for a succession of configurations through which the particles system spends different strata codimension one. The causal structure of the space - time is invariant for every particle along the transformation process. d) Strata Evolution strata of the configuration space C*n, m*, in the space-time E<sup>4</sup> 7 . The translation obeys to

Example **1.** Let S7 9 MoM, be and JR o TC*n, m*, then S ߭ JRoM, describes the curve in 10 the configuration space which describes the sequence of configurations through which the 11 particles system passes to different strata of co-dimension one (see figure 2). Every strata 12 correspond to a phase space of *m*, particles that are moved by curve J and directed from

> I(*x*).

I(*s*))*d*I

O <sup>c</sup>

I

= 6*j*³\*O(*x* – *x*')*x*(*s*)*ds*. Also

 I

 I I

§ · ¨ ¸ ¨ ¸ © ¹ c

 I <sup>c</sup> ( - ') ( ') *<sup>O</sup> x xd d x*

*d*

H

*O s x s ds x x x dx,* ( ( )) ( ) ( - ') ( )

<sup>c</sup> ( ( )) ( ') ( ') *<sup>O</sup> <sup>d</sup> O xs x x*

I

3 a) b) c)

I1, }, I*<sup>n</sup>*).

(*x*), by S, to *n*, particles

14 This defines our intentional conscience. Then are true the following properties:

17 iii. ³Oc(I(s))dI = Oc; dOc(I(s))/dI= Oc(x(s)), in the unlimited space H, 18 iv. Oc = G(s – s'), if and only if Gx(s)/Gx(s') = G(s – s'), s d t, then F(x(s)) = x(s),

I

O

³ ³

then *i*, is satisfied. To the property *ii*. is necessary consider O(*x* – *x*') = (*<sup>x</sup>* + *m*2 23 *i*H)

v. O<sup>1</sup> 19 (x x')Oc(x(s)) = 'F(x x')G(x – x'), x, x'M, and s d t,

ƺ(ƥ)

8 evolution of Lagrangian system given by *L*(

I

16 ii. Oc(x(s))I(x'(s)) = O(x – x'), x, x'M, and s d t,

I(*s*))I(*x*)*dx* =³

15 i. O(x – x')I(x) = G(x – x')I(x), x, x'M,

20 vi. ³H Oc(I(s))dI = ³:O(x – x')x(s)d(x(s)).

H 21 *O*c(

24 [6]. But the operator *O*c, is the defined as

26 Considering the operator O(*x* – *x*'), we have

On the one hand, ³H *O*c(

22

25

27

13 their energy states *d*

22 where the sum is over the edges of \*, J*e* is the one-edge graph, and \*/J*e*, is the quotient 23 (forget about the signs for now).

24 We can give a major generalisation of this graphical homological version of the differential 25 establishing the graduated derivation that comes from considering *H* = T(g), the tensor 26 algebra with reduced co-product

$$
\Delta\Gamma = \sum\_{\mathbf{y} \to \Gamma \to \mathbf{y}'} \mathbf{y} \otimes \mathbf{y}'\_{\prime} \tag{31}
$$

28 Consider the following basic properties of the operators *O*c. Let O(*x* – *x*'), defined in the footnote 1, and I(*x*)H, where the space is the set of points H = {I(*x*)[m] °[m] T\*M} <sup>2</sup> 29 [8].

Here [m] = T\*C*n,m*.

<sup>2</sup> The corresponding cotangent space to vector fields is:

T\*x1(M)= {(I, wPI)H u Tx1(M) ° wPI= P[, [x1(M)}.

1 Points of phase space are called states of the particle system acting in the cotangent space of 2 M. Thus, to give the state of a system, one must specify their *configuration and momentum*.

4 **Figure 3.** a) In every plane there is a particle configuration for a given time. b) The evolution of the particles along 5 everything t, happens for a succession of configurations through which the particles system spends different strata 6 codimension one. The causal structure of the space - time is invariant for every particle along the transformation process. d) Strata Evolution strata of the configuration space C*n, m*, in the space-time E<sup>4</sup> 7 . The translation obeys to 8 evolution of Lagrangian system given by *L*(I1, }, I*<sup>n</sup>*).

Example **1.** Let S7 9 MoM, be and JR o TC*n, m*, then S ߭ JRoM, describes the curve in 10 the configuration space which describes the sequence of configurations through which the 11 particles system passes to different strata of co-dimension one (see figure 2). Every strata 12 correspond to a phase space of *m*, particles that are moved by curve J and directed from 13 their energy states *d*I(*x*), by S, to *n*, particles I(*x*).

14 This defines our intentional conscience. Then are true the following properties:

15 i. O(x – x')I(x) = G(x – x')I(x), x, x'M,

10

222 Advances in Quantum Mechanics

1 closed volume form on (\*), vanishing on the boundaries, i.e. for any subgraph J o \* 2 (viewed as a sub-object) meeting the boundary of \* : [*s*] o [*t*] (viewed as a cobordism), Z(J)

4 Def. **2. 1.** An action on G ("int"), is a character W : *H* o R, which is a cocycle in the

6 Let C*<sup>n</sup>*,*<sup>m</sup>*, be the configuration space of *n*, interior points and *m*, boundary points in the 7 manifold M, with boundary wM (that is to say. [13], upper half-plane H). Its elements will be 8 thought as (geometric) "representations of cobordisms" (enabling degrees of freedom with

^  o ` o  ^ owM` [ ] x(s) [ ] {ǚ DŽ 0} [ ] {ǚ ƥ 0} , *n m* <sup>10</sup>*m* (29)

11 Let *H*, be the Hopf algebra (*associative algebra used to the quantised action in the space-time*), of a 12 class of Feynman graphs G [12]. If \*, is such a graph, then configurations are attached to 13 their vertices, while momentum are attached to edges in the two dual representations 14 (Feynman rules in position and momentum spaces). This duality is represented by a pairing 15 between a "configuration functor" (typically C\*, (configuration space of subgraphs and 16 strings [15], and a "*Lagrangian*" (e.g. Z, determined by its value on an edge, i.e. by a 17 propagator *D*F). Together with the pairing (typically integration) representing the action, 18 they are thought as part of the Feynman model of the state space of a quantum system. The 19 differential (21) considering the DG-structure [12], in the class of Feynman graphs G, can be

> <sup>r</sup> ¦ ƥ

e ƥ ƥ /DŽ , *E* 21 *d* (30)

22 where the sum is over the edges of \*, J*e* is the one-edge graph, and \*/J*e*, is the quotient

24 We can give a major generalisation of this graphical homological version of the differential 25 establishing the graduated derivation that comes from considering *H* = T(g), the tensor

oo ¦ DŽƥDŽ' 27 Ʀƥ DŽ DŽ', (31)

28 Consider the following basic properties of the operators *O*c. Let O(*x* – *x*'), defined in the

<sup>2</sup> 29 [8].

(*x*)H, where the space is the set of points H = {

= P[, [x1(M)}.

e

I

(*x*)[m] °[m] T\*M}

3 = 0. Then an action given by *int*M, is defined through their interior as:

9 constraints). Then the action in (28) takes the form

20 defined as one graph homology differential:

23 (forget about the signs for now).

26 algebra with reduced co-product

I

2 The corresponding cotangent space to vector fields is:

I

)H u Tx1(M) ° wP

footnote 1, and

I, wPI

Here [m] = T\*C*n,m*.

T\*x1(M)= {(

5 associated DG-coalgebra (T(*H*\*), *D*), where G, is a class of Feynman graph.


22

25

27

On the one hand, ³H *O*c(I(*s*))I(*x*)*dx* =³ H 21 *O*c(I(*s*))*d*I= 6*j*³\*O(*x* – *x*')*x*(*s*)*ds*. Also

$$\int\_{\mathfrak{A}(\Gamma)} O\_{\mathfrak{c}}(\phi(\mathbf{s})) \mathfrak{x}(\mathbf{s}) d\mathbf{s} = \int\_{\mathbb{H}} \phi(\mathbf{x} - \mathbf{x}') \phi(\mathbf{x}) d\mathbf{x}',$$

then *i*, is satisfied. To the property *ii*. is necessary consider O(*x* – *x*') = (*<sup>x</sup>* + *m*2 23 *i*H)I(*x x*') 24 [6]. But the operator *O*c, is the defined as

$$O\_{\mathbf{c}}(\mathbf{x}(\mathbf{s})) \phi(\mathbf{x'}) = \left(\frac{d\mathfrak{N}\_{O\_{\mathbf{c}}}}{d\phi}\right) \phi(\mathbf{x'}),$$

26 Considering the operator O(*x* – *x*'), we have

$$
\alpha(\infty \text{ - } \infty \text{'})d\phi = d\mathfrak{N}\_{O\_c}\phi(\infty \text{'}),
$$

12

1 Integrating both members on unlimited space Hu:\*, (applying the principle of Stokes 2 integration given by (10)) we have that integral identity is valid for whole space. Then is 3 verifying *ii*. The property *iii*., is directly consequence of (19), (20) and (21), considering the Stokes 4 theorem given in (10), therefore *O*c(*x*(*s*)), is such that Z*<sup>L</sup>* = (*O*c)\*Z, considering Z = *d*I. Then

$$O\_c = \int\limits\_{\mathfrak{Q}(\Gamma)^{\chi\_{\rm II}}} \mathfrak{o}\_L = \int\limits\_{\mathfrak{Q}(\Gamma)} (O\_c)^\star \mathfrak{o}\_\prime \tag{32}$$

Provisional chapter 13

225

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1 the force F, and this produces the permanent state of energy generated by every component

 ³ ³ <sup>c</sup> ƥ ƥ

*Džx s* 2 which is equivalent to

³ <sup>c</sup>

7 This integral is valid \*M. Thus *O*c(*x*(*s'*)) = G(*s – s'*). With this, the demostration of *iv*., is

10 The identity in *vi*., happens in the phase space created by the cotangent space due to the 11 image of the differential (21). Therefore, both members of integral identity will have to

 
 ³ ³³

c c c c

� �

*Ox x xs m m x dx*

*x x*

H Z H 

2 2

*<sup>Dž</sup>x s ss s t*

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

*x'*) = '*F*(*x* 

I

*x'*), considering the

of the space. On the other side, if <sup>d</sup> ( ) <sup>Dž</sup>( - '), ( ')

3 . But integrating (21) we have

Dž ( ( ')) Dž( - ') ( ') ', *<sup>O</sup>* 4 *x s s s x s ds*

ƥ 6 { ( ( ')) ( ') *O x s x s s s x s ds* Dž( - ') ( ')} ' 0

12 coincide in the intention given by *O*c. Indeed, consider the integral

I

I

H 

*x Ʊ*

*m dd*

³ ³

2

( i )

H H

 I

³³ ³

ƺ(ƥ) ƺ(ƥ) ƺ

17 **3. Quantum intentionality and organized transformations** 

H H

ƺ(ƥ) *O xd O d* ( ( )) <sup>ǚ</sup> *Ʊ Ʊ* 13 (35)

> I

( - ')d ( ) ( i ) ( i )

c

Considering the quantizations of our Lagrangian system describe in (11), (18) and (19) on R*<sup>n</sup>* 18 ,

}, we describe terms of a graded commutative C<sup>f</sup> 19 (M)-algebra *H*, with

22 and the bi-graded differential algebra *H\**, of differential forms (the Chevalley–Eilenberg differential calculus) over *H*<sup>0</sup> 23 , as an R-algebra [1-3]. One can think of generating elements 24 (37) of *H*, as being *sui generis* coordinates of even and odd fields and their partial derivatives.

1 2 <sup>1</sup> { . , , , . . . , ... , . . ,} *aa a a m m l m ll m l lk* 21 ww w w *xx x x* (37)

15 (36)

completed. The identity in *v*, happens because O<sup>1</sup> 8 (*x* 

14 On the other side, inside the quantum wave equation:

�

16 Joining (35) with (36) we have *vi*.

*n* t 2, coordinated by {*xj*

20 generating elements

9 before property (simple conaequence of the property *iv*) [6].

5 which for before implication is G(*s – s'*)G*x*(*s'*). But

³

( <sup>Dž</sup>( - ') ( ') ') Dž( - '), ( ') *<sup>Dž</sup> s s x s ds s s*

ƥ

*Džx s*

6 which is a integral of type (10). Indeed,

$$\mathfrak{T}\_{O\_c} = \int\_{\mathfrak{u} \times \mathfrak{Q}(\Gamma)} \mathfrak{o}\_L = \int\_{\mathfrak{u}} O\_c(\mathfrak{x}(\mathfrak{s})) \,^\* \mathfrak{o} = \int\_{\mathfrak{u}} O\_c(\mathfrak{x}(\mathfrak{s})) d\mathfrak{\phi} = \int\_{\mathfrak{u}} d\mathfrak{N}\_{O\_c}(\mathfrak{\phi}) , \tag{33}$$

8 The derivative in the last integral from (33) is the total differential given by (21) from where 9 we have the derivative formula in the context of the unlimited space H.

10 The property *iv*., require demonstrate two implication where both implications are 11 reciprocates. If *O*c(*x*(*s'*)) = G(*s – s'*), then all intention on trajectory defined \*, its had that

$$\int\_{\Gamma} O\_{\mathbf{c}}(\mathbf{x}(s)) \mathbf{x}(s) \, \mathrm{d}s = \int\_{\Gamma} \delta(\mathbf{s} \cdot \mathbf{s}') \mathbf{r}(s) \, \mathrm{d}s,\tag{34}$$

13 But for the differential (21) and the second member of the integral (34) we have

$$\left\| \mathfrak{Z} \mathfrak{I}\_{O\_c} (\mathfrak{x}(s')) = \mathfrak{G} (\bigcap\_{\Gamma} \mathfrak{f}(s - s') \mathfrak{x}(s') ds'), \right\|$$

since ³ ƥ ( <sup>Dž</sup>( - ') ( ') ') Dž( - '), ( ') *<sup>Dž</sup> s s x s ds s s Džx s* then ³ ƥ 15 *Dž*( Dž( - ') ( ') ' *s s x s ds x s* Dž ), and for other side

$$\delta(\{\delta(s\text{-}s\text{'})\text{x(s')}\text{ds'}\} = \delta(s\text{-}s\text{'})\delta\text{x(s')}\text{'}$$

from where <sup>d</sup> ( ) <sup>Dž</sup>( - '), ( ') *<sup>Dž</sup>x s ss s t Džx s* <sup>3</sup> 17 . But this implies directly F(*x*(*s*)) = *x*(*s*). This property tell

18 us that we can have influence on the space M, considering only a curve any of the space 19 where the influence of the field exists like the force G(*s – s'*), since the space is infiltrated by

3In the general sense the functional derivative I I *<sup>a</sup> <sup>n</sup> ba b <sup>Dž</sup> <sup>y</sup> <sup>Dž</sup> <sup>y</sup> <sup>x</sup> <sup>Dž</sup> <sup>x</sup>* ( ) <sup>Dž</sup> ( - ), ( ) implies

$$\delta\phi\_b(y) = \sum\_a \int \delta''(y \cdot \mathbf{x}) \delta\phi\_a(\mathbf{x}) \delta\_{ba} d\mathbf{x},$$

but does not imply

$$
\delta\phi\_b\left(y\right) = \delta\_{ba}\delta''\left(y - x\right)\delta\phi\_a\left(x\right).
$$

1 the force F, and this produces the permanent state of energy generated by every component of the space. On the other side, if <sup>d</sup> ( ) <sup>Dž</sup>( - '), ( ') *<sup>Dž</sup>x s ss s t Džx s* 2 which is equivalent to ³ ƥ ( <sup>Dž</sup>( - ') ( ') ') Dž( - '), ( ') *<sup>Dž</sup> s s x s ds s s Džx s* 3 . But integrating (21) we have

$$\int\_{\Gamma} \mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathcal{\mathsf{\mathsf{\mathsf{\mathcal{\mathsf{\mathsf{\mathcal{\mathsf{\mathcal{\mathsf{\mathsf{\mathcal{\mathsf{\mathcal{\mathsf{\mathsf{\mathcal{\mathsf{\mathsf{\mathcal{\mathsf{\mathcal{\mathsf{\mathcal{\prime}}\right{\mathcal{\mathsf{\mathcal{\mathsf{\mathcal{\prime}}\right}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} $$
}}}} } 

5 which for before implication is G(*s – s'*)G*x*(*s'*). But

12

1 Integrating both members on unlimited space Hu:\*, (applying the principle of Stokes 2 integration given by (10)) we have that integral identity is valid for whole space. Then is 3 verifying *ii*. The property *iii*., is directly consequence of (19), (20) and (21), considering the Stokes

u ³ ³ H c c ƺ(ƥ) ƺ(ƥ) ǚ ( )\* ǚ, *O O <sup>L</sup>* 5 (32)

> ³³ ³ ³ HH H H <sup>c</sup> <sup>c</sup> c c

<sup>ǚ</sup> ( ( )) \* <sup>ǚ</sup> ( ( )) ( ) *O L O xs O xs d d <sup>O</sup>* 7 (33)

8 The derivative in the last integral from (33) is the total differential given by (21) from where

10 The property *iv*., require demonstrate two implication where both implications are 11 reciprocates. If *O*c(*x*(*s'*)) = G(*s – s'*), then all intention on trajectory defined \*, its had that

> <sup>c</sup> ³ ƥ

> > ƥ

<sup>3</sup> 17 . But this implies directly F(*x*(*s*)) = *x*(*s*). This property tell

18 us that we can have influence on the space M, considering only a curve any of the space 19 where the influence of the field exists like the force G(*s – s'*), since the space is infiltrated by

I

 *<sup>a</sup> <sup>n</sup> ba*

*<sup>Dž</sup> <sup>y</sup> <sup>Dž</sup> <sup>y</sup> <sup>x</sup> <sup>Dž</sup> <sup>x</sup>*

( ) <sup>Dž</sup> ( - ), ( ) implies

I

*b*

then ³

 ³ ³ <sup>c</sup> ƥ ƥ 12 *O x s x s s s s x s ds* ( ( )) ( )d Dž( - ') ( ) , (34)

15 *Dž*( Dž( - ') ( ') ' *s s x s ds x s* Dž ), and for other side

13 But for the differential (21) and the second member of the integral (34) we have

I I

I. Then

4 theorem given in (10), therefore *O*c(*x*(*s*)), is such that Z*<sup>L</sup>* = (*O*c)\*Z, considering Z = *d*

u

Dž ( ( ')) Dž( Dž( - ') ( ') ') *<sup>O</sup>* 14 *x s s s x s ds*

G ( \*

G G <sup>G</sup> ³ <sup>16</sup>( ) ( ) ) ( ) ), *s - s' x s' ds' s - s' x s'*

( <sup>Dž</sup>( - ') ( ') ') Dž( - '), ( ') *<sup>Dž</sup> s s x s ds s s*

*<sup>Dž</sup>x s ss s t*

3In the general sense the functional derivative

 I

 I ƺ ƥ

9 we have the derivative formula in the context of the unlimited space H.

6 which is a integral of type (10). Indeed,

224 Advances in Quantum Mechanics

since ³

I

I

 ¦³ *<sup>n</sup> b a ba*

*a*

but does not imply

 *<sup>n</sup> b ba a Dž* () *y Dž* Dž ( - ) ( ). *y x Dž x*

*Džx s*

ƥ

from where <sup>d</sup> ( ) <sup>Dž</sup>( - '), ( ')

*Džx s*

*Dž* ( ) *y y* Dž ( - ) ( ) , *x Dž x Dž dx*

$$\int\_{\Gamma} \{O\_{\mathbf{c}}(\mathbf{x}(s'))\mathbf{x}(s') + \delta(s - s')\mathbf{x}(s')\} ds' = 0,$$

7 This integral is valid \*M. Thus *O*c(*x*(*s'*)) = G(*s – s'*). With this, the demostration of *iv*., is completed. The identity in *v*, happens because O<sup>1</sup> 8 (*x x'*) = '*F*(*x x'*), considering the 9 before property (simple conaequence of the property *iv*) [6].

10 The identity in *vi*., happens in the phase space created by the cotangent space due to the 11 image of the differential (21). Therefore, both members of integral identity will have to 12 coincide in the intention given by *O*c. Indeed, consider the integral

$$\int\_{\mathfrak{n}} O\_{\mathfrak{c}}(\phi(\mathbf{x}))d\phi = \int\_{\mathfrak{A}(\mathbb{T})} (O\_{\mathfrak{c}}) \* \mathfrak{a} = \int\_{\mathfrak{n}} d\mathfrak{X}\_{\mathbb{O}\mathfrak{c}} = \mathfrak{I}\_{\mathbb{O}\mathfrak{c}},\tag{35}$$

14 On the other side, inside the quantum wave equation:

$$\begin{split} \int\_{\mathfrak{A}(\mathbb{T})} O(\mathbf{x} \cdot \mathbf{z}') d\mathbf{x}(\mathbf{s}) &= \int\_{\mathfrak{A}(\mathbb{T})} (\mathbb{\Box}\_{\mathbf{x}} + m^2 - i\mathbf{z}) d\mathbf{s} = \int\_{\mathfrak{A}} (\Box\_{\mathbf{x}} + m^2 - i\mathbf{z}) \phi(\mathbf{x}) d\mathbf{x} \\ = \int\_{\mathfrak{A}} (\Box\_{\mathbf{x}} + m^2 - i\mathbf{z}) d\phi = \int\_{\mathfrak{A}} d\mathfrak{T}\_{\mathrm{Ox}}(\phi), \end{split} \tag{36}$$

16 Joining (35) with (36) we have *vi*.

### 17 **3. Quantum intentionality and organized transformations**

Considering the quantizations of our Lagrangian system describe in (11), (18) and (19) on R*<sup>n</sup>* 18 , *n* t 2, coordinated by {*xj* }, we describe terms of a graded commutative C<sup>f</sup> 19 (M)-algebra *H*, with 20 generating elements

$$\{\partial\_{\mathbf{m}}\mathbf{x}^{a}, \partial\_{\mathbf{m}}\mathbf{x}^{a}, \partial\_{\mathbf{m}}\mathbf{x}^{a}\_{\,\,\!\!112\,\prime}, \dots, \partial\_{\mathbf{m}}\mathbf{x}^{a}\_{\,\,\!\!11\cdots\to\!\!k}, \dots\},\tag{37}$$

22 and the bi-graded differential algebra *H\**, of differential forms (the Chevalley–Eilenberg differential calculus) over *H*<sup>0</sup> 23 , as an R-algebra [1-3]. One can think of generating elements 24 (37) of *H*, as being *sui generis* coordinates of even and odd fields and their partial derivatives. 14

The graded commutative R-algebra *H*<sup>0</sup> 1 , is provided with the even graded derivations (called 2 total derivatives)

$$d\_{\lambda} = \partial\_{\lambda} + \sum\_{0 \preceq [\lambda]} \hat{\sigma}\_{\lambda + \Lambda}^{a} \hat{\sigma}\_{a}^{\Lambda}, \ \mathbf{d}\_{\Lambda} = \mathbf{d}\_{\lambda\_{1}} \cdots \mathbf{d}\_{\lambda\_{k}} \tag{38}$$

Provisional chapter 15

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 I

1 action T, defined in (28) to any derivation given through their conscience operator (fiber (18)), like the graded derivation w, 2 (considering the derivatives *O*c(*x*) = (*O*c)\*Z, *D*F = *d*I):

4 of the algebra of quantum fields B). With an odd parameter D, let us consider the

^ ` ˆ wD

7 Of the algebra B). This automorphism yields a new state < , >, of B), given by the equality

ˆ ˆ !

 ³ 1 1 1 1 H

8 (43)

*x x* 9 *dU d* . That because the intention is the

11 What happens towards the interior of every particle? what is the field intention mechanism

13 To answer these questions we have to internalise the actions of field *X*, on the particles of 14 the space M, and consider their spin. But for it, it is necessary to do the immersion of the

19 where the image of the 1-form Z, that the Lagrangian defines, Z(*Zi*), is a symplectic form [8], and the variable *Zi*, is constructed through the algebraic equations W*<sup>a</sup>* 20 (*Zi*) = 0 [16].They 21 describe the *k*-dimensional hypersurfaces denoted by S, such that S H, where H, is the 22 phase space defined in the section 2. The index *a* = 1, . . . , *q* runs over the number of polynomials W*<sup>a</sup>* 23 (*Zi*), in the variables Zi and i runs over the dimension of the ambient manifold which is assumed to be C<sup>N</sup> 24 . If the space is a complete intersection, the constraints

4 Having chosen M<sup>2</sup>*<sup>n</sup>*, is to consider the two components of any point in the space CN, (that we are considering isomorfo to the ambient space of any quantum particle *x*(*s*), in the space-time) to have the two components that characterise any quantum particle x(s), that is their spin (direction) and their energy state (density of energy or "living force of the particle"). L, is the corresponding Lagrangian

II

 w w ƭ Ǎ c L <sup>c</sup> <sup>ƭ</sup> : ( , )( ( ),ǚ) ( ( ),<sup>ǚ</sup> ) ()<sup>ǚ</sup> *a a <sup>a</sup> x c <sup>x</sup>* 3 *x x Ox Ox O* (42)

> D

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

 I

<sup>1</sup> ( ) ( )exp N' *k k k k a a a a xx x x U U*

 I

> I M*a a*

I

w U exp Id , 6

where the energy state has survived, since ˆ( )

<sup>2</sup> : , *<sup>n</sup> w* L M o <sup>4</sup> <sup>16</sup>

, 18 *Z wZ i i*

submanifold of the symplectic structure given by (M2n, Z).

10 same. The intention has not changed.

15 Lagrangian Z, defined as the map

17 with rule of correspondence

12 inside every particle?

5 automorphism

4 where / = (O1...O*k*), |/| = *k*, and O+/ = (O, O1, . . . , Ok) are symmetric multi-indices. One can 5 think of even elements

$$\mathbf{L} = (\mathbf{x}^{\dagger}, \partial\_{\mu} \mathbf{x}^{a}) d^{n} \mathbf{x}, \qquad \delta \mathbf{L} = d \partial\_{\mu} \mathbf{x}^{a} \wedge \varepsilon\_{a} d^{n} \mathbf{x} = \sum\_{\mathbf{0} \in \{\mathbf{h}\}} (-1)^{\|\mathbf{h}\|} d \partial\_{\mu} \mathbf{x}^{a} \wedge d\_{\Lambda} (\partial\_{a} {}^{\Lambda} L) d^{n} \mathbf{x}, \tag{39}$$

7 where we observe that G*L*, is the 2-form given by Z*L*, in the formula (20) with *n* = 2, 8 and /= O1O2.

9 Now we consider the dual part of the space (*H*, :(\*)), that is to say, the space (*H*\*, *L*), be

10 We consider quantize this Lagrangian system in the framework of perturbative Euclidean QFT. We suppose that L, is a Lagrangian of Euclidean fields on :(\*) R*<sup>n</sup>* 11 . The key point is that the algebra of Euclidean quantum fields B), as like as *H*<sup>0</sup> 12 , is graded commutative. It is generated by elements IO/ 13 *a*, *x*؏ :(\*). For any *x*؏ :(\*), there is a homomorphism belonging 14 to space *H* o *Hom*(T(*H*), *D*) (with homomorphisms *Hom*(T(*H*), *D*) given for *D*G-algebra of 15 cycles)

$$\chi\_{\mathbf{x}} : \mathbf{i}\_{\mathfrak{a}\_{1} \cdots \mathfrak{a}\_{r}}^{\Lambda\_{1} \cdots \Lambda\_{r}} \hat{\boldsymbol{\alpha}} \mathbf{x}\_{\Lambda\_{1}}^{\mathfrak{a}\_{1}} \cdots \hat{\boldsymbol{\alpha}} \mathbf{x}\_{\Lambda\_{r}}^{\mathfrak{a}\_{r}} \mapsto \mathbf{i}\_{\mathfrak{a}\_{1} \cdots \mathfrak{a}\_{r}}^{\Lambda\_{1} \cdots \Lambda\_{r}} (\mathbf{x}) \boldsymbol{\phi}\_{\mathbf{x}\Lambda\_{1}}^{\mathfrak{a}\_{1}} \cdots \boldsymbol{\phi}\_{\mathbf{x}\Lambda\_{r}}^{\mathfrak{a}\_{r}}, \qquad \mathbf{i}\_{\mathfrak{a}\_{1} \cdots \mathfrak{a}\_{r}}^{\Lambda\_{1} \cdots \Lambda\_{r}} \in \mathbb{C}^{\pi}(\Omega(\Gamma)) \tag{40}$$

Of the algebra *H*<sup>0</sup> 17 , of classical fields to the algebra B), which sends the basic elements w*xa* /*H*0 to the elements IO/*a*B), and replaces coefficient functions I, of elements of *H*<sup>0</sup> 18 , with 19 their values I(*x*) (executions) at a point *x*. Then a state <, > of B) is given by symbolic 20 functional integrals

$$<\boldsymbol{\phi}\_{\boldsymbol{x}\_{1}}^{a\_{1}}\cdots\boldsymbol{\phi}\_{\boldsymbol{x}\_{k}}^{a\_{k}} > \frac{1}{\operatorname{N}} \Big[\boldsymbol{\phi}\_{\boldsymbol{x}\_{1}}^{a\_{1}}\cdots\boldsymbol{\phi}\_{\boldsymbol{x}\_{k}}^{a\_{k}} \exp\left|-\int\_{\boldsymbol{\Omega}(\Gamma)} \boldsymbol{O}\_{\boldsymbol{c}}(\boldsymbol{\phi}\_{\boldsymbol{x}\boldsymbol{\Lambda}}^{a}) d^{\boldsymbol{\ell}}\mathbf{x}\right|\prod\_{\boldsymbol{x}} [d\boldsymbol{\phi}\_{\boldsymbol{x}}^{\boldsymbol{a}}] \,\tag{41}$$

where this is an integral of type I u ³ H c ƺ(ƥ) 22 *O xs* ( ( ))d , as give by the properties. When the

23 intention expands to the whole space, infiltrating their information on the tangent spaces 24 images of the cotangent bundle T\*M, (given by the imagen of I(*x*), under *dO*c).Then their 25 intentionality will be the property of the field to spill or infiltrate their intention from a nano 26 level of strings inside the quantum particles. Then from the energy states of the particles, 27 and considering the intention spilled in them given by *O*c(I), we have the homomorphism 28 (40) that establishes the action from M (# :(\*)), to wM, for their transformation through the 1 action T, defined in (28) to any derivation given through their conscience operator (fiber (18)), like the graded derivation w, 2 (considering the derivatives *O*c(*x*) = (*O*c)\*Z, *D*F = *d*I):

$$\tilde{\mathcal{C}} : \phi^{a}\_{\mathbf{x}\Lambda} \mapsto (\mathbf{x}, \hat{\sigma}^{a}\_{\mathbf{z}} \mathbf{x}) \mapsto (O\_{\mathbf{c}}(\mathbf{x}), \mathbf{a}) \mapsto (O\_{\mathbf{c}}^{\star}(\mathbf{x}), \mathbf{a}\_{\mathbf{L}}) = O\_{\mathbf{c}}^{\star}(\phi^{a}\_{\mathbf{z}\Lambda}) \mathbf{a},\tag{42}$$

4 of the algebra of quantum fields B). With an odd parameter D, let us consider the 5 automorphism

$$
\hat{\mathbf{U}} = \exp\left(\alpha \tilde{\mathcal{O}}\right) = \mathbf{Id} + \alpha \tilde{\mathcal{O}}\,\mu
$$

7 Of the algebra B). This automorphism yields a new state < , >, of B), given by the equality

$$<\phi\_{x\_1}^{a\_1}\cdots\phi\_{x\_k}^{a\_k} > \frac{1}{\mathcal{N}'\_{\mathcal{H}}} \int\_{\mathcal{H}} \hat{\mathcal{U}}(\phi\_{x\_1}^{a\_1})\cdots\hat{\mathcal{U}}(\phi\_{x\_k}^{a\_k}) \exp\tag{43}$$

where the energy state has survived, since ˆ( ) I M *a a x x* 9 *dU d* . That because the intention is the 10 same. The intention has not changed.

11 What happens towards the interior of every particle? what is the field intention mechanism 12 inside every particle?

13 To answer these questions we have to internalise the actions of field *X*, on the particles of 14 the space M, and consider their spin. But for it, it is necessary to do the immersion of the 15 Lagrangian Z, defined as the map

$$w: \mathfrak{n} \to \mathfrak{n}^{2n}, \text{ 4.}$$

17 with rule of correspondence

14

2 total derivatives)

226 Advances in Quantum Mechanics

5 think of even elements

7 where we observe that

8 and /= O1O2.

15 cycles)

w*xa*

20 functional integrals

P

I

where this is an integral of type

I

1 1

 II

*k k k k*

> u ³ H

H

The graded commutative R-algebra *H*<sup>0</sup> 1 , is provided with the even graded derivations (called

4 where / = (O1...O*k*), |/| = *k*, and O+/ = (O, O1, . . . , Ok) are symmetric multi-indices. One can

E

9 Now we consider the dual part of the space (*H*, :(\*)), that is to say, the space (*H*\*, *L*), be

10 We consider quantize this Lagrangian system in the framework of perturbative Euclidean QFT. We suppose that L, is a Lagrangian of Euclidean fields on :(\*) R*<sup>n</sup>* 11 . The key point is that the algebra of Euclidean quantum fields B), as like as *H*<sup>0</sup> 12 , is graded commutative. It is generated by elements IO/ 13 *a*, *x*؏ :(\*). For any *x*؏ :(\*), there is a homomorphism belonging 14 to space *H* o *Hom*(T(*H*), *D*) (with homomorphisms *Hom*(T(*H*), *D*) given for *D*G-algebra of

I

 1 1 11 1 1 1 ƭ ƭ ƭ ƭ ƭ ƭ <sup>x</sup> ƭ ƭ ƭ ƭ DŽ : ( ) , C ( ( )) *r rr r r r rr r r*

Of the algebra *H*<sup>0</sup> 17 , of classical fields to the algebra B), which sends the basic elements

/*H*0 to the elements IO/*a*B), and replaces coefficient functions I, of elements of *H*<sup>0</sup> 18 , with 19 their values I(*x*) (executions) at a point *x*. Then a state <, > of B) is given by symbolic

° ° ! ® ¾

21 (41)

c ƺ(ƥ)

22 *O xs* ( ( ))d , as give by the properties. When the

23 intention expands to the whole space, infiltrating their information on the tangent spaces 24 images of the cotangent bundle T\*M, (given by the imagen of I(*x*), under *dO*c).Then their 25 intentionality will be the property of the field to spill or infiltrate their intention from a nano 26 level of strings inside the quantum particles. Then from the energy states of the particles, 27 and considering the intention spilled in them given by *O*c(I), we have the homomorphism 28 (40) that establishes the action from M (# :(\*)), to wM, for their transformation through the

1 1 c ƭ

³ ³

*a a a a an a xx xx x x*

*a a a a a a aa x x aa* 16 *xx x* (40)

<sup>f</sup> w w : \* II I 11 1

 I

> I

*O d x d*

° ° ¯ ¿

ƺ(ƥ) <sup>1</sup> exp ( ) [ ], <sup>N</sup>

I

½

 I

*x*

*a <sup>a</sup>* 3 *d* (38)

 w w w ¦ <sup>1</sup> ƭ nj nj njƭ ƭ nj nj

<sup>0</sup> ( ( , , ) ) ( 1) , *l an a n <sup>a</sup> <sup>n</sup> a a L x x dx L d x dx d x d Ld x*

/ /

, d d d *<sup>k</sup>*

P

*L*, is the 2-form given by Z*L*, in the formula (20) with *n* = 2,

¦ (39)

d

 P

d / / 6 w w w w

G

G

0 ƭ

$$Z\_i \mapsto w\left(Z\_i\right),$$

19 where the image of the 1-form Z, that the Lagrangian defines, Z(*Zi*), is a symplectic form [8], and the variable *Zi*, is constructed through the algebraic equations W*<sup>a</sup>* 20 (*Zi*) = 0 [16].They 21 describe the *k*-dimensional hypersurfaces denoted by S, such that S H, where H, is the 22 phase space defined in the section 2. The index *a* = 1, . . . , *q* runs over the number of polynomials W*<sup>a</sup>* 23 (*Zi*), in the variables Zi and i runs over the dimension of the ambient manifold which is assumed to be C<sup>N</sup> 24 . If the space is a complete intersection, the constraints

<sup>4</sup> Having chosen M<sup>2</sup>*<sup>n</sup>*, is to consider the two components of any point in the space CN, (that we are considering isomorfo to the ambient space of any quantum particle *x*(*s*), in the space-time) to have the two components that characterise any quantum particle x(s), that is their spin (direction) and their energy state (density of energy or "living force of the particle"). L, is the corresponding Lagrangian submanifold of the symplectic structure given by (M2n, Z).

16

W*<sup>a</sup>* (*Zi*) (there is exact solution to W*<sup>a</sup>* 1 (*Zi*) = 0), are linearly independent and the differential 2 form

$$\Theta^{(n-k)} = \in\_{a\_1 \cdots a\_{N-k}} d\mathcal{W}^{a\_1} \wedge \cdots \wedge d\mathcal{W}^{a\_{N-k}},\tag{44}$$

Provisional chapter 17

229

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1 The latter is independent from the choice of parameters O, (however, some care has to be 2 devoted to the choice of the contour of integration and of the integrand: in the minimal 3 formalism, the presence of delta function G(O), might introduce some singularities which prevent from proving the independence from O, as was pointed out in [18], [19]). Using :(*k*) 4 :(*k*) 5 , one can compute the correlation functions by integrating globally defined functions. 6 When the space is Calabi-Yau, it also exists a globally-defined nowhere vanishing holomorphic form :*hol*(*k*°0), such that :*hol*(*k*°0) :*hol*(*k*°0), is proportional to :(*k*) :(*k*) 7 . The ratio of 8 the two top forms is a globally defined function on the CY-space. In the case of the holomorphic measure :*hol*(*k*°0) 9 , the integration of holomorphic functions is related to the

> ! ³ O O ( ,0)

12 where O(*Zi, pA*), are the vertex operators of the theory localized at the points *pA*, of the 13 Riemann surface and O0(*Zi, pA*), is the zero-mode component of the vertex operators. Newly

15 Example **2.** All Calabi-Yau manifolds are *spin*. In hypothetical quantum process (from point 16 of view QFT), to obtain a Calabi-Yau manifold is necessary add (*or sum*) strings in all 17 directions. In the inverse imaginary process, all these strings define a direction or spin. The 18 strings themselves are Lagrangian submanifolds whose Lagrangian action is a path integral.

19 In mathematics, an isotropic manifold is a manifold in which the geometry doesn't depend 20 on directions. A simple example is the surface of a sphere. This directional independence 21 grants us freedom to generate a quantum dimension process, since it does not import what 22 direction falls ill through a string, the space is the same way affected and it presents the

 = 0, in C<sup>N</sup> 25 . This equation can be put in the form using generalized coordinates S(*ui*) = *wz*, where *i*, runs over *i* = 1, 2, }, *N*-2, coordinates and *w*, *z*, are two combinations of the *Z*'s. S(*ui*), is a polynomial of the coordinates *ui*. For a given N, they are local CY-manifolds (*spin manifolds*) and there exist a globally-defined a organized transformation inside space M [20]. b) Intention inside

30 The importance of this isotropy property in our spin manifold, helps us to establish that the 31 transformations applied to the space that are directed to use (awakening) their nano-32 structure do it through an organized transformation that introduces the time as isotropic

2

(a) (b)

11 *p p* (47)

(Z , ) ƺ (Z , ), *<sup>k</sup> i A i A A*

A DŽ S

23 same aspect in any direction that is observed creating this way their isotropy.

0

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

10 definition of a contour JS, in the complex space

our conscience operator come given by the form :(*k,* 0) 14 .

**Figure 4.** a) For instance, let us consider the hypersurface S : 6*iZi*

24

29 particle Q.

4 is not vanishing. In this case, q = N ƺ *k* and the dimension of the surface is easily determined. 5 For example, if the hypersurface is described by a single algebraic equation W(Z) = 0, the form (44) is given by 4(1) 6 = *d*W. On the other hand, if the hypersurface is not a complete 7 intersection, then there exists a differential form

$$\Theta^{(n-k)} = \mathbf{r}\_{A[a\_1 \cdots a\_{N-k}]} d\mathcal{W}^{a\_1} \wedge \cdots \wedge d\mathcal{W}^{a\_{N-k}} \wedge \eta^{A, (N-k-q)},\tag{45}$$

where K*A*, (*N -k - q*) , is a set of *N – k –q*, forms defined such that 4(*N - k*) 9 , is non-vanishing on the constraints W*<sup>a</sup>* 10 (*Zi*) = 0, and TA, [*a*1 }*aN - k*], is a numerical tensor which is antisymmetric in the indices *a*1 }*aq*. The construction of K*<sup>A</sup>*, (*N -k - q*) 11 , depends upon the precise form of the algebraic manifold (variety of the equations W*<sup>a</sup>* 12 (*Zi*) = 0). In some cases a general form can be given, but 13 in general it is not easy to find it and we did not find a general procedure for that 14 computation.

15 To construct a global form on the space S, one can use a modification of the Griffiths residue 16 method [16], by observing that given the global holomorphic form on the ambient space , *<sup>N</sup> <sup>N</sup> <sup>i</sup>* : *<sup>1</sup> 1 N i i i dZ dZ* we can decompose the {*Zi*}'s, into a set of coordinates Y*<sup>a</sup>* 17 = W*<sup>a</sup>* (*Z*), and the rest. By using the contraction with respect to *q*, vectors {*Za* 18 *i*}, the top form for 19 S, can be written as

$$\mathfrak{Q}^{(k)} = \frac{\mathfrak{l}\_{\underline{Z}\_1^\*} \cdots \mathfrak{l}\_{\underline{Z}\_\theta^\*} \mathfrak{Q}^{(\mathbb{N})}}{\mathfrak{l}\_{\underline{Z}\_1^\*} \cdots \mathfrak{l}\_{\underline{Z}\_\theta^\*} \mathfrak{G}^{(\mathbb{N}-k)}},\tag{46}$$

which is independent from {*Za <sup>i</sup>*}, as can be easily proved by using the constraints W*<sup>a</sup>* 21 (*Zi*) = 0. 22 Notice that this form is nowhere-vanishing and non singular only the case of CY-space 23 (Calabi-Yau manifold). The calabi-Yau manifols is a spin manifold and their existence in the our space M<sup>2</sup>*<sup>n</sup>* 24 , like product of this construction is the first evidence that a spin manifold is the spin of our space-time due to their holomorphicity [17]. The vectors {*Za* 25 *i*}, play the role of gauge 26 fixing parameters needed to choose a polarisation of the space S, into the ambient space.

27 For example, in the case of pure spinor we have: the ambient form 1 <sup>1</sup> , D D  : O O *1 6 i i d d* and  <sup>5</sup> 4 OJ OOJ OOJ O OJ O *m mm mnp* 28 *d d dd d* . From these data, we can get the holomorphic top form :(11) 29 , by introducing 5, independent parameters O, and by 30 using the formula (46).

1 The latter is independent from the choice of parameters O, (however, some care has to be 2 devoted to the choice of the contour of integration and of the integrand: in the minimal 3 formalism, the presence of delta function G(O), might introduce some singularities which prevent from proving the independence from O, as was pointed out in [18], [19]). Using :(*k*) 4 :(*k*) 5 , one can compute the correlation functions by integrating globally defined functions. 6 When the space is Calabi-Yau, it also exists a globally-defined nowhere vanishing holomorphic form :*hol*(*k*°0), such that :*hol*(*k*°0) :*hol*(*k*°0), is proportional to :(*k*) :(*k*) 7 . The ratio of 8 the two top forms is a globally defined function on the CY-space. In the case of the holomorphic measure :*hol*(*k*°0) 9 , the integration of holomorphic functions is related to the 10 definition of a contour JS, in the complex space

16

W*<sup>a</sup>*

228 Advances in Quantum Mechanics

where K*A*, (*N -k - q*)

14 computation.

W*<sup>a</sup>*

2 form

(*Zi*) (there is exact solution to W*<sup>a</sup>* 1 (*Zi*) = 0), are linearly independent and the differential

*N k*

*<sup>k</sup> <sup>a</sup> <sup>a</sup>* 3 *dW dW* (44)

4 is not vanishing. In this case, q = N ƺ *k* and the dimension of the surface is easily determined. 5 For example, if the hypersurface is described by a single algebraic equation W(Z) = 0, the form (44) is given by 4(1) 6 = *d*W. On the other hand, if the hypersurface is not a complete

 4 <sup>T</sup> 1 N

, is a set of *N – k –q*, forms defined such that 4(*N - k*) 9 , is non-vanishing on the constraints W*<sup>a</sup>* 10 (*Zi*) = 0, and TA, [*a*1 }*aN - k*], is a numerical tensor which is antisymmetric in the

*<sup>A</sup>*, (*N -k - q*) 11 , depends upon the precise form of the algebraic manifold (variety of the equations W*<sup>a</sup>* 12 (*Zi*) = 0). In some cases a general form can be given, but 13 in general it is not easy to find it and we did not find a general procedure for that

15 To construct a global form on the space S, one can use a modification of the Griffiths residue 16 method [16], by observing that given the global holomorphic form on the ambient space

*i i dZ dZ* we can decompose the {*Zi*}'s, into a set of coordinates Y*<sup>a</sup>* 17 =

(*Z*), and the rest. By using the contraction with respect to *q*, vectors {*Za* 18 *i*}, the top form for

*<sup>k</sup>* 20 (46)

*<sup>i</sup>*}, as can be easily proved by using the constraints W*<sup>a</sup>* 21 (*Zi*) = 0. 22 Notice that this form is nowhere-vanishing and non singular only the case of CY-space 23 (Calabi-Yau manifold). The calabi-Yau manifols is a spin manifold and their existence in the our space M<sup>2</sup>*<sup>n</sup>* 24 , like product of this construction is the first evidence that a spin manifold is the spin of our space-time due to their holomorphicity [17]. The vectors {*Za* 25 *i*}, play the role of gauge 26 fixing parameters needed to choose a polarisation of the space S, into the ambient space.

27 For example, in the case of pure spinor we have: the ambient form

*i i d d* and  <sup>5</sup> 4 OJ OOJ OOJ O OJ O *m mm mnp* 28 *d d dd d* . From these data, we can get the holomorphic top form :(11) 29 , by introducing 5, independent parameters O, and by

 

NJ NJƺ <sup>ƺ</sup> , NJ NJƪ *a ǂ*

ƨ ƨ

(N)

(N )

 1 lj

ƨ ƨ

1 lj

*a ǂ*

 

*k*

*n k a a A k q Aa a* 8 *dW dW* (45)

[ ] , *<sup>1</sup> N -K*

 4 , *<sup>1</sup> N -K*

K ,(N )

*a a*

*1*

K

*k*

*n*

7 intersection, then there exists a differential form

indices *a*1 }*aq*. The construction of

, *<sup>N</sup> <sup>N</sup> <sup>i</sup>* : *<sup>1</sup> 1 N*

which is independent from {*Za*

<sup>1</sup> ,

 : O O *1 6*

1

D D

30 using the formula (46).

*i*

19 S, can be written as

$$<\prod\_{A} \mathrm{c}(\mathbf{Z}\_{i'} \boldsymbol{\mathcal{p}}\_{A}) > \int\_{\mathrm{V}^{\mathbf{G}}} \mathfrak{Q}^{(k,0)} \prod\_{A} \mathrm{c}\_{0}(\mathbf{Z}\_{i'} \boldsymbol{\mathcal{p}}\_{A}),\tag{47}$$

12 where O(*Zi, pA*), are the vertex operators of the theory localized at the points *pA*, of the 13 Riemann surface and O0(*Zi, pA*), is the zero-mode component of the vertex operators. Newly our conscience operator come given by the form :(*k,* 0) 14 .

15 Example **2.** All Calabi-Yau manifolds are *spin*. In hypothetical quantum process (from point 16 of view QFT), to obtain a Calabi-Yau manifold is necessary add (*or sum*) strings in all 17 directions. In the inverse imaginary process, all these strings define a direction or spin. The 18 strings themselves are Lagrangian submanifolds whose Lagrangian action is a path integral.

19 In mathematics, an isotropic manifold is a manifold in which the geometry doesn't depend 20 on directions. A simple example is the surface of a sphere. This directional independence 21 grants us freedom to generate a quantum dimension process, since it does not import what 22 direction falls ill through a string, the space is the same way affected and it presents the 23 same aspect in any direction that is observed creating this way their isotropy.

**Figure 4.** a) For instance, let us consider the hypersurface S : 6*iZi* 2 = 0, in C<sup>N</sup> 25 . This equation can be put in the form 26 using generalized coordinates S(*ui*) = *wz*, where *i*, runs over *i* = 1, 2, }, *N*-2, coordinates and *w*, *z*, are two 27 combinations of the *Z*'s. S(*ui*), is a polynomial of the coordinates *ui*. For a given N, they are local CY-manifolds (*spin*  28 *manifolds*) and there exist a globally-defined a organized transformation inside space M [20]. b) Intention inside 29 particle Q.

24

30 The importance of this isotropy property in our spin manifold, helps us to establish that the 31 transformations applied to the space that are directed to use (awakening) their nano-32 structure do it through an organized transformation that introduces the time as isotropic 18

1 variable, creating a momentary timelessness in the space where the above mentioned 2 transformation is created. Then the intentionality like a organized transformation is a co-3 action compose by field that act to realise the transformation of space and the field of the 4 proper space that is transformed. Then the symplectic structure subjacent in M, receives 5 sense.

Provisional chapter 19

231

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1 quantum singularity Ï(s), could have an entropy of only 4Δnats; and a Hawking 2 temperature of needing quantum thermal energy comparable approximately to the mass of 3 the finished singularity; and a Compton wavelength equivalent within a radius of 4 Schwarzschild of the singularity in the Universe (this distance being equivalent to the 5 Planck length). This is the point where the classic gravitational description of the object is 6 not valid, being probably very important the quantum effects of the gravity. But there exists 7 another mechanism or thermodynamic limit that is fundamental in the theory of the

**The past and future in the quantum scattering phenomena** 

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

Anti-particle I(1)

xm\$ I(1) Negative Future

\$m x I(1) Negative Past

8 quantum dispersion and of the formation of quantum singularities.

Input xo\$ <sup>I</sup>(1)

Output \$ox <sup>I</sup>(1)

13 *exist*)). Then in this absence of paths, the singularity arises.

Particle I(1)

Positive Future

Positive Past

10 Proposition **1.** The energy of the singularity is born of the proper state of altered energy of 11 the quantum space-time, although without a clear distinction of a suitable path of the 12 particles (*without a normal sequence of superposition of (past and future) particles* (*a path does not* 

14 *Proof*. It is necessary to demonstrate that the past and present particles overlap without a normal sequential order in the causality, passing to the unconscious one (*O*c = 0, since 15 L0 = 0 16 or Z*L* = 0) for fluctuations of energy that have a property of adherence to the transition of 17 energy states forming energy bundles ([23]) that alter the normal behavior of the particles in 18 the atoms. In effect, this happens when the quantum energy fragments and the photons 19 exchange at electromagnetic level do not take in finished form (there is no exchange in the 20 virtual field of the *SO*(3)). For which there is no path that the execution operator I, (see the 21 section 2) could resolve through of a path integral between two photons (*virtual particles*). In 22 this case the path integral does not exist. For the adherent effect, the virtual particles that do 23 not manage to be exchanged accumulate forming the altered energy states (an excess very 24 nearby of particles of the certain class (inclusive anti-particles of certain class) they add to 25 themselves to the adherent space of photon (in this case the adherent photon is an excited 26 photon *int*E G(E), (where E, is the influence of the singularity) with the infinitesimal

Z+/Z-

9 **Table 1.**

6 A transformation T, it is said organized if in whole stage of execution of the transformation, 7 isotropic images are obtained of the original manifold (object space of the transformation), 8 under a finite number of endomorphisms of the underlying group to the manifold that co-9 acts with this transformation.

10 Likewise, if T, is a transformation on the space M, whose subjacent group *G*, have endomorphisms V 11 ,}, Vn, such that VT(M), }VnT(M), are isotropic then the infinite tensor 12 product of isotropic submanifolds is a isotropic manifold, and is a organized transform 13 equivalent to tensor product of spin representations VT(M) }VnT(M) }[2, 5].

### 14 **4. Quantum integral transforms: Elimination of distortions and**  15 **quantum singularities**

16 One of the quantum phenomena that can form or provoke the conditions of formation of 17 singularities at this level is the propensity of a quantum system to develop scattering 18 phenomena for the appearance of the anomalous states of energy, as antimatter energy or 19 energy of particles of matter in the free state [21], which crowds (a big number of particles 20 overlaps) due to the accumulation of the states of energy of the past or future (to see table 1) 21 [21, 22],which on have different existence time and to having met their corresponding pairs 22 of particles (particle/anti-particle pair) provoke bundles of energy that there form in the 23 space time *M* u I*t*, singularities of certain weight (for mechanisms that can be explained 24 inside the actions in *SU*(3) and *SU*(2) [11, 21]) due to its energy charge [21].

25 Studies in astrophysics and experiments in the CERN (Organisation Européenne pour la 26 Recherche Nucléaire) establish that a similar mechanism although with substantial 27 differences (known also like Schwinger mechanism) can explain the formation of a 28 singularity such as the fundamental singularity (big-bang). This one establishes that the 29 gravitational field turns into virtual pairs of particle- antiparticle of an environment of 30 quantum gap in authentic pair's particle-antiparticle. If the black hole (singularity of the 31 Universe) is done of matter (antimatter), it might repel violently to thousands of million 32 antiparticles (particles), expelling them to the space in a second fraction, creating an event of 33 ejection very similar to a Big-Bang. Nevertheless, in case of a singularity in the region of 34 space - time of the particles in a quantum ambience is different in that aspect, since the small 35 mass of a singularity Ï(s), might perform the order of the Planck mass, which is approximately 2 × 10ƺ8 kg or 1,1 × 1019 36 GeV. To this scale, the formulation of the 37 thermodynamic theory of singularities of the space macroscopic time predicts that the 1 quantum singularity Ï(s), could have an entropy of only 4Δnats; and a Hawking 2 temperature of needing quantum thermal energy comparable approximately to the mass of 3 the finished singularity; and a Compton wavelength equivalent within a radius of 4 Schwarzschild of the singularity in the Universe (this distance being equivalent to the 5 Planck length). This is the point where the classic gravitational description of the object is 6 not valid, being probably very important the quantum effects of the gravity. But there exists 7 another mechanism or thermodynamic limit that is fundamental in the theory of the 8 quantum dispersion and of the formation of quantum singularities.


#### 9 **Table 1.**

18

5 sense.

9 acts with this transformation.

230 Advances in Quantum Mechanics

15 **quantum singularities** 

1 variable, creating a momentary timelessness in the space where the above mentioned 2 transformation is created. Then the intentionality like a organized transformation is a co-3 action compose by field that act to realise the transformation of space and the field of the 4 proper space that is transformed. Then the symplectic structure subjacent in M, receives

6 A transformation T, it is said organized if in whole stage of execution of the transformation, 7 isotropic images are obtained of the original manifold (object space of the transformation), 8 under a finite number of endomorphisms of the underlying group to the manifold that co-

10 Likewise, if T, is a transformation on the space M, whose subjacent group *G*, have endomorphisms V 11 ,}, Vn, such that VT(M), }VnT(M), are isotropic then the infinite tensor 12 product of isotropic submanifolds is a isotropic manifold, and is a organized transform

14 **4. Quantum integral transforms: Elimination of distortions and** 

16 One of the quantum phenomena that can form or provoke the conditions of formation of 17 singularities at this level is the propensity of a quantum system to develop scattering 18 phenomena for the appearance of the anomalous states of energy, as antimatter energy or 19 energy of particles of matter in the free state [21], which crowds (a big number of particles 20 overlaps) due to the accumulation of the states of energy of the past or future (to see table 1) 21 [21, 22],which on have different existence time and to having met their corresponding pairs 22 of particles (particle/anti-particle pair) provoke bundles of energy that there form in the 23 space time *M* u I*t*, singularities of certain weight (for mechanisms that can be explained

25 Studies in astrophysics and experiments in the CERN (Organisation Européenne pour la 26 Recherche Nucléaire) establish that a similar mechanism although with substantial 27 differences (known also like Schwinger mechanism) can explain the formation of a 28 singularity such as the fundamental singularity (big-bang). This one establishes that the 29 gravitational field turns into virtual pairs of particle- antiparticle of an environment of 30 quantum gap in authentic pair's particle-antiparticle. If the black hole (singularity of the 31 Universe) is done of matter (antimatter), it might repel violently to thousands of million 32 antiparticles (particles), expelling them to the space in a second fraction, creating an event of 33 ejection very similar to a Big-Bang. Nevertheless, in case of a singularity in the region of 34 space - time of the particles in a quantum ambience is different in that aspect, since the small 35 mass of a singularity Ï(s), might perform the order of the Planck mass, which is approximately 2 × 10ƺ8 kg or 1,1 × 1019 36 GeV. To this scale, the formulation of the 37 thermodynamic theory of singularities of the space macroscopic time predicts that the

13 equivalent to tensor product of spin representations VT(M) }VnT(M) }[2, 5].

24 inside the actions in *SU*(3) and *SU*(2) [11, 21]) due to its energy charge [21].

10 Proposition **1.** The energy of the singularity is born of the proper state of altered energy of 11 the quantum space-time, although without a clear distinction of a suitable path of the 12 particles (*without a normal sequence of superposition of (past and future) particles* (*a path does not*  13 *exist*)). Then in this absence of paths, the singularity arises.

14 *Proof*. It is necessary to demonstrate that the past and present particles overlap without a normal sequential order in the causality, passing to the unconscious one (*O*c = 0, since 15 L0 = 0 16 or Z*L* = 0) for fluctuations of energy that have a property of adherence to the transition of 17 energy states forming energy bundles ([23]) that alter the normal behavior of the particles in 18 the atoms. In effect, this happens when the quantum energy fragments and the photons 19 exchange at electromagnetic level do not take in finished form (there is no exchange in the 20 virtual field of the *SO*(3)). For which there is no path that the execution operator I, (see the 21 section 2) could resolve through of a path integral between two photons (*virtual particles*). In 22 this case the path integral does not exist. For the adherent effect, the virtual particles that do 23 not manage to be exchanged accumulate forming the altered energy states (an excess very 24 nearby of particles of the certain class (inclusive anti-particles of certain class) they add to 25 themselves to the adherent space of photon (in this case the adherent photon is an excited 26 photon *int*E G(E), (where E, is the influence of the singularity) with the infinitesimal 20

3

1 nearby of points given by G(E) [24])) which are the bundles of energy that defines the 2 singularities.

4 **Figure 5.** In a). is the elastic band of the space-time M, and b), result the pulse solutions of the Fokker-Planck 5 Equation. In c), the singularity is being perturbed by states of different particles interacting and creating big scattering 6 (*the red circles are perturbations created by the states of different particles creating scattering with a big level of*  7 *particles pair annihilating. This produce defecting evolution in every sub-particles and increase of inflation in quantum*  8 *level of the space-time*). In d), the singularity is formed.

Let ˆ <sup>0</sup> 9 *U ts* ( , ), the operator of evolution [6], of a particle *x*(*t*), in the space of transition of the 10 levels of conscience operator *O*c, to all time *t* t *s*. Whose operator limits of *s* (*that is to say, coming to the process of understanding of a concept,* (border conditions of ˆ <sup>0</sup> 11 *U ts* (,) )), satisfy

$$\lim\_{t \to s^{+}} \hat{U}\_{0}(t, s) = \mathbf{1}\_{\iota}[\mathbf{31}] \tag{48}$$

4

21

13 comes given for

Provisional chapter 21

233

http://dx.doi.org/10.5772/53439

1 Then a corrective action is the inverse transform that transform the energy load function in 2 energy useful to the process of re-establishment on the quantum space (*remember that it is*  3 *necessary to release the bundle of energy captive*). How this inverse transformation realise?

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

**Figure 6.** a). This is the graph that shows the formation of singularities of quantum type by the energy load (positives and negatives), rational amplitudes (extremes and defects), and bundle energy created and defined by (( 7 . The load is pre-determined by the impulse G(*t s*), which focalize the field action. The adherence zones are propitious to the formation of singularities with all conditions described. The wave function \, stays under constant regime in space evolution, which is \= \(*x*, 0). In the second wave, the corresponding wave function is for other

I

 I

(,)()Ǎ Dž( )()

*w s t x t t s x t dt,*

*t*

*t*

³ ³ ³ ³

f f

*M I M*

23 take the form (that is to say, to past and future particles)

f f

³ ³

11 Lemma **1.** Let M = *M* u *It*, the unlimited space of the quantum space (Fock space [24]). A 12 particle *x*(*t*), that is focalized by a bad evolution given for the energy load function *w*(*t*, *s*),

³ <sup>14</sup>*x s x t dx t t s x t dt* () () () <sup>Dž</sup>( )() , (51)

16 *Proof*. We consider the function *w*(*t*, *s*), like a Green function on the interval *t* t *s*. Given that 17 this function is focalised for the emotional interpretation which is fed by the proper energy 18 of the deep quantum energy (*since it produce an auto-disipant effect that deviates the evolution of*  19 *every particles* [23]), then O*c*(*x*(*t*), *x*'(*t*))*x*(*t*) = 2*w*(*t*, *s*) = G(*t s*) [25]. By the nature of Green

> ª º G « » ¬ ¼

22 Then all particle *x*(*t*), in the space-time *M* u *It*, affected by this regime to time *t* = *s*, and after,

( ') ( ) ( ) ( ) ( - ') ( ) ( )Ǎ ( ( )) ( )Ǎ

*x x t dx t x x x dx t x t O x t x t*

f

f

f

f

c

*t t*

10 transition time (other evolution space), under the same conditions.

15 Then to time *t* = *s*, begin the singularity.

20 function of the weight function *w*(*t*, *s*), we have

2

I

13 But there are waves of certain level (*Table 1*), that act like moderators of wave length to the 14 operator of evolution defined by \(*s*), satisfying that

$$\hat{\mathcal{U}}\_0(t,s)\Big|\mathcal{V}(s) \succ = \begin{cases} \big|\mathcal{V}(s) >, & t \ge s \\ 0, & t < 0 \end{cases} \tag{49}$$

16 having then that the singularity Ï(*s*), that is object of quantum transformation to along of 17 the time, is that product obtained by the integral transform

$$\mathbf{f}^{\mathsf{A}}(\mathbf{s}) = \int\_{\mathrm{X}(\mathbf{C})} O\_{\mathbf{c}}(\mathbf{x}(t)) \mathbf{w}(s, t) dt = (\mathbf{E}^{+} - \mathbf{E}^{-}) \sqrt{\frac{m}{2\pi i n}} \int\_{-\infty}^{\infty} \left[ \int\_{\mathbf{C}} \mathbf{e}^{\frac{i\mathbf{m}(\mathbf{x}-\mathbf{x}')^{2}}{2\pi}} \hat{\mathbf{U}}\_{0}(s, t) d\mathbf{x}' \right] dt,\tag{50}$$

where the singularity changes the time *t* t *s*, for the evolution operator ˆ <sup>0</sup> 19 *U ts* ( , ), [11], due to 20 the evolution of the quantum system in the space-time. But by evolution operator [6, 21], this evolution comes given by the anomalous energy ((( 21 ) [6] (*see figure 3*) , which establishes the energy load functions *w*+(*s*, *t*) *w* 22 (*s*, *t*), that define the energy load function *w*(*s*, *t*), at time *t* t *s*, where *w*(*s*, *t*) = ((( )*U*0(*s*, *t*) (and( 23 are amplitudes of the curves M = I*tx*(*s*), I*t* = M*x*<sup>1</sup> 24 (*s*), and M, I*t*, functions of evolution curves of space-time *see figure 3*). 1 Then a corrective action is the inverse transform that transform the energy load function in 2 energy useful to the process of re-establishment on the quantum space (*remember that it is*  3 *necessary to release the bundle of energy captive*). How this inverse transformation realise?

5 **Figure 6.** a). This is the graph that shows the formation of singularities of quantum type by the energy load 6 (positives and negatives), rational amplitudes (extremes and defects), and bundle energy created and defined by (( 7 . The load is pre-determined by the impulse G(*t s*), which focalize the field action. The adherence zones are 8 propitious to the formation of singularities with all conditions described. The wave function \, stays under constant 9 regime in space evolution, which is \= \(*x*, 0). In the second wave, the corresponding wave function is for other 10 transition time (other evolution space), under the same conditions.

11 Lemma **1.** Let M = *M* u *It*, the unlimited space of the quantum space (Fock space [24]). A 12 particle *x*(*t*), that is focalized by a bad evolution given for the energy load function *w*(*t*, *s*), 13 comes given for

$$\mathbf{x}(s) = \phi \mathbf{x}(t) d\mathbf{x}(t) = \int\_{-\infty}^{+\infty} \mathbb{S}(t-s) \mathbf{x}(t) dt,\tag{51}$$

15 Then to time *t* = *s*, begin the singularity.

4

21

20

3

Let ˆ

2 singularities.

232 Advances in Quantum Mechanics

8 *level of the space-time*). In d), the singularity is formed.

14 operator of evolution defined by \(*s*), satisfying that

17 the time, is that product obtained by the integral transform

*X(C)*

*w*(*s*, *t*), at time *t* t *s*, where *w*(*s*, *t*) = (((

1 nearby of points given by G(E) [24])) which are the bundles of energy that defines the

**Figure 5.** In a). is the elastic band of the space-time M, and b), result the pulse solutions of the Fokker-Planck Equation. In c), the singularity is being perturbed by states of different particles interacting and creating big scattering (*the red circles are perturbations created by the states of different particles creating scattering with a big level of particles pair annihilating. This produce defecting evolution in every sub-particles and increase of inflation in quantum* 

*coming to the process of understanding of a concept,* (border conditions of ˆ <sup>0</sup> 11 *U ts* (,) )), satisfy

ˆ

<sup>0</sup> 9 *U ts* ( , ), the operator of evolution [6], of a particle *x*(*t*), in the space of transition of the 10 levels of conscience operator *O*c, to all time *t* t *s*. Whose operator limits of *s* (*that is to say,* 

> ˆ o

12 *U ts* (48)

13 But there are waves of certain level (*Table 1*), that act like moderators of wave length to the

*t s*

\

*U ts s*

³ ³ ³

where the singularity changes the time *t* t *s*, for the evolution operator ˆ

<sup>0</sup> lim ( , ) 1,[31]

\

! ® ° ¯ <sup>0</sup> () , (,) () , 0, 0

15 (49)

16 having then that the singularity Ï(*s*), that is object of quantum transformation to along of

18 (50)

<sup>0</sup> 19 *U ts* ( , ), [11], due to 20 the evolution of the quantum system in the space-time. But by evolution operator [6, 21], this evolution comes given by the anomalous energy ((( 21 ) [6] (*see figure 3*) , which establishes the energy load functions *w*+(*s*, *t*) *w* 22 (*s*, *t*), that define the energy load function

)*U*0(*s*, *t*) (and( 23 are amplitudes of the curves M = I*tx*(*s*), I*t* = M*x*<sup>1</sup> 24 (*s*), and M, I*t*, functions of evolution curves of space-time *see figure 3*).

° ° ® ¾

*<sup>m</sup> s O <sup>x</sup> <sup>t</sup> <sup>w</sup> <sup>s</sup> <sup>t</sup> dt U <sup>s</sup> <sup>t</sup> dx dt*

c 0

( ) ( ( )) ( , ) (E E ) e ( , ) , 2 i

° ! t

S

*n*

f

*s ts*

*t*

Sˆ

<sup>2</sup> i ( ') 2

*mx x*

½

° ° ¯ ¿

f

C

16 *Proof*. We consider the function *w*(*t*, *s*), like a Green function on the interval *t* t *s*. Given that 17 this function is focalised for the emotional interpretation which is fed by the proper energy 18 of the deep quantum energy (*since it produce an auto-disipant effect that deviates the evolution of*  19 *every particles* [23]), then O*c*(*x*(*t*), *x*'(*t*))*x*(*t*) = 2*w*(*t*, *s*) = G(*t s*) [25]. By the nature of Green 20 function of the weight function *w*(*t*, *s*), we have

$$\begin{aligned} \int\_M \phi(\mathbf{x'}) \mathbf{x}(t) d\mathbf{x}(t) &= \int\_I \left[ \int\_M \phi(\mathbf{x}) \delta(\mathbf{x} - \mathbf{x'}) d\mathbf{x}(t) \right] \mathbf{x}(t) \mathbf{n}\_t = \int\_{-\infty}^{+\infty} O\_\mathbf{c}(\mathbf{x}(t)) \mathbf{x}(t) \mathbf{n}\_t \\\ \overset{+\Leftrightarrow}{=} \int\_{-\infty}^{+\infty} \nabla^2 w(\mathbf{s}, t) \mathbf{x}(t) \mathbf{n}\_t &= \int\_{-\infty}^{+\infty} \delta(t - s) \mathbf{x}(t) dt, \end{aligned}$$

22 Then all particle *x*(*t*), in the space-time *M* u *It*, affected by this regime to time *t* = *s*, and after, 23 take the form (that is to say, to past and future particles)

22

7

$$\int\_{-\alpha}^{s} O\_c(\mathbf{x}(t)) \mathbf{x}(t) dt + \int\_{s}^{+\alpha} O\_c(\mathbf{x}(t)) \mathbf{x}(t) dt = 0 + \int\_{s}^{+\alpha} \delta(t-s) \mathbf{x}(t) dt, \quad s > 0$$

Provisional chapter 23

235

http://dx.doi.org/10.5772/53439

lim i ( , ) ( ) r

*t s*

I

f

*j 2j j j j*

1 thermodynamic limit [23], each transition matrix of energy states has a correspondence with 2 the product of Hermitian matrices of the corresponding evolution operators that to this case

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

XD *e is'h*

° ° ¯ ¿ <sup>³</sup> 0 0 (3) i i 3 3

*x t e Uts drdRe n s* **<sup>r</sup>** 9 **r R** (53)

10 to j*th*-particle in the interaction of a j*th*-thought to time *s*. The diagrams drawn here (figure 3) 11 may be interpreted to represent the evolution of particles contributing the position density 12 matrix *n***r**(**R**, *t*) at *t* = *t*, in the corresponding path integral of correction. The true correction 13 comes given by the evolution created in the quantum process of transformation of the

14 *wts ns* 15 (is to say, when *M* = *It*). If we call *t*j = *s*, (*j-th-time of evolution* in the thought process) then the

¦ ³³ ³ ³

17 *e dt dt dt t t t n F n dx t* (54)

18 where the states I*j*, are established in the density matrix *n*(0), that to a vertex of G, (*w* - 19 diagram), arrange, those perturbations *B* (see figure 3), that they gave origin to the 20 singularity, with the corresponding arrange of those positive perturbations *A*, that will 21 realises the corrective action to transform the singularity signal Ñ(*s*), of an adequate thought 22 given by *x*(*t*). Due to that, the information given by the product *An*(0)*B*, must be changed

23 *wts ns* , then a *w*-diagram must be change for )Ddiagram [27]. But this

24 live in the quantum field of the space-time TM\*, that is to say, in the corresponding zone of 25 the executive operator I. Then in the material space-time (*Einstein universe*), the 26 displacement of energy needs inside this transformation the application of an invariant 27 given in the quantum space that guarantee that the new particle (*boson*) obtained let that 28 correct. This is given by number *dim*/(D), since it depends on the roots system to the 29 representation of the corresponding action group [27], that recover the recognition action. 30 By the integral (4), the transformation due to the new conscience operator created in TM\*-

( ) Dž ( ) (,) ( , ), *j j H t <sup>j</sup> H s <sup>j</sup>*

½ ½ ° ° ° ° ® ¾ ® <sup>¾</sup> ° ° °¯ ¿° ¯ ¿

1 (i)nj ǖ( )ǖ ) ǖ( ) ( ) ( )} ( ) ,

I

f f

f

 ½ ½ § · ° ° °° ¨ ¸ ® ® ¾¾ © ¹ ° ° °° ¯ ¿ ¯ ¿ ³ ³ <sup>c</sup> <sup>³</sup> 1 ( ) <sup>1</sup> ( ) ( ) ( ) (), <sup>ƭ</sup> *j j*

*x t O dx t n F n dx t*

*X(C) X(C) j*

*dim A*

 3 wherewe have used the *lemma 1*, to the 4 arising of the quantum impurity in the space-time M (*quantum singularity*) in the image space of the conscience operator TM\*, located in R<sup>3</sup> 5 u I*t*, in the point or node (# G [12, 23] (*w*-6 diagram)) *t* = *s*, corresponding of root space )D . Then considering a irreducible diagram containing a *w*(*t*, *s*)-part, their contribution will be contained in **r** – **R**-space (which is the E+ 7

X

*C j*

1

D(s') = *eis'h0*

<sup>ˆ</sup> <sup>f</sup>

½ ° ° ® ¾

functional *O*c(Ñ(*s*)), where the information Ñ(*s*), must be changed when <sup>o</sup>

<sup>f</sup> <sup>f</sup> f

12 1 1 00 0 1

j 1 2

*<sup>1</sup> <sup>j</sup> <sup>t</sup> t t*

on a node en *t* = *s*, are given by

16 integral (53), takes the form

j 0 i( )

when <sup>o</sup>

32

*t s*

33 that is the result wanted.

*ttH*

lim i ( , ) ( ) r

31 zone obtained on whole the space-time is,

E 8 -space (see figure 2)) [23] by the function G(*t s*):

2 where the first integral is equal to cero, because there is no singularity before *s*, (the 3 evolution happens after the time *t* t *s* (see (49)). But this evolution is anomalous, since to all *t* 4 < *s*, includes a captive energy not assimilated to *t* = *s* (*this because it does not have a conscience operator at this moment* (*part of* (28) *defined by O*<sup>0</sup> 5 c)). Then

8 **Figure 7.** The surface represents the perturbation given by the existence of a singularity in the space-time of the 9 quantum zone. The *computational model* obeys the solution of Planck-Fokker partial differential equation. The waves 10 of perturbed state begins to s d t, such as it was predicted in the lemma 1. The surface include the waves established in 11 figure 3, given by z = *Plot3D(exp(-1/2(x^2 + y^2))(cos(4x) + sin(2x) + 3sin(1/3y) + cos(5y)ln(2x)))*, through the *space-time 4.0 program*. Observe that is present a kernel of transformation for normal distribution given by *exp*(-1/2(x2 + y2 12 )) that 13 will appear in the transform that defines the singularity when a particle is not appropriately assimilate. The normal 14 distribution kernel is the statistical weight that establishes the appearance of an abnormal evolution created by the 15 existence of the singularity. That is to say, the singularity is detected by the anomalous effects that are glimpsed in the 16 flux of the operator *Oc*, and that are observed in the surface bundle.

17 Theorem. **1. (F. Bulnes**). Let consider a conscience operator with singularity *O*c(Ñ), for the 18 presence of an energy load *w*(*s*, *t*). Then the *elimination* of the singularity Ñ(*s*), comes given for

$$\mathbf{x}(t) = \text{correction} \quad + \text{restoring} \quad = \int\limits\_{\mathbf{x}(\odot)} O\_{\mathbf{c}}(\mathbf{r}(s)) \mathbf{x}(t, s) dt = \dim \Lambda \left[ \left\{ \frac{1}{A} \prod\_{j=1}^{n} \left\{ \int\_{-\alpha}^{+\alpha} \phi(\mathbf{u}\_{j}) \mathbf{F}(\mathbf{u}\_{j}) \right\} d\mathbf{x}(t) \right\} \right] \tag{52}$$

20 where *dim*/(D), is the *Neumann dimension* corresponding to the *Weyl camera* of the roots Dj, 21 [2627@used in the rotation process to eliminate the deviation [11], created by the 22 singularity.

23 *Proof.* Consider an arbitrary irreducible diagram with nodes with *w*-parts (parts of diagrams 24 with nodes of weight *w*(*t*, *s*)). Suppose that this points "nodes" with weight *w*, determines 25 the singularity given by (50). In fact, by the theory of Van Hove on the singularities in the 1 thermodynamic limit [23], each transition matrix of energy states has a correspondence with 2 the product of Hermitian matrices of the corresponding evolution operators that to this case on a node en *t* = *s*, are given by XD(s') = *eis'h0*XD *e is'h* 3 wherewe have used the *lemma 1*, to the 4 arising of the quantum impurity in the space-time M (*quantum singularity*) in the image space of the conscience operator TM\*, located in R<sup>3</sup> 5 u I*t*, in the point or node (# G [12, 23] (*w*-6 diagram)) *t* = *s*, corresponding of root space )D . Then considering a irreducible diagram containing a *w*(*t*, *s*)-part, their contribution will be contained in **r** – **R**-space (which is the E+ 7 E 8 -space (see figure 2)) [23] by the function G(*t s*):

$$\mathbf{x}(t) = \int\_{\mathbb{C}} \left| \prod\_{j=1}^{n} \mathbb{S}^{(3)}(\mathbf{r}^{j}) e^{-iH\_{0}t} \hat{\mathbf{U}}(t,s) \right| d^{3}r^{j} d^{3}R^{'} e^{iH\_{0}s} n\_{\mathbf{r}}(\mathbf{R}, \mathbf{s}^{j}), \tag{53}$$

10 to j*th*-particle in the interaction of a j*th*-thought to time *s*. The diagrams drawn here (figure 3) 11 may be interpreted to represent the evolution of particles contributing the position density 12 matrix *n***r**(**R**, *t*) at *t* = *t*, in the corresponding path integral of correction. The true correction 13 comes given by the evolution created in the quantum process of transformation of the functional *O*c(Ñ(*s*)), where the information Ñ(*s*), must be changed when <sup>o</sup> lim i ( , ) ( ) r *t s* 14 *wts ns* 15 (is to say, when *M* = *It*). If we call *t*j = *s*, (*j-th-time of evolution* in the thought process) then the 16 integral (53), takes the form

$$e^{-i(t-t\_j)H\_0} \left\{ \mathbf{1} + \sum\_{j=1}^{\alpha} (-\mathbf{i})\lambda \left\| \begin{matrix} t\_j \\ dt\_1 \end{matrix} \right\| \begin{matrix} t\_2 \\ dt\_2 \end{matrix} \cdots \int dt\_j \,\mathbf{u}(t\_1)\mathbf{u}(t\_2)\cdots\mathbf{u}(t\_j) \right\} = \prod\_{j=1}^{\alpha} \left\{ \int\_{-\alpha}^{+\alpha} \phi(n\_j) F(n\_j) \right\| \mathbf{x}(t) \right\}, \text{ (54)}$$

18 where the states I*j*, are established in the density matrix *n*(0), that to a vertex of G, (*w* - 19 diagram), arrange, those perturbations *B* (see figure 3), that they gave origin to the 20 singularity, with the corresponding arrange of those positive perturbations *A*, that will 21 realises the corrective action to transform the singularity signal Ñ(*s*), of an adequate thought 22 given by *x*(*t*). Due to that, the information given by the product *An*(0)*B*, must be changed when <sup>o</sup> lim i ( , ) ( ) r *t s* 23 *wts ns* , then a *w*-diagram must be change for )Ddiagram [27]. But this 24 live in the quantum field of the space-time TM\*, that is to say, in the corresponding zone of 25 the executive operator I. Then in the material space-time (*Einstein universe*), the 26 displacement of energy needs inside this transformation the application of an invariant 27 given in the quantum space that guarantee that the new particle (*boson*) obtained let that 28 correct. This is given by number *dim*/(D), since it depends on the roots system to the 29 representation of the corresponding action group [27], that recover the recognition action. 30 By the integral (4), the transformation due to the new conscience operator created in TM\*- 31 zone obtained on whole the space-time is,

$$\int\_{X(\mathbb{C})} O\_{\mathfrak{c}} \left( \frac{\mathfrak{x}(t)}{\dim \Lambda} \right) d\mathfrak{x}(t) = \int\_{X(\mathbb{C})} \left\{ \frac{1}{A} \prod\_{j=1}^{\infty} \left\{ \int\_{-\infty}^{+\infty} \phi(n\_j) F(n\_j) \right\} d\mathfrak{x}(t) \right\},$$

33 that is the result wanted.

32

22

7

22 singularity.

f

*operator at this moment* (*part of* (28) *defined by O*<sup>0</sup> 5 c)). Then

16 flux of the operator *Oc*, and that are observed in the surface bundle.

6 *O x x t dt t s x t dt*

*s*

234 Advances in Quantum Mechanics

f f

*s s*

³³ ³ c c ( ( )) ( ) ( ( )) ( ) 0 <sup>Dž</sup>( )() ,

³ ³ <sup>c</sup>( ) () <sup>Dž</sup>( )() .

2 where the first integral is equal to cero, because there is no singularity before *s*, (the 3 evolution happens after the time *t* t *s* (see (49)). But this evolution is anomalous, since to all *t* 4 < *s*, includes a captive energy not assimilated to *t* = *s* (*this because it does not have a conscience* 

f f

*t t*

**Figure 7.** The surface represents the perturbation given by the existence of a singularity in the space-time of the quantum zone. The *computational model* obeys the solution of Planck-Fokker partial differential equation. The waves of perturbed state begins to s d t, such as it was predicted in the lemma 1. The surface include the waves established in figure 3, given by z = *Plot3D(exp(-1/2(x^2 + y^2))(cos(4x) + sin(2x) + 3sin(1/3y) + cos(5y)ln(2x)))*, through the *space-time 4.0 program*. Observe that is present a kernel of transformation for normal distribution given by *exp*(-1/2(x2 + y2 12 )) that will appear in the transform that defines the singularity when a particle is not appropriately assimilate. The normal distribution kernel is the statistical weight that establishes the appearance of an abnormal evolution created by the existence of the singularity. That is to say, the singularity is detected by the anomalous effects that are glimpsed in the

17 Theorem. **1. (F. Bulnes**). Let consider a conscience operator with singularity *O*c(Ñ), for the 18 presence of an energy load *w*(*s*, *t*). Then the *elimination* of the singularity Ñ(*s*), comes given for

> <sup>1</sup> ( ) ( ( )) ( , ) <sup>ƭ</sup> ( ) ( ) () , *j j X(C) C j*

*x t correction restoring O s w t s dt dim n F n dx t*

19 (52)

20 where *dim*/(D), is the *Neumann dimension* corresponding to the *Weyl camera* of the roots Dj, 21 [2627@used in the rotation process to eliminate the deviation [11], created by the

23 *Proof.* Consider an arbitrary irreducible diagram with nodes with *w*-parts (parts of diagrams 24 with nodes of weight *w*(*t*, *s*)). Suppose that this points "nodes" with weight *w*, determines 25 the singularity given by (50). In fact, by the theory of Van Hove on the singularities in the

I

½ ° ° °° ½ ® ® ¾¾

f f

f

° ° ¯ ¿ ° ° ¯ ¿ ³ ³ <sup>c</sup> <sup>³</sup> <sup>1</sup>

*A*

1 *O x t x t dt O x t x t dt t s x t dt*

24

### 1 **5. Re-composition and determination of the realities**

We consider the space-time M, like space where R<sup>d</sup> 2 u I*t*, is the macroscopic component of the space-time and we called F, the microscopic component of the space-time of ratio 10-33 3 *cm* 4 (*length of a string* [21]). For previously described the quantum zone of the space-time M, is 5 connected with N, which will called virtual zone of the space-time (zone of the space-time 6 where the process and transformation of the virtual particles happen) are connected by 7 possibilities causal space generated by certain class of photons and by the material particles 8 interacting in the material space time, with permanent energy and the material particles recombining their states they become in waves on having moved in R<sup>d</sup> 9 u I*t*, on any path of 10 Feynman. Likewise we can define the space of this double fibration of quantum processing 11 as:

$$\mathcal{L} = \left\{ \mathbf{O}\_c \left( \boldsymbol{\phi}, \boldsymbol{\hat{o}}\_{\mu} \boldsymbol{\phi}, \mathbf{x} \left( t \right), t \right) \in \mathbb{C}^2 \left( \mathbb{R}^d \times I\_t \right) \middle| \frac{\hat{\boldsymbol{o}}^2}{\partial t^2} - \nabla^2 \left( \mathbf{O}\_c \left( \boldsymbol{\phi}, \boldsymbol{\hat{o}}\_{\mu} \boldsymbol{\phi}, \mathbf{x} (t), t \right) \right) = \mathbf{0} \right\}, \tag{55}$$

13 with the states I, of quantum field are in the quantum zone M. Let N, the ambi-space (*set of*  14 *connection and field*) defined as:

$$\mathcal{N} = \left\langle (X, \nabla) \in M \times L \mid \nabla^{X \mathcal{Y}}\_{X^\* \mathcal{Y}} \Psi + \Phi \begin{pmatrix} X \\ \end{pmatrix} = \begin{pmatrix} 0 \\ \end{pmatrix} \right\rangle \tag{56}$$

Provisional chapter 25

237

http://dx.doi.org/10.5772/53439

If M # C<sup>4</sup> 1 , then M = M u Q*x*, is the complete universe (include the cosmogonist perception by 2 the super-symmetry specialist [31]). But, what is there of our quantum universe with regard

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

4 The answer is the same, we have an universe of ten dimensions and M = N u M, where the quantum representation of the object *x*(*s*), is the quantum space-time M = R<sup>3</sup> 5 u I*t*, (*which is the Einstein cosmogonist perception*) then the cosmo-vision of the virtual particles is C<sup>2</sup> 6 u Q*x*, 7 [21], then the execution operator I, that proceeds to connect virtual particles through the 8 paths which have path integrals on double fibration, establishing the *material-quantum-*

10 (59)

11 where *C*, is the material part connected with the quantum zone of the space-time (space 12 taken by atoms) M. The corresponding path integral that connects virtual particles in the

 v³ 1 1 <sup>I</sup> Q c ƥ ( ( ( ))) ( ( ( ( ( ))))<sup>Ǎ</sup> , *<sup>x</sup> <sup>s</sup>* <sup>14</sup>*xs O <sup>x</sup>* (60)

15 always with the space {*x*(*s*)M \_ 4*<sup>x</sup>* N}, to the permanent field actions. Then the *reality*  16 *state* is the obtained through the *integral of perception* (60), considering the fibre of the

20 The integral medicine into of the class of alternative medicine, fundament their methods of 21 cure in to health and reactive the vital field *X*, of the human body *B*, the *regeneration* of the 22 centers of energy of *B*, and the *corrections and restoration* of the flux of energy *Flux*, in and in 23 each organ **B**, of the human body *B*, taking constant of gradient of their electromagnetic 24 current, voltage and resistance, obtaining of this manner, the balance of each organ in 25 sunstone with the other organs to characterize to *B*, like complete synergic system in

27 Now, the cure that is realized to nano-metric scale must be executed with a synergic action 28 of constant field >33@, equal to effect in each atom of our body to unison of *real conscience of*  29 *cure* (*duality mind-body* [11]). Of this way, the conscience of *B*, is the obtained synergy by the 30 atoms in this sense and that will come reflected in the reconstitution of the vital field *X*.

17 corresponding reality in the argument of the operator *Oc*, of the integrating from (60).

TS V U

13 whole fibration is the integral of line type (5) defining feedback connection:

3 to our real universe (included the material part given by the atoms)?

9 *virtual* connection required to a total reality:

18 **6. Applications to the nanosciences** 

19 **6.1. Nanomedicine** 

26 equilibrium and harmony [10].

16 where , is the connection of virtual field *X*, with the quantum field *Y*, and <, is the field 17 whose action is always present to create perceptions in the quantum zone connected with ) 18 (2-form)[28]. Then we can create the correspondence given by the double fibration [29]:

20 This double fibration conformed the interrelation between M, and N. *x*(*t*)M, give 21 beginning to a complex submanifold (that represents the spaces where are the quantum 22 hologram) that includes all these quantum images given by quantum holograms, why? 23 Because this complex submanifolds, considering the causal structure given in the space-time 24 by the light cones (see figure 6 a)) [26], of all trajectories that follow a particle in the space-25 time [29], they can write using (57) as:

$$
\Theta\_{\mathbf{x}} \rightsquigarrow \Theta \Big[ \pi^{-1} \Big( \mathbf{x} \Big) \Big] \Big. \tag{58}
$$

of N, such that 4x # P<sup>1</sup>u P<sup>1</sup> 27 , which by space-time properties to quantum level represent the 28 space of all light rays that transit through *x*, conforming a hypersurface (projective surface) 29 that is a light surface. This surface is called the sky *x* [30]. A sky in this context represents the 30 set of light rays through *x* (*bosons*) that it comes of the virtual field.

If M # C<sup>4</sup> 1 , then M = M u Q*x*, is the complete universe (include the cosmogonist perception by 2 the super-symmetry specialist [31]). But, what is there of our quantum universe with regard 3 to our real universe (included the material part given by the atoms)?

4 The answer is the same, we have an universe of ten dimensions and M = N u M, where the quantum representation of the object *x*(*s*), is the quantum space-time M = R<sup>3</sup> 5 u I*t*, (*which is the Einstein cosmogonist perception*) then the cosmo-vision of the virtual particles is C<sup>2</sup> 6 u Q*x*, 7 [21], then the execution operator I, that proceeds to connect virtual particles through the 8 paths which have path integrals on double fibration, establishing the *material-quantum-*9 *virtual* connection required to a total reality:

11 where *C*, is the material part connected with the quantum zone of the space-time (space 12 taken by atoms) M. The corresponding path integral that connects virtual particles in the 13 whole fibration is the integral of line type (5) defining feedback connection:

$$\mathfrak{a}(\mathfrak{T}\_{Q\_x}(\mathfrak{x}(s))) = \oint\_{\Gamma} O\_{\varepsilon}(\theta(\pi^{-1}(\sigma(\rho^{-1}(\mathfrak{x})))) | \mu\_{s'} $$

15 always with the space {*x*(*s*)M \_ 4*<sup>x</sup>* N}, to the permanent field actions. Then the *reality*  16 *state* is the obtained through the *integral of perception* (60), considering the fibre of the 17 corresponding reality in the argument of the operator *Oc*, of the integrating from (60).

### 18 **6. Applications to the nanosciences**

#### 19 **6.1. Nanomedicine**

24

236 Advances in Quantum Mechanics

11 as:

1 **5. Re-composition and determination of the realities** 

 

I IP

14 *connection and field*) defined as:

25 time [29], they can write using (57) as:

We consider the space-time M, like space where R<sup>d</sup> 2 u I*t*, is the macroscopic component of the space-time and we called F, the microscopic component of the space-time of ratio 10-33 3 *cm* 4 (*length of a string* [21]). For previously described the quantum zone of the space-time M, is 5 connected with N, which will called virtual zone of the space-time (zone of the space-time 6 where the process and transformation of the virtual particles happen) are connected by 7 possibilities causal space generated by certain class of photons and by the material particles 8 interacting in the material space time, with permanent energy and the material particles recombining their states they become in waves on having moved in R<sup>d</sup> 9 u I*t*, on any path of 10 Feynman. Likewise we can define the space of this double fibration of quantum processing

 

*t*

<sup>½</sup> °  w ° w ® ¾ ° ° ¯ ¿ 2 ,, , , *<sup>d</sup> O x c t t t C I* <sup>R</sup> 2

12 L (55)

13 with the states I, of quantum field are in the quantum zone M. Let N, the ambi-space (*set of* 

^() | *X ML* , u < )  *<sup>X</sup>* 0 , ` *XY* <sup>N</sup> *X'Y'* 15 (56)

16 where , is the connection of virtual field *X*, with the quantum field *Y*, and <, is the field 17 whose action is always present to create perceptions in the quantum zone connected with ) 18 (2-form)[28]. Then we can create the correspondence given by the double fibration [29]:

19 (57)

20 This double fibration conformed the interrelation between M, and N. *x*(*t*)M, give 21 beginning to a complex submanifold (that represents the spaces where are the quantum 22 hologram) that includes all these quantum images given by quantum holograms, why? 23 Because this complex submanifolds, considering the causal structure given in the space-time 24 by the light cones (see figure 6 a)) [26], of all trajectories that follow a particle in the space-

  -1 ̋<sup>x</sup> <sup>=</sup>Ό Δ x , ª º 26 ¬ ¼ (58)

of N, such that 4x # P<sup>1</sup>u P<sup>1</sup> 27 , which by space-time properties to quantum level represent the 28 space of all light rays that transit through *x*, conforming a hypersurface (projective surface) 29 that is a light surface. This surface is called the sky *x* [30]. A sky in this context represents the

30 set of light rays through *x* (*bosons*) that it comes of the virtual field.

<sup>w</sup> u

2

I I

w

<sup>2</sup> <sup>c</sup> <sup>Ǎ</sup> *O x* ( ) *t t* 0

20 The integral medicine into of the class of alternative medicine, fundament their methods of 21 cure in to health and reactive the vital field *X*, of the human body *B*, the *regeneration* of the 22 centers of energy of *B*, and the *corrections and restoration* of the flux of energy *Flux*, in and in 23 each organ **B**, of the human body *B*, taking constant of gradient of their electromagnetic 24 current, voltage and resistance, obtaining of this manner, the balance of each organ in 25 sunstone with the other organs to characterize to *B*, like complete synergic system in 26 equilibrium and harmony [10].

27 Now, the cure that is realized to nano-metric scale must be executed with a synergic action 28 of constant field >33@, equal to effect in each atom of our body to unison of *real conscience of*  29 *cure* (*duality mind-body* [11]). Of this way, the conscience of *B*, is the obtained synergy by the 30 atoms in this sense and that will come reflected in the reconstitution of the vital field *X*. 26

1 Then under this reinterpretation, the sickness is only an effect of the fragmentation of this 2 *real conscience of cure* of *B*, that is deduced by disconnections and disparity of atoms [11]. The 3 integral medicine helps to recover the *continuity of this conscience* through of the electronic 4 memory of health of the proper body [11, 22] (see the figure 6 c)).

#### 5 **6.2. Quantology and neurosciences**

6 Let M, the mind space and their organic component (material component) the brain space C. 7 Also we consider the quantum component of the mind given by the space-time M. Studies 8 realised in statistical mechanics have revealed that the *Bose-Einstein statistics* stretches to 9 accentuate the low energy levels. This reflects his closeness to the emission of a *virtual field*, 10 where the virtual particles are not detected in a *virtual energy sea*. This allows to surmise that 11 the radiation that takes place from the virtual field to the quantum field of M, is composed 12 by *photons type bosons* (that is to say it obeys this Bose-Einstein statistics), since the quantum 13 field interacts with the material particles that contains the material field of the mind which is 14 anchored in the brain C, like material organ.

15 Theorem **(F. Bulnes).** [21] The *total Lagrangian of mental field* comes given by the superior 16 action whose total conscience is

$$\mathbf{O}\_{\text{total}} = \mathbf{O}\_{\text{QCD}} \left( \mathbf{O}\_{\text{EM}} \right), \tag{61}$$

1

21 actions [35]

24 integrals [36])

**Figure 8.** a). Concept of the topological space (*sky* [21, 30]) and the cosmogonist perceptions of the mind. The conscience operator of the different realities is given by (60). b) Modelling from boson-shape distribution in localizing of symptoms in the mind by cerebral signals with low interchange electrons [24, 25, 32, 33]. Bose-Einstein distribution was published in [22]. c) Total Action due Gelsem including the action de Bervul and *Ignamara*. The corresponding quantum intelligence code of this total action is given by 011110**101101101101**10111110, [7]. The corresponding code given in black include de codes of *Gelsem* and *Bervul*. The code in orange is the code of *Gelsemium* equal to code

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

10 Considering some applications of quantum electrodynamics in the design of flaying vehicles 11 self-supported and their magnetic levitation, we find the magnetic conscience operator is 12 defined for the transmitting of the diamagnetic property every particle of the ship structure.

13 This vehicle is controlled by one microchip that is programmed by conscience operators 14 algebra of electromagnetic type that leads to the flow of Eddy currents, the iso-rotations and 15 suspension of the special geometrical characteristics vehicle, generating also on the vehicle 16 structure certain "magnetic conscience" that provokes all movements like succeeding the 17 sidereal objects in the universe [26, 34]. This magnetic conscience is generating by the proper 18 particles of the ship structure transmitted for the interaction of superconductor inside the 19 reactor with the magnetic field generated by the rotating rings under the ship. By so doing, 20 the Eddy's currents in the "skin effect" around the structure of the ship are given by the

ship M rot , 22 (63)

23 and using the quantum E H-fields to create (magnetic conscious operator given by the

³ ^

*ship HA L d* 25 (64)

` <sup>2</sup> ( , ) H / 8 V, *M M*

S

8 of *Gelsem*. This quantum code was used to cure a patient with a digestive illness [11, 21, 22].

9 **6.3. Electromagnetic vehicle with levitation magnetic conscience** 

Provisional chapter 27

239

http://dx.doi.org/10.5772/53439

18 to one total action defined by the groups *SU*(3), (*quantum and virtual field*) and *SU*(2), 19 (*material field*).

20 *Proof*. The Lagrangian of the theory is an invariant of Lorentz and invariant under local 21 transformations of phase of the group *SU*(3), (*for the charge of color*) and has the following 22 form [31]:

$$\mathcal{L}\_{\text{TOTAL}} = \left\{ \overline{q}\eta\gamma^{\mu}\partial\_{\mu}q - \overline{q}mq - b\overline{q}\gamma^{\mu}\Gamma\_{a}q\mathbb{b}\_{\mu\nu}^{a} - \frac{1}{4}b\_{\mu\nu}^{a}b\_{a}^{\mu\nu} \right\},\tag{62}$$

24 This corresponds to the space of the mind M =M +C , where *OEM*, put in C, (*neurological*  25 *studies have proved that the process of thought in the level at least visible is of electromagnetic type*) 26 signals through charges in the synapses and neurons of C. Nevertheless these charges produce in one level deeper. The tensor b<sup>a</sup> 27 PQ, is anti-symmetric and represent a *bosonic field* 28 created by the interaction of quarks *q*, *p* = *q* , and [*p*, *q*] = *q* q - q *q.* Whose bosonic field has all the particles of spin 0, and the trace of tensor b<sup>a</sup> 29 PQ, has electromagnetic components 30 conformed by the photons that are stable on the limit of the transformation q(*x*(*s*)) Io *e* iD*a*(*x*(s))*Ta* 31 b(*x*(*s*)), of the thought *x*(*s*), where some D*a*(*x*(*s*))T*a*, is the electromagnetic frequency *k*a, 32 of the term *ik*a*v* [7, 10 26], where in these cases the last term in the second member of (62), is ¼ *F*PQ*F*PQ 33 . Then the *SU*(2)-actions are included in the *SU*(3), actions and their points are 34 electromagnetic particles transformed by these rules like photons (= thoughts in M) [28].

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ... http://dx.doi.org/10.5772/53439 239

Provisional chapter 27

2 **Figure 8.** a). Concept of the topological space (*sky* [21, 30]) and the cosmogonist perceptions of the mind. The 3 conscience operator of the different realities is given by (60). b) Modelling from boson-shape distribution in localizing 4 of symptoms in the mind by cerebral signals with low interchange electrons [24, 25, 32, 33]. Bose-Einstein distribution 5 was published in [22]. c) Total Action due Gelsem including the action de Bervul and *Ignamara*. The corresponding 6 quantum intelligence code of this total action is given by 011110**101101101101**10111110, [7]. The corresponding 7 code given in black include de codes of *Gelsem* and *Bervul*. The code in orange is the code of *Gelsemium* equal to code 8 of *Gelsem*. This quantum code was used to cure a patient with a digestive illness [11, 21, 22].

#### 9 **6.3. Electromagnetic vehicle with levitation magnetic conscience**

1

26

1 Then under this reinterpretation, the sickness is only an effect of the fragmentation of this 2 *real conscience of cure* of *B*, that is deduced by disconnections and disparity of atoms [11]. The 3 integral medicine helps to recover the *continuity of this conscience* through of the electronic

6 Let M, the mind space and their organic component (material component) the brain space C. 7 Also we consider the quantum component of the mind given by the space-time M. Studies 8 realised in statistical mechanics have revealed that the *Bose-Einstein statistics* stretches to 9 accentuate the low energy levels. This reflects his closeness to the emission of a *virtual field*, 10 where the virtual particles are not detected in a *virtual energy sea*. This allows to surmise that 11 the radiation that takes place from the virtual field to the quantum field of M, is composed 12 by *photons type bosons* (that is to say it obeys this Bose-Einstein statistics), since the quantum 13 field interacts with the material particles that contains the material field of the mind which is

15 Theorem **(F. Bulnes).** [21] The *total Lagrangian of mental field* comes given by the superior

18 to one total action defined by the groups *SU*(3), (*quantum and virtual field*) and *SU*(2),

20 *Proof*. The Lagrangian of the theory is an invariant of Lorentz and invariant under local 21 transformations of phase of the group *SU*(3), (*for the charge of color*) and has the following

<sup>1</sup> <sup>L</sup>

23 (62) 24 This corresponds to the space of the mind M =M +C , where *OEM*, put in C, (*neurological*  25 *studies have proved that the process of thought in the level at least visible is of electromagnetic type*) 26 signals through charges in the synapses and neurons of C. Nevertheless these charges produce in one level deeper. The tensor b<sup>a</sup> 27 PQ, is anti-symmetric and represent a *bosonic field* 28 created by the interaction of quarks *q*, *p* = *q* , and [*p*, *q*] = *q* q - q *q.* Whose bosonic field has all the particles of spin 0, and the trace of tensor b<sup>a</sup> 29 PQ, has electromagnetic components 30 conformed by the photons that are stable on the limit of the transformation q(*x*(*s*)) Io

iD*a*(*x*(s))*Ta* 31 b(*x*(*s*)), of the thought *x*(*s*), where some D*a*(*x*(*s*))T*a*, is the electromagnetic frequency *k*a, 32 of the term *ik*a*v* [7, 10 26], where in these cases the last term in the second member of (62), is ¼ *F*PQ*F*PQ 33 . Then the *SU*(2)-actions are included in the *SU*(3), actions and their points are 34 electromagnetic particles transformed by these rules like photons (= thoughts in M) [28].

 ½ w ® ¾ ¯ ¿ b bb Ǎ Ǎ TOTAL <sup>Ǎ</sup> <sup>i</sup>DŽ DŽ T , *a a Ǎǎ <sup>a</sup> Ǎǎ Ǎǎ <sup>a</sup> q qqmq bq q*

4

  O = O O , total QCD EM 17 (61)

4 memory of health of the proper body [11, 22] (see the figure 6 c)).

5 **6.2. Quantology and neurosciences** 

238 Advances in Quantum Mechanics

14 anchored in the brain C, like material organ.

16 action whose total conscience is

19 (*material field*).

22 form [31]:

*e*

10 Considering some applications of quantum electrodynamics in the design of flaying vehicles 11 self-supported and their magnetic levitation, we find the magnetic conscience operator is 12 defined for the transmitting of the diamagnetic property every particle of the ship structure.

13 This vehicle is controlled by one microchip that is programmed by conscience operators 14 algebra of electromagnetic type that leads to the flow of Eddy currents, the iso-rotations and 15 suspension of the special geometrical characteristics vehicle, generating also on the vehicle 16 structure certain "magnetic conscience" that provokes all movements like succeeding the 17 sidereal objects in the universe [26, 34]. This magnetic conscience is generating by the proper 18 particles of the ship structure transmitted for the interaction of superconductor inside the 19 reactor with the magnetic field generated by the rotating rings under the ship. By so doing, 20 the Eddy's currents in the "skin effect" around the structure of the ship are given by the 21 actions [35]

$$
\mathfrak{J}\_{\text{ship}} = \mathfrak{J}\_{\text{M}} + \mathfrak{J}\_{\text{rot}},
\tag{63}
$$

23 and using the quantum E H-fields to create (magnetic conscious operator given by the 24 integrals [36])

$$H(A, \mathfrak{I}\_M) = \int\_{\text{slip}} \left\langle L\_M - \text{H}^2 \, \right\rangle \, 8\pi \Big| \text{dV}\_{\prime} \tag{64}$$

28

1

Provisional chapter 29

241

http://dx.doi.org/10.5772/53439

 <sup>ƥ</sup> ³ ³ <sup>c</sup> ƺ(ƥ) ƺ(ƥ)

2 In the forms language, the conscience operator comes given by the map Z*L* : TM o TM\*, with 3 rule of correspondence given by (19). The quantum conscience shape a continuous flux of 4 energy with an intention, involving a smooth map S (defined in the *example 1*). Then the 5 conscience operator is related with the action , and the trajectories J*t*, through of the

J

8 *O*c(Ñ) – Conscience Operator in the singularity Ñ. This is a kernel of the quantum inverse 9 transfom of path integrals to eliminate singularities. Their direct transform use the kernel

O0(*Zi, pA*) – Vertex operator of the theory given by the equations W*<sup>a</sup>* 11 (Z*i*) = 0, localized at the

O(*x* – *x*') – Is the functional operator O(*x* - *xj*) = (*<sup>x</sup>*+ *m*2 – *i*H)G*<sup>n</sup>* 13 (*x* - *x*j). This operator involves

15 *O*QCD – Quantum chromodynamics conscience operator. Their Lagrangian density using the

PQ 16 , Where *D*, is the QCD gauge covariant derivative (in Feynman notation [V 17 *D*V), *n* = 1, 2, }6 counts the quark

19 *OEM* – Conscience operator defined through of the Lagrangian to quantum electromagnetic 20 field (these like gauge fields). Their Lagrangian density is L*EM* = *ihc*\*D*\*mc2*\\(1/4P0)*F*PQ*F*PQwhere *F*PQ 21 , is the electromagnetic tensor, *D*, is the gauge

23 *O*total – Total quantum conscience operator. This is the composition of operators *OEM*

:(*k,* 0) 26 – Differential form to complex hypersurfaces of dimension k. This form is analogous to

28 4x – Fibers of the topological space Q*x*, called *sky* conformed by the light rays through *x* 29 (*bosons*) that it comes of the virtual field. This is a conscience operator when realises the

quantum chromodynamics is L*EM* = 6*n* (*ihc*\*nD*\*nmnc2*\*n*\*n*)*G*<sup>D</sup>

*t*

M

PQ*G*<sup>D</sup>

o p p o S <sup>C</sup> TM TM \*

*O*

R

1

7

10 *O*c(*x*(*t*)).

24 followed *O*QCD.

6 following diagram:

12 points *pA*, of the Riemann surface S.

14 to electronic propagator in a pulse impulse.

18 types, and is the gluon field strength tensor.

covariant derivative, and *D*, is Feynman notation for[V 22 *D*V .

25 *O*c – Action that involves a conscience operator *O*c.

30 reality transformation by the double fibration.

27 the form Z*L*, and involves the conscience operator (*O*c)\*.

*O vv Lv* ( ) ( ),

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

2 **Figure 9.** a). The green color represents the state of quantum particles in transition to obtain the anti-gravity states 3 through the interaction of the E H-fields used on the structure of the vehicle. The blue flux represents Eddy's 4 currents that interact with the lines of magnetic field to produce an diamagnetic effect in the top part of the vehicle. 5 The red central ring is the magnetic field that generates the twistor surface (this computational simulation is 6 published in [35, 36]). b) Top part of the flying plate showing the generation of the effect skin obtained in the 7 interaction superconductor - magnetic field. The regions in light and obscure blue represent the flow of Eddy's 8 currents, in this top part of the vehicle. The Skin effect is derived by the quantum interaction by the actions of E H 9 [26, 37]. c) The microscopic effects of the electromagnetic fields (quantum densities of field) created inside the 10 algebra E H, create an effect of macro-particle of the vehicle [36] (the vehicle and their electromagnetic revetment 11 behaves like a particle) where their displacement is realized in instantaneous form and their direction it is a macro-spin 12 projected from the *magnetic conscious operator* (from M, from (63)) of the ship which defines their angular moment 13 [35, 36].

### 14 **Apendix**

#### 15 **Technical notation**

16 *O*c – Is an operator that involves the Lagrangian but directing this Lagrangian in one specific 17 fiber (direction) *prefixing tha Lagrangian action in one direction*. This is defined as the map: *O*<sup>c</sup> : TM o TM\*, with rule of correspondence <sup>c</sup> 18 *w O vw* ( ) , where *w* = *L*(*v*), with *L*, the classic 19 Lagrangian. *This defines the quantum conscience*. If we locally restrict to *O*c, that is to say, on 20 the tangent space T*x*M u T*x*M, *x*M (# :(\*)), we have that

$$O\_c: \mathrm{T\_xM} \times \mathrm{T\_xM} \left(\cong\underset{locally}{\mathrm{TM}}\right) \xrightarrow{\longrightarrow} \mathrm{TM^\*},$$

22 with rule of correspondence

$$(v, w) \xrightarrow{\quad} \longrightarrow O\_{\mathfrak{c}}(v)w\_{\vee}$$

24 *O*c(*v*)*w*, generalise the means of *O*c(*v*)*v* = *L*(*v*), *v*T*x*M, x:(\*). Likewise, if : TM o R, with rule of correspondence ( ) ( ( )) ( ) c 25 *Lv Lv O vv* , then the total action along the 26 trajectory \*, will be

Provisional chapter 29 Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ... http://dx.doi.org/10.5772/53439 241

$$\mathfrak{J}\_{\Gamma} = \int\_{\mathfrak{Q}(\Gamma)} O\_{\mathfrak{c}}(\upsilon) \upsilon = \int\_{\mathfrak{Q}(\Gamma)} L(\upsilon) \, \iota$$

1

7

28

240 Advances in Quantum Mechanics

1

13 [35, 36].

14 **Apendix** 

15 **Technical notation** 

22 with rule of correspondence

26 trajectory \*, will be

**Figure 9.** a). The green color represents the state of quantum particles in transition to obtain the anti-gravity states through the interaction of the E H-fields used on the structure of the vehicle. The blue flux represents Eddy's currents that interact with the lines of magnetic field to produce an diamagnetic effect in the top part of the vehicle. The red central ring is the magnetic field that generates the twistor surface (this computational simulation is published in [35, 36]). b) Top part of the flying plate showing the generation of the effect skin obtained in the interaction superconductor - magnetic field. The regions in light and obscure blue represent the flow of Eddy's currents, in this top part of the vehicle. The Skin effect is derived by the quantum interaction by the actions of E H [26, 37]. c) The microscopic effects of the electromagnetic fields (quantum densities of field) created inside the algebra E H, create an effect of macro-particle of the vehicle [36] (the vehicle and their electromagnetic revetment behaves like a particle) where their displacement is realized in instantaneous form and their direction it is a macro-spin projected from the *magnetic conscious operator* (from M, from (63)) of the ship which defines their angular moment

(a) (b) (c)

16 *O*c – Is an operator that involves the Lagrangian but directing this Lagrangian in one specific 17 fiber (direction) *prefixing tha Lagrangian action in one direction*. This is defined as the map: *O*<sup>c</sup> : TM o TM\*, with rule of correspondence <sup>c</sup> 18 *w O vw* ( ) , where *w* = *L*(*v*), with *L*, the classic 19 Lagrangian. *This defines the quantum conscience*. If we locally restrict to *O*c, that is to say, on

24 *O*c(*v*)*w*, generalise the means of *O*c(*v*)*v* = *L*(*v*), *v*T*x*M, x:(\*). Likewise, if : TM o R,

25 *Lv Lv O vv* , then the total action along the

u # o

c

20 the tangent space T*x*M u T*x*M, *x*M (# :(\*)), we have that

o <sup>c</sup> ( , ) *v w O vw* () , <sup>23</sup>

with rule of correspondence ( ) ( ( )) ( )

*locally* <sup>c</sup> :T T ( T ) T \*, <sup>21</sup>*<sup>O</sup> x x* M M M M

2 In the forms language, the conscience operator comes given by the map Z*L* : TM o TM\*, with 3 rule of correspondence given by (19). The quantum conscience shape a continuous flux of 4 energy with an intention, involving a smooth map S (defined in the *example 1*). Then the 5 conscience operator is related with the action , and the trajectories J*t*, through of the 6 following diagram:

$$\begin{array}{ccc} \mathsf{TM} & \stackrel{\textstyle \begin{array}{c} \Box\_{\mathsf{C}} \longrightarrow} \mathsf{TM}^{\*} \\ \mathsf{T} \downarrow \\ \mathsf{R} & \stackrel{\textstyle \begin{array}{c} \Box\_{\mathsf{T}} \end{array}}{\end{array} \end{array} \begin{array}{c} \mathsf{TM}^{\*} \\ \downarrow \\ \mathsf{M} \end{array} \end{array}$$

8 *O*c(Ñ) – Conscience Operator in the singularity Ñ. This is a kernel of the quantum inverse 9 transfom of path integrals to eliminate singularities. Their direct transform use the kernel 10 *O*c(*x*(*t*)).

O0(*Zi, pA*) – Vertex operator of the theory given by the equations W*<sup>a</sup>* 11 (Z*i*) = 0, localized at the 12 points *pA*, of the Riemann surface S.

O(*x* – *x*') – Is the functional operator O(*x* - *xj*) = (*<sup>x</sup>*+ *m*2 – *i*H)G*<sup>n</sup>* 13 (*x* - *x*j). This operator involves 14 to electronic propagator in a pulse impulse.

15 *O*QCD – Quantum chromodynamics conscience operator. Their Lagrangian density using the quantum chromodynamics is L*EM* = 6*n* (*ihc*\*nD*\*nmnc2*\*n*\*n*)*G*<sup>D</sup> PQ*G*<sup>D</sup> PQ 16 , Where *D*, is the QCD gauge covariant derivative (in Feynman notation [V 17 *D*V), *n* = 1, 2, }6 counts the quark 18 types, and is the gluon field strength tensor.

19 *OEM* – Conscience operator defined through of the Lagrangian to quantum electromagnetic 20 field (these like gauge fields). Their Lagrangian density is L*EM* = *ihc*\*D*\*mc2*\\(1/4P0)*F*PQ*F*PQwhere *F*PQ 21 , is the electromagnetic tensor, *D*, is the gauge covariant derivative, and *D*, is Feynman notation for[V 22 *D*V .

23 *O*total – Total quantum conscience operator. This is the composition of operators *OEM* 24 followed *O*QCD.

25 *O*c – Action that involves a conscience operator *O*c.

:(*k,* 0) 26 – Differential form to complex hypersurfaces of dimension k. This form is analogous to 27 the form Z*L*, and involves the conscience operator (*O*c)\*.

28 4x – Fibers of the topological space Q*x*, called *sky* conformed by the light rays through *x* 29 (*bosons*) that it comes of the virtual field. This is a conscience operator when realises the 30 reality transformation by the double fibration.

30

1 wM- Action that have codimension strata *n* – *k*. This action is due to the differential *d*(I)*h*.

Provisional chapter 31

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1 [9] Sokolnikoff, IS. Tensor Analysis: Theory and Applications. New York: Wiley and Sons;

Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals ...

3 [10] Bulnes F; Bulnes H. F; Hernandez E; Maya J. Integral Medicine: New Methods of 4 Organ-Regeneration by Cellular Encoding through Path Integrals applied to the 5 Quantum Medicine. Journal of Nanotechnology in Engineering and Medicine ASME

7 [11] Bulnes F; Bulnes H. F; Hernandez E; Maya J. Diagnosis and Spectral Encoding in 8 Integral Medicine through Electronic Devices Designed and Developed by Path 9 Integrals. Journal of Nanotechnology in Engineering and Medicine ASME 2011;

11 [12] Ionescu LM. Considerations on some algebraic properties of Feynman integrals.

13 [13] Kontsevich M, Deformation quantization of Poisson manifolds I. In: Joseph A, Mignot F, 14 Murat F, Prum B, Rentschler R.(eds.) Proceedings of the First European Congress of 15 Mathematics, Vol. II, 6–10 July 1992, Paris, France. Progr. Math., 120, Birkhäuser, Basel, 1994. 16 [14] Kontsevich M. Feynman diagrams and low-dimensional topology. In: Joseph A, Mignot 17 F, Murat F, Prum B, Rentschler R.(eds.) Proceedings of the first European congress of 18 mathematics (ECM), Vol. II, 6–10 July 1992, Paris, France, Progr. Math., 120, Birkhäuser,

20 [15] Bulnes F. Cohomology of Cycles and Integral Topology. In: Bulnes, F. (ed.) Meeting of 21 27- 29 Autumn 2008, Mexico City, Appliedmath 4, IM-UNAM, Mexico, 2008.

23 [16] Griffiths P; Harris J. Principles of Algebraic Geometry. USA: Wiley-Interscience; 1994. 24 [17] Gross M; Huybrechts D; Joyce D. Calabi-Yau Manifolds and Related Geometries.

26 [18] Hoogeveen J; Skenderis K. Decoupling of unphysical states in the minimal pure spinor

28 [19] Berkovits N; Hoogeveen J; Skenderis K. Decoupling of unphysical states in the minimal

30 [20] Kibbe TWB. Geometrization of Quantum Mechanics. Springer Online Journal Archives

32 [21] Bulnes F; Bulnes HF; Cote D. Symptom Quantum Theory: Loops and Nodes in 33 Psychology and Nanometric Actions by Quantum Medicine on the Mind Mechanisms 34 Programming Path Integrals. Journal of Smart Nanosystems in Engineering and

36 [22] Bulnes F; Bulnes HF. Quantum Medicine Actions: Programming Path Integrals on 37 Integral Mono-Pharmacists for Strengthening and Arranging of the Mind on Body.

12 Surveys in Mathematics and Its Applications. Vol. 3 (2008). p79-110.

2 1951.

6 2010; 030019(1) 7.

10 021009(2)10.

19 Basel, 1994.

22 www.Appliedmath4.ipn.mx

25 Norway: Springer, 2001.

31 1860-2000 1979; 65(2) 189-201.

35 Medicine 2012; 1(1) 97-121.

27 formalism I. JHEP 1001 (2010) 041. [arXiv:0906.3368].

38 Journal of Frontiers of Public Health 2012; accepted.

29 pure spinor formalism II. JHEP 0909 (2009) 035. [arXiv:0906.3371]

2 *int*M- Action that have codimension strata *k*. This action is due by (I).

### 3 **Author details**

4 Francisco Bulnes

5 Department of Research in Mathematics and Engineering, TESCHA, Mexico

### 6 **Acknowledgements**

7 I am grateful with Carlos Sotero, Eng., for the help offered for the digital process of the 8 images that were included in this chapter.

### 9 **7. References**


1 [9] Sokolnikoff, IS. Tensor Analysis: Theory and Applications. New York: Wiley and Sons; 2 1951.

30

3 **Author details** 

242 Advances in Quantum Mechanics

4 Francisco Bulnes

9 **7. References** 

6 **Acknowledgements** 

8 images that were included in this chapter.

16 December, 2008, Havana, Cuba.

22 by-path-integrals-and-the

24 Addison-Wesley; 1993.

27 Publishing Company; 1992.

29 Addison-Wesley; 1964.

31 Springer, 1983.

10 [1] Sobreiro R., editor. Quantum Gravity. Rijeka: InTech; 2012.

1 wM- Action that have codimension strata *n* – *k*. This action is due to the differential *d*(I)*h*.

7 I am grateful with Carlos Sotero, Eng., for the help offered for the digital process of the

11 http://www.intechopen.com/books/quantum-gravity- (accessed 20 January 2012).

12 [2] Bulnes F. Analysis of prospective and development of effective technologies through 13 integral synergic operators of the mechanics. In: ISPJAE, Superior Education Ministry of Cuba (eds.) 14th 14 Scientific Convention of Engineering and Arquitecture: proceedings of 15 the 5th Cuban Congress of Mechanical Engineering, December 2-5, 2008, CCIA2008, 2-5

17 [3] Bulnes F. Theoretical Concepts of Quantum Mechanics. In: Mohammad Reza Pahlavani 18 (ed.) Correction, Alignment, Restoration and Re-Composition of Quantum Mechanical 19 Fields of Particles by Path Integrals and Their Applications. Rijeka: InTech; 2012. p 20 Available from http://www.intechopen.com/books/theoretical-concepts-of-quantum-21 mechanics/correction-alignment-restoration-and-re-composition-of-fields-of-particles-

23 [4] Marsden, JE.; Abraham, R. Manifolds, tensor analysis and applications. Massachusetts:

26 [6] Holstein, BR. Topics in Advanced Quantum Mechanics. CA, USA: Addison-Wesley

28 [7] Feynman, RP; Leighton, RB; Sands, M. Electromagnetism and matter (Vol. II). USA:

30 [8] Warner, FW. Foundations of Differential Manifolds and Lie Groups, New York:

25 [5] Lawson, HB.; Michelsohn, ML. Spin Geometry, Princeton University Press; 1989.

2 *int*M- Action that have codimension strata *k*. This action is due by (I).

5 Department of Research in Mathematics and Engineering, TESCHA, Mexico


32

1 [23] Fujita S. Introduction to non-equilibrium quantum statistical mechanics. Malabar, Fla: 2 W. Krieger Pub. Co. 1983.

**Section 4**

**Perturbation Theory**


**Section 4**

**Perturbation Theory**

32

12 2000.

2 W. Krieger Pub. Co. 1983.

244 Advances in Quantum Mechanics

4 York: Academic Press, 1972.

6 Berlin Germany: Springer, 2006.

14 Phys. Rev. 115 (1959) 485-491.

21 (December 8). UK, 1987.

26 registered SEP, 1996.

24 III/1). Berlin: Springer-Verlag. 1960.

1 [23] Fujita S. Introduction to non-equilibrium quantum statistical mechanics. Malabar, Fla:

3 [24] Simon B; Reed M. Mathematical methods for physics, Vol. I (functional analysis). New

5 [25] Eleftherios N; Economou E; Economou N. Green's Functions in Quantum Physics.

7 [26] Bulnes F, Doctoral course of mathematical electrodynamics. In: National Polytechnique 8 Institute (ed.) Appliedmath3: Advanced Courses: Proceedings of the Applied Mathematics

10 [27] Bulnes F, Conferences of Mathematics: Seminar of Representation Theory of Reductive 11 Lie Groups. Mexico: Compilation of Institute of Mathematics (ed.), UNAM Publications,

13 [28] Aharonov Y; Bohm D. Significance of electromagnetic potentials in quantum theory.

15 [29] Bulnes F; Shapiro M. On general theory of integral operators to analysis and geometry

17 [30] LeBrun ER, Twistors, Ambitwistors and Conformal Gravity. In: Bailey TN, Baston RJ 18 (ed.) Twistors in Mathematics and Physics, UK: Cambridge University; 1990. p71-86. 19 [31] Hughston LP; Shaw WT. Classical Strings in Ten Dimensions. Proceedings of the Royal 20 Society of London. Series A, Mathematical and Physical Sciences, Vol 414. No. 1847

23 [33] Truesdell C; Topin RA. The classical fields theories (in encyclopedia of physics, Vol.

25 [34] Bulnes F. Special dissertations of Maxwell equations. Mexico: unpublished. Only

27 [35] Bulnes F; Hernández E; Maya J. Design and Development of an Impeller Synergic 28 System of Electromagnetic Type for Levitation, Suspension and Movement of 29 Symmetrical Body. Imece2010: Fluid Flow, Heat Transfer and Thermal Systems Part A and B: Proceedings of 11th 30 Symposium on Advances in Materials Processing Science and

32 [36] Bulnes F; Maya J; Martínez I. Design and Development of Impeller Synergic Systems of 33 Electromagnetic Type to Levitation/Suspension Flight of Symmetrical Bodies. Journal of 34 Electromagnetic Analysis and Applications 2012 1(4) 42-52. See in:

36 [37] Bulnes F. Foundations on possible technological applications of the mathematical 37 electrodynamics. Masterful Conference in Section of Postgraduate Studies and Research

9 International Congress, Appliedmath3, 25-29 October SEPI-IPN, México, 2006.

16 (Monograph in Mathematics). Mexico: SEPI-IPN, IMUMAM, 2007.

22 [32] Dossey L. Space, Time and Medicine. Boston: Shambhala, USA, 1982.

31 Manufacturing, 12-18 November2010, British Columbia, Canada, 2010.

35 http://www.scirp.org/journal/PaperInformation.aspx?paperID=17151

38 (SEPI), National Polytechnic Institute, Federal District, Mexico 2007.

**Chapter 11**

**Provisional chapter**

**Convergence of the Neumann Series for the**

**Schrödinger Equation and General Volterra**

**Convergence of the Neumann Series for the**

**in Banach Spaces**

http://dx.doi.org/10.5772/52489

10.5772/52489

**1. Introduction**

Fernando D. Mera and Stephen A. Fulling

Additional information is available at the end of the chapter

Fernando D. Mera and Stephen A. Fulling

**Equations in Banach Spaces**

Additional information is available at the end of the chapter

**Schrödinger Equation and General Volterra Equations**

The time-dependent Schrödinger equation, like many other time-evolution equations, can be converted along with its initial data into a linear integral equation of Volterra type (defined below). Such an equation can be solved formally by iteration (the Picard algorithm), which produces a Neumann series whose *j*th term involves the *j*th power of an integral operator. The Volterra structure of the integral operator ensures that the time integration in this term is over a *j*-simplex, so that its size is of the order of 1/*j*!. One would therefore expect to be able to prove that the series converges, being bounded by an exponential series. The difficulty in implementing this idea is that the integrand usually is itself an operator in an infinite-dimensional vector space (for example, representing integration over the spatial variables of a wave function). If one can prove that this operator is bounded, uniformly in its time variables, with respect to some Banach-space norm, then one obtains a convergence theorem for the Neumann series. This strategy is indeed implemented for the heat equation in the books of the Rubinsteins [1] and Kress [2]. The objective of the thesis [3] was to treat the Schrödinger equation as much as possible in parallel with this standard treatment of the heat equation. This article reports from the thesis a summary of the rigorous framework of the problem, the main theorem, and the most elementary applications of the theorem.

We stress that the situation for time-evolution equations is different (in this respect, nicer) than for the Laplace and Poisson equations, which are the problems studied in most detail in most graduate textbooks on partial differential equations, such as [4]. In that *harmonic potential theory* the problem is similarly reduced to an integral equation, but the integral equation is not of Volterra type and therefore the Neumann series does not converge

> ©2012 Mera and Fulling, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 D. Mera and A. Fulling; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 D. Mera and A. Fulling; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Provisional chapter**

#### **Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces**

Fernando D. Mera and Stephen A. Fulling Fernando D. Mera and Stephen A. Fulling

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52489 10.5772/52489

### **1. Introduction**

The time-dependent Schrödinger equation, like many other time-evolution equations, can be converted along with its initial data into a linear integral equation of Volterra type (defined below). Such an equation can be solved formally by iteration (the Picard algorithm), which produces a Neumann series whose *j*th term involves the *j*th power of an integral operator. The Volterra structure of the integral operator ensures that the time integration in this term is over a *j*-simplex, so that its size is of the order of 1/*j*!. One would therefore expect to be able to prove that the series converges, being bounded by an exponential series. The difficulty in implementing this idea is that the integrand usually is itself an operator in an infinite-dimensional vector space (for example, representing integration over the spatial variables of a wave function). If one can prove that this operator is bounded, uniformly in its time variables, with respect to some Banach-space norm, then one obtains a convergence theorem for the Neumann series. This strategy is indeed implemented for the heat equation in the books of the Rubinsteins [1] and Kress [2]. The objective of the thesis [3] was to treat the Schrödinger equation as much as possible in parallel with this standard treatment of the heat equation. This article reports from the thesis a summary of the rigorous framework of the problem, the main theorem, and the most elementary applications of the theorem.

We stress that the situation for time-evolution equations is different (in this respect, nicer) than for the Laplace and Poisson equations, which are the problems studied in most detail in most graduate textbooks on partial differential equations, such as [4]. In that *harmonic potential theory* the problem is similarly reduced to an integral equation, but the integral equation is not of Volterra type and therefore the Neumann series does not converge

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 D. Mera and A. Fulling; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 D. Mera and A. Fulling; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Mera and Fulling, licensee InTech. This is an open access chapter distributed under the terms of the

automatically. The terms are bounded by a geometric series but not an exponential one, so to prove convergence it is not enough to show that the Banach-space operator has finite norm; the norm would need to be less than unity, whereas in the PDE application it turns out to be exactly unity. Therefore, in the theory of elliptic PDEs the Neumann series is not used to prove existence of a solution; instead, the Fredholm theory is used to prove existence more abstractly. In time-evolution problems the concrete convergence of the series gives rigorous meaning to formal constructions used by physicists, such as path integrals and perturbation series.

10.5772/52489

249

http://dx.doi.org/10.5772/52489

bounded potential function. In that case the unitarity of the free Schrödinger evolution operator between fixed times is the key to proving boundedness of the integral operator, and the resulting Neumann series is a standard form of time-dependent perturbation theory.

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

The wavefunction Ψ(*x*, *t*) of a nonrelativistic particle in **R***<sup>n</sup>* is a solution to the Schrödinger

<sup>2</sup>*<sup>m</sup> <sup>p</sup>*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*(*x*, *<sup>t</sup>*) ≡ − *<sup>h</sup>*¯ <sup>2</sup>

*<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>

Unlike the corresponding operator for the heat equation, *L* is formally self-adjoint with

again in all of space-time. If the source term *F*(*x*, *t*) is prescribed, (6) is a nonhomogeneous version of the free Schrödinger equation. In order to get an integral equation for the homogeneous problem with a potential *V*(*x*, *t*), however, we will later take *F*(*x*, *t*) to be

For the differential operator appearing in (3) we introduce the notation

*H*Ψ(*x*, *t*) = *ih*¯ *∂t*Ψ(*x*, *t*), (1)

∆*<sup>x</sup>* + *V*(*x*, *t*). (2)

<sup>2</sup>*<sup>m</sup>* . (4)

*L* = *a*2∆*<sup>x</sup>* + *ih*¯ *∂<sup>t</sup>* . (5)

2*m*

*ih*¯ *∂t*Ψ(*x*, *t*) = −*a*2∆*x*Ψ(*x*, *t*), ∀(*x*, *t*) ∈ **R***<sup>n</sup>* × **R**, (3)

*Lu*(*x*, *t*) ≡ *a*2∆*xu*(*x*, *t*) + *ih*¯ *∂tu*(*x*, *t*) = *F*(*x*, *t*), (6)

*u*(*x*, 0) = *f*(*x*), ∀(*x*, *t*)=(*x*, 0) ∈ **R***<sup>n</sup>* × {*t* = 0} (7)

**2. The Poisson integral and source integral theorems**

*<sup>H</sup>* <sup>=</sup> *<sup>H</sup>*<sup>0</sup> <sup>+</sup> *<sup>V</sup>* <sup>≡</sup> <sup>1</sup>

In the "free"case, *V*(*x*, *t*) = 0, the equation becomes

equation,

where

where *H* is the Hamiltonian, given by

respect to the usual *L*<sup>2</sup> inner product.

We now consider the more general equation

*V*(*x*, *t*)*u*(*x*, *t*). In any case, one imposes the initial condition

and usually concentrates attention tacitly on *t* > 0.

The similarities between the Schrödinger equation and the heat equation were used in [3] to create a theoretical framework for representing and studying the solutions to the Schrödinger problem, which is summarized here. As much as possible, we use the books [1, 2] as guides to treat the quantum problem like a heat problem. However, the parallel between the heat equation and the Schrödinger equation is a limited one, because the exponential decay of the heat equation's fundamental solution is not available here. Therefore, different formulations and proofs needed to be constructed for the basic representation theorems in section 2, as well as for the main theorem in section 4 . For example, the Poisson integral formula (14) with the Schrödinger kernel (11) is shown to hold in the "Abel summable" sense [5, Sec. 1.5][6, Sec. 6.2].

Section 2 is devoted to the basic integral representation of a solution of the Schrödinger equation in terms of prescribed data and the fundamental solution (11). Here, unlike [3], we do not consider boundary-value problems, so the representation consists of two terms, a *Poisson integral* incorporating the initial data and a *source integral*. (In a boundary-value problem there is a third term incorporating boundary data.) For the free Schrödinger equation (6) with a known nonhomogeneous term *F*(*x*, *t*), the source integral (10) simply gives the contribution of *F* to the solution. In the more interesting case of a homogeneous equation including a potential, *F* involves the unknown function (multiplied by the potential), so the representation theorem yields an integral equation that must be solved. The crucial feature of the integral operator in (10) is that the upper limit of the time integration is *t*, the time variable of the solution, rather than +∞ or some constant. This is the Volterra structure that causes the iterative solution of the equation to converge exponentially. Thus the initial-value problem for the Schrödinger PDE has been expressed as a Volterra integral equation of the second kind with respect to time. Our main task is to use the Picard–Neumann method of successive approximation to construct the unique solution of this integral equation. The abstract theory of such iterative solutions for linear operators in arbitrary Banach spaces is outlined in section 3 .

The main theorem is proved in section 4 . It treats a Volterra integral equation for a function of *t* taking values at each *t* in some Banach space, B, such as *L*2(**R**3). More precisely, one has bounded operators *A*(*t*, *τ*) : B→B, with the bound independent of the time variables, that satisfy the Volterra property that *A*(*t*, *τ*) = 0 unless *τ* < *t*. It can then be proved inductively that the *j*th term of the Neumann series has norm proportional to *t<sup>j</sup>* /*j*!. The conclusion is that the series converges in the topology of *L*∞((0, *T*); B) for *t* < *T*. A variant with *L*<sup>∞</sup> replaced by *L<sup>p</sup>* is also given.

In section 5 the main theorem is applied to some simple and familiar cases. First, we consider classical integral equations, such as one with a kernel that is Hilbert–Schmidt in space and Volterra in time. Then we return to the Schrödinger problem set up in section 2, with a bounded potential function. In that case the unitarity of the free Schrödinger evolution operator between fixed times is the key to proving boundedness of the integral operator, and the resulting Neumann series is a standard form of time-dependent perturbation theory.

### **2. The Poisson integral and source integral theorems**

The wavefunction Ψ(*x*, *t*) of a nonrelativistic particle in **R***<sup>n</sup>* is a solution to the Schrödinger equation,

$$H\Psi(\mathbf{x},t) = i\hbar\partial\_t\Psi(\mathbf{x},t),\tag{1}$$

where *H* is the Hamiltonian, given by

$$H = H\_0 + V \equiv \frac{1}{2m}p^2 + V(\mathbf{x}, t) \equiv -\frac{\hbar^2}{2m}\Delta\_\mathbf{x} + V(\mathbf{x}, t). \tag{2}$$

In the "free"case, *V*(*x*, *t*) = 0, the equation becomes

$$i\hbar \partial\_t \Psi(\mathbf{x}, t) = -a^2 \Delta\_\mathbf{x} \Psi(\mathbf{x}, t), \quad \forall (\mathbf{x}, t) \in \mathbb{R}^n \times \mathbb{R},\tag{3}$$

where

2 Advances in Quantum Mechanics

series.

Sec. 6.2].

arbitrary Banach spaces is outlined in section 3 .

by *L<sup>p</sup>* is also given.

that the *j*th term of the Neumann series has norm proportional to *t<sup>j</sup>*

automatically. The terms are bounded by a geometric series but not an exponential one, so to prove convergence it is not enough to show that the Banach-space operator has finite norm; the norm would need to be less than unity, whereas in the PDE application it turns out to be exactly unity. Therefore, in the theory of elliptic PDEs the Neumann series is not used to prove existence of a solution; instead, the Fredholm theory is used to prove existence more abstractly. In time-evolution problems the concrete convergence of the series gives rigorous meaning to formal constructions used by physicists, such as path integrals and perturbation

The similarities between the Schrödinger equation and the heat equation were used in [3] to create a theoretical framework for representing and studying the solutions to the Schrödinger problem, which is summarized here. As much as possible, we use the books [1, 2] as guides to treat the quantum problem like a heat problem. However, the parallel between the heat equation and the Schrödinger equation is a limited one, because the exponential decay of the heat equation's fundamental solution is not available here. Therefore, different formulations and proofs needed to be constructed for the basic representation theorems in section 2, as well as for the main theorem in section 4 . For example, the Poisson integral formula (14) with the Schrödinger kernel (11) is shown to hold in the "Abel summable" sense [5, Sec. 1.5][6,

Section 2 is devoted to the basic integral representation of a solution of the Schrödinger equation in terms of prescribed data and the fundamental solution (11). Here, unlike [3], we do not consider boundary-value problems, so the representation consists of two terms, a *Poisson integral* incorporating the initial data and a *source integral*. (In a boundary-value problem there is a third term incorporating boundary data.) For the free Schrödinger equation (6) with a known nonhomogeneous term *F*(*x*, *t*), the source integral (10) simply gives the contribution of *F* to the solution. In the more interesting case of a homogeneous equation including a potential, *F* involves the unknown function (multiplied by the potential), so the representation theorem yields an integral equation that must be solved. The crucial feature of the integral operator in (10) is that the upper limit of the time integration is *t*, the time variable of the solution, rather than +∞ or some constant. This is the Volterra structure that causes the iterative solution of the equation to converge exponentially. Thus the initial-value problem for the Schrödinger PDE has been expressed as a Volterra integral equation of the second kind with respect to time. Our main task is to use the Picard–Neumann method of successive approximation to construct the unique solution of this integral equation. The abstract theory of such iterative solutions for linear operators in

The main theorem is proved in section 4 . It treats a Volterra integral equation for a function of *t* taking values at each *t* in some Banach space, B, such as *L*2(**R**3). More precisely, one has bounded operators *A*(*t*, *τ*) : B→B, with the bound independent of the time variables, that satisfy the Volterra property that *A*(*t*, *τ*) = 0 unless *τ* < *t*. It can then be proved inductively

the series converges in the topology of *L*∞((0, *T*); B) for *t* < *T*. A variant with *L*<sup>∞</sup> replaced

In section 5 the main theorem is applied to some simple and familiar cases. First, we consider classical integral equations, such as one with a kernel that is Hilbert–Schmidt in space and Volterra in time. Then we return to the Schrödinger problem set up in section 2, with a

/*j*!. The conclusion is that

$$a^2 = \frac{\hbar^2}{2m}.\tag{4}$$

For the differential operator appearing in (3) we introduce the notation

$$L = a^2 \Delta\_{\mathfrak{X}} + i\hbar \partial\_t \,. \tag{5}$$

Unlike the corresponding operator for the heat equation, *L* is formally self-adjoint with respect to the usual *L*<sup>2</sup> inner product.

We now consider the more general equation

$$L u(\mathbf{x}, t) \equiv a^2 \Delta\_\mathbf{x} u(\mathbf{x}, t) + i \hbar \partial\_t u(\mathbf{x}, t) = F(\mathbf{x}, t), \tag{6}$$

again in all of space-time. If the source term *F*(*x*, *t*) is prescribed, (6) is a nonhomogeneous version of the free Schrödinger equation. In order to get an integral equation for the homogeneous problem with a potential *V*(*x*, *t*), however, we will later take *F*(*x*, *t*) to be *V*(*x*, *t*)*u*(*x*, *t*). In any case, one imposes the initial condition

$$
\mu(\mathbf{x}, \mathbf{0}) = f(\mathbf{x}), \quad \forall (\mathbf{x}, t) = (\mathbf{x}, \mathbf{0}) \in \mathbb{R}^n \times \{t = \mathbf{0}\} \tag{7}
$$

and usually concentrates attention tacitly on *t* > 0.

The initial-value problem for (6) with the nonhomogeneous initial condition (7) can be reduced to the analogous problem with homogeneous initial condition by decomposing the solution *u* into two integral representations:

$$
\mu(\mathbf{x},t) = \Phi(\mathbf{x},t) + \Pi(\mathbf{x},t), \tag{8}
$$

10.5772/52489

251

<sup>2</sup>)*f*(*y*) ∈ *L*1(**R***n*)*. Then the Poisson*

http://dx.doi.org/10.5772/52489

*<sup>K</sup>*f(*<sup>x</sup>* − *<sup>y</sup>*, *<sup>t</sup>*)*f*(*y*) *dy* (14)

*f*(*x* + *γz*) *dz* (16)

*f*(*x* + *γz*) *dz* = *I*<sup>1</sup> + *I*<sup>2</sup> + *I*<sup>3</sup> , (17)

{ *f*(*x* + *γz*) − *f*(*x*)} *dz*, (18)

*f*(*x* + *γz*) *dz*, (19)

*f*(*x*) *dz*. (20)

**Theorem 1.** *Let f*(*x*) *be a function on* **R***<sup>n</sup> such that* (1 + |*y*|

*exists and is a solution of the equation*

hypothesis is obtained from [7, Chapter IV].

2¯*ht*/*m* ; then we can rewrite the Poisson integral as

*<sup>u</sup>*(*x*, *<sup>t</sup>*) = <sup>1</sup>

where |*z*| = |*x* − *y*|/*γ*. Let *ǫ* be any positive number. Then

(*πi*)*n*/2*u*(*x*, *<sup>t</sup>*) =

*I*<sup>1</sup> = |*z*|≤*ǫ e i*|*z*| 2

> *I*<sup>2</sup> = |*z*|≥*ǫ e i*|*z*| 2

> > *I*<sup>3</sup> = |*z*|≤*ǫ e i*|*z*| 2

To dispose of *<sup>I</sup>*<sup>1</sup> , let *<sup>x</sup>* be a point in **<sup>R</sup>***<sup>n</sup>* where *<sup>f</sup>* is continuous: ∀ *<sup>η</sup>* > <sup>0</sup> ∃*<sup>δ</sup>* > 0 such that ∀*y* ∈ **R***<sup>n</sup>* with |*y* − *x*| < *δ* one has | *f*(*y*) − *f*(*x*)| < *η*. Given *ǫ* (however large) and *η* (however small), choose *t* (hence *γ*) so small that *γǫ* < *δ*; then | *f*(*x* + *γz*) − *f*(*x*)| < *η* for all *z* such

*πi*

**R***n e i*|*z*| 2

*<sup>u</sup>*(*x*, *<sup>t</sup>*) = *<sup>K</sup>*<sup>f</sup> ∗ *<sup>f</sup>* =

 **R***n*

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

*and it satisfies the initial condition* (7) *in the sense of Abel summability. The Poisson integral defines a solution of the free Schrödinger equation in* **R***<sup>n</sup>* × {*t* �= 0} *(including negative t). This solution is extended into* **R***<sup>n</sup>* × **R** *by the initial condition u*(*x*, 0) = *f*(*x*) *at all points x at which f is continuous.*

be interchanged to verify that the Poisson integral solves the Schrödinger equation. This

The harder part is verifying the initial value. Assuming *t* > 0, let *y* = *x* + *γz*, where *γ*<sup>2</sup> =

**R***n e i*|*z*| 2

*<sup>n</sup>*/2

*Lu*(*x*, *t*) = *a*2∆*xu*(*x*, *t*) + *ih*¯ *∂tu*(*x*, *t*) = 0, ∀(*x*, *t*) ∈ **R***<sup>n</sup>* × **R**, (15)

<sup>2</sup> *f*(*y*) ∈ *L*1(**R***n*), then the order of differentiation and integration in (15), (14) can

*integral*

*Proof.* If |*y*|

where

that |*z*| ≤ *ǫ*. Therefore,

where Φ(*x*, *t*), called the source term, contains the effects of *F* and has null initial data, while Π(*x*, *t*), the Poisson integral term, solves the homogeneous equation (3) with the data (7). We shall show (Theorem 2) that

$$\Pi(\mathbf{x},t) = e^{-i t H\_0/\hbar} f(\mathbf{x}) = \int\_{\mathbb{R}^n} \mathcal{K}\_\mathbf{f}(\mathbf{x},y,t) f(y) \, dy \tag{9}$$

and

$$\Phi(\mathbf{x},t) = \int\_0^t e^{-it\mathbf{H}\_0/\hbar} e^{i\mathbf{\tau}\mathbf{H}\_0/\hbar} \mathbf{F}(\cdot,\mathbf{r}) \,d\mathbf{\tau} = -\frac{\mathrm{i}}{\hbar} \int\_0^t \int\_{\mathbb{R}^n} \mathbf{K}\_{\mathbf{f}}(\mathbf{x},y,t-\tau) \mathbf{F}(y,\tau) \,dy \,d\tau. \tag{10}$$

Here *K*f(*x*, *y*, *t*) is the fundamental solution (free propagator) to the Schrödinger equation (3) in **R***n*, which is given by

$$\mathbf{K}\_{\mathbf{f}}(\mathbf{x}, y, t) \equiv \mathbf{K}\_{\mathbf{f}}(\mathbf{x} - y, t) = \left(\frac{m}{2\pi\hbar it}\right)^{n/2} e^{im|\mathbf{x} - y|^2 / 2\hbar t}, \quad \forall \mathbf{x}, y \in \mathbb{R}^n, t \neq 0. \tag{11}$$

The formula (9) is equivalent to the statement that *K*f(*x*, *y*, *t*) as a function of (*x*, *t*) satisfies the homogeneous free Schrödinger equation and the initial condition

$$K\_{\mathbf{f}}(\mathbf{x}, y, \mathbf{0}) = \lim\_{t \downarrow \tau} K\_{\mathbf{f}}(\mathbf{x}, y, t - \tau) = \delta(\mathbf{x} - y). \tag{12}$$

Thus *<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*) vanishes as a distribution as *<sup>t</sup>* → 0 in the region *<sup>x</sup>* �= *<sup>y</sup>*, even though as a function it does not approach pointwise limits there. The formula (10) is equivalent to the alternative characterization that *K*<sup>f</sup> is the causal solution of the nonhomogeneous equation

$$LK\_{\mathbf{f}}(\mathbf{x}, y, t - \tau) = \delta(\mathbf{x} - y)\delta(t - \tau), \tag{13}$$

where *L* acts on the (*x*, *t*) variables.

The following theorem introduces the Poisson integral, which gives the solution of the initial-value problem for the free Schrödinger equation. Our discussion of the Poisson integral is somewhat more detailed than that of Evans [7], especially concerning the role of Abel summability.

**Theorem 1.** *Let f*(*x*) *be a function on* **R***<sup>n</sup> such that* (1 + |*y*| <sup>2</sup>)*f*(*y*) ∈ *L*1(**R***n*)*. Then the Poisson integral*

$$\mu(\mathbf{x},t) = \mathbf{K}\_{\mathbf{f}} \* f = \int\_{\mathbb{R}^n} \mathbf{K}\_{\mathbf{f}}(\mathbf{x} - y, t) f(y) \, dy \tag{14}$$

*exists and is a solution of the equation*

$$L\mu(\mathbf{x},t) = a^2 \Delta\_{\mathbf{x}} \mu(\mathbf{x},t) + i\hbar \partial\_t \mu(\mathbf{x},t) = 0, \quad \forall (\mathbf{x},t) \in \mathbb{R}^n \times \mathbb{R}, \tag{15}$$

*and it satisfies the initial condition* (7) *in the sense of Abel summability. The Poisson integral defines a solution of the free Schrödinger equation in* **R***<sup>n</sup>* × {*t* �= 0} *(including negative t). This solution is extended into* **R***<sup>n</sup>* × **R** *by the initial condition u*(*x*, 0) = *f*(*x*) *at all points x at which f is continuous.*

*Proof.* If |*y*| <sup>2</sup> *f*(*y*) ∈ *L*1(**R***n*), then the order of differentiation and integration in (15), (14) can be interchanged to verify that the Poisson integral solves the Schrödinger equation. This hypothesis is obtained from [7, Chapter IV].

The harder part is verifying the initial value. Assuming *t* > 0, let *y* = *x* + *γz*, where *γ*<sup>2</sup> = 2¯*ht*/*m* ; then we can rewrite the Poisson integral as

$$u(\mathbf{x},t) = \left(\frac{1}{\pi\mathrm{i}}\right)^{n/2} \int\_{\mathbb{R}^n} e^{i|z|^2} f(\mathbf{x} + \gamma z) \, dz \tag{16}$$

where |*z*| = |*x* − *y*|/*γ*. Let *ǫ* be any positive number. Then

$$(\pi \mathbf{i})^{n/2} u(\mathbf{x}, t) = \int\_{\mathbb{R}^n} e^{i|z|^2} f(\mathbf{x} + \gamma z) \, dz = I\_1 + I\_2 + I\_3 \,. \tag{17}$$

where

4 Advances in Quantum Mechanics

shall show (Theorem 2) that

Φ(*x*, *t*) =

in **R***n*, which is given by

 *<sup>t</sup>* 0 *e* <sup>−</sup>*itH*0/¯*he*

where *L* acts on the (*x*, *t*) variables.

of Abel summability.

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*) ≡ *<sup>K</sup>*f(*<sup>x</sup>* − *<sup>y</sup>*, *<sup>t</sup>*) =

and

solution *u* into two integral representations:

Π(*x*, *t*) = *e*

The initial-value problem for (6) with the nonhomogeneous initial condition (7) can be reduced to the analogous problem with homogeneous initial condition by decomposing the

where Φ(*x*, *t*), called the source term, contains the effects of *F* and has null initial data, while Π(*x*, *t*), the Poisson integral term, solves the homogeneous equation (3) with the data (7). We

> **R***n*

> > *h*¯ *<sup>t</sup>* 0 **R***n*

Here *K*f(*x*, *y*, *t*) is the fundamental solution (free propagator) to the Schrödinger equation (3)

*<sup>n</sup>*/2 *e im*|*x*−*y*|

The formula (9) is equivalent to the statement that *K*f(*x*, *y*, *t*) as a function of (*x*, *t*) satisfies

Thus *<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*) vanishes as a distribution as *<sup>t</sup>* → 0 in the region *<sup>x</sup>* �= *<sup>y</sup>*, even though as a function it does not approach pointwise limits there. The formula (10) is equivalent to the alternative characterization that *K*<sup>f</sup> is the causal solution of the nonhomogeneous equation

The following theorem introduces the Poisson integral, which gives the solution of the initial-value problem for the free Schrödinger equation. Our discussion of the Poisson integral is somewhat more detailed than that of Evans [7], especially concerning the role

<sup>−</sup>*itH*0/¯*<sup>h</sup> <sup>f</sup>*(*x*) =

*<sup>i</sup>τH*0/¯*hF*(·, *<sup>τ</sup>*) *<sup>d</sup><sup>τ</sup>* <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

 *m* 2*πhit* ¯

the homogeneous free Schrödinger equation and the initial condition

*K*f(*x*, *y*, 0) = lim

*u*(*x*, *t*) = Φ(*x*, *t*) + Π(*x*, *t*), (8)

*K*f(*x*, *y*, *t*)*f*(*y*) *dy* (9)

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*) *dy dτ*. (10)

2/2¯*ht*, ∀*x*, *y* ∈ **R***n*, *t* �= 0. (11)

*<sup>t</sup>*↓*<sup>τ</sup> <sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*) = *<sup>δ</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*). (12)

*LK*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*) = *<sup>δ</sup>*(*<sup>x</sup>* − *<sup>y</sup>*)*δ*(*<sup>t</sup>* − *<sup>τ</sup>*), (13)

$$I\_1 = \int\_{|z| \le \varepsilon} e^{i|z|^2} \left\{ f(\mathbf{x} + \gamma z) - f(\mathbf{x}) \right\} dz,\tag{18}$$

$$I\_2 = \int\_{|z| \ge \varepsilon} e^{i|z|^2} f(x + \gamma z) \, dz,\tag{19}$$

$$I\_3 = \int\_{|z| \le \mathfrak{e}} \mathfrak{e}^{i|z|^2} f(\mathfrak{x}) \, dz. \tag{20}$$

To dispose of *<sup>I</sup>*<sup>1</sup> , let *<sup>x</sup>* be a point in **<sup>R</sup>***<sup>n</sup>* where *<sup>f</sup>* is continuous: ∀ *<sup>η</sup>* > <sup>0</sup> ∃*<sup>δ</sup>* > 0 such that ∀*y* ∈ **R***<sup>n</sup>* with |*y* − *x*| < *δ* one has | *f*(*y*) − *f*(*x*)| < *η*. Given *ǫ* (however large) and *η* (however small), choose *t* (hence *γ*) so small that *γǫ* < *δ*; then | *f*(*x* + *γz*) − *f*(*x*)| < *η* for all *z* such that |*z*| ≤ *ǫ*. Therefore,

$$|I\_1| \le \eta \int\_{|z| \le \varepsilon} dz,\tag{21}$$

253

*it dt*. (27)

http://dx.doi.org/10.5772/52489

<sup>−</sup>*<sup>z</sup> dz*.

= (*πi*)*<sup>n</sup>*/2. (28)

which technically is not convergent. Therefore. we insert the Abel factor *<sup>e</sup>*−*α<sup>t</sup>* (*<sup>α</sup>* <sup>&</sup>gt; 0) into

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

 <sup>∞</sup> 0 *e* −*αt t me*

*it dt* <sup>=</sup> lim*r*→<sup>∞</sup> *<sup>i</sup>*

The path of integration can be moved back to the positive real axis, because the integral over

 *ir* 0 *e*

<sup>−</sup>*iαzz*(*n*/2)−1*<sup>e</sup>*

<sup>−</sup>*iαz*(*iz*)*me*

<sup>−</sup>*<sup>z</sup> dz*,

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>H</sup>*0*u*(*t*) + *<sup>F</sup>*(*t*). (29)

*<sup>A</sup>*(*α*) <sup>≡</sup> *<sup>ω</sup><sup>n</sup>*

This integral is convergent, and it can be transformed as

<sup>=</sup> lim*r*→<sup>∞</sup>

 *<sup>r</sup>* 0 *e* −*αt t me*

2*A*(*α*) *ωn*

= *i*

*A*(0) = <sup>1</sup>

*ih*¯ *∂u*(*t*)

space B, equipped with the norm (cf. Definition 5)

*<sup>n</sup>*/2 <sup>∞</sup> 0 *e*

<sup>2</sup>*ωni n*/2Γ *n* 2 

This analysis confirms (23) in an alternative way and gives it a rigorous meaning.

considers the nonhomogeneous Schrödinger equation (6) in the more abstract form

This completes the proof that the Poisson integral has the initial value *u*(*x*, 0) = *f*(*x*) at all

Theorem 2 establishes formula (10) rigorously. Our proof is partly based on [9], which

Here and later, *I* will denote the time interval (0, *T*), where *T* is a positive constant. In Theorem 2 we deal with the space *L*∞(*I*; B) of functions *u*(*t*) taking values in the Banach

**Theorem 2.** *Let f*(*x*) *belong to some Banach space* B *of functions on* **R***<sup>n</sup> that includes those for*

�*u*�*L*<sup>∞</sup>(*I*;B) <sup>=</sup> inf{*<sup>M</sup>* <sup>≥</sup> 0 : �*u*(*t*)�B <sup>≤</sup> *<sup>M</sup>* for almost all *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*]}. (30)

<sup>2</sup>)*f*(*y*) ∈ *L*1(**R***n*)*. Furthermore, suppose that the source term F*(*x*, *t*) *is continuous in*

�*F*(·, *<sup>t</sup>*)�*L*<sup>1</sup>(**R***<sup>n</sup>*) <sup>≤</sup> *<sup>ξ</sup>*(*t*), �*ξ*�*L*<sup>∞</sup>(*I*) <sup>≤</sup> *<sup>M</sup>* (31)

2*A*(*α*) *ωn*

the arc of radius *r* tends to 0. Thus

points *x* where *f* is continuous.

2

(26) to get

and in the limit

*which* (1 + |*y*|

*t and satisfies the condition*

which can be made arbitrarily small in the limit *t* → 0.

On the other hand, since *f* ∈ *L*1(**R***n*),

$$|I\_2| \le \int\_{|z| \ge \varepsilon} |f(\mathfrak{x} + \gamma z)| \, dz \to 0 \tag{22}$$

(not necessarily uniformly in *x*) as *ǫ* → ∞. Thus the initial value *u*(*x*, 0+) comes entirely from *I*3 .

To evaluate *I*<sup>3</sup> we use the Fresnel integral formula

$$\int\_{\mathbb{R}^n} e^{i|z|^2} dz = (\pi i)^{n/2}. \tag{23}$$

A proof of (23) with *n* = 1, which converges classically, appears in [8, pp. 82–83]. The one-dimensional formula appears to imply the product version by

$$\int\_{\mathbb{R}^n} e^{i|z|^2} \, dz = \int\_{\mathbb{R}^n} \exp\left(i \sum\_{k=1}^n z\_k^2\right) dz = \prod\_{k=1}^n \int\_{-\infty}^{\infty} e^{i z\_k^2} \, dz\_k = \prod\_{k=1}^n (\pi i)^{1/2} = (\pi i)^{n/2}.$$

Therefore, we have

$$\lim\_{\varepsilon \to \infty} I\_3 = (\pi i)^{n/2} f(\mathbf{x}) \,\tag{24}$$

which is what we want to prove.

However, the integral on the left side of (23) is rather questionable when *n* > 1, so we reconsider it in polar coordinates:

$$\int\_{\mathbb{R}^n} e^{i|z|^2} \, dz = \int\_0^\infty \int\_{S^{n-1}} e^{i\rho^2} \rho^{n-1} \, d\rho \, d\Omega \equiv \omega\_n \int\_0^\infty \rho^{n-1} e^{i\rho^2} \, d\rho \dots$$

The surface area of the unit *n*-sphere is

$$
\omega\_n = 2\pi^{n/2} / \Gamma\left(\frac{n}{2}\right) \,. \tag{25}
$$

With the substitutions *t* = *ρ*2, *m* = (*n* − 2)/2, we obtain

$$\int\_{\mathbb{R}^n} e^{i|z|^2} \, dz = \frac{\omega\_n}{2} \int\_0^\infty t^m e^{it} \, dt,\tag{26}$$

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> which technically is not convergent. Therefore. we insert the Abel factor *<sup>e</sup>*−*α<sup>t</sup>* (*<sup>α</sup>* <sup>&</sup>gt; 0) into (26) to get

$$A(a) \equiv \frac{\omega\_n}{2} \int\_0^\infty e^{-at} t^m e^{it} \, dt. \tag{27}$$

This integral is convergent, and it can be transformed as

$$\frac{2A(\alpha)}{\omega\_{\text{fl}}} = \lim\_{r \to \infty} \int\_0^r e^{-at} t^m e^{it} \, dt = \lim\_{r \to \infty} i \int\_0^{ir} e^{-iaz} (iz)^m e^{-z} \, dz.$$

The path of integration can be moved back to the positive real axis, because the integral over the arc of radius *r* tends to 0. Thus

$$\frac{2A(\alpha)}{\omega\_{\text{fl}}} = i^{n/2} \int\_0^\infty e^{-i\alpha z} z^{(n/2)-1} e^{-z} \, dz,$$

and in the limit

6 Advances in Quantum Mechanics

*I*3 .

 **R***<sup>n</sup> e i*|*z*| 2 *dz* = **R***<sup>n</sup>* exp *i n* ∑ *k*=1 *z*2 *k dz* =

Therefore, we have

which is what we want to prove.

reconsider it in polar coordinates:

 **R***<sup>n</sup> e i*|*z*| 2 *dz* =

The surface area of the unit *n*-sphere is


which can be made arbitrarily small in the limit *t* → 0.

To evaluate *I*<sup>3</sup> we use the Fresnel integral formula


> **R***<sup>n</sup> e i*|*z*| 2

one-dimensional formula appears to imply the product version by

 <sup>∞</sup> 0 *Sn*−<sup>1</sup> *e iρ*2

With the substitutions *t* = *ρ*2, *m* = (*n* − 2)/2, we obtain

 **R***<sup>n</sup> e i*|*z*| 2

On the other hand, since *f* ∈ *L*1(**R***n*),

 |*z*|≤*ǫ*

(not necessarily uniformly in *x*) as *ǫ* → ∞. Thus the initial value *u*(*x*, 0+) comes entirely from

A proof of (23) with *n* = 1, which converges classically, appears in [8, pp. 82–83]. The

*n* ∏ *k*=1

However, the integral on the left side of (23) is rather questionable when *n* > 1, so we

*<sup>ω</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>*π<sup>n</sup>*/2/<sup>Γ</sup>

*dz* <sup>=</sup> *<sup>ω</sup><sup>n</sup>* 2

*<sup>ρ</sup>n*−<sup>1</sup> *<sup>d</sup><sup>ρ</sup> <sup>d</sup>*<sup>Ω</sup> <sup>≡</sup> *<sup>ω</sup><sup>n</sup>*

 *n* 2 

 <sup>∞</sup> 0 *t me*

 <sup>∞</sup> −∞ *e iz*2 *<sup>k</sup> dzk* =

*dz*, (21)


*dz* = (*πi*)*<sup>n</sup>*/2. (23)

(*πi*)1/2 = (*πi*)*<sup>n</sup>*/2.

*n* ∏ *k*=1

lim*ǫ*→<sup>∞</sup> *<sup>I</sup>*<sup>3</sup> = (*πi*)*n*/2 *<sup>f</sup>*(*x*), (24)

 <sup>∞</sup> 0

*<sup>ρ</sup>n*−1*<sup>e</sup> iρ*2 *dρ*.

. (25)

*it dt*, (26)

$$A(0) = \frac{1}{2} \omega\_n \mathbf{i}^{n/2} \Gamma\left(\frac{n}{2}\right) = (\pi i)^{n/2}. \tag{28}$$

This analysis confirms (23) in an alternative way and gives it a rigorous meaning.

This completes the proof that the Poisson integral has the initial value *u*(*x*, 0) = *f*(*x*) at all points *x* where *f* is continuous.

Theorem 2 establishes formula (10) rigorously. Our proof is partly based on [9], which considers the nonhomogeneous Schrödinger equation (6) in the more abstract form

$$i\hbar\frac{\partial u(t)}{\partial t} = H\_0 u(t) + F(t). \tag{29}$$

Here and later, *I* will denote the time interval (0, *T*), where *T* is a positive constant. In Theorem 2 we deal with the space *L*∞(*I*; B) of functions *u*(*t*) taking values in the Banach space B, equipped with the norm (cf. Definition 5)

$$\|u\|\_{L^{\infty}(I;\mathcal{B})} = \inf\{M \ge 0 : \|u(t)\|\_{\mathcal{B}} \le M \quad \text{for almost all } t \in [0, T]\}.\tag{30}$$

**Theorem 2.** *Let f*(*x*) *belong to some Banach space* B *of functions on* **R***<sup>n</sup> that includes those for which* (1 + |*y*| <sup>2</sup>)*f*(*y*) ∈ *L*1(**R***n*)*. Furthermore, suppose that the source term F*(*x*, *t*) *is continuous in t and satisfies the condition*

$$\|\|F(\cdot, t)\|\|\_{L^1(\mathbb{R}^n)} \le \tilde{\xi}(t), \qquad \|\tilde{\xi}\|\|\_{L^\infty(I)} \le M \tag{31}$$

*for some positive constant M. The solution of the initial-value problem for the nonhomogeneous Schrödinger equation* (6) *can be represented in the form u* = Π + Φ *of* (8)*, where the initial term is*

$$\Pi(\mathbf{x},t) = \int\_{\mathbb{R}^n} \mathbf{K}\_{\mathbf{f}}(\mathbf{x},y,t) f(y) \, dy \tag{32}$$

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255

(38)

(40)

*<sup>i</sup>τH*0/¯*hF*(*τ*) *dτ*. (37)

http://dx.doi.org/10.5772/52489

*∂u*(*τ*) *∂τ dτ*

*<sup>i</sup>τH*0/¯*hLu*(*τ*) *dτ*, (39)

*∂u ∂τ dτ*

−*a*∆*u*(*τ*) + *ih*¯

−*H*0*u*(*τ*) + *ih*¯

 *dτ*

*u*(*t*) = *e*

Consider the integral

<sup>−</sup>*ih*¯ <sup>−</sup><sup>1</sup>

This calculation implies that

*L*2(**R***n*)*, not pointwise.*

<sup>Ψ</sup>(*x*, *<sup>t</sup>*) =

**R***n*

have

*u*(*t*) = *e*

�Φ(*t*)� =

 *ih*¯ <sup>−</sup>1*<sup>e</sup>*

Π(*x*, *t*), we have again established that (8) is the desired solution.

*is equivalent to a nonhomogeneous Volterra integral equation of the second kind,*

*h*¯ *<sup>t</sup>* 0 **R***n*

≤ 1 *h <sup>t</sup>* 0

**Corollary 4.** *The homogeneous Schrödinger initial-value problem,*

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*)*f*(*y*) *dy* <sup>−</sup> *<sup>i</sup>*

*Proof.* (42) is (34) with the source *F* in (6) identified with *V*Ψ.

 *<sup>t</sup>* 0 *e* <sup>−</sup>*itH*0/¯*<sup>h</sup> <sup>f</sup>*(*x*) <sup>−</sup> *<sup>i</sup>*

<sup>=</sup> <sup>−</sup>*ih*¯ <sup>−</sup><sup>1</sup>

= *<sup>t</sup>* 0 *∂ ∂τ e*

= *e*

<sup>−</sup>*itH*0/¯*hu*(0) <sup>−</sup> *ih*¯ <sup>−</sup>1*<sup>e</sup>*

*<sup>i</sup>τH*0/¯*hLu*(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>=</sup> <sup>−</sup>*ih*¯ <sup>−</sup><sup>1</sup>

*h*¯ *<sup>t</sup>* 0 *e* <sup>−</sup>*itH*0/¯*he*

 *<sup>t</sup>* 0 *e iτH*+0/¯*h* 

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

 *<sup>t</sup>* 0 *e iτH*0/¯*h* 

*itH*0/¯*hu*(*t*) − *u*(0).

−*itH*0/¯*h*

which is equivalent to (37) and to the Volterra integral formula (34). The expression *u*(*t*) − *<sup>e</sup>*−*itH*0/¯*hu*(0) is simply the source term <sup>Φ</sup>(*x*, *<sup>t</sup>*). Taking its Banach space norm and using the unitarity of the evolution operator *<sup>e</sup>*−*itH*0/¯*<sup>h</sup>* and the fundamental theorem of calculus, we

> *<sup>t</sup>* 0 *e*

> > 1 *h*¯ *<sup>t</sup>* 0

*ih*¯ *∂t*Ψ(*x*, *t*) = −*a*2∆Ψ(*x*.*t*) + *V*(*x*, *t*)Ψ(*x*, *t*), Ψ(*x*, 0) = *f*(*x*), (41)

−*itH*0/¯*h*

�*Lu*� *dτ* ≤

because of (31). Therefore, <sup>Φ</sup> → 0 when *<sup>t</sup>* → 0. Since *<sup>e</sup>*−*itH*0/¯*hu*(0) is another way of writing

**Remark 3.** *The L*<sup>1</sup> *condition of Theorem 1 has not been used in the second, more abstract proof of Theorem 2, because the limits (t* ↓ 0*) are being taken in the topology of the quantum Hilbert space*

 *<sup>t</sup>* 0 *e*

*<sup>i</sup>τH*0/¯*hLu*(*τ*) *dτ*

*<sup>ξ</sup>*(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>≤</sup> *Mt*

 

*h*¯ ,

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*V*(*y*, *<sup>τ</sup>*)Ψ(*y*, *<sup>τ</sup>*) *dy dτ*. (42)

*<sup>i</sup>τH*0/¯*hu*(*τ*)

*and the source term is*

$$\Phi(\mathbf{x},t) = -\frac{i}{\hbar} \int\_0^t \int\_{\mathbb{R}^n} \mathbf{K}\_{\mathbf{f}}(\mathbf{x}, y, t-\tau) \mathbf{F}(y, \tau) \, dy \, d\tau. \tag{33}$$

*Here K*f(*x*, *y*, *t*) *is the fundamental solution* (11) *and u*(*x*, 0) = *f*(*x*)*. The solution u belongs to the Banach space L*∞(*I*; B)*.*

*Proof.* Theorem 1 shows that the Poisson integral Π solves the initial-value problem for the homogeneous Schrödinger equation. We claim that the solution of the full problem has the Volterra integral representation

$$\mu(\mathbf{x},t) = \int\_{\mathbb{R}^n} \mathcal{K}\_{\mathbf{f}}(\mathbf{x},y,t) f(y) \, dy - \frac{i}{\hbar} \int\_0^t \int\_{\mathbb{R}^n} \mathcal{K}\_{\mathbf{f}}(\mathbf{x},y,t-\tau) \mathcal{F}(y,\tau) \, dy \, d\tau$$

$$\equiv \Pi(\mathbf{x},t) + \Phi(\mathbf{x},t). \tag{34}$$

By applying the Schrödinger operator (5) to *u*(*t*), we have

$$\begin{split} Lu &= L\Pi + L\Phi = a^2 \Delta\_\mathbf{X} \Phi + i\hbar \frac{\partial \Phi}{\partial t} \\ &= a^2 \left( -\frac{i}{\hbar} \right) \Delta\_\mathbf{x} \int\_{\mathbb{R}^n} K\_\mathbf{f}(\mathbf{x}, y, t - \tau) F(y, \tau) \, dy \, d\tau \\ &\quad + i\hbar \frac{\partial}{\partial t} \left( -\frac{i}{\hbar} \int\_0^t \int\_{\mathbb{R}^n} K\_\mathbf{f}(\mathbf{x}, y, t - \tau) F(y, \tau) \, dy \, d\tau \right) \\ &= \int\_0^t \int\_{\mathbb{R}^n} \mathcal{L}K\_\mathbf{f}(\mathbf{x}, y, t - \tau) F(y, \tau) \, dy \, d\tau \\ &\quad + \lim\_{t \downarrow \tau} \int\_{\mathbb{R}^n} K\_\mathbf{f}(\mathbf{x}, y, t - \tau) F(y, \tau) \, dy. \end{split} \tag{35}$$

But *LK*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*) = 0 for all *<sup>t</sup>* > *<sup>τ</sup>*, and Theorem 1 shows that *<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*) → *<sup>δ</sup>*(*<sup>x</sup>* − *<sup>y</sup>*). Therefore, we have

$$L\Phi = F(\mathfrak{x}, t). \tag{36}$$

Furthermore, it is clear that *F*(*x*, 0) = 0. Therfore, by linearity the sum *u* = Π + Φ solves the problem.

Another way to express (34) is via unitary operators:

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$$u(t) = e^{-i t H\_0/\hbar} f(\mathbf{x}) - \frac{\mathbf{i}}{\hbar} \int\_0^t e^{-i t H\_0/\hbar} e^{i \tau H\_0/\hbar} F(\mathbf{r}) \, d\tau. \tag{37}$$

Consider the integral

8 Advances in Quantum Mechanics

*and the source term is*

*Banach space L*∞(*I*; B)*.*

Therefore, we have

problem.

Volterra integral representation

*u*(*x*, *t*) =

*for some positive constant M. The solution of the initial-value problem for the nonhomogeneous Schrödinger equation* (6) *can be represented in the form u* = Π + Φ *of* (8)*, where the initial term is*

*Here K*f(*x*, *y*, *t*) *is the fundamental solution* (11) *and u*(*x*, 0) = *f*(*x*)*. The solution u belongs to the*

*Proof.* Theorem 1 shows that the Poisson integral Π solves the initial-value problem for the homogeneous Schrödinger equation. We claim that the solution of the full problem has the

> *h*¯ *t* 0

*LK*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*) *dy d<sup>τ</sup>*

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*)*dy*.

But *LK*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*) = 0 for all *<sup>t</sup>* > *<sup>τ</sup>*, and Theorem 1 shows that *<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*) → *<sup>δ</sup>*(*<sup>x</sup>* − *<sup>y</sup>*).

Furthermore, it is clear that *F*(*x*, 0) = 0. Therfore, by linearity the sum *u* = Π + Φ solves the

*∂*Φ *∂t*

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*) *dy d<sup>τ</sup>*

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*) *dy d<sup>τ</sup>*

*K*f(*x*, *y*, *t*)*f*(*y*) *dy* (32)

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*) *dy dτ*. (33)

**<sup>R</sup>***<sup>n</sup> <sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*F*(*y*, *<sup>τ</sup>*) *dy d<sup>τ</sup>* ≡ Π(*x*, *t*) + Φ(*x*, *t*). (34)

*L*Φ = *F*(*x*, *t*). (36)

(35)

Π(*x*, *t*) =

*h*¯ *<sup>t</sup>* 0 **R***n*

**<sup>R</sup>***<sup>n</sup> <sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*)*f*(*y*) *dy* − *<sup>i</sup>*

*Lu* = *L*Π + *L*Φ = *a*2∆*x*Φ + *ih*¯

By applying the Schrödinger operator (5) to *u*(*t*), we have

= *a*<sup>2</sup> − *i h*¯ ∆*x* **R***n*

= *<sup>t</sup>* 0 **R***n*

<sup>+</sup> *ih*¯ *<sup>∂</sup> ∂t* − *i h*¯ *<sup>t</sup>* 0 **R***n*

+ lim *t*↓*τ* **R***n*

Another way to express (34) is via unitary operators:

<sup>Φ</sup>(*x*, *<sup>t</sup>*) = <sup>−</sup> *<sup>i</sup>*

 **R***n*

$$\begin{split} -i\hbar^{-1} \int\_{0}^{t} e^{i\tau H\_{0}/\hbar} L u(\tau) \,d\tau &= -i\hbar^{-1} \int\_{0}^{t} e^{i\tau H\_{+}/\hbar} \left( -a\Delta u(\tau) + i\hbar \frac{\partial u(\tau)}{\partial \tau} \right) d\tau \\ &= -i\hbar^{-1} \int\_{0}^{t} e^{i\tau H\_{0}/\hbar} \left( -H\_{0}u(\tau) + i\hbar \frac{\partial u}{\partial \tau} \right) d\tau \\ &= \int\_{0}^{t} \frac{\partial}{\partial \tau} \left( e^{i\tau H\_{0}/\hbar} u(\tau) \right) d\tau \\ &= e^{i t H\_{0}/\hbar} u(t) - u(0). \end{split} \tag{38}$$

This calculation implies that

$$u(t) = e^{-i t H\_0/\hbar} u(0) - i\hbar^{-1} e^{-i t H\_0/\hbar} \int\_0^t e^{i\tau H\_0/\hbar} L u(\tau) \,d\tau,\tag{39}$$

which is equivalent to (37) and to the Volterra integral formula (34). The expression *u*(*t*) − *<sup>e</sup>*−*itH*0/¯*hu*(0) is simply the source term <sup>Φ</sup>(*x*, *<sup>t</sup>*). Taking its Banach space norm and using the unitarity of the evolution operator *<sup>e</sup>*−*itH*0/¯*<sup>h</sup>* and the fundamental theorem of calculus, we have

$$\begin{split} \|\Phi(t)\| &= \left\| i\hbar^{-1} e^{-itH\_0/\hbar} \int\_0^t e^{i\tau H\_0/\hbar} L u(\tau) \, d\tau \right\| \\ &\le \frac{1}{\hbar} \int\_0^t \|Lu\| \, d\tau \le \frac{1}{\hbar} \int\_0^t \tilde{\xi}(\tau) \, d\tau \le \frac{Mt}{\hbar} \, \end{split} \tag{40}$$

because of (31). Therefore, <sup>Φ</sup> → 0 when *<sup>t</sup>* → 0. Since *<sup>e</sup>*−*itH*0/¯*hu*(0) is another way of writing Π(*x*, *t*), we have again established that (8) is the desired solution.

**Remark 3.** *The L*<sup>1</sup> *condition of Theorem 1 has not been used in the second, more abstract proof of Theorem 2, because the limits (t* ↓ 0*) are being taken in the topology of the quantum Hilbert space L*2(**R***n*)*, not pointwise.*

**Corollary 4.** *The homogeneous Schrödinger initial-value problem,*

$$i\hbar \partial\_l \Psi(\mathbf{x}, t) = -a^2 \Delta \Psi(\mathbf{x}, t) + V(\mathbf{x}, t) \Psi(\mathbf{x}, t), \qquad \Psi(\mathbf{x}, 0) = f(\mathbf{x}), \tag{41}$$

*is equivalent to a nonhomogeneous Volterra integral equation of the second kind,*

$$\Psi(\mathbf{x},t) = \int\_{\mathbb{R}^n} \mathcal{K}\_{\mathbf{f}}(\mathbf{x},y,t) f(y) \, dy - \frac{i}{\hbar} \int\_0^t \int\_{\mathbb{R}^n} \mathcal{K}\_{\mathbf{f}}(\mathbf{x},y,t-\tau) V(y,\tau) \Psi(y,\tau) \, dy \, d\tau. \tag{42}$$

*Proof.* (42) is (34) with the source *F* in (6) identified with *V*Ψ.

### **3. Integral equations and Neumann series**

In this section we introduce integral operators in arbitrary Banach spaces in order to set up a framework for constructing solutions to the Schrödinger equation. This section is a preliminary to the general Volterra theorems that are proved in section 4. It uses as a foundation Kress's treatment of linear integral equations [2].

In operator notation, an integral equation of the second kind has the structure

$$
\boldsymbol{\phi} - \boldsymbol{\hat{Q}} \boldsymbol{\phi} = \boldsymbol{f}\_{\prime} \tag{43}
$$

10.5772/52489

257

<sup>0</sup> ··· *<sup>d</sup><sup>τ</sup> if one has defined A*(*t*, *<sup>τ</sup>*) *to be* <sup>0</sup>

http://dx.doi.org/10.5772/52489

*Q*ˆ *<sup>j</sup> f* (48)

*<sup>j</sup>*=<sup>0</sup> *<sup>Q</sup>*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* is a convergent series with respect to the

**Remark 8.** *In* (47) *one may write the integration as <sup>T</sup>*

*whenever τ* > *t. In that case A is called a Volterra kernel.*

1. the function *<sup>φ</sup>*<sup>0</sup> ≡ *<sup>f</sup>* belongs to a Banach space B,

**4. Volterra kernels and successive approximations**

• B is a Banach space, and *I* = (0, *T*) is an interval, with closure ¯*I*.

• For all (*t*, *τ*) ∈ ¯*I*2, *A*(*t*, *τ*) is a linear operator from B to B.

• *A*(*t*, *τ*) satisfes the Volterra condition, *A*(*t*, *τ*) = 0 if *τ* > *t*.

Definition 6, with bound �*A*�*L*<sup>∞</sup>(*I*2;B→B) <sup>=</sup> *<sup>D</sup>*.

3. the infinite (Neumann) series *<sup>φ</sup>* <sup>=</sup> <sup>∑</sup><sup>∞</sup>

topology of *L*∞(*I*; B).

detail in the next two sections.

**Hypotheses**

Neumann series. The successive approximations (Picard's algorithm)

In Corollary 4 we have reformulated the Schrödinger equation as an integral equation of the second kind. The existence and uniqueness of its solution can be found by analysis of the

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

converge to the exact solution of the integral equation (43), if some technical conditions are

If these three conditions are satisfied, then the Neumann series provides the exact solution to the integral equation (43). In the Schrödinger case, therefore, it solves the original initial-value problem for the Schrödinger equation. This program will be implemented in

In this section we implement the method of successive approximations set forth in section 3. The Volterra operator has a nice property, known as the simplex structure, which makes its

It follows from the convergence of the Neumann series that the spectral radius of the Volterra integral operator of the second kind is zero. In Kress's treatment of the heat equation [2] the logic runs in the other direction — convergence follows from a theorem on spectral radius. For the Schrödinger equation we find it more convenient to prove convergence directly.

• the operator kernel *A*(*t*, *τ*) is measurable and uniformly bounded, in the sense of

infinite Neumann series converge. This claim is made precise in our main theorems.

*N* ∑ *j*=0

*φ<sup>N</sup>* = *Q*ˆ *φN*−<sup>1</sup> + *f* =

satisfied. In the terminology of an arbitrary Banach space, one must establish that

2. the integral operator *Q*ˆ is a bounded Volterra operator on *L*∞(*I*; B), and

where *Q*ˆ is a bounded linear operator from a Banach space W to itself, and *φ* and *f* are in W. A solution *<sup>φ</sup>* exists and is unique for each *<sup>f</sup>* if and only if the inverse operator (1<sup>−</sup> *<sup>Q</sup>*ˆ)−<sup>1</sup> exists (where 1 indicates the identity operator). For Volterra operators, the focus of our attention, the existence of the inverse operator will become clear below. Equivalently, the theorems of the next section will prove that the spectral radius of a Volterra operator is zero. For these purposes we need to work in Lebesgue spaces W = *Lp*(*I*; B) (including, especially, *p* = ∞) of functions of *t* to obtain useful estimates.

**Definition 5.** *Let* (Ω, Σ, *µ*) *be a measure space and* B *be a Banach space. The collection of all essentially bounded measurable functions on* <sup>Ω</sup> *taking values in* B *is denoted L*∞(Ω, *<sup>µ</sup>*; B)*, the reference to µ being omitted when there is no danger of confusion. The essential supremum of a function ϕ*: Ω → B *is given by*

$$\|\|\varphi\|\|\_{L^{\infty}(\Omega; \mathbb{B})} = \inf \{ M \ge 0 : \|\|\varphi(\mathbf{x})\|\|\_{\mathcal{B}} \le M \text{ for almost all } \mathbf{x} \}. \tag{44}$$

**Definition 6.** *Let* B<sup>1</sup> *and* B<sup>2</sup> *be Banach spaces and* <sup>Ω</sup> *be some measurable space. For each* (*x*, *<sup>y</sup>*) ∈ <sup>Ω</sup> × <sup>Ω</sup> ≡ <sup>Ω</sup><sup>2</sup> *let A*(*x*, *<sup>y</sup>*) : B<sup>1</sup> → B<sup>2</sup> *be a bounded linear operator, and suppose that the function A*(·, ·) *is measurable. At each* (*x*, *y*) *define its norm*

$$\|A(\mathbf{x}, y)\|\_{\mathcal{B}\_1 \to \mathcal{B}\_2} = \inf\{M \ge 0 : \|A(\mathbf{x}, y)\phi\| \le M \|\phi\|\}, \quad \forall \phi \in \mathcal{B}\_1\}.\tag{45}$$

*If* <sup>B</sup><sup>1</sup> <sup>=</sup> <sup>B</sup><sup>2</sup> <sup>=</sup> <sup>B</sup>*, then one abbreviates* �*A*(*x*, *<sup>y</sup>*)�B1→B<sup>2</sup> *as* �*A*(*x*, *<sup>y</sup>*)�B *or even* �*A*(*x*, *<sup>y</sup>*)�*. Now define the uniform norm*

$$\begin{split} \|A\|\_{L^{\infty}(\Omega^{2};\mathbb{B}\_{1}\to\mathbb{B}\_{2})} \equiv \inf\{M \ge 0 : \|A(\mathbf{x},\mathbf{y})\| \le M \text{ for almost all } (\mathbf{x},\mathbf{y}) \in \Omega^{2}\} \\ \equiv \operatorname{ess\,sup}\_{(\mathbf{x},\mathbf{y})\in\Omega^{2}} \|A(\mathbf{x},\mathbf{y})\|\_{\mathcal{B}\_{1}\to\mathcal{B}\_{2}} \end{split} \tag{46}$$

*and call A*(·, ·) *a uniformly bounded operator kernel if* �*A*�*L*<sup>∞</sup>(Ω2;B1→B2) *is finite.*

**Definition 7.** *In Definition 6 let* Ω = *I* = (0, *T*)*. If A is a uniformly bounded operator kernel, the operator Q defined by* ˆ

$$
\hat{Q}f(t) = \int\_0^t A(t,\tau)f(\tau) \,d\tau \tag{47}
$$

*is called a bounded Volterra operator on L*∞(*I*; B) *with kernel A.*

**Remark 8.** *In* (47) *one may write the integration as <sup>T</sup>* <sup>0</sup> ··· *<sup>d</sup><sup>τ</sup> if one has defined A*(*t*, *<sup>τ</sup>*) *to be* <sup>0</sup> *whenever τ* > *t. In that case A is called a Volterra kernel.*

In Corollary 4 we have reformulated the Schrödinger equation as an integral equation of the second kind. The existence and uniqueness of its solution can be found by analysis of the Neumann series. The successive approximations (Picard's algorithm)

$$\mathfrak{d}\_{N} = \hat{\mathbb{Q}}\mathfrak{d}\_{N-1} + f = \sum\_{j=0}^{N} \hat{\mathbb{Q}}^{j} f \tag{48}$$

converge to the exact solution of the integral equation (43), if some technical conditions are satisfied. In the terminology of an arbitrary Banach space, one must establish that


If these three conditions are satisfied, then the Neumann series provides the exact solution to the integral equation (43). In the Schrödinger case, therefore, it solves the original initial-value problem for the Schrödinger equation. This program will be implemented in detail in the next two sections.

#### **4. Volterra kernels and successive approximations**

In this section we implement the method of successive approximations set forth in section 3. The Volterra operator has a nice property, known as the simplex structure, which makes its infinite Neumann series converge. This claim is made precise in our main theorems.

It follows from the convergence of the Neumann series that the spectral radius of the Volterra integral operator of the second kind is zero. In Kress's treatment of the heat equation [2] the logic runs in the other direction — convergence follows from a theorem on spectral radius. For the Schrödinger equation we find it more convenient to prove convergence directly.

#### **Hypotheses**

10 Advances in Quantum Mechanics

**3. Integral equations and Neumann series**

of functions of *t* to obtain useful estimates.

*A*(·, ·) *is measurable. At each* (*x*, *y*) *define its norm*

*function ϕ*: Ω → B *is given by*

*define the uniform norm*

*operator Q defined by* ˆ

foundation Kress's treatment of linear integral equations [2].

In this section we introduce integral operators in arbitrary Banach spaces in order to set up a framework for constructing solutions to the Schrödinger equation. This section is a preliminary to the general Volterra theorems that are proved in section 4. It uses as a

where *Q*ˆ is a bounded linear operator from a Banach space W to itself, and *φ* and *f* are in W. A solution *<sup>φ</sup>* exists and is unique for each *<sup>f</sup>* if and only if the inverse operator (1<sup>−</sup> *<sup>Q</sup>*ˆ)−<sup>1</sup> exists (where 1 indicates the identity operator). For Volterra operators, the focus of our attention, the existence of the inverse operator will become clear below. Equivalently, the theorems of the next section will prove that the spectral radius of a Volterra operator is zero. For these purposes we need to work in Lebesgue spaces W = *Lp*(*I*; B) (including, especially, *p* = ∞)

**Definition 5.** *Let* (Ω, Σ, *µ*) *be a measure space and* B *be a Banach space. The collection of all essentially bounded measurable functions on* <sup>Ω</sup> *taking values in* B *is denoted L*∞(Ω, *<sup>µ</sup>*; B)*, the reference to µ being omitted when there is no danger of confusion. The essential supremum of a*

**Definition 6.** *Let* B<sup>1</sup> *and* B<sup>2</sup> *be Banach spaces and* <sup>Ω</sup> *be some measurable space. For each* (*x*, *<sup>y</sup>*) ∈ <sup>Ω</sup> × <sup>Ω</sup> ≡ <sup>Ω</sup><sup>2</sup> *let A*(*x*, *<sup>y</sup>*) : B<sup>1</sup> → B<sup>2</sup> *be a bounded linear operator, and suppose that the function*

*If* <sup>B</sup><sup>1</sup> <sup>=</sup> <sup>B</sup><sup>2</sup> <sup>=</sup> <sup>B</sup>*, then one abbreviates* �*A*(*x*, *<sup>y</sup>*)�B1→B<sup>2</sup> *as* �*A*(*x*, *<sup>y</sup>*)�B *or even* �*A*(*x*, *<sup>y</sup>*)�*. Now*

�*A*�*L*<sup>∞</sup>(Ω2;B1→B2) <sup>≡</sup> inf{*<sup>M</sup>* <sup>≥</sup> 0 : �*A*(*x*, *<sup>y</sup>*)� ≤ *M for almost all* (*x*, *<sup>y</sup>*) <sup>∈</sup> <sup>Ω</sup>2}

**Definition 7.** *In Definition 6 let* Ω = *I* = (0, *T*)*. If A is a uniformly bounded operator kernel, the*

 *<sup>t</sup>* 0

*and call A*(·, ·) *a uniformly bounded operator kernel if* �*A*�*L*<sup>∞</sup>(Ω2;B1→B2) *is finite.*

*Q f* ˆ (*t*) =

*is called a bounded Volterra operator on L*∞(*I*; B) *with kernel A.*

�*ϕ*�*L*<sup>∞</sup>(Ω;B) <sup>=</sup> inf{*<sup>M</sup>* <sup>≥</sup> 0 : �*ϕ*(*x*)�B <sup>≤</sup> *M for almost all x*}. (44)

�*A*(*x*, *<sup>y</sup>*)�B1→B<sup>2</sup> <sup>=</sup> inf{*<sup>M</sup>* <sup>≥</sup> 0 : �*A*(*x*, *<sup>y</sup>*)*φ*� ≤ *<sup>M</sup>*�*φ*�, <sup>∀</sup>*<sup>φ</sup>* ∈ B1}. (45)

<sup>≡</sup> ess sup(*x*,*y*)∈Ω<sup>2</sup> �*A*(*x*, *<sup>y</sup>*)�B1→B<sup>2</sup> (46)

*A*(*t*, *τ*)*f*(*τ*) *dτ* (47)

*φ* − *Q*ˆ *φ* = *f* , (43)

In operator notation, an integral equation of the second kind has the structure


Our primary theorem, like the definitions in section 3, deals with the space *L*∞(*I*; B). We also provide variants of the theorem and the key lemma for other Lebesgue spaces, *L*1(*I*; B) and *Lp*(*I*; B). In each case, the space B is likely, in applications, to be itself a Lebesgue space of functions of a spatial variable, *Lm*(**R***n*), with no connection between *m* and *p*.

The first step of the proof is a fundamental lemma establishing a bound on the Volterra operator that fully exploits its simplex structure. This argument inductively establishes the norm of each term in the Neumann series, from which the convergence quickly follows. In the lemmas, *j* (the future summation index) is understood to be an arbitrary nonnegative integer (or even a real positive number).

**Lemma 9.** *Let the Volterra integral operator, Q*ˆ : *L*∞(*I*; B) → *L*∞(*I*; B)*, be defined by*

$$
\hat{Q}\phi(t) = \int\_0^T A(t,\tau)\phi(\tau) \,d\tau = \int\_0^t A(t,\tau)\phi(\tau) \,d\tau. \tag{49}
$$

*Let <sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*∞(*I*; <sup>B</sup>) *and assume that* <sup>∃</sup>*<sup>C</sup>* <sup>&</sup>gt; <sup>0</sup> *such that for each subinterval Jt of the form* (0, *<sup>t</sup>*)*, we have* �*φ*�*L*<sup>∞</sup>(*Jt*;B) <sup>≡</sup> ess sup0<sup>&</sup>lt;*τ*<*<sup>t</sup>* �*φ*(*τ*)�B <sup>≤</sup> *Ct<sup>j</sup> . Assume that the Hypotheses are satisfied. Then it follows that*

$$\|\dot{\mathbb{Q}}\phi\|\_{L^{\infty}(I;\mathcal{B})} \leq \frac{D\mathbb{C}}{j+1}t^{j+1}.\tag{50}$$

10.5772/52489

259

(54)

*Proof.* The argument is the same as before, except that the *L*1(*Jt*; B) norm of *Q*ˆ *φ*(·) is

�*φ*(*τ*)� *<sup>d</sup>τdt*<sup>1</sup> ≤

 *dt*<sup>1</sup> ≤

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

*<sup>j</sup>* <sup>+</sup> <sup>1</sup> .

**Corollary 11.** *Let the Volterra integral operator, Q*ˆ : *Lp*(*I*; B) → *Lp*(*I*; B)*, where* 1 < *p* < ∞*, be defined by* (49)*. Let φ* ∈ *Lp*(*I*; B) *and assume that* ∃*C* > 0 *such that for each subinterval Jt* = (0, *t*)*,*

�*φ*(*τ*)�*<sup>p</sup>*

B *dτ*

*<sup>j</sup>* <sup>+</sup> <sup>1</sup> *<sup>t</sup>*

1/*<sup>p</sup>*

 *<sup>t</sup>* 0

 *<sup>t</sup>* 0

 *<sup>t</sup>*<sup>1</sup> 0

*<sup>D</sup>*�*φ*�*L*<sup>1</sup>(*Jt*

�*A*(*t*, *τ*)� �*φ*(*τ*)� *dτ*

http://dx.doi.org/10.5772/52489

≤ *Ctn*. (55)

*<sup>j</sup>*<sup>+</sup>1. (56)

≤ *Ctn*. (57)

[*p*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>)]1/*<sup>p</sup>* . (58)

<sup>1</sup> ;B) *dt*<sup>1</sup>

*A*(*t*, *τ*)*φ*(*τ*) *dτ*

<sup>1</sup> *dt*<sup>1</sup> <sup>=</sup> *DCtj*+<sup>1</sup>

 *<sup>t</sup>* 0

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L<sup>p</sup>* (*Jt*,B) <sup>≤</sup> *DC*

*Proof.* This follows from Lemmas 9 and 10 by the Riesz–Thorin theorem [10, pp. 27–28].

It may be of some interest to see how the *L<sup>p</sup>* theorem can be proved directly. The proof of

**Lemma 12.** *Let the Volterra integral operator, Q*ˆ : *Lp*(*I*; B) → *Lp*(*I*; B)*, where* 1 < *p* < ∞*, be defined by* (49)*. Let φ* ∈ *Lp*(*I*; B) *and assume that* ∃*C* > 0 *such that for each subinterval Jt* = (0, *t*)*,*

�*φ*(*τ*)�*<sup>p</sup>*

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L<sup>p</sup>* (*Jt*,B) <sup>≤</sup> *DCtj*+<sup>1</sup>

*Proof.* Let *<sup>q</sup>* be the conjugate exponent (*p*−<sup>1</sup> <sup>+</sup> *<sup>q</sup>*−<sup>1</sup> <sup>=</sup> 1). The Banach-space norm of *<sup>Q</sup>*<sup>ˆ</sup> *<sup>φ</sup>*(*t*)

B *dτ*

1/*<sup>p</sup>*

the needed lemma uses Folland's proof of Young's inequality [4] as a model.

 *<sup>t</sup>* 0

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L*<sup>1</sup>(*Jt*;B) <sup>=</sup>

*we have*

*we have*

satisfies

 *<sup>t</sup>* 0 *<sup>t</sup>*<sup>1</sup> 0

�*φ*�*L<sup>p</sup>* (*Jt*;B) <sup>≡</sup>

�*φ*�*L<sup>p</sup>* (*Jt*;B) <sup>≡</sup>

*Assume that the Hypotheses are satisfied. Then it follows that*

*Assume that the Hypotheses are satisfied. Then it follows that*

≤ *<sup>t</sup>* 0 *D <sup>t</sup>*<sup>1</sup> 0

≤ *<sup>t</sup>* 0 *DCt<sup>j</sup>*

*Proof.* Recall that *<sup>D</sup>* is defined so that �*A*(*t*, *<sup>τ</sup>*)� ≤ *<sup>D</sup>* <sup>&</sup>lt; <sup>∞</sup> for all (*t*, *<sup>τ</sup>*) <sup>∈</sup> ¯*I*2. The *<sup>L</sup>*∞(*Jt*; <sup>B</sup>) norm of the function *Q*ˆ *φ*(·) is

$$\begin{split} \|\bigotimes\!\!\|\bigotimes\!\|\_{L^{\alpha}(I\_{i};\mathcal{B})} &= \sup\_{t\_{1}\leq t} \left\| \int\_{0}^{t\_{1}} A(t,\tau)\phi(\tau) \,d\tau \right\| \leq \sup\_{t\_{1}\leq t} \int\_{0}^{t\_{1}} \|A(t,\tau)\phi(\tau)\| \,d\tau \\ &\leq \sup\_{t\_{1}\leq t} \int\_{0}^{t\_{1}} \|A(t,\tau)\| \,\|\big|\phi(\tau)\| \,d\tau \leq \sup\_{t\_{1}\leq t} \int\_{0}^{t\_{1}} D\mathbb{C}\tau^{j} \,d\tau \\ &= \sup\_{t\_{1}\leq t} D\mathbb{C} \frac{l\_{1}^{j+1}}{j+1} = \frac{D\mathcal{C}t^{j+1}}{j+1} .\end{split} \tag{51}$$

**Lemma 10.** *Let the Volterra integral operator, Q*ˆ : *L*1(*I*; B) → *L*1(*I*; B)*, be defined by* (49)*. Let φ* ∈ *L*1(*I*; B)*, and assume that* ∃*C* > 0 *such that for each subinterval Jt* = (0, *t*)*, we have*

$$\|\|\phi\|\|\_{L^1(I;\mathcal{B})} \equiv \int\_0^t \|\|\phi(\tau)\|\,d\tau \le \mathcal{C}t^j. \tag{52}$$

*Assume that the Hypotheses are satisfied. Then it follows that*

$$\|\hat{\mathbb{Q}}\Phi\|\|\_{L^{1}(I\_{\nu}\mathcal{B})} \leq \frac{DC}{j+1}t^{j+1}.\tag{53}$$

*Proof.* The argument is the same as before, except that the *L*1(*Jt*; B) norm of *Q*ˆ *φ*(·) is

$$\begin{split} \|\bigotimes \Phi\|\|\_{L^{1}(I;\mathfrak{E})} &= \int\_{0}^{t} \left\| \int\_{0}^{t\_{1}} A(t,\tau)\phi(\tau) \,d\tau \right\| \,dt\_{1} \leq \int\_{0}^{t} \int\_{0}^{t\_{1}} \|A(t,\tau)\| \, \|\phi(\tau)\| \,d\tau \\ &\leq \int\_{0}^{t} D \int\_{0}^{t\_{1}} \|\phi(\tau)\| \,d\tau dt\_{1} \leq \int\_{0}^{t} D \|\phi\|\_{L^{1}(I\_{1};\mathfrak{E})} \,dt\_{1} \\ &\leq \int\_{0}^{t} D \mathsf{C}t\_{1}^{j} \,dt\_{1} = \frac{D \mathsf{C}t^{j+1}}{j+1} . \end{split} \tag{54}$$

**Corollary 11.** *Let the Volterra integral operator, Q*ˆ : *Lp*(*I*; B) → *Lp*(*I*; B)*, where* 1 < *p* < ∞*, be defined by* (49)*. Let φ* ∈ *Lp*(*I*; B) *and assume that* ∃*C* > 0 *such that for each subinterval Jt* = (0, *t*)*, we have*

$$\|\|\phi\|\|\_{L^p(I\_i;\mathcal{B})} \equiv \left(\int\_0^t \|\|\phi(\tau)\|\|\_{\mathcal{B}}^p d\tau\right)^{1/p} \le Ct^n. \tag{55}$$

*Assume that the Hypotheses are satisfied. Then it follows that*

12 Advances in Quantum Mechanics

integer (or even a real positive number).

*Q*ˆ *φ*(*t*) =

*have* �*φ*�*L*<sup>∞</sup>(*Jt*;B) <sup>≡</sup> ess sup0<sup>&</sup>lt;*τ*<*<sup>t</sup>* �*φ*(*τ*)�B <sup>≤</sup> *Ct<sup>j</sup>*

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L*<sup>∞</sup>(*Jt*;B) <sup>=</sup> sup

*t*1≤*t <sup>t</sup>*<sup>1</sup> 0

> *<sup>t</sup>*<sup>1</sup> 0

�*φ*�*L*<sup>1</sup>(*Jt*;B) <sup>≡</sup>

≤ sup *t*1≤*t*

= sup *t*1≤*t DC t j*+1 1

*Assume that the Hypotheses are satisfied. Then it follows that*

norm of the function *Q*ˆ *φ*(·) is

*follows that*

 *<sup>T</sup>* 0

Our primary theorem, like the definitions in section 3, deals with the space *L*∞(*I*; B). We also provide variants of the theorem and the key lemma for other Lebesgue spaces, *L*1(*I*; B) and *Lp*(*I*; B). In each case, the space B is likely, in applications, to be itself a Lebesgue space

The first step of the proof is a fundamental lemma establishing a bound on the Volterra operator that fully exploits its simplex structure. This argument inductively establishes the norm of each term in the Neumann series, from which the convergence quickly follows. In the lemmas, *j* (the future summation index) is understood to be an arbitrary nonnegative

of functions of a spatial variable, *Lm*(**R***n*), with no connection between *m* and *p*.

**Lemma 9.** *Let the Volterra integral operator, Q*ˆ : *L*∞(*I*; B) → *L*∞(*I*; B)*, be defined by*

*A*(*t*, *τ*)*φ*(*τ*) *dτ* =

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L*<sup>∞</sup>(*Jt*,B) <sup>≤</sup> *DC*

*Let <sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*∞(*I*; <sup>B</sup>) *and assume that* <sup>∃</sup>*<sup>C</sup>* <sup>&</sup>gt; <sup>0</sup> *such that for each subinterval Jt of the form* (0, *<sup>t</sup>*)*, we*

*Proof.* Recall that *<sup>D</sup>* is defined so that �*A*(*t*, *<sup>τ</sup>*)� ≤ *<sup>D</sup>* <sup>&</sup>lt; <sup>∞</sup> for all (*t*, *<sup>τ</sup>*) <sup>∈</sup> ¯*I*2. The *<sup>L</sup>*∞(*Jt*; <sup>B</sup>)

*A*(*t*, *τ*)*φ*(*τ*) *dτ*

*<sup>j</sup>* <sup>+</sup> <sup>1</sup> <sup>=</sup> *DCtj*+<sup>1</sup>

*φ* ∈ *L*1(*I*; B)*, and assume that* ∃*C* > 0 *such that for each subinterval Jt* = (0, *t*)*, we have*

 *<sup>t</sup>* 0

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L*<sup>1</sup>(*Jt*,B) <sup>≤</sup> *DC*

**Lemma 10.** *Let the Volterra integral operator, Q*ˆ : *L*1(*I*; B) → *L*1(*I*; B)*, be defined by* (49)*. Let*

 *<sup>t</sup>* 0

*<sup>j</sup>* <sup>+</sup> <sup>1</sup> *<sup>t</sup>*

 

*<sup>j</sup>* <sup>+</sup> <sup>1</sup> .

�*φ*(*τ*)� *dτ* ≤ *Ct<sup>j</sup>*

*j* + 1 *t*

�*A*(*t*, *τ*)� �*φ*(*τ*)� *dτ* ≤ sup

≤ sup *t*1≤*t*  *<sup>t</sup>*<sup>1</sup> 0

> *<sup>t</sup>*<sup>1</sup> 0

*t*1≤*t*

*A*(*t*, *τ*)*φ*(*τ*) *dτ*. (49)

*<sup>j</sup>*<sup>+</sup>1. (50)

�*A*(*t*, *τ*)*φ*(*τ*)� *dτ*

*DCτ<sup>j</sup> dτ*

. (52)

*<sup>j</sup>*<sup>+</sup>1. (53)

(51)

*. Assume that the Hypotheses are satisfied. Then it*

$$\|\dot{Q}\Phi\|\_{L^p(I\_\nu\mathcal{B})} \le \frac{D\mathbb{C}}{j+1} t^{j+1}.\tag{56}$$

*Proof.* This follows from Lemmas 9 and 10 by the Riesz–Thorin theorem [10, pp. 27–28]. 

It may be of some interest to see how the *L<sup>p</sup>* theorem can be proved directly. The proof of the needed lemma uses Folland's proof of Young's inequality [4] as a model.

**Lemma 12.** *Let the Volterra integral operator, Q*ˆ : *Lp*(*I*; B) → *Lp*(*I*; B)*, where* 1 < *p* < ∞*, be defined by* (49)*. Let φ* ∈ *Lp*(*I*; B) *and assume that* ∃*C* > 0 *such that for each subinterval Jt* = (0, *t*)*, we have*

$$\|\|\phi\|\|\_{L^p(I\_i;\mathcal{B})} \equiv \left(\int\_0^t \|\|\phi(\tau)\|\|\_{\mathcal{B}}^p d\tau\right)^{1/p} \le Ct^n. \tag{57}$$

*Assume that the Hypotheses are satisfied. Then it follows that*

$$\|\vert \hat{Q}\phi\vert\|\_{L^{p}(\mathfrak{J}\_{\ell}, \mathfrak{B})} \leq \frac{D\mathfrak{C}t^{j+1}}{[p(j+1)]^{1/p}}.\tag{58}$$

*Proof.* Let *<sup>q</sup>* be the conjugate exponent (*p*−<sup>1</sup> <sup>+</sup> *<sup>q</sup>*−<sup>1</sup> <sup>=</sup> 1). The Banach-space norm of *<sup>Q</sup>*<sup>ˆ</sup> *<sup>φ</sup>*(*t*) satisfies

$$\begin{split} \|\bigotimes \Phi(t\_1)\|\_{\mathcal{B}} &\leq \left(\int\_0^{t\_1} \|A(t\_1, \tau)\| \right) d\tau \Big)^{1/q} \left(\int\_0^{t\_1} \|A(t\_1, \tau)\| \, \|\phi(\tau)\| \|^p d\tau\right)^{1/p} \\ &\leq \mathcal{D}^{1/q} \left(\int\_0^{t\_1} d\tau \right)^{1/q} \left(\int\_0^{t\_1} \mathcal{D} \|\phi(\tau)\| \|^p d\tau \right)^{1/p} \\ &\leq \mathcal{D}^{1/q} \mathcal{D}^{1/p} \mathbf{1}\_1^{1/q} \left(\int\_0^{t\_1} \|\phi(\tau)\| \|^p d\tau \right)^{1/p} \\ &\leq \mathcal{D} \mathbf{1}\_1^{1/q} \left(\int\_0^{t\_1} \|\phi(\tau)\| ^p d\tau \right)^{1/p} .\end{split} \tag{59}$$

261

�*Q*<sup>ˆ</sup> <sup>1</sup> *<sup>f</sup>* �*L*<sup>∞</sup>(*Jt*;B) <sup>≤</sup> *DC*0*t*. (64)

*<sup>j</sup>*! <sup>=</sup> � *<sup>f</sup>* �*L*<sup>∞</sup>(*I*;B)*eDt* (66)

. (65)

http://dx.doi.org/10.5772/52489

Then by inductively applying Lemma 9 with *<sup>C</sup>* <sup>=</sup> *<sup>D</sup>j*−1*C*0/(*<sup>j</sup>* <sup>−</sup> <sup>1</sup>)! , we see that the *<sup>j</sup>*th term

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

for all *t* ∈ (0, *T*]. Therefore, the Neumann series converges in the topology of *L*∞(*I*; B).

The *L*<sup>∞</sup> norm on the time behavior is the most natural and likely one to apply to solutions of a time-evolution equation (especially for the Schrödinger equation with B a Hilbert space, because of the unitary of the evolution). However, the other *L<sup>p</sup>* norms may prove to be useful, and it is easy to generalize the theorem to them. Note that the appropriate condition

**Theorem 14. (***L*<sup>1</sup> **Volterra Theorem)** *Let the Hypotheses be satisfied, and let Q be defined by* ˆ (49)*. Let f belong to L*1(*I*; B)*. Then the Volterra integral equation φ* = *Q*ˆ *φ* + *f can be solved by successive approximations. That is, the Neumann series for φ,* (63)*, converges in the topology of L*1(*I*; B)*.*

*Proof.* Let � *<sup>f</sup>* �*L*<sup>1</sup>(*I*;B) <sup>=</sup> *<sup>C</sup>*<sup>0</sup> and argue as before, except that Lemma 10 is used to bound all

**Theorem 15. (***L<sup>p</sup>* **Volterra Theorem)** *Let the Hypotheses be satisfied, and let Q be defined by* ˆ (49)*. Let f belong to Lp*(*I*; B)*. Then the Volterra integral equation φ* = *Q*ˆ *φ* + *f can be solved by successive*

*Proof.* The proof based on the Riesz–Thorin theorem goes exactly like the previous two, using Corollary 11. To prove the theorem directly, let � *<sup>f</sup>* �*L<sup>p</sup>* (*I*;B) <sup>=</sup> *<sup>C</sup>*<sup>0</sup> and use Lemma 12

*Dj*

�*Q*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* �*L<sup>p</sup>* (*I*;B) <sup>=</sup> � *<sup>f</sup>* �*L<sup>p</sup>* (*I*;B)

*<sup>p</sup>j*/*<sup>p</sup>* � *<sup>f</sup>* �*L<sup>p</sup>* (*I*;B)

∞ ∑ *j*=0

*Dj pj*/*<sup>p</sup>* *tj*

*tj*

(*j*!)1/*<sup>p</sup>* . (67)

(*j*!)1/*<sup>p</sup>* (68)

*approximations. That is, the Neumann series converges in the topology of Lp*(*I*; B)*.*

�*Q*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* �*L<sup>p</sup>* (*Jt*;B) <sup>≤</sup>

∞ ∑ *j*=0 *C*0 *tj j*!

�*Q*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* �*L*<sup>∞</sup>(*Jt*;B) <sup>≤</sup> *<sup>D</sup><sup>j</sup>*

*Dj tj*

of the Neumann series, *Q*ˆ *<sup>j</sup> f* , has, because of its simplex structure, the bound

*C*0 ∞ ∑ *n*=0

on *A*(*t*, *τ*) is still the uniform boundedness of Definition 6.

Therefore, the series (63) is majorized by

the terms �*Q*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* �*L*<sup>1</sup>(*Jt*;B) .

inductively to show

To see whether the series

Then we must raise both sides to the *p*th power and integrate, seeing by Fubini's theorem that

$$\begin{split} \int\_{0}^{t\_{1}} \|\hat{\mathbb{Q}}\phi(t\_{1})\|^{p} \, dt\_{1} &\leq \int\_{0}^{t} \mathbb{D}^{p}t\_{1}^{p/q} \int\_{0}^{t\_{1}} \|\phi(\tau)\|^{p} \, d\tau \, dt\_{1} \leq \int\_{0}^{t} \mathbb{D}^{p} \int\_{0}^{t} t\_{1}^{p/q} \|\phi\|\_{L^{p}(\varprojlim \mathcal{B})}^{p} \, dt\_{1} \\ &\leq D^{p} \int\_{0}^{t} \mathbb{C}^{p} t\_{1}^{np+p/q} \, dt\_{1} \leq D^{p} \mathbb{C}^{p} \frac{t^{np+p/q+1}}{np+\frac{p}{q}+1} = D^{p} \mathbb{C}^{p} \frac{t^{(j+1)p}}{jp+p} \,. \end{split} \tag{60}$$

since 1 + *<sup>p</sup> <sup>q</sup>* = *p*. Now take the *p*th root, getting

$$\|\hat{\mathbb{Q}}\Phi\|\_{L^p(I\_i;\mathbb{B})} \le D\mathbb{C} \frac{t^{j+1}}{[p(j+1)]^{1/p}}.\qquad\square\tag{61}$$

The following theorem is the main theorem of [3]; its proof is corrected here.

**Theorem 13. (***L*<sup>∞</sup> **Volterra Theorem)** *Let the Hypotheses be satisfied, and let Q be defined by* ˆ (49)*. Let f belong to L*∞(*I*; B)*. Then the Volterra integral equation*

$$
\phi = \bar{Q}\phi + f \tag{62}
$$

*can be solved by successive approximations. That is, the Neumann series for φ,*

$$\boldsymbol{\phi} = \sum\_{j=0}^{\infty} \hat{\mathbb{Q}}^j \boldsymbol{f}\_{\prime} \tag{63}$$

*converges in the topology of L*∞(*I*; B)*.*

*Proof.* Let � *<sup>f</sup>* �*L*<sup>∞</sup>(*I*;B) <sup>=</sup> *<sup>C</sup>*<sup>0</sup> . Of course, � *<sup>f</sup>* �*L*<sup>∞</sup>(*Jt*;B) <sup>≤</sup> *<sup>C</sup>*<sup>0</sup> on a smaller interval, *Jt* = (0, *<sup>t</sup>*), so by Lemma 9 with *j* = 0,

<sup>260</sup> Advances in Quantum Mechanics Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces 15 10.5772/52489 Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces http://dx.doi.org/10.5772/52489 261

$$\|\|\hat{Q}^1 f\|\|\_{L^\infty(\mathbb{J}; \mathcal{B})} \le D\mathbb{C}\_0 \mathfrak{t}.\tag{64}$$

Then by inductively applying Lemma 9 with *<sup>C</sup>* <sup>=</sup> *<sup>D</sup>j*−1*C*0/(*<sup>j</sup>* <sup>−</sup> <sup>1</sup>)! , we see that the *<sup>j</sup>*th term of the Neumann series, *Q*ˆ *<sup>j</sup> f* , has, because of its simplex structure, the bound

$$\|\|\dot{\mathbb{Q}}^j f\|\|\_{L^\infty(I; \mathbb{B})} \le D^j \mathbb{C}\_0 \frac{t^j}{j!} \,. \tag{65}$$

Therefore, the series (63) is majorized by

14 Advances in Quantum Mechanics

that

 *<sup>t</sup>*<sup>1</sup> 0

since 1 + *<sup>p</sup>*

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*(*t*1)�B ≤

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*(*t*1)�*<sup>p</sup> dt*<sup>1</sup> ≤

*converges in the topology of L*∞(*I*; B)*.*

by Lemma 9 with *j* = 0,

 *<sup>t</sup>*<sup>1</sup> 0

> *<sup>t</sup>*<sup>1</sup> 0 *dτ*

> > *<sup>t</sup>*<sup>1</sup> 0

≤ *D*1/*<sup>q</sup>*

≤ *Dt*1/*<sup>q</sup>* 1

> *<sup>t</sup>* 0 *Dpt p*/*q* 1

≤ *D<sup>p</sup> <sup>t</sup>* 0 *Cpt*

*<sup>q</sup>* = *p*. Now take the *p*th root, getting

*Let f belong to L*∞(*I*; B)*. Then the Volterra integral equation*

≤ *D*1/*qD*1/*pt*

�*A*(*t*1, *<sup>τ</sup>*)�) *<sup>d</sup><sup>τ</sup>*

1/*q* 1

 *<sup>t</sup>*<sup>1</sup> 0

*np*+*p*/*q*

�*Q*<sup>ˆ</sup> *<sup>φ</sup>*�*L<sup>p</sup>* (*Jt*;B) <sup>≤</sup> *DC <sup>t</sup>j*+<sup>1</sup>

The following theorem is the main theorem of [3]; its proof is corrected here.

*can be solved by successive approximations. That is, the Neumann series for φ,*

*φ* =

1/*<sup>q</sup> <sup>t</sup>*<sup>1</sup>

 *<sup>t</sup>*<sup>1</sup> 0

�*φ*(*τ*)�*<sup>p</sup> dτ*

0

Then we must raise both sides to the *p*th power and integrate, seeing by Fubini's theorem

�*φ*(*τ*)�*<sup>p</sup> <sup>d</sup><sup>τ</sup> dt*<sup>1</sup> ≤

<sup>1</sup> *dt*<sup>1</sup> <sup>≤</sup> *<sup>D</sup>pCp <sup>t</sup>*

**Theorem 13. (***L*<sup>∞</sup> **Volterra Theorem)** *Let the Hypotheses be satisfied, and let Q be defined by* ˆ (49)*.*

∞ ∑ *j*=0

*Proof.* Let � *<sup>f</sup>* �*L*<sup>∞</sup>(*I*;B) <sup>=</sup> *<sup>C</sup>*<sup>0</sup> . Of course, � *<sup>f</sup>* �*L*<sup>∞</sup>(*Jt*;B) <sup>≤</sup> *<sup>C</sup>*<sup>0</sup> on a smaller interval, *Jt* = (0, *<sup>t</sup>*), so

1/*<sup>q</sup> <sup>t</sup>*<sup>1</sup>

�*φ*(*τ*)�*<sup>p</sup> dτ*

1/*<sup>p</sup>* .

0

*D*�*φ*(*τ*)�*<sup>p</sup> dτ*

1/*<sup>p</sup>*

 *<sup>t</sup>* 0 *Dp <sup>t</sup>* 0 *t p*/*q* <sup>1</sup> �*φ*�*<sup>p</sup>*

*np*+*p*/*q*+1 *np* + *<sup>p</sup>*

*<sup>q</sup>* <sup>+</sup> <sup>1</sup> <sup>=</sup> *<sup>D</sup>pCp <sup>t</sup>*

[*p*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>)]1/*<sup>p</sup>* . (61)

*φ* = *Q*ˆ *φ* + *f* (62)

*Q*ˆ *<sup>j</sup> f* , (63)

�*A*(*t*1, *<sup>τ</sup>*)� �*φ*(*τ*)�*<sup>p</sup> <sup>d</sup><sup>τ</sup>*

1/*<sup>p</sup>*

1/*<sup>p</sup>*

*L<sup>p</sup>* (*Jt*

(*j*+1)*p jp* <sup>+</sup> *<sup>p</sup>* ,

<sup>1</sup> ;B) *dt*<sup>1</sup>

(59)

(60)

$$\mathbb{C}\_0 \sum\_{n=0}^{\infty} \frac{D^j t^j}{j!} = \|f\|\_{L^\infty(I; \mathcal{B})} e^{Dt} \tag{66}$$

for all *t* ∈ (0, *T*]. Therefore, the Neumann series converges in the topology of *L*∞(*I*; B). 

The *L*<sup>∞</sup> norm on the time behavior is the most natural and likely one to apply to solutions of a time-evolution equation (especially for the Schrödinger equation with B a Hilbert space, because of the unitary of the evolution). However, the other *L<sup>p</sup>* norms may prove to be useful, and it is easy to generalize the theorem to them. Note that the appropriate condition on *A*(*t*, *τ*) is still the uniform boundedness of Definition 6.

**Theorem 14. (***L*<sup>1</sup> **Volterra Theorem)** *Let the Hypotheses be satisfied, and let Q be defined by* ˆ (49)*. Let f belong to L*1(*I*; B)*. Then the Volterra integral equation φ* = *Q*ˆ *φ* + *f can be solved by successive approximations. That is, the Neumann series for φ,* (63)*, converges in the topology of L*1(*I*; B)*.*

*Proof.* Let � *<sup>f</sup>* �*L*<sup>1</sup>(*I*;B) <sup>=</sup> *<sup>C</sup>*<sup>0</sup> and argue as before, except that Lemma 10 is used to bound all the terms �*Q*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* �*L*<sup>1</sup>(*Jt*;B) .

**Theorem 15. (***L<sup>p</sup>* **Volterra Theorem)** *Let the Hypotheses be satisfied, and let Q be defined by* ˆ (49)*. Let f belong to Lp*(*I*; B)*. Then the Volterra integral equation φ* = *Q*ˆ *φ* + *f can be solved by successive approximations. That is, the Neumann series converges in the topology of Lp*(*I*; B)*.*

*Proof.* The proof based on the Riesz–Thorin theorem goes exactly like the previous two, using Corollary 11. To prove the theorem directly, let � *<sup>f</sup>* �*L<sup>p</sup>* (*I*;B) <sup>=</sup> *<sup>C</sup>*<sup>0</sup> and use Lemma 12 inductively to show

$$\|\vert \hat{\mathbb{Q}}^j f\|\|\_{L^p(I\_!; \mathcal{B})} \le \frac{D^j}{p^{j/p}} \|f\|\|\_{L^p(I; \mathcal{B})} \frac{t^j}{(j!)^{1/p}}.\tag{67}$$

To see whether the series

$$\sum\_{j=0}^{\infty} \|\hat{\mathcal{Q}}^j f\|\_{L^p(I; \mathcal{B})} = \|f\|\_{L^p(I; \mathcal{B})} \sum\_{j=0}^{\infty} \frac{D^j}{p^{j/p}} \frac{t^j}{(j!)^{1/p}} \tag{68}$$

is convergent, we use the ratio test. Let

$$L = \lim\_{j \to \infty} \left| \frac{a\_{j+1}}{a\_j} \right|, \qquad a\_{\bar{j}} = \||\hat{\mathcal{Q}}^{\bar{j}} f||\_{L^p(I; \mathcal{B})}.\tag{69}$$

*Lm*(**R***n*) is defined by

Therefore, if

That is,

uniformly in (*x*, *t*, *τ*), then

*<sup>A</sup>*(*t*, *<sup>τ</sup>*)*φ*(*x*) =

<sup>|</sup>[*A*(*t*, *<sup>τ</sup>*)*φ*](*x*, *<sup>t</sup>*, *<sup>τ</sup>*)| ≤

*<sup>Q</sup>*<sup>ˆ</sup> *<sup>φ</sup>* + *<sup>f</sup>* can be solved by iteration within *<sup>L</sup>*∞,∞(*I*; **<sup>R</sup>***n*).

This time we need the condition that

uniformly in (*y*, *t*, *τ*); then

<sup>|</sup>*A*(*t*, *<sup>τ</sup>*)*φ*(*x*, *<sup>t</sup>*, *<sup>τ</sup>*)| ≤

 **R***n*

 **R***n* **R***n*

**R***n*

≤ �*φ*�*L*∞,<sup>∞</sup>(*I*;**R***<sup>n</sup>*)

and Lemma 9 applies. Theorem 13 therefore proves that the Volterra integral equation *φ* =

Now suppose instead that we want to work in *L*∞,1(*I*; **R***n*). In place of (73) we have

**R***n*

≤ �*φ*�*L*∞,1(*I*;**R***<sup>n</sup>*) sup


 **R***n*

for *φ* ∈ *Lm*(**R***n*). (We shall be using this equation for functions *φ* that depend on *τ* as well as *y*, so that *A*(*t*, *τ*)*φ* is a function of (*x*, *t*, *τ*).) To assure that *A*(*t*, *τ*) is a bounded Banach-space operator, we need to impose an additional technical condition on the kernel function *K*. The simplest possibility is to exploit the generalized Young inequality [4, Theorem (0.10)]. Suppose that we wish to treat functions *<sup>φ</sup>*(*y*, *<sup>τ</sup>*) ∈ *<sup>L</sup>*∞,∞(*I*; **<sup>R</sup>***n*). Then (72) leads to

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

10.5772/52489

263

http://dx.doi.org/10.5772/52489

(73)

*K*(*x*, *t*; *y*, *τ*)*φ*(*y*) *dy*, (72)



�*A*�*L*<sup>∞</sup>(*I*2;B→B) <sup>≤</sup> *<sup>D</sup>*, (76)


�*A*(*t*, *<sup>τ</sup>*)*φ*�*L*∞,1(*I*;**R***<sup>n</sup>*) <sup>≤</sup> *<sup>D</sup>*�*φ*�*L*∞,1(*I*;**R***<sup>n</sup>*) (79)

<sup>|</sup>*K*(*x*, *<sup>t</sup>*; *<sup>y</sup>*, *<sup>τ</sup>*)|. (77)

�*A*(*t*, *<sup>τ</sup>*)*φ*�*L*∞,<sup>∞</sup>(*I*;**R***<sup>n</sup>*) <sup>≤</sup> *<sup>D</sup>*�*φ*�*L*∞,<sup>∞</sup>(*I*;**R***<sup>n</sup>*) . (75)


*y*

Let *<sup>M</sup>*(*p*) = *Dp*−1/*p*. Then

$$\begin{split} L &= \lim\_{n \to \infty} \frac{\|f\|\_{L^p(I; \mathcal{B})} \mathcal{M}(p)^{j+1} t^{j+1}}{[(j+1)!]^{1/p}} \cdot \frac{(j!)^{1/p}}{\|f\|\_{L^p(I; \mathcal{B})} \mathcal{M}(p)^j t^j} \\ &= \mathcal{M}(p) t \lim\_{j \to \infty} (j+1)^{-1/p} = 0, \end{split} \tag{70}$$

Thus *L* < 1, and by the ratio-test and series-majorization theorems, the Neumann series converges absolutely in the topology of *Lp*(*I*; B).

#### **5. Applications of the Volterra theorem**

In this section we present some quick applications of the general Volterra theorem of section 4. The conclusions are already well known, or are obvious generalizations of those that are, so these examples just show how they fit into the general framework. More serious applications are delayed to later papers. The first set of examples comprises some of the standard elementary types of Volterra integral equations [11–13], generalized to vector-valued functions and functions of additional variables. The second application is to the Schrödinger problem set up in section 2 ; the result is essentially what is known in textbooks of quantum mechanics as "time-dependent perturbation theory".

Although we use the *L*<sup>∞</sup> version of the theorem, Theorem 13, one could easily apply Theorems 14 and 15 as well. Thus a general setting for many examples is the generic double Lebesgue space defined as follows. As usual, let *I* = (0, *T*) be the maximal time interval considered. In the role of B, consider the Lebesgue space *Lm*(**R***n*) of functions of an *n*-dimensional spatial variable. Then *Lp*,*m*(*I*; **R***n*) is the Banach space of functions on *I* taking values in *Lm*(**R***n*) and subjected to the *L<sup>p</sup>* norm as functions of *t*. Thus

$$L^{p,m}(I; \mathbb{R}^n) = \left\{ \phi : \left( \int\_I \left[ \int\_{\mathbb{R}^n} |\phi(y\_\prime, \tau)|^m dy \right]^{p/m} d\tau \right)^{1/p} \equiv \|\phi\|\_{L^{p,\mathfrak{m}}(I; \mathbb{R}^n)} < \infty \right\}.\tag{71}$$

When either *p* or *m* is ∞, the Lebesgue norm is replaced by the essential supremum in the obvious way.

#### **5.1. Classical integral equations**

#### *5.1.1. Spatial variables*

For *x* and *y* in **R***n*, let *K*(*x*, *t*; *y*, *τ*) be a uniformly bounded complex-valued function, satisfying the Volterra condition in (*t*, *τ*). The Volterra operator kernel *A*(*t*, *τ*) : *Lm*(**R***n*) →

<sup>262</sup> Advances in Quantum Mechanics Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces 17 10.5772/52489 Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces http://dx.doi.org/10.5772/52489 263

*Lm*(**R***n*) is defined by

16 Advances in Quantum Mechanics

Let *<sup>M</sup>*(*p*) = *Dp*−1/*p*. Then

*Lp*,*m*(*I*; **R***n*) =

**5.1. Classical integral equations**

obvious way.

*5.1.1. Spatial variables*

 *φ* : *I* **R***n*

is convergent, we use the ratio test. Let

*L* = lim *j*→∞ 

*<sup>L</sup>* <sup>=</sup> lim*n*→<sup>∞</sup>

= *M*(*p*)*t* lim

converges absolutely in the topology of *Lp*(*I*; B).

**5. Applications of the Volterra theorem**

*j*→∞

textbooks of quantum mechanics as "time-dependent perturbation theory".

values in *Lm*(**R***n*) and subjected to the *L<sup>p</sup>* norm as functions of *t*. Thus


*aj*+<sup>1</sup> *aj* 

� *<sup>f</sup>* �*L<sup>p</sup>* (*I*;B)*M*(*p*)*j*+1*tj*+<sup>1</sup> [(*j* + 1)!]

(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>)−1/*<sup>p</sup>* <sup>=</sup> 0,

Thus *L* < 1, and by the ratio-test and series-majorization theorems, the Neumann series

In this section we present some quick applications of the general Volterra theorem of section 4. The conclusions are already well known, or are obvious generalizations of those that are, so these examples just show how they fit into the general framework. More serious applications are delayed to later papers. The first set of examples comprises some of the standard elementary types of Volterra integral equations [11–13], generalized to vector-valued functions and functions of additional variables. The second application is to the Schrödinger problem set up in section 2 ; the result is essentially what is known in

Although we use the *L*<sup>∞</sup> version of the theorem, Theorem 13, one could easily apply Theorems 14 and 15 as well. Thus a general setting for many examples is the generic double Lebesgue space defined as follows. As usual, let *I* = (0, *T*) be the maximal time interval considered. In the role of B, consider the Lebesgue space *Lm*(**R***n*) of functions of an *n*-dimensional spatial variable. Then *Lp*,*m*(*I*; **R***n*) is the Banach space of functions on *I* taking

*<sup>m</sup> dy*

When either *p* or *m* is ∞, the Lebesgue norm is replaced by the essential supremum in the

For *x* and *y* in **R***n*, let *K*(*x*, *t*; *y*, *τ*) be a uniformly bounded complex-valued function, satisfying the Volterra condition in (*t*, *τ*). The Volterra operator kernel *A*(*t*, *τ*) : *Lm*(**R***n*) →

*p*/*<sup>m</sup> dτ* 1/*<sup>p</sup>*

1/*<sup>p</sup>* · (*j*!)1/*<sup>p</sup>*

� *<sup>f</sup>* �*L<sup>p</sup>* (*I*;B)*M*(*p*)*<sup>j</sup>*

, *aj* <sup>=</sup> �*Q*<sup>ˆ</sup> *<sup>j</sup> <sup>f</sup>* �*L<sup>p</sup>* (*I*;B) . (69)

*tj*

≡ �*φ*�*Lp*,*<sup>m</sup>*(*I*;**R***<sup>n</sup>*) <sup>&</sup>lt; <sup>∞</sup>

. (71)

(70)

$$A(t,\tau)\phi(\mathbf{x}) = \int\_{\mathbb{R}^n} K(\mathbf{x}, t; y, \tau)\phi(y) \, dy,\tag{72}$$

for *φ* ∈ *Lm*(**R***n*). (We shall be using this equation for functions *φ* that depend on *τ* as well as *y*, so that *A*(*t*, *τ*)*φ* is a function of (*x*, *t*, *τ*).) To assure that *A*(*t*, *τ*) is a bounded Banach-space operator, we need to impose an additional technical condition on the kernel function *K*. The simplest possibility is to exploit the generalized Young inequality [4, Theorem (0.10)].

Suppose that we wish to treat functions *<sup>φ</sup>*(*y*, *<sup>τ</sup>*) ∈ *<sup>L</sup>*∞,∞(*I*; **<sup>R</sup>***n*). Then (72) leads to

$$\begin{split} \left| \left[ A(t,\tau)\phi \right](\mathbf{x},t,\tau) \right| &\leq \int\_{\mathbb{R}^n} \left| K(\mathbf{x},t;\mathbf{y},\tau) \right| \left| \phi(\mathbf{y},\tau) \right| d\mathbf{y} \\ &\leq \left\| \phi \right\|\_{L^{\infty,\infty}(I;\mathbb{R}^n)} \int\_{\mathbb{R}^n} \left| K(\mathbf{x},t;\mathbf{y},\tau) \right| d\mathbf{y}. \end{split} \tag{73}$$

Therefore, if

$$\int\_{\mathbb{R}^n} |\mathcal{K}(\mathbf{x}, t; y, \tau)| \, dy \le D \tag{74}$$

uniformly in (*x*, *t*, *τ*), then

$$\|\|A(t,\tau)\phi\|\|\_{L^{\infty\infty}(I;\mathbb{R}^n)} \le D \|\|\phi\|\|\_{L^{\infty\infty}(I;\mathbb{R}^n)}.\tag{75}$$

That is,

$$\|A\|\_{L^{\infty}(I^{2};\mathcal{B}\to\mathcal{B})} \leq D\_{\prime} \tag{76}$$

and Lemma 9 applies. Theorem 13 therefore proves that the Volterra integral equation *φ* = *<sup>Q</sup>*<sup>ˆ</sup> *<sup>φ</sup>* + *<sup>f</sup>* can be solved by iteration within *<sup>L</sup>*∞,∞(*I*; **<sup>R</sup>***n*).

Now suppose instead that we want to work in *L*∞,1(*I*; **R***n*). In place of (73) we have

$$\begin{split} |A(t,\tau)\phi(\mathbf{x},t,\tau)| &\leq \int\_{\mathbb{R}^n} |\mathcal{K}(\mathbf{x},t;\mathbf{y},\tau)| |\phi(\mathbf{y},\tau)| \, d\mathbf{y} \\ &\leq \|\phi\|\_{L^{\infty 1}(I;\mathbb{R}^n)} \sup\_{\mathbf{y}} |\mathcal{K}(\mathbf{x},t;\mathbf{y},\tau)|. \end{split} \tag{77}$$

This time we need the condition that

$$\int\_{\mathbb{R}^n} |K(\mathbf{x}, t; y, \tau)| \, d\mathbf{x} \le D \tag{78}$$

uniformly in (*y*, *t*, *τ*); then

$$\|\|A(t,\tau)\phi\|\|\_{L^{\infty,1}(I;\mathbb{R}^n)} \le D \|\|\phi\|\|\_{L^{\infty,1}(I;\mathbb{R}^n)}\tag{79}$$

in place of (75). The argument concludes as before, using Lemma 10 and Theorem 14.

For *L*∞,*p*(*I*; **R***n*), the generalized Young inequality [4] assumes both (74) and (78) and assures that

$$\|\|A(t,\tau)\phi\|\|\_{L^{\infty,p}(I;\mathbb{R}^n)} \le D \|\|\phi\|\|\_{L^{\infty,p}(I;\mathbb{R}^n)}\,. \tag{80}$$

10.5772/52489

265

�*A*�*L*<sup>∞</sup>(*I*2;B→B) <sup>≤</sup> *<sup>D</sup>*, (84)

<sup>2</sup>)−<sup>1</sup> *dx*).) In order for our method to work simply, we

*<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>* − *<sup>τ</sup>*)*V*(*y*, *<sup>τ</sup>*). (86)

*<sup>U</sup>*f(*<sup>t</sup>* − *<sup>τ</sup>*)*V*ˆ(*τ*), (87)

*<sup>L</sup>*<sup>2</sup>(**R***<sup>n</sup>*) ,

(88)

. The solution of that equation

http://dx.doi.org/10.5772/52489

and hence Lemma 9 and Theorem 13 apply as usual, establishing convergence of the Picard

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

In Corollary 4 we converted the time-dependent Schrödinger problem to a Volterra integral

by iteration (successive approximations, Picard algorithm, Neumann series) is effectively a power series in the potential *V*, so it is the same thing as a perturbation calculation with

In this problem the Banach space B is the Hilbert space *L*2(**R***n*). (To assure pointwise convergence to the initial data, according to Theorem 2 and Remark 3, we should also take

must assume that *V*(*x*, *t*) is a bounded potential. It may be time-dependent, but in that case

Note that the role of *f* in the abstract Volterra equation (62) is played by the entire first

*h*¯

It satisfies neither the Hilbert–Schmidt condition (81) nor the generalized Young conditions

*h*¯

where *U*<sup>f</sup> is the free time evolution (9) implemented by the kernel *K*<sup>f</sup> , and *V*ˆ is the operator of pointwise multiplication by the potential *V*(*y*, *τ*). It is well known [7, Chapter 4] that *<sup>U</sup>*f(*t*) = *<sup>e</sup>*−*itH*0/¯*<sup>h</sup>* is unitary, and hence its norm as an operator on *<sup>L</sup>*2(**R***n*) is 1. On the other



<sup>2</sup> *dy*

<sup>2</sup> *dy* = *D*2� *f*(*τ*)�<sup>2</sup>

2/4*t*

*<sup>V</sup>* ∈ *<sup>L</sup>*∞(**R***<sup>n</sup>* × *<sup>I</sup>*); |*V*(*y*, *<sup>t</sup>*)| ≤ *<sup>D</sup>* (almost everywhere). (85)

solution of *<sup>φ</sup>* = *<sup>Q</sup>*<sup>ˆ</sup> *<sup>φ</sup>* + *<sup>f</sup>* in the topology of *<sup>L</sup>*∞,2(*I*, **<sup>R</sup>***n*).

equation, (42), wherein *<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*)=(4*πit*)−*n*/2*ei*|*x*−*y*<sup>|</sup>

its bound should be independent of *t*. That is, we assume

From the other term of (42) we extract the kernel function

*<sup>K</sup>*(*x*, *<sup>t</sup>*; *<sup>y</sup>*, *<sup>τ</sup>*) = <sup>−</sup> *<sup>i</sup>*

(73) and (77). However, the resulting operator kernel can be factored as

*<sup>L</sup>*<sup>2</sup>(**R***<sup>n</sup>*) =

*<sup>A</sup>*(*t*, *<sup>τ</sup>*) = <sup>−</sup> *<sup>i</sup>*

 **R***n*

≤ *D*<sup>2</sup> **R***n*

respect to a coupling constant multiplying *V*.

the intersection with *L*1(**R***n*;(1 + |*x*|

integral term in (42), *K*f(*x*, *y*, *t*)*f*(*y*) *dy*.

�*V*(*τ*)*f*(*τ*)�<sup>2</sup>

hand,

**5.2. Perturbation theory for the Schrödinger equation**

The argument concludes as before, using Lemma 12 and Theorem 15, proving convergence of the Neumann series within *L*∞,*p*(*I*; **R***n*).

#### *5.1.2. Vector-valued functions*

A similar but simpler situation is where B is finite-dimensional, say **C***n*. Then *A*(*t*, *τ*) is an *n* × *n* matrix. Boundedness of *A* as an operator is automatic, but uniformity in the time variables is still a nontrivial condition. The theorem then gives a vectorial generalization of the usual Neumann series for a scalar Volterra equation.

#### *5.1.3. Hilbert–Schmidt operators*

Although the generalized Young approach yields a theorem for B = *L*2(**R***n*), the boundedness of operators on that space is often proved from a stronger condition on their kernels. In our context a Hilbert–Schmidt kernel is a function *K* : **R***<sup>n</sup>* × **R** × **R***<sup>n</sup>* × **R** → **C** for which

$$\left(\int\_{\mathbb{R}^n \times \mathbb{R}^n} |\mathcal{K}(\mathbf{x}, t; y, \tau)|^2 \, d\mathbf{x} \, dy\right)^{1/2} \equiv \|\mathcal{K}(t, \tau)\|\_{L^2(\mathbb{R}^{2n})} \le D < \infty,\tag{81}$$

and, of course, we also want it to be Volterra in (*t*, *τ*). In other words, *K*(*x*, *t*; *y*, *τ*) belongs to *<sup>L</sup>*∞,2(*I*2; **<sup>R</sup>**2*n*) and vanishes (or is ignored in the integrals) when *<sup>τ</sup>* > *<sup>t</sup>*. Then *A*(*t*, *τ*) : *L*2(**R***n*) → *L*2(**R***n*) defined by (72) is a Hilbert–Schmidt operator, which under our assumptions is uniformly bounded with norm at most *D*.

In parallel with (73) or (77) one has

$$\begin{split} |A(t,\tau)\phi(\mathbf{x},t,\tau)| &\leq \int\_{\mathbb{R}^n} |\mathcal{K}(\mathbf{x},t;y,\tau)| |\phi(y,\tau)| dy \\ &\leq \left(\int\_{\mathbb{R}^n} |\mathcal{K}(\mathbf{x},t;y,\tau)|^2 dy\right)^{1/2} \left(\int\_{\mathbb{R}^n} |\phi(y,\tau)|^2 dy\right)^{1/2} \end{split} \tag{82}$$

and hence

$$\|A(t,\tau)\phi(\cdot)\|\_{L^2(\mathbb{R}^d)} \le \|K(t,\tau)\|\_{L^2(\mathbb{R}^{2n})} \|\phi(\tau)\|\_{L^2(\mathbb{R}^n)} \le \|K\|\_{L^{\infty2}(I^2; \mathbb{R}^{2n})} \|\phi(\tau)\|\_{L^2(\mathbb{R}^n)}.\tag{83}$$

Therefore,

<sup>264</sup> Advances in Quantum Mechanics Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces 19 10.5772/52489 Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces http://dx.doi.org/10.5772/52489 265

$$\|A\|\_{L^{\infty}(I^{2};\mathcal{B}\to\mathcal{B})} \leq D\_{\prime} \tag{84}$$

and hence Lemma 9 and Theorem 13 apply as usual, establishing convergence of the Picard solution of *<sup>φ</sup>* = *<sup>Q</sup>*<sup>ˆ</sup> *<sup>φ</sup>* + *<sup>f</sup>* in the topology of *<sup>L</sup>*∞,2(*I*, **<sup>R</sup>***n*).

#### **5.2. Perturbation theory for the Schrödinger equation**

18 Advances in Quantum Mechanics

*5.1.2. Vector-valued functions*

*5.1.3. Hilbert–Schmidt operators*

In parallel with (73) or (77) one has


**R***n*×**R***<sup>n</sup>*

of the Neumann series within *L*∞,*p*(*I*; **R***n*).

the usual Neumann series for a scalar Volterra equation.


assumptions is uniformly bounded with norm at most *D*.

 **R***n*

≤ **R***n*

that

which

and hence

Therefore,

in place of (75). The argument concludes as before, using Lemma 10 and Theorem 14.

For *L*∞,*p*(*I*; **R***n*), the generalized Young inequality [4] assumes both (74) and (78) and assures

The argument concludes as before, using Lemma 12 and Theorem 15, proving convergence

A similar but simpler situation is where B is finite-dimensional, say **C***n*. Then *A*(*t*, *τ*) is an *n* × *n* matrix. Boundedness of *A* as an operator is automatic, but uniformity in the time variables is still a nontrivial condition. The theorem then gives a vectorial generalization of

Although the generalized Young approach yields a theorem for B = *L*2(**R***n*), the boundedness of operators on that space is often proved from a stronger condition on their kernels. In our context a Hilbert–Schmidt kernel is a function *K* : **R***<sup>n</sup>* × **R** × **R***<sup>n</sup>* × **R** → **C** for

1/2

and, of course, we also want it to be Volterra in (*t*, *τ*). In other words, *K*(*x*, *t*; *y*, *τ*) belongs to *<sup>L</sup>*∞,2(*I*2; **<sup>R</sup>**2*n*) and vanishes (or is ignored in the integrals) when *<sup>τ</sup>* > *<sup>t</sup>*. Then *A*(*t*, *τ*) : *L*2(**R***n*) → *L*2(**R***n*) defined by (72) is a Hilbert–Schmidt operator, which under our


�*A*(*t*, *<sup>τ</sup>*)*φ*(·)�*L*<sup>2</sup>(**R***<sup>n</sup>*) ≤ �*K*(*t*, *<sup>τ</sup>*)�*L*<sup>2</sup>(**R**2*<sup>n</sup>*)�*φ*(*τ*)�*L*<sup>2</sup>(**R***<sup>n</sup>*) ≤ �*K*�*L*∞,2(*I*2;**R**2*<sup>n</sup>*)�*φ*(*τ*)�*L*<sup>2</sup>(**R***<sup>n</sup>*) . (83)

<sup>2</sup>*dy*

1/2

**R***n*


<sup>2</sup>*dy*

1/2 (82)


<sup>2</sup> *dx dy*

�*A*(*t*, *<sup>τ</sup>*)*φ*�*L*∞,*<sup>p</sup>* (*I*;**R***<sup>n</sup>*) <sup>≤</sup> *<sup>D</sup>*�*φ*�*L*∞,*<sup>p</sup>* (*I*;**R***<sup>n</sup>*) . (80)

≡ �*K*(*t*, *<sup>τ</sup>*)�*L*<sup>2</sup>(**R**2*<sup>n</sup>*) <sup>≤</sup> *<sup>D</sup>* <sup>&</sup>lt; <sup>∞</sup>, (81)

In Corollary 4 we converted the time-dependent Schrödinger problem to a Volterra integral equation, (42), wherein *<sup>K</sup>*f(*x*, *<sup>y</sup>*, *<sup>t</sup>*)=(4*πit*)−*n*/2*ei*|*x*−*y*<sup>|</sup> 2/4*t* . The solution of that equation by iteration (successive approximations, Picard algorithm, Neumann series) is effectively a power series in the potential *V*, so it is the same thing as a perturbation calculation with respect to a coupling constant multiplying *V*.

In this problem the Banach space B is the Hilbert space *L*2(**R***n*). (To assure pointwise convergence to the initial data, according to Theorem 2 and Remark 3, we should also take the intersection with *L*1(**R***n*;(1 + |*x*| <sup>2</sup>)−<sup>1</sup> *dx*).) In order for our method to work simply, we must assume that *V*(*x*, *t*) is a bounded potential. It may be time-dependent, but in that case its bound should be independent of *t*. That is, we assume

$$V \in L^{\infty}(\mathbb{R}^{\mathbb{N}} \times I); \qquad |V(y, t)| \le D \text{ (almost everywhere)}.\tag{85}$$

Note that the role of *f* in the abstract Volterra equation (62) is played by the entire first integral term in (42), *K*f(*x*, *y*, *t*)*f*(*y*) *dy*.

From the other term of (42) we extract the kernel function

$$K(\mathbf{x}, t; y, \tau) = -\frac{i}{\hbar} \mathcal{K}\_{\mathbf{f}}(\mathbf{x}, y, t - \tau) V(y, \tau). \tag{86}$$

It satisfies neither the Hilbert–Schmidt condition (81) nor the generalized Young conditions (73) and (77). However, the resulting operator kernel can be factored as

$$A(t,\tau) = -\frac{i}{\hbar} \mathcal{U}\_{\mathbf{f}}(t-\tau)\hat{\mathcal{V}}(\tau),\tag{87}$$

where *U*<sup>f</sup> is the free time evolution (9) implemented by the kernel *K*<sup>f</sup> , and *V*ˆ is the operator of pointwise multiplication by the potential *V*(*y*, *τ*). It is well known [7, Chapter 4] that *<sup>U</sup>*f(*t*) = *<sup>e</sup>*−*itH*0/¯*<sup>h</sup>* is unitary, and hence its norm as an operator on *<sup>L</sup>*2(**R***n*) is 1. On the other hand,

$$\begin{split} \|V(\tau)f(\tau)\|\_{L^{2}(\mathbb{R}^{n})}^{2} &= \int\_{\mathbb{R}^{n}} |V(y,\tau)f(y,\tau)|^{2} \, dy \\ &\leq D^{2} \int\_{\mathbb{R}^{n}} |f(y,\tau)|^{2} \, dy = D^{2} \|f(\tau)\|\_{L^{2}(\mathbb{R}^{n})}^{2} \end{split} \tag{88}$$

so the operator norm of *V*ˆ(*τ*) is

$$\|\|\hat{V}(\mathbf{\tau})\| = \|V(\cdot, \mathbf{\tau})\|\_{L^{\infty}(\mathbb{R}^{n})} \le \|V\|\_{L^{\infty}(I \times \mathbb{R}^{n})} \le D. \tag{89}$$

10.5772/52489

267

http://dx.doi.org/10.5772/52489

coupling constant as in Theorem 16. The domain of validity of the construction in its simplest form is limited because the caustic structure of *K*scl can spoil the uniform boundedness;

Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces

Chapter 8 dealt with the application of the Volterra method to boundary-value problems for the Schrödinger equation. Following the heat-equation theory [1, 2, 18], the solutions were formally represented as single-layer and double-layer potentials, giving rise to Volterra integral equations on the boundary. Unfortunately, the proof in [3] of the existence and boundedness of the resulting operators is defective. The problem remains under investigation, and we hope that generalizing the Volterra theorem to a less obvious space

We are grateful to Ricardo Estrada, Arne Jensen, Peter Kuchment, and Tetsuo Tsuchida for various pieces of helpful advice. The paper [18] originally led us to the mathematical literature [1] on the series solution of the Schrödinger equation. This research was supported

Departments of Mathematics and Physics, Texas A&M University, College Station, TX, USA

[1] I. Rubinstein and L. Rubinstein. *Partial Differential Equations in Classical Mathematical*

[3] F. D. Mera. The Schrödinger equation as a Volterra problem. Master's thesis, Texas

[4] G. B. Folland. *Introduction to Partial Differential Equations*. Princeton University Press,

[5] G. H. Hardy. *Divergent Series*. Chelsea Publishing Co., New York, second edition, 1991.

[6] R. R. Estrada and R. P. Kanwal. *A Distributional Approach to Asymptotics: Theory and*

[7] L.C. Evans. *Partial Differential Equations, Graduate Studies in Mathematics*, volume 19.

[8] G. F. Carrier, M. Krook, and C. E. Pearson. *Functions of a Complex Variable: Theory and Technique*. Society for Industrial and Applied Mathematics, New York, 1966.

American Mathematical Society, Providence, RI, second edition, 2010.

[2] R. Kress. *Linear Integral Equations*. Springer-Verlag, New York, second edition, 1999.

improvements are an open field of research.

Fernando D. Mera and Stephen A. Fulling<sup>⋆</sup>

<sup>⋆</sup> Address all correspondence to: fulling@math.tamu.edu

*Physics*. Cambridge University Press, New York, 1998.

*Applications*. Birkhäuser, Boston, second edition, 2002.

A&M University, College Station, TX, May 2011.

Princeton, New Jersey, second edition, 1995.

**Acknowledgements**

**Author details**

**References**

(similarly to Theorems 14 and 15) will provide the answer.

by National Science Foundation Grants PHY-0554849 and PHY-0968269.

Therefore, the norm of the product operator is

$$\|\|A(t,\tau)\|\| = \|(i\hbar)^{-1}\mathcal{U}\_{\mathbf{f}}(t-\tau)\hat{\mathcal{V}}(\tau)\| \le D/\hbar. \tag{90}$$

Therefore, Lemma 9 and Theorem 13 apply to the integral equation (42), and we reach the desired conclusion:

**Theorem 16.** *If the potential V*(*x*, *t*) *is uniformly bounded, then the time-dependent Schrödinger problem described in Corollary 4 can be solved by iteration. That is, the perturbation (Neumann) series converges in the topology of L*∞((0, *T*), *L*2(**R***n*)) *for any finite, positive T.*

### **6. Concluding remarks**

Most mathematical physics literature on the Schrödinger equation (for example, [14]) works in an abstract Hilbert-space framework and concentrates on proving that particular second-order elliptic Hamiltonian operators are self-adjoint, then describing their spectra and other properties. Here we have investigated a different aspect of the subject; we regard the time-dependent Schrödinger equation as a classical partial differential equation analogous to the heat or wave equation and study it by classical analysis.

The similarities between the Schrödinger and heat equations were exploited to create the theoretical framework, and then their technical differences were addressed. In section 2 the structure of solutions in terms of the free propagator *K*<sup>f</sup> was worked out, and thereby the initial-value problem was recast as an integral equation.

The key feature of that equation is its Volterra character: It involves integration only up to the time in question. In this respect it is like the heat equation and unlike, for instance, the Poisson equation. The consequence of the Volterra property is that when the equation is solved by iteration, the *j*th iterate involves integration over a *j*-dimensional simplex (not a hypercube). The resulting volume factor of (*j*!)−<sup>1</sup> suggests that the series should converge.

The implementation of that idea in any particular case requires some technical work to prove that the operators *A*(*t*, *τ*) connecting any two times are bounded, and uniformly so. In section 4 we showed, in the setting of any Banach space, that that hypothesis is sufficient to establish the convergence of the Neumann series. In section 5 we verified the hypothesis in several simple examples, including the Schrödinger problem with a bounded potential.

In future work we hope to apply the Volterra theorem in contexts more complicated than the simple examples presented here. Preliminary work on those applications appears in Chapters 8 and 9 of [3]. Chapter 9 and [15] (see also [16]) implement an idea due to Balian and Bloch [17] to use a semiclassical Green function to construct a perturbation expansion for a smooth potential *V*(*x*, *t*). The solution of the Schrödinger equation is approximated in terms of classical paths, and the resulting semiclassical propagator *K*scl = *AeiS*/¯*<sup>h</sup>* is used as the building block for the exact propagator. The result is a series in ¯*h*, rather than in a coupling constant as in Theorem 16. The domain of validity of the construction in its simplest form is limited because the caustic structure of *K*scl can spoil the uniform boundedness; improvements are an open field of research.

Chapter 8 dealt with the application of the Volterra method to boundary-value problems for the Schrödinger equation. Following the heat-equation theory [1, 2, 18], the solutions were formally represented as single-layer and double-layer potentials, giving rise to Volterra integral equations on the boundary. Unfortunately, the proof in [3] of the existence and boundedness of the resulting operators is defective. The problem remains under investigation, and we hope that generalizing the Volterra theorem to a less obvious space (similarly to Theorems 14 and 15) will provide the answer.

### **Acknowledgements**

20 Advances in Quantum Mechanics

desired conclusion:

**6. Concluding remarks**

so the operator norm of *V*ˆ(*τ*) is

Therefore, the norm of the product operator is

�*V*ˆ(*τ*)� <sup>=</sup> �*V*(·, *<sup>τ</sup>*)�*L*<sup>∞</sup>(**R***<sup>n</sup>*) ≤ �*V*�*L*<sup>∞</sup>(*I*×**R***<sup>n</sup>*) <sup>≤</sup> *<sup>D</sup>*. (89)

�*A*(*t*, *<sup>τ</sup>*)� <sup>=</sup> �(*ih*¯)−1*U*f(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*V*ˆ(*τ*)� ≤ *<sup>D</sup>*/¯*h*. (90)

Therefore, Lemma 9 and Theorem 13 apply to the integral equation (42), and we reach the

**Theorem 16.** *If the potential V*(*x*, *t*) *is uniformly bounded, then the time-dependent Schrödinger problem described in Corollary 4 can be solved by iteration. That is, the perturbation (Neumann)*

Most mathematical physics literature on the Schrödinger equation (for example, [14]) works in an abstract Hilbert-space framework and concentrates on proving that particular second-order elliptic Hamiltonian operators are self-adjoint, then describing their spectra and other properties. Here we have investigated a different aspect of the subject; we regard the time-dependent Schrödinger equation as a classical partial differential equation analogous to

The similarities between the Schrödinger and heat equations were exploited to create the theoretical framework, and then their technical differences were addressed. In section 2 the structure of solutions in terms of the free propagator *K*<sup>f</sup> was worked out, and thereby the

The key feature of that equation is its Volterra character: It involves integration only up to the time in question. In this respect it is like the heat equation and unlike, for instance, the Poisson equation. The consequence of the Volterra property is that when the equation is solved by iteration, the *j*th iterate involves integration over a *j*-dimensional simplex (not a hypercube). The resulting volume factor of (*j*!)−<sup>1</sup> suggests that the series should converge.

The implementation of that idea in any particular case requires some technical work to prove that the operators *A*(*t*, *τ*) connecting any two times are bounded, and uniformly so. In section 4 we showed, in the setting of any Banach space, that that hypothesis is sufficient to establish the convergence of the Neumann series. In section 5 we verified the hypothesis in several simple examples, including the Schrödinger problem with a bounded potential.

In future work we hope to apply the Volterra theorem in contexts more complicated than the simple examples presented here. Preliminary work on those applications appears in Chapters 8 and 9 of [3]. Chapter 9 and [15] (see also [16]) implement an idea due to Balian and Bloch [17] to use a semiclassical Green function to construct a perturbation expansion for a smooth potential *V*(*x*, *t*). The solution of the Schrödinger equation is approximated in terms of classical paths, and the resulting semiclassical propagator *K*scl = *AeiS*/¯*<sup>h</sup>* is used as the building block for the exact propagator. The result is a series in ¯*h*, rather than in a

*series converges in the topology of L*∞((0, *T*), *L*2(**R***n*)) *for any finite, positive T.*

the heat or wave equation and study it by classical analysis.

initial-value problem was recast as an integral equation.

We are grateful to Ricardo Estrada, Arne Jensen, Peter Kuchment, and Tetsuo Tsuchida for various pieces of helpful advice. The paper [18] originally led us to the mathematical literature [1] on the series solution of the Schrödinger equation. This research was supported by National Science Foundation Grants PHY-0554849 and PHY-0968269.

### **Author details**

Fernando D. Mera and Stephen A. Fulling<sup>⋆</sup>

<sup>⋆</sup> Address all correspondence to: fulling@math.tamu.edu

Departments of Mathematics and Physics, Texas A&M University, College Station, TX, USA

### **References**


[9] G. A. Hagedorn and A. Joye. Semiclassical dynamics with exponentially small error estimates. *Commun. Math. Phys.*, 207:439–465, 1999.

**Chapter 12**

**Quantum Perturbation Theory in Fluid Mixtures**

Experimental assessment of macroscopic thermo-dynamical parameters under extreme conditions is almost impossible and very expensive. Therefore, theoretical EOS for further experiments or evaluation is inevitable. In spite of other efficient methods of calculation such as integral equations and computer simulations, we have used perturbation theory because of its extensive qualities. Moreover, other methods are more time consuming than perturbation theories. When one wants to deal with realistic intermolecular interactions, the problem of deriving the thermodynamic and structural properties of the system becomes rather formida‐ ble. Thus, perturbation theories of liquid have been devised since the mid-20th century. Thermodynamic perturbation theory offers a molecular, as opposed to continuum approach to the prediction of fluid thermodynamic properties. Although, perturbation predictions are not expected to rival those of advanced integral-equations or large scale computer simulations methods, they are far more numerically efficient than the latter approaches and often produced

Dealing with light species such as *He* and *H*2 at low temperature and high densities makes it necessary taking into account quantum mechanical effects. Quantum rules and shapes related

Furthermore, for this fluid mixture, the quantum effect has been exerted in terms of first order quantum mechanical correction term in the Wigner-Kirkwood expansion. This term by generalizing the Wigner-Kirkwood correction for one component fluid to binary mixture produce acceptable results in comparison with simulation and other experimental data. Since utilizing Wigner-Kirkwood expansion in temperatures below 50 K bears diverges, we preferred to restrict our investigations in ranges above those temperatures from 50 to 4000 degrees. In these regions our calculations provide more acceptable results in comparison with

> © 2013 Motevalli and Azimi; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Motevalli and Azimi; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

with the electronic orbital change completely the macroscopic properties.

S. M. Motevalli and M. Azimi

http://dx.doi.org/10.5772/54056

comparably accurate results.

other studies.

**1. Introduction**

Additional information is available at the end of the chapter

