4. The extended Feynman path integral and quantum measurement

#### 4.1. Why is it concerning with the Feynman path integral?

As we know, in the history of the quantum theory, there are three equivalent expressions, namely, the differential equation of Schrödinger, the matrix algebra of Heisenberg and the path integral formulation of Feynman. However, these three expressions have their own focuses. The Schrodinger and Heisenberg expressions focus on the evolution of states and operations, respectively, whereas the path integral formulation of Feynman on the "correlation" of point to point as states is evolving [11]. On the other hand, in quantum mechanics, when do a measurement on a wave function diffusing in all of space, such as the measurement of the position of an electron in the experiment of double-slit interference, we will find that the whole wave function will instantaneously collapse to this position measured with some probabilities. Obviously there may be some inner "correlation" in wave function transferring the action of the measurement from local part to whole. These two "correlations" have common characters and may be unified to be one.

Moreover, we notice that the action integral in Feynman path integral formulation is the classical form. The classical physics is born to be a local theory and of course cannot exhibit the character of nonlocality. However, the relativity theory is different. In relativity theory, the time and space are coupling. Beyond the light cones in Minkowski space, the space-time causality is broken, and this may cause the nonlocality. The superluminal velocities are forbidden in real world, but for a connection description of virtual paths in the path integral theory, it might be practicable. What will happen when we extend the classical action to relativistic action? Could the superluminal trajectories included in possible paths to calculate quantum amplitude in the Feynman theory cause the nonlocality? How is the relationship between "unitary evolution operation" and "quantum measurement"? These questions will be revealed when we extend the Feynman path integral.

#### 4.2. How to extend the Feynman path integral?

The formulation for Feynman path integral can be written as

$$K(r, r\_0; t, t\_0) = \mathbb{C} \sum\_{\text{all paths}} \exp\left(i\mathbb{S}/\hbar\right) \tag{1}$$

where the coefficient C is a constant independent of paths and S is the action with classical form

$$S(t\_0, t\_1) = \int\_{t\_0}^{t\_1} L(\dot{r}(t), r(t))dt\tag{2}$$

K rð Þ ;r0; t; t<sup>0</sup> in Eq. (1) is the propagator and defined into

$$K(r, r\_0; t, t\_0) = \left\langle \left\langle r \middle| \hat{\mathcal{U}}(t, t\_0) \middle| r\_0 \right\rangle \right\rangle \tag{3}$$

Eq. (1) reveals an important assumption in Feynman path integral: the weights of different paths for propagator are the same. This assumption makes Feynman path integral very successful in nonrelativistic quantum theory, but it is also the top offender that impedes the integration between Feynman path integral and relativity in non-field theory. Why should this be?

For the extension, it is necessary to break up this assumption, and Eq. (1) should be written into a more general formulation in the following:

$$F(r, r\_0; t, t\_0) = R \sum\_{\text{all paths}} W(\wp) \exp\left(i S/\hbar\right) \tag{4}$$

where R is the parameter that is independent of paths and Wð Þ ℘ is the weight function with paths [13]. Additionally, some rules should be set to limit the range of choices for R and Wð Þ ℘ :


Under these four limitations, the forms of R and W pð Þ are very few. The final forms of R and W pð Þ chosen in extended Feynman path integral are

$$R = \frac{1}{\sqrt{2i\pi\hbar c^2}} \frac{H'}{\sqrt{mc^2 + H'}} ; W(\wp) = \frac{\mathbb{P}(\wp)}{\mathcal{P}(\wp)} (\Delta \tau)^{-1/2} \tag{5}$$

The H<sup>0</sup> in Eq. (5) is the main Hamiltonian:

$$H' = \sqrt{m^2 c^4 + \left(p - A\_0\right)^2 c^2} \tag{6}$$

and

3.4. Dynamical reduction models

140 Advanced Technologies of Quantum Key Distribution

problem.

The theory of dynamical reduction models is a nonlinear and stochastic modification of the Schrödinger equation. It is proposed by Bassia and Ghirardia [10]. They integrated the master equation and linear Schrödinger equation and proposed a new nonlinear differential equation. This theory successfully solves the problems of "stochastic output" and "preferred basis" in quantum measurement and deduced the Born probability rule basing on the white noise model. However, it is still a nonrelativistic theory and remains the nonlocality

4. The extended Feynman path integral and quantum measurement

As we know, in the history of the quantum theory, there are three equivalent expressions, namely, the differential equation of Schrödinger, the matrix algebra of Heisenberg and the path integral formulation of Feynman. However, these three expressions have their own focuses. The Schrodinger and Heisenberg expressions focus on the evolution of states and operations, respectively, whereas the path integral formulation of Feynman on the "correlation" of point to point as states is evolving [11]. On the other hand, in quantum mechanics, when do a measurement on a wave function diffusing in all of space, such as the measurement of the position of an electron in the experiment of double-slit interference, we will find that the whole wave function will instantaneously collapse to this position measured with some probabilities. Obviously there may be some inner "correlation" in wave function transferring the action of the measurement from local part to whole. These two "correlations" have common

Moreover, we notice that the action integral in Feynman path integral formulation is the classical form. The classical physics is born to be a local theory and of course cannot exhibit the character of nonlocality. However, the relativity theory is different. In relativity theory, the time and space are coupling. Beyond the light cones in Minkowski space, the space-time causality is broken, and this may cause the nonlocality. The superluminal velocities are forbidden in real world, but for a connection description of virtual paths in the path integral theory, it might be practicable. What will happen when we extend the classical action to relativistic action? Could the superluminal trajectories included in possible paths to calculate quantum amplitude in the Feynman theory cause the nonlocality? How is the relationship between "unitary evolution operation" and "quantum measurement"? These questions will be revealed

4.1. Why is it concerning with the Feynman path integral?

characters and may be unified to be one.

when we extend the Feynman path integral.

4.2. How to extend the Feynman path integral?

The formulation for Feynman path integral can be written as

$$\mathbb{P}(\mathfrak{g}\mathfrak{o}) = \int\_{t\_0}^{t} |P| d\tau; \mathcal{P}(\mathfrak{g}\mathfrak{o}) = \int\_{t\_0}^{t} \left| \sqrt{2mT} \right| d\tau \tag{7}$$

calculation in 1D space for simplification. The methods of the calculation in 2D and 3D are the same. Before this calculation, we define two parameters as <sup>τ</sup><sup>0</sup> <sup>¼</sup> <sup>ℏ</sup><sup>=</sup> mc<sup>2</sup> � � and <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>ε</sup>=τ0:

> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v<sup>2</sup>=c<sup>2</sup> � � p <sup>1</sup>=<sup>2</sup>

> > 0 @

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup>

s

X <sup>m</sup> i pε ℏ

<sup>2</sup>m! du !exp ð Þ ipx

0 u �1

Figure 1. Contour integral. This figure shows the contour integral in a complex plane. The black line in figure denotes the

this contour integral, there is no singular point, and of course the total integral value is zero. Therefore,

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup> c2

<sup>2</sup> 1 � ð Þ 1 � u <sup>2</sup> � �<sup>m</sup>

ε0

1

� �<sup>2</sup><sup>m</sup> <sup>1</sup> � <sup>u</sup><sup>2</sup> � �<sup>m</sup>

exp ð Þ iuε<sup>0</sup> � <sup>i</sup>ε<sup>0</sup> du � �

<sup>0</sup> ⋯du. The integral on the red line is always zero when j j!z ∞. For

<sup>ε</sup>0exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Stochastic Quantum Potential Noise and Quantum Measurement

Aexp ð Þ �ipvε=ℏ dv

<sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> <sup>ε</sup><sup>0</sup> � �ψð Þ <sup>r</sup>0; <sup>t</sup><sup>0</sup> dv

http://dx.doi.org/10.5772/intechopen.74253

1

CCA

exp ð Þ ipx=ℏ

(12)

143

ε 1=2 0

<sup>r</sup> <sup>ε</sup>0exp �<sup>i</sup>

<sup>0</sup> exp ð Þ �iuε<sup>0</sup>

<sup>φ</sup>pexp ð Þ ipx dp <sup>ð</sup><sup>1</sup>

I<sup>0</sup> ¼ ðct �ct

> ¼ ð∞ �∞

> ¼ ð∞ �∞

<sup>¼</sup> <sup>X</sup>

integral � <sup>Ð</sup> <sup>1</sup>þi<sup>∞</sup>

Ð <sup>1</sup>þi<sup>∞</sup> <sup>1</sup> <sup>⋯</sup>du <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

<sup>1</sup> <sup>⋯</sup>du � <sup>Ð</sup> <sup>1</sup>

<sup>1</sup> ⋯du:

⋯dr<sup>0</sup> ¼ ε

φpdp 2Rτ

φpdp 2Rτ

<sup>m</sup>2R i pc ℏ

BB@

0

ðc �ct

> 1=2 0 ðc 0

1=2 0 ð1 0

� �<sup>2</sup><sup>m</sup> cε<sup>2</sup>mþ1=<sup>2</sup>

2m!

Similarly, we can also get the expression of I0:

⋯dv ¼ 2Rτ

1 þ

ε 1=2 0

ð Þ <sup>1</sup> � <sup>u</sup> �1=<sup>2</sup>

ð∞ �∞

The contour integral is used in the last step as shown in Figure 1.

<sup>0</sup> <sup>⋯</sup>du; the blue line denotes <sup>Ð</sup> <sup>∞</sup>

1=2 0 ðct �ct

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup> c2 � � q <sup>1</sup>=<sup>2</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup> c2

> > ε 1=2

P, T and Δτ are called the momentum, kinetic energy and proper time in terms of fourdimensional space–time, respectively:

$$|P| = \frac{mv}{\sqrt{1 - v^2/c^2}},\\ T = \frac{mc^2}{\sqrt{1 - v^2/c^2}} - mc^2,\\ \Delta\tau = \int\_0^t \frac{1}{\sqrt{1 - v^2/c^2}}d\tau \tag{8}$$

The expressions of Wð Þ ℘ and R are very interesting. As we can see, under the low-energy and low-velocity condition, H<sup>0</sup> <sup>≪</sup> mc<sup>2</sup> and <sup>v</sup> <sup>≪</sup> <sup>c</sup>, then <sup>R</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffi <sup>2</sup>iπℏc<sup>2</sup> <sup>p</sup> and <sup>W</sup>ð Þ¼ <sup>℘</sup> ð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup> 1=2 because j j <sup>P</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi <sup>2</sup>mT <sup>p</sup> in classical physical theory. This means Eq. (4) can be transformed into the Feynman path integral if we choose the formulations of W pð Þ and R as shown in Eq. (5). What is concerning then for us is what we can get from Eq. (4) under very high energy and velocity.

#### 4.3. The new differential equation and Klein-Gordon equation

It is hard to directly calculate the value of Eq. (4) because the path integral is not normal integral term and the normal integral method is invalid for Eq. (4). A way to get some results from Eq. (4) is to follow the method that Feynman used [11, 12]. We consider a minimal evolution time process, t ¼ t<sup>0</sup> þ ε, where ε ! 0. In this process:

$$
\psi(r, t\_0 + \varepsilon) = \int\_{-\infty}^{\infty} \psi(r\_0, t\_0) F(r, r\_0; t\_0 + \varepsilon, t\_0) dr\_0 = R \int\_{-\infty}^{\infty} \psi(r\_0, t\_0) W(\mathfrak{g}) dr\_0 \tag{9}
$$

When ε ! 0, the weight function Wð Þ ℘ can be simply expressed the term of

$$W(\wp) = \frac{\left(1 + \sqrt{1 - v^2/c^2}\right)^{1/2}}{\varepsilon^{1/2}\sqrt{1 - v^2/c^2}}\tag{10}$$

where v ¼ ð Þ r � r<sup>0</sup> =ε. This value can be greater than the superluminal velocity, and F rð Þ ;r0; t<sup>0</sup> þ ε; t<sup>0</sup> therefore will become the complex function when v > c. The integral form should be departed into two parts: the part that contains the low-velocity paths and the part that contains superluminal-velocity paths:

$$I = \int\_{-\infty}^{\infty} \psi(r\_0, t\_0) F(r, r\_0; t\_0 + \varepsilon, t\_0) dr\_0 = \int\_{-ct}^{ct} \cdots dr\_0 + \left(\int\_{ct}^{\infty} \cdots dr\_0 + \int\_{-\infty}^{-ct} \cdots dr\_0\right) = I\_0 + I\_1 \tag{11}$$

This can be exactly calculated. The amazing thing is the final result calculated for I that contains the term ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∇ þ A<sup>0</sup> 2 c2 q . In the following context, we will detail this calculation in 1D space for simplification. The methods of the calculation in 2D and 3D are the same. Before this calculation, we define two parameters as <sup>τ</sup><sup>0</sup> <sup>¼</sup> <sup>ℏ</sup><sup>=</sup> mc<sup>2</sup> � � and <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>ε</sup>=τ0:

I<sup>0</sup> ¼ ðct �ct ⋯dr<sup>0</sup> ¼ ε ðc �ct ⋯dv ¼ 2Rτ 1=2 0 ðct �ct <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v<sup>2</sup>=c<sup>2</sup> � � p <sup>1</sup>=<sup>2</sup> ε 1=2 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> <sup>ε</sup>0exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> <sup>ε</sup><sup>0</sup> � �ψð Þ <sup>r</sup>0; <sup>t</sup><sup>0</sup> dv ¼ ð∞ �∞ φpdp 2Rτ 1=2 0 ðc 0 1 þ ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup> c2 � � q <sup>1</sup>=<sup>2</sup> ε 1=2 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup> c2 <sup>r</sup> <sup>ε</sup>0exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup> c2 s ε0 0 @ 1 Aexp ð Þ �ipvε=ℏ dv 0 BB@ 1 CCA exp ð Þ ipx=ℏ ¼ ð∞ �∞ φpdp 2Rτ 1=2 0 ð1 0 ð Þ <sup>1</sup> � <sup>u</sup> �1=<sup>2</sup> ε 1=2 <sup>0</sup> exp ð Þ �iuε<sup>0</sup> X <sup>m</sup> i pε ℏ � �<sup>2</sup><sup>m</sup> <sup>1</sup> � <sup>u</sup><sup>2</sup> � �<sup>m</sup> <sup>2</sup>m! du !exp ð Þ ipx <sup>¼</sup> <sup>X</sup> <sup>m</sup>2R i pc ℏ � �<sup>2</sup><sup>m</sup> cε<sup>2</sup>mþ1=<sup>2</sup> 2m! ð∞ �∞ <sup>φ</sup>pexp ð Þ ipx dp <sup>ð</sup><sup>1</sup> 0 u �1 <sup>2</sup> 1 � ð Þ 1 � u <sup>2</sup> � �<sup>m</sup> exp ð Þ iuε<sup>0</sup> � <sup>i</sup>ε<sup>0</sup> du � � (12)

Similarly, we can also get the expression of I0:

Pð Þ¼ ℘

<sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> , T <sup>¼</sup> mc<sup>2</sup>

4.3. The new differential equation and Klein-Gordon equation

evolution time process, t ¼ t<sup>0</sup> þ ε, where ε ! 0. In this process:

ð∞ �∞

dimensional space–time, respectively:

142 Advanced Technologies of Quantum Key Distribution

and low-velocity condition, H<sup>0</sup>

ψð Þ¼ r; t<sup>0</sup> þ ε

that contains superluminal-velocity paths:

q

ψð Þ r0; t<sup>0</sup> F rð Þ ;r0; t<sup>0</sup> þ ε; t<sup>0</sup> dr<sup>0</sup> ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∇ þ A<sup>0</sup>

because j j <sup>P</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi

velocity.

I ¼ ð∞ �∞

contains the term

j j <sup>P</sup> <sup>¼</sup> mv ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðt t0

j j P dτ;Pð Þ¼ ℘

P, T and Δτ are called the momentum, kinetic energy and proper time in terms of four-

The expressions of Wð Þ ℘ and R are very interesting. As we can see, under the low-energy

the Feynman path integral if we choose the formulations of W pð Þ and R as shown in Eq. (5). What is concerning then for us is what we can get from Eq. (4) under very high energy and

It is hard to directly calculate the value of Eq. (4) because the path integral is not normal integral term and the normal integral method is invalid for Eq. (4). A way to get some results from Eq. (4) is to follow the method that Feynman used [11, 12]. We consider a minimal

ψð Þ r0; t<sup>0</sup> F rð Þ ;r0; t<sup>0</sup> þ ε; t<sup>0</sup> dr<sup>0</sup> ¼ R

<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v<sup>2</sup>=c<sup>2</sup> � � p <sup>1</sup>=<sup>2</sup>

where v ¼ ð Þ r � r<sup>0</sup> =ε. This value can be greater than the superluminal velocity, and F rð Þ ;r0; t<sup>0</sup> þ ε; t<sup>0</sup> therefore will become the complex function when v > c. The integral form should be departed into two parts: the part that contains the low-velocity paths and the part

⋯dr<sup>0</sup> þ

This can be exactly calculated. The amazing thing is the final result calculated for I that

ð∞ ct

⋯dr<sup>0</sup> þ

ð�ct �∞

. In the following context, we will detail this

� �

⋯dr<sup>0</sup>

¼ I<sup>0</sup> þ I<sup>1</sup> (11)

ðct �ct

> 2 c2

ε<sup>1</sup>=<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

When ε ! 0, the weight function Wð Þ ℘ can be simply expressed the term of

Wð Þ¼ ℘

<sup>≪</sup> mc<sup>2</sup> and <sup>v</sup> <sup>≪</sup> <sup>c</sup>, then <sup>R</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffi

<sup>2</sup>mT <sup>p</sup> in classical physical theory. This means Eq. (4) can be transformed into

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> � mc<sup>2</sup>

ðt t0

ffiffiffiffiffiffiffiffiffi 2mT � � p �

, Δτ ¼

� � �

ðt t0

ð∞ �∞

<sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> (10)

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dτ (7)

<sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> <sup>d</sup><sup>τ</sup> (8)

1=2

<sup>2</sup>iπℏc<sup>2</sup> <sup>p</sup> and <sup>W</sup>ð Þ¼ <sup>℘</sup> ð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup>

ψð Þ r0; t<sup>0</sup> Wð Þ ℘ dr<sup>0</sup> (9)

The contour integral is used in the last step as shown in Figure 1.

Figure 1. Contour integral. This figure shows the contour integral in a complex plane. The black line in figure denotes the integral � <sup>Ð</sup> <sup>1</sup>þi<sup>∞</sup> <sup>1</sup> <sup>⋯</sup>du � <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>⋯</sup>du; the blue line denotes <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> ⋯du. The integral on the red line is always zero when j j!z ∞. For this contour integral, there is no singular point, and of course the total integral value is zero. Therefore, Ð <sup>1</sup>þi<sup>∞</sup> <sup>1</sup> <sup>⋯</sup>du <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup> <sup>1</sup> ⋯du:

$$\begin{split} I\_{0} &= \int\_{t}^{\pi} \cdots dr\_{0} + \varepsilon \int\_{-\pi}^{-\varepsilon t} \cdots dr\_{0} = 2R\tau\_{0}^{1/2} \int\_{t}^{\pi} \frac{\left(1 + \sqrt{1 - v^{2}/c^{2}}\right)^{1/2}}{\varepsilon\_{0}^{1/2} \sqrt{1 - v^{2}/c^{2}}} \cdot \varepsilon\_{0} \exp\left(-i\sqrt{1 - v^{2}/c^{2}}\varepsilon\_{0}\right) \\ &\times \left(\psi(r\_{0}, t\_{0}) + \psi(-r\_{0}, t\_{0})\right) du = \sum\_{m} 2R\left(i\frac{\nu\varepsilon}{\hbar}\right)^{2m} \frac{c\varepsilon^{2m + 1/2}}{2m!} \int\_{-\infty}^{\infty} \varphi\_{p} \exp\left(i\mathrm{p}x\right) \\ &\times dp \left(\int\_{1}^{1+i\nu} u^{\frac{\pi}{2}} \left(1 - \left(1 - u\right)^{2}\right)^{m} \exp\left(i\nu\varepsilon\_{0} - i\varepsilon\_{0}\right) du\right) \\ &= \sum\_{m} 2R\left(i\frac{\nu\varepsilon}{\hbar}\right)^{2m} \frac{c\varepsilon^{2m + 1/2}}{2m!} \int\_{-\infty}^{\infty} \varphi\_{p} \exp\left(i\mathrm{p}x\right) dp \left(\int\_{1}^{\nu} u^{\frac{\pi}{2}} \left(1 - \left(1 - u\right)^{2}\right)^{m} \exp\left(i\nu\varepsilon\_{0} - i\varepsilon\_{0}\right) du\right) \end{split} \tag{13}$$

iℏ d

dtψð Þ¼ <sup>r</sup>; <sup>t</sup>

iℏ d

iℏ d

iℏ d

interpretation for Klein-Cordon equation by EFPI theory.

4.4. The extended Feynman path integral and density-flux equation

equation.

Combining Eqs. (20) and (21), we get these two equations:

dt � V rð Þ � �ϕ� ¼ �

dt � V rð Þ � �ψ<sup>þ</sup> ¼ �

dt � V rð Þ � �ψ� ¼ �

The more general formulation in 3D is

iℏ d

dtψð Þ¼ <sup>x</sup>; <sup>t</sup>

q

deduction can be seen in supplementary online material of the reference [13].

function <sup>ϕ</sup>þ. <sup>ϕ</sup><sup>þ</sup> satisfied the relation that Eq. (20) has shown and <sup>ϕ</sup>� is satisfied

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>m</sup><sup>2</sup>c<sup>4</sup> þ �ð Þ icℏ∂<sup>x</sup> <sup>2</sup>

> 2 c2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∇ � A<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∇ � A<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∇ � A<sup>0</sup>

2 c2

2 c2

2 c2

<sup>þ</sup> V rð Þ � �ψð Þ <sup>r</sup>; <sup>t</sup> (20)

Stochastic Quantum Potential Noise and Quantum Measurement

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∇ � A<sup>0</sup>

It is more complicated to get Eq. (20), and we will not detail it in this chapter. The detailed

It should be mentioned that Eq. (20) is not a covariant equation under the Lorentz transformation. To construct a Lorentz covariant, the antiparticle wave function should be introduced. The antiparticle wave function is denoted as <sup>ϕ</sup>� to be distinguished from the particle wave

q

q

q

where and . Eqs. (22) and (23) are the Klein-Gordon

In 1926, Oskar Klein and Walter Gordon proposed this relativistic wave equation. However, it was found later that this equation is not suitable for one particle because the probability density is not a positive quantity, which means the particle can be created and annihilated arbitrarily in Klein-Gordon equation [14]. The extended Feynman path integral shows the explanation for this non-positive probability density here. The wave function that is determined by Klein-Gordon equation is the mixed state of the particle and its antiparticle. Because particles and antiparticles can be annihilated each other to a vacuum state, and the vacuum state can produce particles and antiparticles, so the mixed state with superposition state of a particle and an antiparticle is a matter of course of a non-positive quantity. This is the physical

In quantum mechanics, the continuity equation describes the conservation of probability density in the transport process. It is a local form of conservation laws. It says the probability cannot be created or annihilated and, at the same time, also cannot be teleported from one

ψð Þ x; t (19)

http://dx.doi.org/10.5772/intechopen.74253

145

<sup>ϕ</sup>� (21)

<sup>ψ</sup><sup>þ</sup> (22)

<sup>ψ</sup>� (23)

Integrating Eq. (12) and Eq. (13), we get the conclusion finally:

$$\begin{split} I &= \sum\_{m} 2R\left(l\frac{\mu}{\hbar}\right)^{2m} \frac{c\mathcal{E}^{2m+1/2}}{2m!} \int\_{0}^{\infty} \wp\_{p} \exp\left(ip\mathbf{x}\right) dp \left(\int\_{0}^{u} u^{\frac{1}{2}} \left(1 - \left(1 - u\right)^{2}\right)^{m} \exp\left(i\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}\_{0} - i\boldsymbol{\varepsilon}\_{0}\right) du\right) \\ &= \int\_{0}^{\infty} \sum\_{m} 2R\left(l\frac{\mu}{\hbar}\right)^{2m} \frac{c\mathcal{E}^{2m + \frac{1}{2}}}{2m!} \Gamma\left(2m + \frac{1}{2}\right) M\left(-m, \frac{1}{2} - 2m, -2i\boldsymbol{\varepsilon}\_{0}\right) \wp\_{p} \exp\left(i\boldsymbol{p}\boldsymbol{x}\right) dp \end{split} \tag{14}$$

The function M að Þ ; b; z is the Kummer's function (confluent hypergeometric function) and equals

$$M\left(-m, \frac{1}{2} - 2m, -2i\varepsilon\_0\right) = \sum\_{n} \frac{m!}{n!} \frac{(4m-1)!}{n!} \left(-i\varepsilon\_0\right)^n \tag{15}$$

Summation in Eq. (14) is then

$$\sum\_{m} \mathcal{R}\left(i\frac{pc}{\hbar}\right)^{2m} \frac{c\varepsilon^{2m+\frac{1}{2}}}{2m!} \Gamma\left(2m+\frac{1}{2}\right) M\left(-m, \frac{1}{2} - 2m, -2i\varepsilon\_0\right) = \exp\left(\frac{-i\sqrt{m^2c^4 + p^2c^2}\varepsilon}{\hbar}\right) \tag{16}$$

And Eq. (14) can be further simplified:

$$I = \int\_0^\infty \exp\left(\frac{-i\sqrt{m^2c^4 + p^2c^2}\varepsilon}{\hbar}\right) \varphi\_p \exp\left(ipx\right) dp$$

$$\exp\left(\frac{-i\sqrt{m^2c^4 + (-ic\hbar\nabla\_x)\varepsilon}}{\hbar}\right) \int\_0^\infty \varphi\_p \exp(ipx) dp = \exp\left(\frac{-i\sqrt{m^2c^4 + (-ic\hbar\partial\_x)\varepsilon}}{\hbar}\right) \psi(x, t\_0)$$

It is, namely:

$$\psi(\mathbf{x}, t\_0 + \varepsilon) = \exp\left(\frac{-i\sqrt{m^2c^4 + (-ic\hbar\mathfrak{d}\_\mathbf{x})^2}\varepsilon}{\hbar}\right)\psi(\mathbf{x}, t\_0) \tag{18}$$

Hence, the new differential equation we get in this extended Feynman path integral is

Stochastic Quantum Potential Noise and Quantum Measurement http://dx.doi.org/10.5772/intechopen.74253 145

$$i\hbar\frac{d}{dt}\psi(\mathbf{x},t) = \sqrt{m^2c^4 + \left(-ic\hbar\partial\_x\right)^2}\psi(\mathbf{x},t)\tag{19}$$

The more general formulation in 3D is

<sup>I</sup><sup>0</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

ct ⋯dr<sup>0</sup> þ ε

144 Advanced Technologies of Quantum Key Distribution

<sup>m</sup>2R i pc ℏ

<sup>m</sup>2R i pc ℏ

> <sup>m</sup>2R i pc ℏ � �<sup>2</sup><sup>m</sup> <sup>c</sup>ε<sup>2</sup>mþ<sup>1</sup>

Summation in Eq. (14) is then

� �<sup>2</sup><sup>m</sup> cε<sup>2</sup>mþ<sup>1</sup>

And Eq. (14) can be further simplified:

m<sup>2</sup>c<sup>4</sup> þ �ð Þ icℏ∇<sup>x</sup> p ε ℏ ! <sup>Ð</sup> <sup>∞</sup>

exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>m</sup>2R i pc ℏ

�dp <sup>Ð</sup> <sup>1</sup>þi<sup>∞</sup> <sup>1</sup> u �1

<sup>¼</sup> <sup>P</sup>

<sup>I</sup> <sup>¼</sup> <sup>P</sup>

equals

X

It is, namely:

<sup>¼</sup> <sup>Ð</sup> <sup>∞</sup> 0 P Ð �ct

�ð Þ <sup>ψ</sup>ð Þþ <sup>r</sup>0; <sup>t</sup><sup>0</sup> <sup>ψ</sup>ð Þ �r0; <sup>t</sup><sup>0</sup> du <sup>¼</sup> <sup>P</sup>

� �<sup>2</sup><sup>m</sup> <sup>c</sup>ε<sup>2</sup>mþ1=<sup>2</sup> 2m!

� �<sup>2</sup><sup>m</sup> <sup>c</sup>ε<sup>2</sup>mþ1=<sup>2</sup> 2m!

<sup>2</sup> 1 � ð Þ 1 � u <sup>2</sup> � �<sup>m</sup>

�<sup>∞</sup> <sup>⋯</sup>dr<sup>0</sup> <sup>¼</sup> <sup>2</sup>R<sup>τ</sup>

� �

ð∞ �∞

Integrating Eq. (12) and Eq. (13), we get the conclusion finally:

ð∞ 0

2 <sup>2</sup>m! <sup>Γ</sup> <sup>2</sup><sup>m</sup> <sup>þ</sup>

1

<sup>2</sup> � <sup>2</sup>m; �2iε<sup>0</sup> � �

> 1 2 � �

<sup>ψ</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>ε</sup> exp �<sup>i</sup>

M �m;

<sup>0</sup> exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

<sup>m</sup><sup>2</sup>c<sup>4</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>c<sup>2</sup> <sup>p</sup> <sup>ε</sup> ℏ !

q

Hence, the new differential equation we get in this extended Feynman path integral is

0 @

M �m;

2 <sup>2</sup>m! <sup>Γ</sup> <sup>2</sup><sup>m</sup> <sup>þ</sup>

<sup>I</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

1=2 0 Ð ∞ ct

> <sup>m</sup>2R i pc ℏ

φpexp ð Þ ipx dp

φpexp ð Þ ipx dp

1 2 � �

<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v<sup>2</sup>=c<sup>2</sup> � � p <sup>1</sup>=<sup>2</sup>

� �<sup>2</sup><sup>m</sup> <sup>c</sup>ε<sup>2</sup>mþ1=<sup>2</sup> 2m!

> ð∞ 1 u �1

ð∞ 0 u �1

1

M �m;

The function M að Þ ; b; z is the Kummer's function (confluent hypergeometric function) and

<sup>¼</sup> <sup>X</sup> n m! n!

<sup>2</sup> � <sup>2</sup>m; �2iε<sup>0</sup> � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup>

> ð∞ �∞

<sup>2</sup> 1 � ð Þ 1 � u <sup>2</sup> � �<sup>m</sup>

<sup>2</sup> 1 � ð Þ 1 � u <sup>2</sup> � �<sup>m</sup>

ð Þ 4m � 1 !

φpexp ð Þ ipx dp

ε

1

<sup>0</sup> <sup>φ</sup>pexp ð Þ ipx dp <sup>¼</sup> exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>m</sup><sup>2</sup>c<sup>4</sup> þ �ð Þ icℏ∂<sup>x</sup> <sup>2</sup>

ℏ

<sup>2</sup> � <sup>2</sup>m; �2iε<sup>0</sup> � �

<sup>ε</sup>0exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

φpexp ð Þ ipx

� �

φpexp ð Þ ipx dp

<sup>¼</sup> exp �<sup>i</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m<sup>2</sup>c<sup>4</sup> þ �ð Þ icℏ∂<sup>x</sup> p ε ℏ !

� �

<sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> <sup>ε</sup><sup>0</sup> � �

exp ð Þ iuε<sup>0</sup> � iε<sup>0</sup> du

exp ð Þ iuε<sup>0</sup> � iε<sup>0</sup> du

<sup>n</sup>! ð Þ �iε<sup>0</sup> <sup>n</sup> (15)

<sup>m</sup><sup>2</sup>c<sup>4</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>c<sup>2</sup> <sup>p</sup> <sup>ε</sup> ℏ !

ψð Þ x; t<sup>0</sup>

Aψð Þ x; t<sup>0</sup> (18)

(13)

(14)

(16)

(17)

ε 1=2 0

exp ð Þ iuε<sup>0</sup> � iε<sup>0</sup> du

$$i\hbar\frac{d}{dt}\psi(r,t) = \left(\sqrt{m^2c^4 + \left(-i\hbar\nabla - A\_0\right)^2c^2} + V(r)\right)\psi(r,t)\tag{20}$$

It is more complicated to get Eq. (20), and we will not detail it in this chapter. The detailed deduction can be seen in supplementary online material of the reference [13].

It should be mentioned that Eq. (20) is not a covariant equation under the Lorentz transformation. To construct a Lorentz covariant, the antiparticle wave function should be introduced. The antiparticle wave function is denoted as <sup>ϕ</sup>� to be distinguished from the particle wave function <sup>ϕ</sup>þ. <sup>ϕ</sup><sup>þ</sup> satisfied the relation that Eq. (20) has shown and <sup>ϕ</sup>� is satisfied

$$\left(i\hbar\frac{d}{dt} - V(r)\right)\phi\_- = -\sqrt{m^2c^4 + \left(-i\hbar\nabla - A\_0\right)^2c^2}\phi\_-\tag{21}$$

Combining Eqs. (20) and (21), we get these two equations:

$$\left(i\hbar\frac{d}{dt} - V(r)\right)\psi\_{+} = -\sqrt{m^{2}c^{4} + \left(-i\hbar\nabla - A\_{0}\right)^{2}c^{2}}\psi\_{+}\tag{22}$$

$$\left(i\hbar\frac{d}{dt} - V(r)\right)\psi\_- = -\sqrt{m^2c^4 + \left(-i\hbar\nabla - A\_0\right)^2c^2}\psi\_-\tag{23}$$

where and . Eqs. (22) and (23) are the Klein-Gordon equation.

In 1926, Oskar Klein and Walter Gordon proposed this relativistic wave equation. However, it was found later that this equation is not suitable for one particle because the probability density is not a positive quantity, which means the particle can be created and annihilated arbitrarily in Klein-Gordon equation [14]. The extended Feynman path integral shows the explanation for this non-positive probability density here. The wave function that is determined by Klein-Gordon equation is the mixed state of the particle and its antiparticle. Because particles and antiparticles can be annihilated each other to a vacuum state, and the vacuum state can produce particles and antiparticles, so the mixed state with superposition state of a particle and an antiparticle is a matter of course of a non-positive quantity. This is the physical interpretation for Klein-Cordon equation by EFPI theory.

#### 4.4. The extended Feynman path integral and density-flux equation

In quantum mechanics, the continuity equation describes the conservation of probability density in the transport process. It is a local form of conservation laws. It says the probability cannot be created or annihilated and, at the same time, also cannot be teleported from one place to another. However, in the extended Feynman path integral, the density-flux equation will be revised, and the local conservation is broken.

In extended Feynman path integral, the density-flux equation can be written as the following formula:

$$\frac{\partial \rho(r,t)}{\partial t} + \nabla \cdot \dot{\boldsymbol{j}} + \sum\_{n=2}^{\text{res}} \boldsymbol{B}\_{n} \nabla^{n} \cdot \boldsymbol{Q}\_{n}(r,t) = \mathbf{0} \tag{24}$$

Considering a minimum time-evolution process, the propagator is

A0ðx0, tÞd<sup>0</sup> exists in the integral formula of Eq. (28), then lim<sup>ε</sup>!<sup>0</sup>

Stochastic Quantum Potential Noise and Quantum Measurement

http://dx.doi.org/10.5772/intechopen.74253

F xð Þ <sup>1</sup>; x0; t<sup>0</sup> þ ε; t<sup>0</sup> 6¼ δð Þ x<sup>1</sup> � x<sup>0</sup> . This is different from the normal propagator K xð Þ ; x0; t<sup>0</sup> þ ε; t<sup>0</sup> shown in Eq. (2), because lim<sup>ε</sup>!<sup>0</sup> K xð ; x0; t<sup>0</sup> þ ε; t0Þ ¼ δð Þ x � x<sup>0</sup> . This difference, caused by relativistic effect of paths, is the root that produces the nonlocality in quantum measurement

lim<sup>ε</sup>!<sup>0</sup> F xð Þ <sup>1</sup>; x0; t<sup>0</sup> þ ε; t<sup>0</sup> 6¼ δð Þ x<sup>0</sup> � x<sup>0</sup> means the change of arbitrary point should spend time to propagate the other point and exhibit stronge nonlocal space-time character. If the value of wave function at x ¼ x<sup>0</sup> changes, the whole wave function will change for the nonlocal prop-

<sup>R</sup><sup>b</sup> <sup>≈</sup>Rb<sup>0</sup> <sup>1</sup> � AIc<sup>2</sup>ð Þ <sup>b</sup><sup>p</sup> � <sup>A</sup><sup>0</sup>

After this definition, we will show how the measurement happens under the potential noise.

H0, if we put the potential noise in this system, the initial state will change. We denote the evolution state in arbitrary time t as ψðx, tÞ. The ψðx, tÞ can be expanded with basis states

H<sup>0</sup>

P

<sup>m</sup>amφm. The task for us is to find out the varying value of am under each

� � (30)

<sup>m</sup>amφmð Þx , where φ<sup>m</sup> is the eigenstate of

Because the term Ð <sup>x</sup>

process. In fact:

Therefore

We define <sup>R</sup>b<sup>0</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffi

<sup>φ</sup><sup>m</sup> as <sup>ψ</sup>ðx, tÞ ¼ <sup>P</sup>

perturbational noise:

<sup>2</sup>iπℏc<sup>2</sup> <sup>p</sup>

x�η

agator. In the followings, we will detail this character.

<sup>H</sup><sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffi mc2þH<sup>0</sup> p ; then

Considering an initial state with the form ψð Þ¼ x; t<sup>0</sup>

ð28Þ

147

ð29Þ

where Qnð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>ψ</sup>∗∇<sup>n</sup><sup>ψ</sup> � <sup>ψ</sup>∇<sup>n</sup>ψ<sup>∗</sup> and Bn ¼��ð Þ <sup>i</sup><sup>ℏ</sup> <sup>2</sup>n�<sup>1</sup> c<sup>2</sup>n= mc<sup>2</sup> � �<sup>2</sup>n�<sup>1</sup> . The last term in the right of Eq. (24) is caused by relativistic effect and breaks the local conservation.

#### 4.5. The wave function collapse in extended Feynman path integral

From the theory of Neumann, the difficulties of understanding collapse are the probability, which seems incompatible with the deterministic time-evolution equation, and the instantaneity, which seems that it breaks the special relativity theory. In this section, we will show that these puzzling characters are due to the potential noise and nonlocal correlation (or relativistic effect).

Let us return to Eq. (9). The superluminal paths are included when we calculate the propagator. The superluminal paths will support complex phases in Eq. (9), and these phases cannot be canceled by each other like the real phases in Feynman path integral theory. These complex phases are the main culprits that cause the nonlocal correlation.

To describe this mechanism concisely, the nonlocal correlation produced in 1D space is just detailed here. Assume a system in the potential field with the scalar potential U xð Þ and vector A0ð Þx . A potential noise AIð Þt is under this system and satisfies the white noise equations, namely:

$$
\langle A\_I(t\_1) A\_I(t\_0) \rangle = \frac{2mk\_bT}{\eta} \delta(t\_1 - t\_0); \langle A\_I(t) \rangle = 0 \tag{25}
$$

The Hamiltonian of this system is then

$$H = \sqrt{m^2 c^4 + \left(-i\hbar \partial\_x - (A\_0 + A\_I)\right)^2 c^2} + V(\mathbf{x}) \tag{26}$$

And we define a new Hamiltonian without potential noise as

$$H\_0 = \sqrt{m^2c^4 + \left(-i\hbar\partial\_\mathbf{x} - A\_0\right)^2c^2} + V(\mathbf{x})\tag{27}$$

We will see later that H<sup>0</sup> is very important in quantum measurement, because it determines the basis-state-space that the wave function collapses into. The basis-preferred problem puzzles us for many years; we do not know why the system measured prefers to collapse into some set of basis state. According to the extended Feynman path integral theory, the preferred basis is depended by the Hamiltonian H0. This will be detailed in the following.

Considering a minimum time-evolution process, the propagator is

$$F(\mathbf{x}\_1, \mathbf{x}\_0; t\_0 + \varepsilon, \mathbf{t}\_0) = \mathcal{R} \sqrt{\frac{c}{l\eta}} \exp(-mc|\eta|\hbar^{-1} + i\hbar^{-1} \int\_{\mathbf{x}-\eta}^{\mathbf{x}} A\_0(\mathbf{x}\_0, t) d\mathbf{x}\_0) \tag{28}$$

Because the term Ð <sup>x</sup> x�η A0ðx0, tÞd<sup>0</sup> exists in the integral formula of Eq. (28), then lim<sup>ε</sup>!<sup>0</sup> F xð Þ <sup>1</sup>; x0; t<sup>0</sup> þ ε; t<sup>0</sup> 6¼ δð Þ x<sup>1</sup> � x<sup>0</sup> . This is different from the normal propagator K xð Þ ; x0; t<sup>0</sup> þ ε; t<sup>0</sup> shown in Eq. (2), because lim<sup>ε</sup>!<sup>0</sup> K xð ; x0; t<sup>0</sup> þ ε; t0Þ ¼ δð Þ x � x<sup>0</sup> . This difference, caused by relativistic effect of paths, is the root that produces the nonlocality in quantum measurement process.

In fact:

place to another. However, in the extended Feynman path integral, the density-flux equation

In extended Feynman path integral, the density-flux equation can be written as the following

n¼2

From the theory of Neumann, the difficulties of understanding collapse are the probability, which seems incompatible with the deterministic time-evolution equation, and the instantaneity, which seems that it breaks the special relativity theory. In this section, we will show that these puzzling characters are due to the potential noise and nonlocal correlation (or relativistic

Let us return to Eq. (9). The superluminal paths are included when we calculate the propagator. The superluminal paths will support complex phases in Eq. (9), and these phases cannot be canceled by each other like the real phases in Feynman path integral theory. These complex

To describe this mechanism concisely, the nonlocal correlation produced in 1D space is just detailed here. Assume a system in the potential field with the scalar potential U xð Þ and vector A0ð Þx . A potential noise AIð Þt is under this system and satisfies the white noise equations,

η

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∂<sup>x</sup> � ð Þ A<sup>0</sup> þ AI

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ �ð Þ iℏ∂<sup>x</sup> � A<sup>0</sup>

We will see later that H<sup>0</sup> is very important in quantum measurement, because it determines the basis-state-space that the wave function collapses into. The basis-preferred problem puzzles us for many years; we do not know why the system measured prefers to collapse into some set of basis state. According to the extended Feynman path integral theory, the preferred basis is

2 c2

2 c2

Bn∇<sup>n</sup> � Qnð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>0</sup> (24)

δð Þ t<sup>1</sup> � t<sup>0</sup> ; Ah i <sup>I</sup>ð Þt ¼ 0 (25)

þ V xð Þ (26)

þ V xð Þ (27)

. The last term in the right

c<sup>2</sup>n= mc<sup>2</sup> � �<sup>2</sup>n�<sup>1</sup>

will be revised, and the local conservation is broken.

146 Advanced Technologies of Quantum Key Distribution

∂rð Þ r; t

where Qnð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>ψ</sup>∗∇<sup>n</sup><sup>ψ</sup> � <sup>ψ</sup>∇<sup>n</sup>ψ<sup>∗</sup> and Bn ¼��ð Þ <sup>i</sup><sup>ℏ</sup> <sup>2</sup>n�<sup>1</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>∇</sup> � <sup>j</sup> <sup>þ</sup>X<sup>∞</sup>

of Eq. (24) is caused by relativistic effect and breaks the local conservation.

4.5. The wave function collapse in extended Feynman path integral

phases are the main culprits that cause the nonlocal correlation.

The Hamiltonian of this system is then

H ¼

q

And we define a new Hamiltonian without potential noise as

H<sup>0</sup> ¼

q

depended by the Hamiltonian H0. This will be detailed in the following.

h i AIð Þ <sup>t</sup><sup>1</sup> AIð Þ <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>mkbT

formula:

effect).

namely:

$$\int\_{t\_0}^{t\_0+\varepsilon} -mc^2\sqrt{1-v^2/c^2} \, dt = \int\_{t\_0}^{t\_0+\varepsilon} -mc^2\sqrt{(dt)^2 - dx^2/c^2} = imc\{\Delta x\};$$

$$\int\_{t\_0}^{t\_0+\varepsilon} (-U(\varkappa) + A\nu)dt = A\Delta\varkappa$$

Therefore

$$F(\mathbf{x}\_1, \mathbf{x}\_0; \mathbf{t}\_0 + \varepsilon, \mathbf{t}\_0) = \frac{1}{\sqrt{2\pi\hbar c^2}} \frac{\hbar^{\prime}}{\sqrt{mc^2 + \hbar^{\prime}}} \sqrt{\frac{c}{\hbar}} \exp\{-mc|\eta|\hbar^{-1} + i\hbar^{-1} \int\_{\mathbf{x} - \eta}^{\mathbf{x}} A\_0(\mathbf{x}\_0, \mathbf{t}) d\mathbf{x}\_0\} \tag{29}$$
 
$$\psi\_- = 1/\sqrt{2} (\phi\_+ - \phi\_-)$$

lim<sup>ε</sup>!<sup>0</sup> F xð Þ <sup>1</sup>; x0; t<sup>0</sup> þ ε; t<sup>0</sup> 6¼ δð Þ x<sup>0</sup> � x<sup>0</sup> means the change of arbitrary point should spend time to propagate the other point and exhibit stronge nonlocal space-time character. If the value of wave function at x ¼ x<sup>0</sup> changes, the whole wave function will change for the nonlocal propagator. In the followings, we will detail this character.

We define <sup>R</sup>b<sup>0</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffi <sup>2</sup>iπℏc<sup>2</sup> <sup>p</sup> <sup>H</sup><sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffi mc2þH<sup>0</sup> p ; then

$$
\widehat{R} \approx \widehat{R}\_0 \left( 1 - \frac{A\_I c^2 (\widehat{p} - A\_0)}{H\_0} \right) \tag{30}
$$

After this definition, we will show how the measurement happens under the potential noise. Considering an initial state with the form ψð Þ¼ x; t<sup>0</sup> P <sup>m</sup>amφmð Þx , where φ<sup>m</sup> is the eigenstate of H0, if we put the potential noise in this system, the initial state will change. We denote the evolution state in arbitrary time t as ψðx, tÞ. The ψðx, tÞ can be expanded with basis states <sup>φ</sup><sup>m</sup> as <sup>ψ</sup>ðx, tÞ ¼ <sup>P</sup> <sup>m</sup>amφm. The task for us is to find out the varying value of am under each perturbational noise:

$$\begin{split} a\_n(t+\delta) &= \int\_{-\infty}^{\infty} \wp\_n(u,t) \* \phi(u,t) du \\ &= \int\_{-\infty}^{\infty} \wp\_n(u,t) \* c^{\frac{1}{2}} \Re(u,t) \* \int\_{-\infty}^{\infty} (i\eta)^{-\frac{1}{2}} \exp\left(-\frac{mc|\eta|}{\hbar}\right) \exp(\xi\_l)\phi(u-\eta,t) \,d\eta \\ &= \sum\_m a\_m(t)\,\lambda\_{n,m}(t-\delta) \left(1 + \frac{A\_lc^2 p\_n}{E\_n}\delta\_{n,m}\right) \end{split}$$

After rearranging the equation above, we get

$$a\_n(t+\delta) = \sum\_m a\_m(t)D\_{m,n}(t-\delta)\tag{31}$$

attentions of physicists since the beginning of the quantum theory establishment, but there is still no consensus. The measurement problem blocks up the way for us to understand the nonlocality and manipulate quantum state. Can the quantum measurement be controlled? Can we get the definite output we want under every measurement? If the quantum measurement can be controlled, the teleportation without classical communication channel can be realized, and the aim of superfast manipulation for quantum state will arrive. We can even transfer the energy thought nonlocality under controlled quantum measurement and make more novel encryption scheme for quantum communication. However, the key problem is "can we control

Figure 2. The process of collapse under a "potential noise". (a) The red line denotes the absolute value of probability

line is the function of potential. The different sets of noise cause the different collapse results. According the simulation, the process time of collapse is 0.3 ns in the top picture and 0.1 ns in the bottom picture. (b) The function of AI shown in

amplitude <sup>a</sup>0ð Þ<sup>t</sup> with the initial value 1/2, and the blue one denotes <sup>a</sup>1ð Þ<sup>t</sup> with the initial value ffiffiffi

The extended Feynman path integral mechanism answered this question. According to this mechanism, the character, "stochastic output" and "instantaneous collapse process" of quantum measurement are rooted in the "random" potential noise and "nonlocal" wave function inner correlation. The "nonlocality" is caused by the "relativistic effect" of superluminal paths in path integral theory. The superluminal paths will support a complex action function S in

theory cannot be canceled and makes F xð Þ <sup>1</sup>; x0; t0; t<sup>0</sup> 6¼ δð Þ x<sup>1</sup> � x<sup>0</sup> . This relation reveals that the propagator is no longer a local correlation. All points in space are correlated simultaneously, and any local perturbation will simultaneously transfer into the whole space. The extended Feynman path integral gives a simulation for two-energy-level system and exhibits that the

<sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> of <sup>S</sup>. This complex action that acted as a phase in integral

3

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<sup>p</sup> <sup>=</sup>2. The black oscillatory

the quantum measurement?" If yes, how? If no, why?

Eq. (4) for the expression ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eq. (32).

where

$$D\_{m,n}(t-\delta) = \lambda\_{n,m}(t-\delta)\left(1 + \frac{A\_I c^2 p\_n}{E\_n} \delta\_{n,m}\right)$$

$$\lambda\_{n,m}(t-\delta) = \int\_{-\infty}^{+\infty} \varphi\_n(\mathbf{x}) \mathbf{R}(\mathbf{x}, t-\delta) \* \widehat{R}\_0^{-1} \varphi\_m(\mathbf{x}) d\mathbf{x}$$

$$\mathbf{R}(\mathbf{x}, t) = \frac{\psi(\mathbf{x}, t)}{\widehat{R}\_0^{-1} \psi(\mathbf{x}, t)}$$

δ is the time interval of the neighbor potential noise pulses. In fact, to simulate the process of quantum measurement under potential noise, we let

$$A\_{I} = \sum\_{n=0}^{\infty} \left(\frac{2mk\_{b}T}{\eta \Delta}\right)^{1/2} \text{Random}(n) \left(\Theta(t - n\delta) - \Theta(t - (n-1)\delta)\right) \tag{32}$$

We simulate the collapse process of a wave function with the form j i <sup>ψ</sup> <sup>¼</sup> <sup>1</sup>=2 0j i <sup>þ</sup> ffiffiffi 3 <sup>p</sup> <sup>=</sup>2 1j i, where 0j i and 1j i are the harmonic-oscillator basis. According the simulation, we show the j i ψ will randomly collapse into 0j i or 1j i quickly (Figure 2).

#### 5. Conclusions

Measurement, in quantum theory, is not just a theory concerning the Schrödinger cat that is alive or dead, or the moon being here or not, but also the key and basis to the problem of the interpretation of quantum mechanics. In fact, the different views for the quantum measurement yield different interpretation for quantum mechanics, such as the Copenhagen interpretation, relative-state interpretation, Bohmian mechanics and so on. It has attracted many

After rearranging the equation above, we get

148 Advanced Technologies of Quantum Key Distribution

where

anð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>δ</sup> <sup>X</sup>

Dm,nð Þ¼ t � δ λn,mð Þ t � δ 1 þ

ðþ<sup>∞</sup> �∞

λn,mð Þ¼ t � δ

quantum measurement under potential noise, we let

n¼0

will randomly collapse into 0j i or 1j i quickly (Figure 2).

2mkbT ηΔ � �<sup>1</sup>=<sup>2</sup>

AI <sup>¼</sup> <sup>X</sup><sup>∞</sup>

5. Conclusions

<sup>m</sup>amð Þt Dm,nð Þ t � δ (31)

AIc<sup>2</sup>pn En

<sup>φ</sup>nð Þ<sup>x</sup> R xð Þ ; <sup>t</sup> � <sup>δ</sup> <sup>∗</sup>Rb�<sup>1</sup>

R xð Þ¼ ; <sup>t</sup> <sup>ψ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>R</sup>b�<sup>1</sup> <sup>0</sup> ψð Þ x; t

δ is the time interval of the neighbor potential noise pulses. In fact, to simulate the process of

We simulate the collapse process of a wave function with the form j i <sup>ψ</sup> <sup>¼</sup> <sup>1</sup>=2 0j i <sup>þ</sup> ffiffiffi

where 0j i and 1j i are the harmonic-oscillator basis. According the simulation, we show the j i ψ

Measurement, in quantum theory, is not just a theory concerning the Schrödinger cat that is alive or dead, or the moon being here or not, but also the key and basis to the problem of the interpretation of quantum mechanics. In fact, the different views for the quantum measurement yield different interpretation for quantum mechanics, such as the Copenhagen interpretation, relative-state interpretation, Bohmian mechanics and so on. It has attracted many

� �

δn,m

<sup>0</sup> φmð Þx dx

Random nð Þð Þ θð Þ� t � nδ θð Þ t � ð Þ n � 1 δ (32)

3 <sup>p</sup> <sup>=</sup>2 1j i,

Figure 2. The process of collapse under a "potential noise". (a) The red line denotes the absolute value of probability amplitude <sup>a</sup>0ð Þ<sup>t</sup> with the initial value 1/2, and the blue one denotes <sup>a</sup>1ð Þ<sup>t</sup> with the initial value ffiffiffi 3 <sup>p</sup> <sup>=</sup>2. The black oscillatory line is the function of potential. The different sets of noise cause the different collapse results. According the simulation, the process time of collapse is 0.3 ns in the top picture and 0.1 ns in the bottom picture. (b) The function of AI shown in Eq. (32).

attentions of physicists since the beginning of the quantum theory establishment, but there is still no consensus. The measurement problem blocks up the way for us to understand the nonlocality and manipulate quantum state. Can the quantum measurement be controlled? Can we get the definite output we want under every measurement? If the quantum measurement can be controlled, the teleportation without classical communication channel can be realized, and the aim of superfast manipulation for quantum state will arrive. We can even transfer the energy thought nonlocality under controlled quantum measurement and make more novel encryption scheme for quantum communication. However, the key problem is "can we control the quantum measurement?" If yes, how? If no, why?

The extended Feynman path integral mechanism answered this question. According to this mechanism, the character, "stochastic output" and "instantaneous collapse process" of quantum measurement are rooted in the "random" potential noise and "nonlocal" wave function inner correlation. The "nonlocality" is caused by the "relativistic effect" of superluminal paths in path integral theory. The superluminal paths will support a complex action function S in Eq. (4) for the expression ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup><sup>2</sup>=c<sup>2</sup> <sup>p</sup> of <sup>S</sup>. This complex action that acted as a phase in integral theory cannot be canceled and makes F xð Þ <sup>1</sup>; x0; t0; t<sup>0</sup> 6¼ δð Þ x<sup>1</sup> � x<sup>0</sup> . This relation reveals that the propagator is no longer a local correlation. All points in space are correlated simultaneously, and any local perturbation will simultaneously transfer into the whole space. The extended Feynman path integral gives a simulation for two-energy-level system and exhibits that the potential noise can indeed lead to the collapse state randomly and rapidly. Therefore, the key to control the quantum measurement is to control the potential noise exactly. "Potential noise" is caused by thermal fluctuation of potential filed or irregularity potential boundary. How to control this potential noise is still an unsolved topic.

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The extended Feynman path integral mechanism also solves the "basis-preferred" problem in quantum measurement. It exhibits the reason that the state prefers to collapse some set of basis states, which is due to the main Hamiltonian H<sup>0</sup> defined in Eq. (27). H<sup>0</sup> is the Hamiltonian that contains no noise. The eigenstates are the basis state that wave function prefers to collapse into.

The extended Feynman path integral mechanism shows the relation between "quantum measurement" and "unitary evolution operation". They are one and the same thing but are departed by jumpy potential noise. In mathematics, the function of potential noise is nowhere differentiable functions, and therefore, the path integral shown in Eq. (4) is not the regular path integral function under a noised potential. This is the main difference between "quantum measurement" and "unitary evolution operation" in mathematics. In physics, each potential noise point can be quickly absorbed by wave function through the nonlocality correlation, and the amounts of noise points will quickly accumulate to be a big quantity to change the whole wave function jRbj.

Additionally, besides the potential noise, the condition that the quantum measurement happens is that the interaction of system and environment should be big enough to distinguish the preferred basis state " φ<sup>n</sup> � �". If the interaction is not big enough, <sup>φ</sup>njRbjφ<sup>n</sup> D E <sup>≈</sup> <sup>φ</sup>mjRbjφ<sup>m</sup> D E and then Dmn ! δm,n in Eq. (31), then the collapse will not happen. In other words, the instrument that can realize the quantum measurement should be "macro" enough to produce enough noise and have big enough energy gaps of a system measured.
