**4. Dirac spacetime matrix equation**

�*∂*<sup>4</sup> <sup>0000</sup> �*∂*<sup>3</sup> <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> �*∂*<sup>4</sup> 0 0 <sup>þ</sup>*∂*<sup>3</sup> <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> 0 0 �*∂*<sup>4</sup> <sup>0</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>0</sup> �*∂*<sup>3</sup> �*∂*<sup>4</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>4</sup> <sup>000</sup> �*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>4</sup> 0 0 <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> 0 0 <sup>þ</sup>*∂*<sup>4</sup> <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> �*∂*<sup>3</sup> <sup>0000</sup> <sup>þ</sup>*∂*<sup>4</sup>

The compact matrix form of Eq. (26) is given by

∇ � **A***<sup>e</sup>* þ

*c ∂ ∂t*

**<sup>E</sup>** ¼ �∇*ϕ<sup>e</sup>* � <sup>1</sup>

 *c ∂ ∂t*

**3.5 Electromagnetic potential wave equations**

□2

**<sup>A</sup>***<sup>e</sup>* ¼ � <sup>4</sup>*<sup>π</sup> c*

□2

potential wave equations.

The ket vector ∣*a*i corresponds to the eight-by-one column vector on the left-hand side of Eq. (26). Equation (26), when expanded, yields the Lorentz conditions and the relationship between electromagnetic fields and potentials:

**<sup>A</sup>***<sup>e</sup>* � <sup>∇</sup> � **<sup>A</sup>***<sup>m</sup>* and **<sup>B</sup>** ¼ �∇*ϕ<sup>m</sup>* � <sup>1</sup>

electric vector potential **A***<sup>e</sup>* ¼ ð Þ *Ae*<sup>1</sup> *Ae*<sup>2</sup> *Ae*<sup>3</sup> , the magnetic vector potential **A***<sup>m</sup>* ¼ ð Þ *Am*<sup>1</sup> *Am*<sup>2</sup> *Am*<sup>3</sup> , the electric scalar potential *ϕe*, and the magnetic scalar potential *ϕm*. So again we see how the eight-by-eight spacetime matrix operator *M*^

plays a central role in tying together important electromagnetic relations.

sides of Eq. (27) by the spacetime matrix operator *M*^ . This gives

*ϕ<sup>e</sup>* ¼ 0 and ∇ � **A***<sup>m</sup>* þ

The new scalar and vector quantities appearing in the above equations are the

It is well-known that the electromagnetic vector and scalar potentials satisfy wave equations (see [9], pp. 179–181). This can be easily shown by multiplying both

Next replace the term *<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*<sup>i</sup> by the ket vector <sup>∣</sup> *<sup>j</sup>*<sup>i</sup> using Eq. (20). This yields

Expanding this single matrix equation yields eight partial differential equations which can be easily combined to form the following four potential wave equations:

**<sup>J</sup>***<sup>e</sup>* and □<sup>2</sup>

The single compact matrix (Eq. (31)) is therefore equivalent to these four

*<sup>ϕ</sup><sup>e</sup>* ¼ �4*πρ<sup>e</sup>* and □<sup>2</sup>

�*Ae*<sup>1</sup> �*Ae*<sup>2</sup> �*Ae*<sup>3</sup> �*ϕ<sup>m</sup>* �*iAm*<sup>1</sup> �*iAm*<sup>2</sup> �*iAm*<sup>3</sup> *iϕe*

*<sup>M</sup>*^ <sup>∣</sup>*a*i ¼ <sup>∣</sup> *<sup>f</sup>*i*:* (27)

 *c ∂ ∂t*

> *c ∂ ∂t*

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup>*a*i ¼ *<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i*:* (30)

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup>*a*i ¼ <sup>∣</sup> *<sup>j</sup>*i*:* (31)

**<sup>A</sup>***<sup>m</sup>* ¼ � <sup>4</sup>*<sup>π</sup>*

*c*

*ϕ<sup>m</sup>* ¼ �4*πρ<sup>m</sup>* (32)

**J***m:* (33)

*iE*<sup>1</sup> *iE*<sup>2</sup> *iE*<sup>3</sup> *B*1 *B*2 *B*3 

*ϕ<sup>m</sup>* ¼ 0 (28)

**A***<sup>m</sup>* þ ∇ � **A***e:* (29)

*:* (26)

*Progress in Relativity*

The nonrelativistic Schrödinger wave equation (see [10], pp. 143–146) plays a fundamental role in quantum mechanical phenomena where the spin property of nonrelativistic particles may be ignored. This equation is usually first met in modern physics textbooks. However, when a particle with half-integer spin and/or moving at relativistic speeds is involved, the relativistic Dirac equation [11] comes into play.

#### **4.1 Dirac spacetime matrix equation for free space**

Using the spacetime matrix operator *M*^ , the authors introduced in their most recent publication [1] a modified version of the traditional Dirac equation, referred to as the Dirac spacetime matrix equation. In the absence of electromagnetic potentials [11], the Dirac spacetime matrix equation for free space is given by

$$
\begin{bmatrix}
\mathbf{0} & -\partial\_{4} & \mathbf{0} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & -\partial\_{1} & -\partial\_{2} \\
\mathbf{0} & \mathbf{0} & -\partial\_{4} & \mathbf{0} & -\partial\_{2} & +\partial\_{1} & \mathbf{0} & -\partial\_{3} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -\partial\_{4} & +\partial\_{1} & +\partial\_{2} & +\partial\_{3} & \mathbf{0} \\
\mathbf{0} & +\partial\_{3} & -\partial\_{2} & +\partial\_{1} & +\partial\_{4} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
+\partial\_{2} & -\partial\_{1} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & \mathbf{0} & +\partial\_{4} & \mathbf{0} \\
\end{bmatrix}
\begin{bmatrix}
iU\_{1} \\
iU\_{2} \\
0 \\
L\_{1} \\
L\_{2} \\
L\_{3} \\
0
\end{bmatrix} + \kappa \begin{bmatrix}
0 \\
0 \\
0 \\
0 \\
L\_{1} \\
L\_{2} \\
0
\end{bmatrix} = \begin{bmatrix}
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0
\end{bmatrix}.
\tag{34}
$$

The compact matrix form of Eq. (34) is given by

$$
\hat{M}|\phi\rangle + \kappa|\phi\rangle = |\sigma\rangle. \tag{35}
$$

The wave function ∣*ϕ*i is an eight-by-one ket vector containing, in general, six nonzero scalar components associated with two vector quantities **U** ¼ ð Þ *U*<sup>1</sup> *U*<sup>2</sup> *U*<sup>3</sup> and **L** ¼ ð Þ *L*<sup>1</sup> *L*<sup>2</sup> *L*<sup>3</sup> . The elements (4,1) and (8,1) in ∣*ϕ*i have purposely been set equal to zero. The case when these two elements are nonzero will also be considered when the generalized spacetime matrix equation for free space is discussed later in this chapter. The ket vector ∣*o*i represents the eight-by-one null vector. The constant *κ* is defined by

$$\kappa \equiv m\_{\mathfrak{o}}c/\hbar. \tag{36}$$

Here *mo* represents the rest mass of the matter-wave particle under consideration, *c* again is the speed of light in free space, and ℏ is equal to the Planck constant *h* divided by 2*π*.

The Dirac spacetime matrix equation (34) when expanded is equivalent to eight partial differential equations. These eight equations can be rewritten as two divergence and two curl equations [1], namely,

$$
\nabla \cdot \mathbf{U} = \mathbf{0} \qquad \text{and} \qquad \nabla \cdot \mathbf{L} = \mathbf{0} \tag{37}
$$

$$\nabla \times \mathbf{U} = -\frac{1}{c} \frac{\partial}{\partial t} \mathbf{L} - i\kappa \mathbf{L} \qquad \text{and} \qquad \nabla \times \mathbf{L} = +\frac{1}{c} \frac{\partial}{\partial t} \mathbf{U} - i\kappa \mathbf{U}. \tag{38}$$

We refer to these equations as the Dirac spacetime vector equations for free space. It is noted that these equations resemble the four Maxwell field equations for free space in the absence of charge, current, and ordinary matter terms.

The simplest solutions of these vector equations are time-harmonic plane-wave solutions of the form

$$\mathbf{U}(\mathbf{r},t) = \mathbf{U\_o} \exp\left\{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar\right\} \qquad \text{and} \qquad \mathbf{L}(\mathbf{r},t) = \mathbf{L\_o} \exp\left\{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar\right\}. \tag{39}$$

The quantities **p** and *E* correspond to the linear momentum and the total energy of the associated matter-wave particle; **r** and *t* represent the position vector and the instantaneous time. For particles with nonzero rest mass *mo*, the following special theory of relativity equations (see [10], pp. 21–25) may also be useful:

$$E = \gamma m\_o c^2 \qquad p = \gamma m\_o v \qquad \text{where} \qquad \gamma = \mathbf{1}/\sqrt{\mathbf{1} - \boldsymbol{\beta}^2} \qquad \boldsymbol{\beta} = \boldsymbol{v}/c. \tag{40}$$

The quantities *γ* and *β* are known as the Lorentz factor and the speed parameter, respectively. The symbol *v* represents the relativistic speed of the matter-wave particle. Substitution of the preceding time-harmonic plane-wave solutions back into the Dirac spacetime vector equations yield the following set of vector equations for matter waves:

$$\mathbf{p}\mathbf{c} \cdot \mathbf{U\_o} = 0 \qquad \text{and} \qquad \mathbf{p}\mathbf{c} \cdot \mathbf{L\_o} = 0 \tag{41}$$

with spin-less (spin-0) particles. An example of a spin-less particle is the recently

with the compact matrix form of the Dirac spacetime matrix equation for free space, namely, Eq. (35). Multiply both sides by the spacetime matrix operator *M*^ .

Next replace the term *<sup>M</sup>*^ <sup>∣</sup>*ϕ*<sup>i</sup> with �*κ*∣*ϕ*<sup>i</sup> using Eq. (35). We obtain

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup>*ϕ*i � *<sup>κ</sup>*<sup>2</sup>

We refer to this equation as the Klein-Gordon spacetime matrix equation for free space. Using the fourth property of the spacetime matrix operator *M*^ , it can be easily shown that Eq. (47) is equivalent to the following two equations involving the

**<sup>U</sup>** <sup>¼</sup> 0 and □<sup>2</sup>

Therefore, the vectors **U** and **L** also satisfy Klein-Gordon type equations.

**5.1 Big unanswered questions and mysteries in physics and astronomy**

from the smallest to the largest, in the fields of physics and astronomy, is

The number of unanswered questions and mysteries regarding the universe

unimaginable. There are many references, too numerous to list here, which address this topic. However, an excellent comprehensive list of unsolved problems in physics appears in [13] for various broad areas of physics. These areas include general physics, quantum physics, cosmology, general relativity, quantum gravity, highenergy physics, particle physics, astronomy, astrophysics, nuclear physics, atomic physics, molecular physics, optical physics, classical mechanics, condensed matter physics, plasma physics, and biophysics. The following is a partial list of some of the most important questions and mysteries being addressed today by physicists and

How did the universe begin and what is the ultimate fate of the universe?

Why is there more matter than antimatter in the universe?

Why are the galaxies distributed in clumps and filaments?

Do dark gravity, dark charge, and dark antimatter exist?

Is spacetime fundamentally continuous or discrete? How can we create a quantum theory of gravity?

In this section, we will introduce for the first time a new matrix equation where again the spacetime operator *M*^ plays a central role. We will refer to this equation as

A version of the Klein-Gordon equation can be easily derived by simply starting

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup>*ϕ*i þ *<sup>κ</sup>M*^ <sup>∣</sup>*ϕ*i ¼ <sup>∣</sup>*o*i*:* (46)

**<sup>L</sup>** � *<sup>κ</sup>*<sup>2</sup>

∣*ϕ*i ¼ ∣*o*i*:* (47)

**L** ¼ 0*:* (48)

discovered Higgs boson.

This gives

vectors **U** and **L**:

□2

astronomers around the globe:

**25**

Is the universe infinite or just very big?

What came before the big bang?

Are there additional dimensions?

What is dark energy and dark matter?

**<sup>U</sup>** � *<sup>κ</sup>*<sup>2</sup>

*Eight-by-Eight Spacetime Matrix Operator and Its Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.86982*

**5. Generalized spacetime matrix equation**

the generalized spacetime matrix equation for free space.

$$\mathbf{p}\mathbf{c} \times \mathbf{U}\_{\mathbf{o}} = +E \begin{array}{c} \frac{(\boldsymbol{\chi} - \mathbf{1})}{\boldsymbol{\chi}} \end{array} \mathbf{L}\_{\mathbf{o}} \qquad \text{and} \qquad \mathbf{p}\boldsymbol{c} \times \mathbf{L}\_{\mathbf{o}} = -E \begin{array}{c} (\boldsymbol{\chi} + \mathbf{1}) \\ \boldsymbol{\chi} \end{array} \mathbf{U}\_{\mathbf{o}}.\tag{42}$$

From the previous equations we find the three vectors **U***o*, **L***o*, and **p***c* are mutually perpendicular. That is,

$$\mathbf{p}c \perp \mathbf{U}\_{\bullet} \qquad \mathbf{U}\_{\bullet} \perp \mathbf{L}\_{\bullet} \qquad \mathbf{p}c \perp \mathbf{L}\_{\bullet} \tag{43}$$

These properties represent transverse waves. In addition, we also obtain the important result:

$$\left(\boldsymbol{\chi} + \mathbf{1}\right) \boldsymbol{U}\_o^2 = \left(\boldsymbol{\chi} - \mathbf{1}\right) \boldsymbol{L}\_o^2. \tag{44}$$

The magnitudes of the vectors **U***<sup>o</sup>* and **L***<sup>o</sup>* are related through the Lorentz factor *γ*, which depends on the speed parameter *β*, which ultimately depends on the speed *v* of the nonzero rest-mass particle. Note, for *γ* much greater than unity, characteristic of a relativistic particle, the magnitudes of the vectors **U***<sup>o</sup>* and **L***<sup>o</sup>* are nearly equal. On the other hand, for *γ* close to unity, characteristic of a nonrelativistic particle, the magnitude of the vector **L***<sup>o</sup>* is much greater than the magnitude of the vector **U***o*. One other important result is

$$E^2 = p^2c^2 + m\_o^2c^4 \qquad \text{which implies} \qquad E = \pm \sqrt{p^2c^2 + m\_o^2c^4}.\tag{45}$$

The � sign is associated with the quantum mechanical energy *E* of a matterwave particle, like a half-integer spin electron. This was first interpreted by Paul A. M. Dirac. He recognized the negative energy levels predicted by his relativistic equation could not be ignored. This led to his concept of a hole theory of positrons. For a detailed discussion on negative energy states (see [11]).

#### **4.2 Klein-Gordon spacetime matrix equation**

The Klein-Gordon equation (see [12], pp. 118–129) is yet another quantum mechanical relativistic equation which is the field equation of the quanta associated The simplest solutions of these vector equations are time-harmonic plane-wave

The quantities **p** and *E* correspond to the linear momentum and the total energy of the associated matter-wave particle; **r** and *t* represent the position vector and the instantaneous time. For particles with nonzero rest mass *mo*, the following special

The quantities *γ* and *β* are known as the Lorentz factor and the speed parameter,

respectively. The symbol *v* represents the relativistic speed of the matter-wave particle. Substitution of the preceding time-harmonic plane-wave solutions back into the Dirac spacetime vector equations yield the following set of vector equations

From the previous equations we find the three vectors **U***o*, **L***o*, and **p***c* are

These properties represent transverse waves. In addition, we also obtain the

*<sup>o</sup>* <sup>¼</sup> ð Þ *<sup>γ</sup>* � <sup>1</sup> *<sup>L</sup>*<sup>2</sup>

The magnitudes of the vectors **U***<sup>o</sup>* and **L***<sup>o</sup>* are related through the Lorentz factor *γ*, which depends on the speed parameter *β*, which ultimately depends on the speed *v* of the nonzero rest-mass particle. Note, for *γ* much greater than unity, characteristic of a relativistic particle, the magnitudes of the vectors **U***<sup>o</sup>* and **L***<sup>o</sup>* are nearly equal. On the other hand, for *γ* close to unity, characteristic of a nonrelativistic particle, the magnitude of the vector **L***<sup>o</sup>* is much greater than the magnitude of the

<sup>4</sup> which implies *<sup>E</sup>* ¼ �

For a detailed discussion on negative energy states (see [11]).

**4.2 Klein-Gordon spacetime matrix equation**

The � sign is associated with the quantum mechanical energy *E* of a matterwave particle, like a half-integer spin electron. This was first interpreted by Paul A. M. Dirac. He recognized the negative energy levels predicted by his relativistic equation could not be ignored. This led to his concept of a hole theory of positrons.

The Klein-Gordon equation (see [12], pp. 118–129) is yet another quantum mechanical relativistic equation which is the field equation of the quanta associated

ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> *<sup>U</sup>*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> q

**p***c* � **Uo** ¼ 0 and **p***c* � **Lo** ¼ 0 (41)

**p***c*⊥ **Uo Uo** ⊥ **Lo p***c*⊥**Lo** (43)

*γ*

*<sup>o</sup>:* (44)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *p*<sup>2</sup>*c*<sup>2</sup> þ *m*<sup>2</sup>

q

*oc*4

*:* (45)

**Lo** and **<sup>p</sup>***<sup>c</sup>* � **Lo** ¼ �*<sup>E</sup>* ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup>

*β* ¼ *v=c:* (40)

**Uo***:* (42)

**U r**ð Þ¼ *; t* **Uo** exp f g *i*ð Þ **p** � **r** � *Et =*ℏ and **L r**ð Þ¼ *; t* **Lo** exp f g *i*ð Þ **p** � **r** � *Et =*ℏ *:* (39)

theory of relativity equations (see [10], pp. 21–25) may also be useful:

<sup>2</sup> *<sup>p</sup>* <sup>¼</sup> *<sup>γ</sup>mov* where *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>*

solutions of the form

*Progress in Relativity*

*E* ¼ *γmoc*

for matter waves:

important result:

**<sup>p</sup>***<sup>c</sup>* � **Uo** ¼ þ*<sup>E</sup>* ð Þ *<sup>γ</sup>* � <sup>1</sup>

mutually perpendicular. That is,

vector **U***o*. One other important result is

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> *c* <sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *oc*

**24**

*γ*

with spin-less (spin-0) particles. An example of a spin-less particle is the recently discovered Higgs boson.

A version of the Klein-Gordon equation can be easily derived by simply starting with the compact matrix form of the Dirac spacetime matrix equation for free space, namely, Eq. (35). Multiply both sides by the spacetime matrix operator *M*^ . This gives

$$
\hat{M}\hat{M}|\phi\rangle + \kappa \hat{M}|\phi\rangle = |\sigma\rangle. \tag{46}
$$

Next replace the term *<sup>M</sup>*^ <sup>∣</sup>*ϕ*<sup>i</sup> with �*κ*∣*ϕ*<sup>i</sup> using Eq. (35). We obtain

$$
\hat{M}\hat{M}|\phi\rangle - \kappa^2|\phi\rangle = |\phi\rangle. \tag{47}
$$

We refer to this equation as the Klein-Gordon spacetime matrix equation for free space. Using the fourth property of the spacetime matrix operator *M*^ , it can be easily shown that Eq. (47) is equivalent to the following two equations involving the vectors **U** and **L**:

$$
\Box^2 \mathbf{U} - \kappa^2 \mathbf{U} = \mathbf{0} \qquad \text{and} \qquad \Box^2 \mathbf{L} - \kappa^2 \mathbf{L} = \mathbf{0}.\tag{48}
$$

Therefore, the vectors **U** and **L** also satisfy Klein-Gordon type equations.

### **5. Generalized spacetime matrix equation**

In this section, we will introduce for the first time a new matrix equation where again the spacetime operator *M*^ plays a central role. We will refer to this equation as the generalized spacetime matrix equation for free space.

#### **5.1 Big unanswered questions and mysteries in physics and astronomy**

The number of unanswered questions and mysteries regarding the universe from the smallest to the largest, in the fields of physics and astronomy, is unimaginable. There are many references, too numerous to list here, which address this topic. However, an excellent comprehensive list of unsolved problems in physics appears in [13] for various broad areas of physics. These areas include general physics, quantum physics, cosmology, general relativity, quantum gravity, highenergy physics, particle physics, astronomy, astrophysics, nuclear physics, atomic physics, molecular physics, optical physics, classical mechanics, condensed matter physics, plasma physics, and biophysics. The following is a partial list of some of the most important questions and mysteries being addressed today by physicists and astronomers around the globe:

How did the universe begin and what is the ultimate fate of the universe? Is the universe infinite or just very big? Why is there more matter than antimatter in the universe? What came before the big bang? Why are the galaxies distributed in clumps and filaments? Are there additional dimensions? Is spacetime fundamentally continuous or discrete? How can we create a quantum theory of gravity? What is dark energy and dark matter? Do dark gravity, dark charge, and dark antimatter exist?

What happens inside a black hole and do naked singularities exist? Why does time seem to flow only in one direction? Is time travel really possible? Is string theory or M-theory a viable theory of everything? What kind of physics underlies the standard model? Are there really just three generations of leptons and quarks? Do gravitons exist? Are protons unstable? Do magnetic monopoles exist? What are the masses of neutrinos? Do the quarks or leptons have any substructure? Do tachyons exist and can information travel faster than light? Why do the particles have the precise masses they do? Do fundamental physical constants vary over time? Why are the strengths of the fundamental forces what they are? Do parallel universes exist and is there a multiverse? Was our spatially 3-D universe formed out of a vacuum by a 2-D hologram? Was the hologram formed by a flow of information? If so, what form? Does pair production formed, spontaneously, out of a vacuum? Are they likewise formed out of a flow of information? Do life processes, such as ion flows through cell membranes, form likewise as flows of information?

The compact matrix form of Eq. (49) is given by

*Eight-by-Eight Spacetime Matrix Operator and Its Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.86982*

**5.3 Eigenvalue spacetime matrix equations**

restrictions.

ℏ*c*

spacetime matrix (Eq. (49)).

This is the eighth compact matrix equation in **Table 1**. Note the similarity between the generalized spacetime matrix equation for free space and the Dirac spacetime matrix equation for free space (34) when *κ* ¼ *moc=*ℏ and the Maxwell spacetime matrix equation for free space (9) when *κ* ¼ 0. In those equations we purposely set the (4,1) and (8,1) elements in the ket vectors identically equal to zero. Doing so allowed us to convert those matrix equations to vector equations (involving

three-dimensional vectors only) which are described in greater detail in [1].

The wave function ∣*ψ*i can be thought of as being composed of two four-

In Eq. (49), we no longer restrict elements (4,1) and (8,1) to be equal to zero.

dimensional vectors Δ ¼ ð Þ Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> and Ω ¼ ð Þ Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup> . The implications by avoiding the earlier restrictions on elements (4,1) and (8,1) will be investigated shortly. We will find some new predictions and surprises by removing these

Our primary goal now is to determine the properties of time-harmonic plane-wave solutions satisfying the generalized spacetime matrix (Eq. (49)) for free space. The approach we will take is to cast Eq. (49) into an eigenvalue equation and use the methods of linear algebra to determine the set of orthonormal eigenvectors and corresponding eigenvalues satisfying this eigenvalue equation. (For an excellent book on linear algebra and the solution of eigenvalue equations; see [14], pp. 189–190.) For now let *κ* ¼ *moc=*ℏ, the same constant in the Dirac spacetime

matrix equation. Later on we will look at the special case when *κ* ¼ 0.

manipulation, we obtain the following matrix equation:

�*<sup>κ</sup>* <sup>0000</sup> <sup>þ</sup>*∂*<sup>3</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> �*<sup>κ</sup>* 0 0 �*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> 0 0 �*<sup>κ</sup>* <sup>0</sup> <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> �*<sup>κ</sup>* �*∂*<sup>1</sup> �*∂*<sup>2</sup> �*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*<sup>κ</sup>* <sup>000</sup> �*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>0</sup> <sup>þ</sup>*<sup>κ</sup>* 0 0 <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> 0 0 <sup>þ</sup>*<sup>κ</sup>* <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> �*∂*<sup>3</sup> <sup>0000</sup> <sup>þ</sup>*<sup>κ</sup>*

The compact matrix form of this equation is given by

*<sup>H</sup>*^ <sup>∣</sup>*ψ*i ¼ *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup>*

This equation has the same identical form as the nonrelativistic Schrödinger equation (see [12], pp. 118–129). However, the Hamiltonian matrix operator *H*^ is entirely different. This equation represents the canonical form of the generalized

*∂t*

We first multiply Eq. (49) by the factor ℏ*cM*4. The matrix *M*<sup>4</sup> is the fourth of the spacetime matrices first introduced in Eq. (3). After doing so, with minor algebraic

Δ1 Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup>

 *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup> ∂t*

∣*ψ*i*:* (52)

Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup>

*:* (51)

*<sup>M</sup>*^ <sup>∣</sup>*ψ*i þ *<sup>κ</sup>*∣*ψ*i ¼ <sup>∣</sup>*o*i*:* (50)

As we can see, even with all of the discoveries made over the past several hundred years, there is so much we do not understand and so much yet to be discovered about our universe and possibly beyond.

So far we have described the first seven compact matrix equations listed in **Table 1** where the spacetime matrix operator *M*^ plays a fundamental role. We found that each of these seven equations correspond to a variety of fundamental equations, in both classical electrodynamics and relativistic quantum mechanics. In the next subsection, we will discuss in detail the eighth compact matrix equation listed in **Table 1**. This eighth equation is associated with a new matrix equation which we will refer to as the generalized spacetime matrix equation for free space. As we will see, there are several theoretical implications resulting from our study of the generalized spacetime matrix equation which perhaps may be added as unanswered questions or mysteries to the preceding list.

#### **5.2 Generalized spacetime matrix equation for free space**

We define the generalized spacetime matrix equation for free space by the following equation:

�*∂*<sup>4</sup> <sup>0000</sup> �*∂*<sup>3</sup> <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> �*∂*<sup>4</sup> 0 0 <sup>þ</sup>*∂*<sup>3</sup> <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> 0 0 �*∂*<sup>4</sup> <sup>0</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>0</sup> �*∂*<sup>3</sup> �*∂*<sup>4</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>4</sup> <sup>000</sup> �*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>4</sup> 0 0 <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> 0 0 <sup>þ</sup>*∂*<sup>4</sup> <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> �*∂*<sup>3</sup> <sup>0000</sup> <sup>þ</sup>*∂*<sup>4</sup> Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup> þ *κ* Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup> *:* (49)

*Eight-by-Eight Spacetime Matrix Operator and Its Applications DOI: http://dx.doi.org/10.5772/intechopen.86982*

The compact matrix form of Eq. (49) is given by

What happens inside a black hole and do naked singularities exist?

Why does time seem to flow only in one direction?

Do the quarks or leptons have any substructure?

Why do the particles have the precise masses they do? Do fundamental physical constants vary over time?

Do parallel universes exist and is there a multiverse?

Are they likewise formed out of a flow of information?

discovered about our universe and possibly beyond.

swered questions or mysteries to the preceding list.

�*∂*<sup>4</sup> <sup>0000</sup> �*∂*<sup>3</sup> <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> �*∂*<sup>4</sup> 0 0 <sup>þ</sup>*∂*<sup>3</sup> <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> 0 0 �*∂*<sup>4</sup> <sup>0</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>0</sup> �*∂*<sup>3</sup> �*∂*<sup>4</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> �*∂*<sup>2</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>4</sup> <sup>000</sup> �*∂*<sup>3</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>1</sup> <sup>þ</sup>*∂*<sup>2</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>4</sup> 0 0 <sup>þ</sup>*∂*<sup>2</sup> �*∂*<sup>1</sup> <sup>0</sup> <sup>þ</sup>*∂*<sup>3</sup> 0 0 <sup>þ</sup>*∂*<sup>4</sup> <sup>0</sup> �*∂*<sup>1</sup> �*∂*<sup>2</sup> �*∂*<sup>3</sup> <sup>0000</sup> <sup>þ</sup>*∂*<sup>4</sup>

**5.2 Generalized spacetime matrix equation for free space**

Is string theory or M-theory a viable theory of everything? What kind of physics underlies the standard model?

Are there really just three generations of leptons and quarks?

Do tachyons exist and can information travel faster than light?

Why are the strengths of the fundamental forces what they are?

Was our spatially 3-D universe formed out of a vacuum by a 2-D hologram? Was the hologram formed by a flow of information? If so, what form? Does pair production formed, spontaneously, out of a vacuum?

Do life processes, such as ion flows through cell membranes, form likewise as

As we can see, even with all of the discoveries made over the past several hundred years, there is so much we do not understand and so much yet to be

So far we have described the first seven compact matrix equations listed in **Table 1** where the spacetime matrix operator *M*^ plays a fundamental role. We found that each of these seven equations correspond to a variety of fundamental equations, in both classical electrodynamics and relativistic quantum mechanics. In the next subsection, we will discuss in detail the eighth compact matrix equation listed in **Table 1**. This eighth equation is associated with a new matrix equation which we will refer to as the generalized spacetime matrix equation for free space. As we will see, there are several theoretical implications resulting from our study of the generalized spacetime matrix equation which perhaps may be added as unan-

We define the generalized spacetime matrix equation for free space by the

Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup>

þ *κ*

Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup>

*:* (49)

Is time travel really possible?

Do magnetic monopoles exist? What are the masses of neutrinos?

Do gravitons exist? Are protons unstable?

*Progress in Relativity*

flows of information?

following equation:

$$
\hat{M}|\psi\rangle + \kappa|\psi\rangle = |\sigma\rangle. \tag{50}
$$

This is the eighth compact matrix equation in **Table 1**. Note the similarity between the generalized spacetime matrix equation for free space and the Dirac spacetime matrix equation for free space (34) when *κ* ¼ *moc=*ℏ and the Maxwell spacetime matrix equation for free space (9) when *κ* ¼ 0. In those equations we purposely set the (4,1) and (8,1) elements in the ket vectors identically equal to zero. Doing so allowed us to convert those matrix equations to vector equations (involving three-dimensional vectors only) which are described in greater detail in [1].

In Eq. (49), we no longer restrict elements (4,1) and (8,1) to be equal to zero. The wave function ∣*ψ*i can be thought of as being composed of two fourdimensional vectors Δ ¼ ð Þ Δ<sup>1</sup> Δ<sup>2</sup> Δ<sup>3</sup> Δ<sup>4</sup> and Ω ¼ ð Þ Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup> . The implications by avoiding the earlier restrictions on elements (4,1) and (8,1) will be investigated shortly. We will find some new predictions and surprises by removing these restrictions.

#### **5.3 Eigenvalue spacetime matrix equations**

Our primary goal now is to determine the properties of time-harmonic plane-wave solutions satisfying the generalized spacetime matrix (Eq. (49)) for free space. The approach we will take is to cast Eq. (49) into an eigenvalue equation and use the methods of linear algebra to determine the set of orthonormal eigenvectors and corresponding eigenvalues satisfying this eigenvalue equation. (For an excellent book on linear algebra and the solution of eigenvalue equations; see [14], pp. 189–190.) For now let *κ* ¼ *moc=*ℏ, the same constant in the Dirac spacetime matrix equation. Later on we will look at the special case when *κ* ¼ 0.

We first multiply Eq. (49) by the factor ℏ*cM*4. The matrix *M*<sup>4</sup> is the fourth of the spacetime matrices first introduced in Eq. (3). After doing so, with minor algebraic manipulation, we obtain the following matrix equation:

$$
\hbar c \begin{bmatrix} -\kappa & 0 & 0 & 0 & 0 & +\partial\_3 & -\partial\_2 & +\partial\_1 \\ 0 & -\kappa & 0 & 0 & -\partial\_3 & 0 & +\partial\_1 & +\partial\_2 \\ 0 & 0 & -\kappa & 0 & +\partial\_2 & -\partial\_1 & 0 & +\partial\_3 \\ 0 & 0 & 0 & -\kappa & -\partial\_1 & -\partial\_2 & -\partial\_3 & 0 \\ 0 & +\partial\_3 & -\partial\_2 & +\partial\_1 & +\kappa & 0 & 0 & 0 \\ -\partial\_3 & 0 & +\partial\_1 & +\partial\_2 & 0 & +\kappa & 0 & 0 \\ +\partial\_2 & -\partial\_1 & 0 & +\partial\_3 & 0 & 0 & +\kappa & 0 \\ -\partial\_1 & -\partial\_2 & -\partial\_3 & 0 & 0 & 0 & 0 & +\kappa \end{bmatrix} \begin{bmatrix} \Delta\_1 \\ \Delta\_2 \\ \Delta\_3 \\ \Delta\_4 \\ \Delta\_5 \\ \Delta\_6 \\ \Delta\_7 \\ \Delta\_8 \end{bmatrix} = i\hbar \frac{\partial}{\partial t} \begin{bmatrix} \Delta\_1 \\ \Delta\_2 \\ \Delta\_3 \\ \Delta\_4 \\ \Delta\_5 \\ \Delta\_6 \\ \Delta\_7 \\ \Delta\_8 \end{bmatrix}. \tag{51}$$

The compact matrix form of this equation is given by

$$
\hat{H}|\psi\rangle = i\hbar \frac{\partial}{\partial t}|\psi\rangle. \tag{52}
$$

This equation has the same identical form as the nonrelativistic Schrödinger equation (see [12], pp. 118–129). However, the Hamiltonian matrix operator *H*^ is entirely different. This equation represents the canonical form of the generalized spacetime matrix (Eq. (49)).

For time-harmonic plane-wave solutions, the ket vector ∣*ψ*i may be expressed as

$$|\psi\rangle = |\psi\_o\rangle \exp\left[+i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar\right].\tag{53}$$

where

may also be of use:

matrix (Eq. (58)).

*a* �

ffiffi p 

s

ffiffiffiffiffiffiffiffiffiffi *γ* þ 1 *γ*

*Eo* � *moc*

The matrix in Eq. (58) is an eight-by-eight square matrix. A compact matrix

At this point we are now in a position to determine eight eigenvectors ∣*ψn*i and the corresponding eigenvalues *En* satisfying the eigenvalue (Eq. (58)). We chose to use the matrix software program MATLAB [15] for determining the eigenvalues and eigenvectors. As it turns out, there are only two unique eigenvalues given by

From the special theory of relativity (see [10], pp. 21–25), the following relations

As before, *γ* and *β* are referred to as the Lorentz factor and speed parameter, respectively. For each of the two eigenvalues, there are four unique eigenvectors.

The symbol *δmn* represents the Kronecker delta. In **Table 2** is a summary of the

*<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *<sup>b</sup>* �

Inspection of the contents of **Table 2** reveals the following important results:

1. ∣*ψ*1i and ∣*ψ*2i represent transverse waves with positive energy þ*γEo*.

2. ∣*ψ*3i and ∣*ψ*4i represent transverse waves with negative energy �*γEo*.

3. ∣*ψ*5i and ∣*ψ*6i represent non-transverse waves with positive energy þ*γEo*.

4.∣*ψ*7i and ∣*ψ*8i represent non-transverse waves with negative energy �*γEo*.

For wave propagation in the +*z* direction, the transverse waves have eigenvector solutions ∣*ψ*i where elements (3,1), (4,1), (7,1), and (8,1) are identically equal to zero. In other words, Δ ¼ ð Þ Δ<sup>1</sup> Δ<sup>2</sup> 0 0 and Ω ¼ ð Þ Ω<sup>1</sup> Ω<sup>2</sup> 0 0 . For this case, Δ1, Δ<sup>2</sup> and Ω1, Ω<sup>2</sup> correspond to the *x* and *y* components. Thus, for wave propagation in the +*z* direction, the transverse wave solutions only have *x* and *y* vector components

eigenvalues and orthonormal eigenvectors satisfying the eigenvalue spacetime

version of Eq. (58) may be expressed as follows:

*Eight-by-Eight Spacetime Matrix Operator and Its Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.86982*

*E*<sup>þ</sup> ¼ þ*E* and *E*� ¼ �*E* where *E* ¼

*E* ¼ *γEo p* ¼ *γmov pc* ¼ *γβEo γ* ¼ 1*=*

The eight eigenvectors ∣*ψn*i form an orthonormal set, that is,

The constants *a* and *b* appearing in **Table 2** are defined by

characteristic of a transverse wave in three dimensions.

*H*∣*ψn*i ¼ *En*∣*ψn*i *n* ¼ 1*,* 2*,* 3*, :::*8*:* (60)

*:* (59)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *E*2 *<sup>o</sup>* þ *p*<sup>2</sup>*c*<sup>2</sup>

*:* (61)

*β* ¼ *v=c:* (62)

*:* (64)

q

ffiffiffiffiffiffiffiffiffiffiffiffiffi � *<sup>β</sup>*<sup>2</sup> q

*ψ <sup>m</sup>*j*ψ<sup>n</sup>* h i ¼ *δmn:* (63)

ffiffi p 

s

ffiffiffiffiffiffiffiffiffiffi *γ* � 1 *γ*

Again the quantities **p** and *E* correspond to the linear momentum vector and the total energy; **r** and *t* represent the position vector and the instantaneous time. After substituting the eight-by-one ket vector ∣*ψ*i back into Eq. (51), we obtain the following eigenvalue equation:

$$pc \begin{bmatrix} pc \\ 0 & -\mu & 0 & 0 & -ia\_3 & 0 & +ia\_1 & +ia\_1 \\ 0 & 0 & -\mu & 0 & +ia\_2 & -ia\_1 & 0 & +ia\_3 \\ 0 & 0 & 0 & -\mu & -ia\_1 & -ia\_2 & -ia\_3 & 0 \\ 0 & 0 & 0 & -\mu & -ia\_1 & -ia\_2 & -ia\_3 & 0 \\ 0 & +ia\_3 & -ia\_2 & +ia\_1 & +\mu & 0 & 0 & 0 \\ -ia\_3 & 0 & +ia\_1 & +ia\_2 & 0 & +\mu & 0 & 0 \\ +ia\_2 & -ia\_1 & 0 & +ia\_3 & 0 & 0 & +\mu & 0 \\ -ia\_1 & -ia\_2 & -ia\_3 & 0 & 0 & 0 & 0 & +\mu \end{bmatrix} \begin{bmatrix} \Delta\_1 \\ \Delta\_2 \\ \Delta\_3 \\ \Delta\_4 \\ \Delta\_5 \\ \Delta\_6 \\ \Delta\_7 \\ \Delta\_8 \end{bmatrix} = E \begin{bmatrix} \Delta\_1 \\ \Delta\_2 \\ \Delta\_3 \\ \Delta\_4 \\ \Delta\_5 \\ \Delta\_6 \\ \Delta\_7 \\ \Delta\_8 \end{bmatrix}. \tag{54}$$

We will refer to Eq. (54) as the eigenvalue spacetime matrix equation. The compact matrix form of Eq. (54) is represented by

$$H|\psi\rangle = E|\psi\rangle. \tag{55}$$

The eight-by-eight matrix *H* is Hermitian which implies the eigenvalues *E* are real (see [14], p. 222). The following equations define various quantities appearing in Eq. (54):

$$
\mu \equiv m\_o c^2 / pc \qquad \text{and} \qquad \mathbf{p} \equiv p \ (a\_1 \ a\_2 \ a\_3). \tag{56}
$$

The quantity *p* is the magnitude of the linear momentum vector **p**, and *α*1*, α*2*, α*<sup>3</sup> represent the direction cosines, associated with the direction of the linear momentum vector **p**.
