Open access peer-reviewed chapter

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

By Richard P. Bocker and B. Roy Frieden

Submitted: February 20th 2019Reviewed: May 21st 2019Published: June 19th 2019

DOI: 10.5772/intechopen.86982

## Abstract

A recent journal article by the authors introduced the eight-by-eight spacetime matrix operator M̂ which played a key role in the formulation of Lorentz invariant matrix equations for both the classical electrodynamic Maxwell field equations and the quantum mechanical relativistic Dirac equation for free space. Those new equations we referred to as the Maxwell spacetime matrix and the Dirac spacetime matrix equations. These matrix equations will be briefly reviewed at the beginning of this chapter. Next we will show how the same matrix operator M̂ plays a central role in the matrix formulation of other fundamental equations in both electromagnetic and quantum theories. These include the electromagnetic wave and charge continuity equations, the Lorentz conditions and electromagnetic potentials, the electromagnetic potential wave equations, and the quantum mechanical Klein-Gordon equation. In addition, a new generalized spacetime matrix equation, again employing the operator M̂, will be described which is a generalization of the Maxwell and Dirac spacetime matrix equations. We will explore time-harmonic plane-wave solutions of this equation as well as the properties of these solutions.

### Keywords

• special theory of relativity
• matrix operators
• classical electrodynamics
• relativistic quantum mechanics
• matter waves
• electromagnetic waves
• optics
• applied mathematics

## 1. Introduction

The eight-by-eight spacetime matrix operator M̂plays a key role in the matrix formulation of a number of well-known fundamental equations in both the fields of classical electrodynamics and relativistic quantum mechanics (see [1]). The spacetime matrix operator is defined by Eq. (1):

M̂400003+210400+301200402+1030004+1+2+300+32+1+400030+1+20+400+210+300+401230000+4.E1

The partial derivative symbols are defined by the following:

1x2y3z41ict.E2

The imaginary quantity irepresents the square root of minus one, and the physical quantity ccorresponds to the speed of light in free space.

Eight compact matrix equations are listed in Table 1, each containing the spacetime matrix operator M̂. Each of these equations, as well as the ket vector appearing in these equations, will be discussed in greater detail in the following sections of this chapter. An excellent introduction to bra and ket vector notation may be found in [2]. The Gaussian system of units (see [3], p. 781) is employed throughout this chapter.

Compact matrix equationCompact matrix equation description
M̂f=jMaxwell matrix equation with charges and currents
M̂M̂f=M̂jCharge continuity and electromagnetic wave equations
M̂a=fLorentz conditions and electromagnetic potentials
M̂M̂a=jElectromagnetic potential wave equations

### Table 1.

Compact matrix equations where the spacetime matrix operator M̂plays a central role.

## 2. Eight-by-eight spacetime matrix operator properties

The spacetime matrix operator M̂, defined in Eq. (1), may also be expressed by the following equation:

M̂=M11+M22+M33+M44.E3

The four eight-by-eight matrices Mμ, where μ=1,2,3,4, are simply referred to as the spacetime matrices. These matrices have the following properties:

1. Each matrix Mμis equal to its own multiplicative inverse

Mμ=Mμ1.E4

• These matrices satisfy the anti-commutation relation

MμMν+MνMμ=2δμνI.E5

• Each matrix Mμis Hermitian

Mμ=Mμ.E6

M̂M̂=M̂2=I2.E7

• The symbol δμνis the Kronecker delta, and Irepresents the eight-by-eight identity matrix. The d’Alembertian (see [4], p. 290) and the Laplacian (see [4], p. 15) operators are defined by

221c22t2and22x2+2y2+2z2.E8

Some authors use the symbol to represent the d’Alembertian operator.

## 3. Maxwell spacetime matrix equation

The Maxwell field equations play a fundamental role in both classical electrodynamics and physical optics. The propagations of electromagnetic waves through free space (see [4], pp. 514–522), nonconducting media (see [3], pp. 295–309), thin-film optical filters [5], and solid-state crystalline materials [6] are just a few examples where the Maxwell field equations play an important role.

An earlier eight-by-eight matrix representation of the Maxwell field equations was first introduced by the authors back in 1993 [7]. An improved updated version using the spacetime matrix operator M̂was published recently [1]. For free space, in the absence of charges and currents, this later version is given by

400003+210400+301200402+1030004+1+2+300+32+1+400030+1+20+400+210+300+401230000+4iE1iE2iE30B1B2B30=00000000.E9

The compact matrix form of Eq. (9) is given by

M̂f=o.E10

The wave function fis an eight-by-one ket vector containing, in general, six nonzero scalar components associated with the electric field vector E=E1E2E3and the magnetic induction vector B=B1B2B3. The elements (4,1) and (8,1) in fhave purposely been set equal to zero. The case when these two elements are nonzero will be considered when the generalized spacetime matrix equation for free space is discussed. The ket vector orepresents the eight-by-one null vector.

The Maxwell spacetime matrix equation (9) when expanded is equivalent to two divergences and two curl equations, namely,

E=0andB=0E11
×E+1ctB=0and×B1ctE=0.E12

We recognize these four equations as the traditional Maxwell field equations (Gaussian units) for free space in the absence of charges, currents, and ordinary matter terms (see [8], pp. 362–368).

For electromagnetic waves, time-harmonic plane-wave solutions of the form

Ert=EoexpikrωtandBrt=BoexpikrωtE13

will next be substituted back into the previous four vector equations. This yields the following set of equations:

kEo=0andkBo=0E14
k×Eo=+ωcBoandk×Bo=ωcEo.E15

The quantities kand ωcorrespond to the wave vector and the angular frequency associated with the electromagnetic wave; rand trepresent the position vector and the instantaneous time. From the preceding equations, we find the vectors Eo, Bo, and kare mutually perpendicular. That is,

kEoEoBokBo.E16

These properties represent transverse electromagnetic waves. We also obtain the important results

Eo=BoE17

and

ω=kcλf=cwhereω=2πfk=2π/λ.E18

The quantities k, f, and λrepresent the wave number, the frequency, and the wavelength, respectively, associated with the electromagnetic wave. So for free space, the magnitudes of the electromagnetic field vectors Eoand Boare equal, a well-known result in electromagnetic wave propagation. Recall we are using Gaussian units.

### 3.2 Maxwell spacetime matrix equation with charges and currents

The Maxwell spacetime matrix equation, with the addition of charge and current terms [1], is given by

400003+210400+301200402+1030004+1+2+300+32+1+400030+1+20+400+210+300+401230000+4iE1iE2iE30B1B2B30=4πcJe1Je2Je3cρmiJm1iJm2iJm3icρe.E19

The compact matrix form of the Maxwell spacetime matrix equation is given by

M̂f=j.E20

Eq. (19), when expanded, is equivalent to two divergences and two curl equations. The resulting four vector equations are referred to as the microscopic Maxwell field equations (see [8], pp. 283–290). They are given by

E=+4πρeandB=+4πρmE21
×E+1ctB=4πcJmand×B1ctE=+4πcJe.E22

The various scalar and vector quantities appearing in the microscopic Maxwell vector equations are the electric field vector E=E1E2E3, the magnetic induction vector B=B1B2B3, the electric current density vector Je=Je1Je2Je3, the magnetic current density vector Jm=Jm1Jm2Jm3, the electric charge density ρe, the magnetic charge density ρm, and the speed of light cin free space. Both magnetic charge and magnetic current density (see [8], pp. 283–290) have been included in the Maxwell vector equations for purposes of completeness. They, of course, may be set equal to zero since hypothetical magnetic monopoles have not been discovered in nature. The ket vector frepresents the eight-by-one column vector on the left-hand side of Eq. (19). The ket vector jcorresponds to the eight-by-one column vector on the right-hand side of Eq. (19) multiplied by the factor 4π/c.

### 3.3 Charge continuity and electromagnetic wave equations

Charge continuity equations for electric (see [8], p. 15) and magnetic charges as well as the electromagnetic wave equations involving electric and magnetic charges and currents may be easily obtained by simply multiplying both sides of the Maxwell spacetime matrix equation in compact form (20) by the spacetime matrix operator M̂. That is,

M̂M̂f=M̂j.E23

Expanding this single matrix equation yields the charge continuity and electromagnetic wave equations:

Je+tρe=0andJm+tρm=0E24
2E=4πc2tJe+4πρe+4πc×Jmand2B=4πc2tJm+4πρm4πc×Je.E25

### 3.4 Lorentz conditions and electromagnetic potentials

By using the spacetime matrix operator M̂, we can determine the relationship between electromagnetic fields and vector-scalar potentials as well as determine expressions for the Lorentz conditions (see [9], pp. 179–181) in a single matrix equation. The following matrix equation provides the desired relation:

400003+210400+301200402+1030004+1+2+300+32+1+400030+1+20+400+210+300+401230000+4Ae1Ae2Ae3ϕmiAm1iAm2iAm3iϕe=iE1iE2iE30B1B2B30.E26

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

M̂a=f.E27

The ket vector acorresponds 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:

Ae+1ctϕe=0andAm+1ctϕm=0E28
E=ϕe1ctAe×AmandB=ϕm1ctAm+×Ae.E29

The new scalar and vector quantities appearing in the above equations are the electric vector potential Ae=Ae1Ae2Ae3, the magnetic vector potential Am=Am1Am2Am3, the electric scalar potential ϕe, and the magnetic scalar potential ϕm. So again we see how the eight-by-eight spacetime matrix operator M̂plays a central role in tying together important electromagnetic relations.

### 3.5 Electromagnetic potential wave equations

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 sides of Eq. (27) by the spacetime matrix operator M̂. This gives

M̂M̂a=M̂f.E30

Next replace the term M̂fby the ket vector jusing Eq. (20). This yields

M̂M̂a=j.E31

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

2ϕe=4πρeand2ϕm=4πρmE32
2Ae=4πcJeand2Am=4πcJm.E33

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

## 4. Dirac spacetime matrix equation

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.

Using the spacetime matrix operator M̂, 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

400003+210400+301200402+1030004+1+2+300+32+1+400030+1+20+400+210+300+401230000+4iU1iU2iU30L1L2L30+κiU1iU2iU30L1L2L30=00000000.E34

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

M̂ϕ+κϕ=o.E35

The wave function ϕis an eight-by-one ket vector containing, in general, six nonzero scalar components associated with two vector quantities U=U1U2U3and L=L1L2L3. The elements (4,1) and (8,1) in ϕhave 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 orepresents the eight-by-one null vector. The constant κis defined by

κmoc/.E36

Here morepresents the rest mass of the matter-wave particle under consideration, cagain is the speed of light in free space, and is equal to the Planck constant hdivided by 2π.

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,

U=0andL=0E37
×U=1ctLLand×L=+1ctUU.E38

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

Urt=UoexpiprEt/andLrt=LoexpiprEt/.E39

The quantities pand Ecorrespond to the linear momentum and the total energy of the associated matter-wave particle; rand trepresent 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=γmoc2p=γmovwhereγ=1/1β2β=v/c.E40

The quantities γand βare known as the Lorentz factor and the speed parameter, respectively. The symbol vrepresents 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:

pcUo=0andpcLo=0E41
pc×Uo=+Eγ1γLoandpc×Lo=Eγ+1γUo.E42

From the previous equations we find the three vectors Uo, Lo, and pcare mutually perpendicular. That is,

pcUoUoLopcLoE43

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

γ+1Uo2=γ1Lo2.E44

The magnitudes of the vectors Uoand Loare related through the Lorentz factor γ, which depends on the speed parameter β, which ultimately depends on the speed vof the nonzero rest-mass particle. Note, for γmuch greater than unity, characteristic of a relativistic particle, the magnitudes of the vectors Uoand Loare nearly equal. On the other hand, for γclose to unity, characteristic of a nonrelativistic particle, the magnitude of the vector Lois much greater than the magnitude of the vector Uo. One other important result is

E2=p2c2+mo2c4which impliesE=±p2c2+mo2c4.E45

The ±sign is associated with the quantum mechanical energy Eof a matter-wave 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 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 M̂. This gives

M̂M̂ϕ+κM̂ϕ=o.E46

Next replace the term M̂ϕwith κϕusing Eq. (35). We obtain

M̂M̂ϕκ2ϕ=o.E47

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

2Uκ2U=0and2Lκ2L=0.E48

Therefore, the vectors Uand Lalso 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 M̂plays 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, high-energy 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:

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

2. Is the universe infinite or just very big?

3. Why is there more matter than antimatter in the universe?

4. What came before the big bang?

5. Why are the galaxies distributed in clumps and filaments?

7. Is spacetime fundamentally continuous or discrete?

8. How can we create a quantum theory of gravity?

9. What is dark energy and dark matter?

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

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

12. Why does time seem to flow only in one direction?

13. Is time travel really possible?

14. Is string theory or M-theory a viable theory of everything?

15. What kind of physics underlies the standard model?

16. Are there really just three generations of leptons and quarks?

17. Do gravitons exist?

18. Are protons unstable?

19. Do magnetic monopoles exist?

20. What are the masses of neutrinos?

21. Do the quarks or leptons have any substructure?

22. Do tachyons exist and can information travel faster than light?

23. Why do the particles have the precise masses they do?

24. Do fundamental physical constants vary over time?

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

26. Do parallel universes exist and is there a multiverse?

27. Was our spatially 3-D universe formed out of a vacuum by a 2-D hologram?

28. Was the hologram formed by a flow of information? If so, what form?

29. Does pair production formed, spontaneously, out of a vacuum?

30. Are they likewise formed out of a flow of information?

31. Do life processes, such as ion flows through cell membranes, form likewise as flows of information?

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 M̂plays 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.

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

400003+210400+301200402+1030004+1+2+300+32+1+400030+1+20+400+210+300+401230000+4Δ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4+κΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4=00000000.E49

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

M̂ψ+κψ=o.E50

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=moc/and the Maxwell spacetime matrix equation for free space (9) when κ=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 ψcan be thought of as being composed of two four-dimensional vectors Δ=Δ1Δ2Δ3Δ4and Ω=Ω1Ω2Ω3Ω4. 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=moc/, the same constant in the Dirac spacetime matrix equation. Later on we will look at the special case when κ=0.

We first multiply Eq. (49) by the factor cM4. The matrix M4is the fourth of the spacetime matrices first introduced in Eq. (3). After doing so, with minor algebraic manipulation, we obtain the following matrix equation:

cκ0000+32+10κ0030+1+200κ0+210+3000κ12300+32+1+κ00030+1+20+κ00+210+300+κ01230000+κΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4=itΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4.E51

The compact matrix form of this equation is given by

Ĥψ=itψ.E52

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

For time-harmonic plane-wave solutions, the ket vector ψmay be expressed as

ψ=ψoexp+iprEt/.E53

Again the quantities pand Ecorrespond to the linear momentum vector and the total energy; rand trepresent the position vector and the instantaneous time. After substituting the eight-by-one ket vector ψback into Eq. (51), we obtain the following eigenvalue equation:

pcμ0000+iα3iα2+iα10μ00iα30+iα1+iα200μ0+iα2iα10+iα3000μiα1iα2iα300+iα3iα2+iα1+μ000iα30+iα1+iα20+μ00+iα2iα10+iα300+μ0iα1iα2iα30000+μΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4=EΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4.E54

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

Hψ=Eψ.E55

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

μmoc2/pcandppα1α2α3.E56

The quantity pis the magnitude of the linear momentum vector p, and α1,α2,α3represent the direction cosines, associated with the direction of the linear momentum vector p.

### 5.4 Wave propagation along the +zdirection for κ=moc/ℏ

Without loss of generality, let us consider matter-wave propagation along the +zdirection, that is,

p=p001.E57

Eq. (54) reduces to the following simplified form:

Eo0000+ipc000Eo00ipc00000Eo0000+ipc000Eo00ipc00+ipc00+Eo000ipc0000+Eo00000+ipc00+Eo000ipc0000+EoΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4=EΔ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4E58

where

Eomoc2.E59

The matrix in Eq. (58) is an eight-by-eight square matrix. A compact matrix version of Eq. (58) may be expressed as follows:

Hψn=Enψnn=1,2,3,...8.E60

At this point we are now in a position to determine eight eigenvectors ψnand the corresponding eigenvalues Ensatisfying 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

E+=+EandE=EwhereE=Eo2+p2c2.E61

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

E=γEop=γmovpc=γβEoγ=1/1β2β=v/c.E62

As before, γand βare referred to as the Lorentz factor and speed parameter, respectively. For each of the two eigenvalues, there are four unique eigenvectors. The eight eigenvectors ψnform an orthonormal set, that is,

ψmψn=δmn.E63

The symbol δmnrepresents the Kronecker delta. In Table 2 is a summary of the eigenvalues and orthonormal eigenvectors satisfying the eigenvalue spacetime matrix (Eq. (58)).

EnE1E2E3E4E5E6E7E8
E+γEo+γEoγEoγEo+γEo+γEoγEoγEo
ψnψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8
Δ10+b0+a0000
Δ2+ib0+ia00000
Δ30000+b0+a0
Δ400000+ib0+ia
Ω1a0+b00000
Ω20ia0+ib0000
Ω300000a0+b
Ω40000ia0+ib0

### Table 2.

Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave propagation in the +z direction when κ=moc/.

The constants aand bappearing in Table 2 are defined by

a22γ+1γa2+b2=1b22γ1γ.E64

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

1. ψ1and ψ2represent transverse waves with positive energy +γEo.

2. ψ3and ψ4represent transverse waves with negative energy γEo.

3. ψ5and ψ6represent non-transverse waves with positive energy +γEo.

4. ψ7and ψ8represent non-transverse waves with negative energy γEo.

For wave propagation in the +zdirection, the transverse waves have eigenvector solutions ψwhere elements (3,1), (4,1), (7,1), and (8,1) are identically equal to zero. In other words, Δ=Δ1Δ200and Ω=Ω1Ω200. For this case, Δ1, Δ2and Ω1, Ω2correspond to the xand ycomponents. Thus, for wave propagation in the +zdirection, the transverse wave solutions only have xand yvector components characteristic of a transverse wave in three dimensions.

On the other hand, for wave propagation in the +zdirection, the non-transverse waves have eigenvector solutions ψwhere elements (1,1), (2,1), (5,1), and (6,1) are identically equal to zero. That is to say, Δ=00Δ3Δ4and Ω=00Ω3Ω4. This implies, Δ3and Ω3represent z-components. Δ4and Ω4represent the fourth components (unknown origin) in a four-dimensional space. Thus, for wave propagation in the +zdirection, the non-transverse wave solutions have a zvector component (longitudinal in nature) and a fourth vector component (neither transverse nor longitudinal in nature, perhaps a “time” component) of a non-transverse wave in four dimensions.

The authors, in their most recent publication [1], indicated solutions of their Dirac spacetime matrix equation for free space could be mapped into solutions satisfying the traditional Dirac matrix equation. We wish to explore this in greater detail. The traditional Dirac equation, in the absence of electromagnetic potential terms, is given by

+40i32i10+4+2i1+i3+i3+2+i1402+i1i304Σ1Σ2Σ3Σ4+κΣ1Σ2Σ3Σ4=0000E65

This equation corresponds to the special case employing the Dirac representation (see [12], pp. 694–706) for details. The compact matrix form of Eq. (65) is given by

D̂σ+κσ=o.E66

The Dirac matrix operator D̂represents the four-by-four matrix operator on the left-hand side of Eq. (65), σis the four-by-one ket vector appearing twice on the left-hand side, and ois the four-by-one null ket vector appearing on the right-hand side. For time-harmonic plane-wave solutions, the ket vector σmay be expressed as

σ=σoexp+iprEt/.E67

Substituting this time-harmonic plane-wave solution back into the traditional Dirac equation (65) ultimately leads to the corresponding eigenvalue equation:

pc+μ0+α3iα2+α10+μ+iα2+α1α3+α3iα2+α1μ0+iα2+α1α30μΣ1Σ2Σ3Σ4=EΣ1Σ2Σ3Σ4.E68

For the special case of wave propagation in the +zdirection, the preceding eigenvalue equation reduces to the following simplified form:

+Eo0+pc00+Eo0pc+pc0Eo00pc0EoΣ1Σ2Σ3Σ4=EΣ1Σ2Σ3Σ4.E69

Again using the matrix software MATLAB, the four orthonormal eigenvectors and corresponding eigenvalues satisfying Eq. (69) are listed in the Table 3.

EnE1E2E3E4
E+γEo+γEoγEoγEo
σnσ1σ2σ3σ4
Σ122a22a+22b+22b
Σ222a+22a+22b22b
Σ322b22b22a22a
Σ4+22b22b+22a22a

### Table 3.

Eigenvalues and orthonormal eigenvectors associated with the traditional Dirac equation for wave propagation in the +z direction when κ=moc/.

The quantities aand bappearing in Table 3 are defined by

a22γ+1γa2+b2=1b22γ1γ.E70

Note, the quantities aand bappearing in the traditional Dirac equation eigenvectors listed in Table 3 are the same aand bquantities appearing in the generalized spacetime matrix equation eigenvectors listed in Table 2 for κ=moc/.

### 5.6 Linear transformation equation

For the special case of a matter wave traveling through free space in the + zdirection, we found the orthonormal set of eigenvectors and corresponding eigenvalues, for both the generalized spacetime matrix (Eq. (49)) and the traditional Dirac equation (65), when κ=moc/. These two sets of orthonormal eigenvectors are related [1] through the following linear transformation matrix equation:

Σ1Σ2Σ3Σ4=220000+1i+1i0000+1+i1i1+i1+i00001i+1+i0000Δ1Δ2Δ3Δ4Ω1Ω2Ω3Ω4.E71

The compact matrix form of Eq. (71) is given by

σ=Zψ.E72

When we substitute each the eight eigenvectors ψnfrom Table 2 back into Eq. (71), we obtain the following results:

1. The four transverse eigenvectors in Table 2 map into the four eigenvectors in Table 3:

σ1=Zψ1σ2=Zψ2σ3=Zψ3σ4=Zψ4.E73

• The four non-transverse eigenvectors in Table 2 map into the same four eigenvectors in Table 3:

σ1=Zψ5σ2=Zψ6σ3=Zψ7σ4=Zψ8.E74

• Therefore, whether we use the four transverse eigenvector solutions or the four non-transverse eigenvector solutions satisfying the generalized spacetime matrix (Eq. (49)), the same four eigenvector solutions satisfying the traditional Dirac equation (65) are obtained using Eq. (71). It is noted the four transverse eigenvector solutions could have been obtained from the four Dirac vector equations (37) and (38).

### 5.7 Wave propagation along the +zdirection for κ=0

For the special case of wave propagation in the +zdirection, when κ=0, time-harmonic plane-wave solutions satisfying the generalized spacetime matrix equation for free space (49) yield the set of eigenvectors and eigenvalues presented in Table 4. The eight eigenvectors ψnalso form an orthonormal set, that is,

EnE1E2E3E4E5E6E7E8
pnc+pc+pcpcpc+pc+pcpcpc
ψnψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8
Δ10+b0+a0000
Δ2+ib0+ia00000
Δ30000+b0+a0
Δ400000+ib0+ia
Ω1a0+b00000
Ω20ia0+ib0000
Ω300000a0+b
Ω40000ia0+ib0
vnv1v2v3v4v5v6v7v8
+c+ccc+c+ccc

### Table 4.

Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave propagation in the +z direction when κ=0.

ψmψn=δmn.E75

The constants aand bappearing in Table 4 are now defined by

a22a2+b2=1b22E76

Inspection of the contents of Table 4 reveals the following important results:

1. ψ1and ψ2represent transverse waves moving with speed +c.

2. ψ3and ψ4represent transverse waves moving with speed c.

3. ψ5and ψ6represent non-transverse waves moving with speed +c.

4. ψ7and ψ8represent non-transverse waves moving with speed c.

For wave propagation in the +zdirection, the transverse waves have eigenvector solutions ψwhere elements (3,1), (4,1), (7,1), and (8,1) are identically equal to zero. In other words, Δ=Δ1Δ200and Ω=Ω1Ω200. For this case, Δ1, Δ2and Ω1, Ω2correspond to the xand ycomponents. Thus, for wave propagation in the +zdirection, the transverse wave solutions only have xand yvector components, characteristic of a transverse wave in three dimensions. Only those waves propagating at a speed in free space of +crepresent real electromagnetic waves.

On the other hand, for wave propagation in the +zdirection, the non-transverse waves have eigenvector solutions ψwhere elements (1,1), (2,1), (5,1), and (6,1) are identically equal to zero. That is to say, Δ=00Δ3Δ4and Ω=00Ω3Ω4. This implies, Δ3and Ω3represent z-components. Δ4and Ω4represent the fourth components in a four-dimensional space. Thus, for wave propagation in the +zdirection, the non-transverse wave solutions have a zvector component (longitudinal in nature) and a fourth vector component (neither transverse nor longitudinal in nature) of a non-transverse wave in four dimensions. Perhaps there is new physics regarding these additional solutions.

### 5.8 Unresolved issues regarding the generalized spacetime matrix equation

The eigenvectors and eigenvalues associated with the generalized spacetime matrix equation, for the special case of a time-harmonic plane-wave propagating in free space in the +zdirection, have been determined for both κ=moc/and κ=0. The following are the key points found in this analysis:

1. For the case when κ=moc/, we found there were four orthonormal eigenvectors (two having positive energy eigenvalues +γEoand two having negative energy eigenvalues γEo) describing waves having transverse properties. From Table 2, each of these four eigenvectors have components Δ3=Δ4=Ω3=Ω4=0.Using the linear transformation equation (71), these eigenvectors map nicely into four orthonormal eigenvectors satisfying the traditional Dirac equation.

2. For the case when κ=moc/, we found there were also four orthonormal eigenvectors (again two having positive energy eigenvalues +γEoand two having negative energy eigenvalues γEo) describing waves having non-transverse properties. From Table 2, each of these four eigenvectors have components Δ1=Δ2=Ω1=Ω2=0.Again, using the linear transformation equation (71), these four eigenvectors map nicely into the same four orthonormal eigenvectors satisfying the traditional Dirac equation as mentioned in Case 1.

3. Therefore, for the case when κ=moc/, the generalized spacetime matrix equation (49) for free space provides eight orthonormal eigenvector solutions (both transverse and non-transverse) which map into four orthonormal eigenvector solutions satisfying the traditional Dirac equation (65).

4. For the case when κ=0, we found there were four orthonormal eigenvectors (two associated with waves propagating in free space with speed +cand two associated with waves propagating in free space with speed -c) describing waves having transverse properties. From Table 4, each of these four eigenvectors have components Δ3=Δ4=Ω3=Ω4=0.For the case of transverse waves propagating with +c, these eigenvectors are associated with real electromagnetic waves predicted by the traditional Maxwell equations.

5. For the case when κ=0, we found there were also four orthonormal eigenvectors (two associated with waves propagating in free space with speed +cand two associated with waves propagating in free space with speed -c) describing waves having non-transverse properties. From Table 4, each of these four eigenvectors has components Δ1=Δ2=Ω1=Ω2=0.

In the de Broglie-Bohm picture of quantum mechanics, Hardy [16] and Bell [17] suggest empty waves represented by wave functions propagating in spacetime, but not carrying energy or momentum, can exist. This same concept was called ghost waves or ghost fields by Albert Einstein (see [18]). The controversy as to whether matter waves correspond to real waves or ghost waves has been and is still a subject of debate and controversy.

In Section 5.1, we mentioned that 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. Allowing the elements Δ4and Ω4to have nonzero values in the generalized spacetime matrix equation certainly raises a number of unanswered questions. The following is the author’s list of 12 unanswered questions and mysteries regarding our analysis of the generalized spacetime matrix equation for free space:

For relativistic quantum mechanics—matter waves:

1. What class of particles do the transverse eigenvectors represent?

2. Do the transverse eigenvectors represent real or ghost waves?

3. What class of particles do the non-transverse eigenvectors represent?

4. Do non-transverse eigenvectors represent real or ghost waves?

5. Are the transverse and non-transverse eigenvectors equivalent in some way?

For classical electrodynamics—electromagnetic waves:

1. What can be said about those waves propagating with speed -c?

2. Do these represent a new type of electromagnetic wave?

3. What can be said about those waves having a longitudinal component?

4. What can be said about those waves having a fourth component?

5. Could these be associated with undiscovered electromagnetic waves?

And two last questions:

1. Why do the Diracand Maxwellvector equations resemble each other?

2. Does the spacetime matrix operator M̂have more surprises in store?

## 6. Conclusions

1. The four classical electromagnetic microscopic Maxwell field equations have been rewritten as a single matrix equation, referred to as the Maxwell spacetime matrix equation, using the spacetime matrix operator M̂. The Maxwell spacetime matrix equation is relativistic invariant under a Lorenz transformation.

2. The square eight-by-eight matrix operator M̂has several benefits as summarized next. Other fundamental equations of electromagnetic theory have also been expressed as single matrix equations using the spacetime matrix operator M̂, namely, the electromagnetic wave and charge continuity equations, the Lorentz conditions and electromagnetic potentials, and the electromagnetic potential wave equations.

3. The traditional relativistic Dirac equation for free space has been expressed as a new matrix equation, referred to as the Dirac spacetime matrix equation for free space, using the same spacetime matrix operator M̂. The Dirac spacetime matrix equation is also relativistic invariant under a Lorenz transformation.

4. Solutions of the new Dirac spacetime matrix equation can be easily transformed into solutions satisfying the traditional relativistic Dirac equation using the linear transformation matrix Z.

5. The Dirac spacetime matrix equation is equivalent to four new relativistic quantum mechanical vector equations. We referred to these equations as the Dirac spacetime vector equations. In the absence of electromagnetic potentials, these vector equations resemble the four classical electromagnetic microscopic Maxwell field vector equations in the absence of charge and current densities.

6. Multiplication of the Dirac spacetime matrix equation by the spacetime matrix operator M̂leads to the relativistic Klein-Gordon spacetime matrix equation.

7. Four transverse orthonormal eigenvectors as well as the four non-transverse orthonormal eigenvectors satisfying the Dirac spacetime matrix equation map, via the linear transformation matrix Z, into the same set of four orthonormal eigenvectors satisfying the traditional Dirac equation.

8. A new generalized spacetime matrix equation employing the operator M̂was introduced. This equation is a generalization of the Maxwell and Dirac spacetime matrix equations for free space. We explored time-harmonic plane-wave solutions of this equation as well as their properties. Some of results obtained may suggest new physics.

## Acknowledgments

We are most appreciative of the help by Ms. Trin Riojas of the Optical Sciences Center in coordinating computer station inputs/outputs between authors and publishers. The past informal discussions with Dr. Arvind S. Marathay of the Optical Sciences Center are also greatly appreciated. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Richard P. Bocker and B. Roy Frieden (June 19th 2019). Eight-by-Eight Spacetime Matrix Operator and Its Applications, Progress in Relativity, Calin Gheorghe Buzea, Maricel Agop and Leo Butler, IntechOpen, DOI: 10.5772/intechopen.86982. Available from:

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