Eight-by-Eight Spacetime Matrix Operator and Its Applications

*Richard P. Bocker and B. Roy Frieden*

#### **Abstract**

A recent journal article by the authors introduced the eight-by-eight spacetime matrix operator *M*^ 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 *M*^ 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 *M*^ , 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 *M*^ plays 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):

$$
\hat{M} \equiv \begin{bmatrix} -\partial\_4 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\partial\_3 & +\partial\_2 & -\partial\_1 \\ \mathbf{0} & -\partial\_4 & \mathbf{0} & \mathbf{0} & +\partial\_3 & \mathbf{0} & -\partial\_1 & -\partial\_2 \\ \mathbf{0} & \mathbf{0} & -\partial\_4 & \mathbf{0} & -\partial\_2 & +\partial\_1 & \mathbf{0} & -\partial\_3 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\partial\_4 & +\partial\_1 & +\partial\_2 & +\partial\_3 & \mathbf{0} \\ \mathbf{0} & +\partial\_3 & -\partial\_2 & +\partial\_1 & +\partial\_4 & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ -\partial\_3 & \mathbf{0} & +\partial\_1 & +\partial\_2 & \mathbf{0} & +\partial\_4 & \mathbf{0} & \mathbf{0} \\ +\partial\_2 & -\partial\_1 & \mathbf{0} & +\partial\_3 & \mathbf{0} & \mathbf{0} & +\partial\_4 & \mathbf{0} \\ -\partial\_1 & -\partial\_2 & -\partial\_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & +\partial\_4 \end{bmatrix}. \tag{1}
$$


#### **Table 1.**

*Compact matrix equations where the spacetime matrix operator M*^ *plays a central role*.

The partial derivative symbols are defined by the following:

$$
\partial\_1 \equiv \frac{\partial}{\partial x} \qquad \partial\_2 \equiv \frac{\partial}{\partial y} \qquad \partial\_3 \equiv \frac{\partial}{\partial z} \qquad \partial\_4 \equiv \frac{1}{ic} \frac{\partial}{\partial t} . \tag{2}
$$

4.In addition

p. 15) operators are defined by

□<sup>2</sup> � <sup>∇</sup><sup>2</sup> � <sup>1</sup>

**3. Maxwell spacetime matrix equation**

*c*2

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

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

**3.1 Maxwell spacetime matrix equation for free space**

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

divergences and two curl equations, namely,

**19**

in the absence of charges and currents, this later version is given by

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

*∂*2

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>¼</sup> *<sup>M</sup>*^ <sup>2</sup> <sup>¼</sup> *<sup>I</sup>* □<sup>2</sup>

*<sup>∂</sup>t*<sup>2</sup> and <sup>∇</sup><sup>2</sup> � *<sup>∂</sup>*<sup>2</sup>

Some authors use the □ symbol to represent the d'Alembertian operator.

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 *M*^ was published recently [1]. For free space,

The wave function ∣ *f*i is an eight-by-one ket vector containing, in general, six nonzero scalar components associated with the electric field vector **E** ¼ ð Þ *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup> and the magnetic induction vector **B** ¼ ð Þ *B*<sup>1</sup> *B*<sup>2</sup> *B*<sup>3</sup> . The elements (4,1) and (8,1) in ∣ *f*i have 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 ∣*o*i represents the eight-by-one null vector.

The Maxwell spacetime matrix equation (9) when expanded is equivalent to two

*∂x*<sup>2</sup> þ

*<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i ¼ <sup>∣</sup>*o*i*:* (10)

∇ � **E** ¼ 0 and ∇ � **B** ¼ 0 (11)

*iE*<sup>1</sup> *iE*<sup>2</sup> *iE*<sup>3</sup> 0 *B*1 *B*2 *B*3 0

*:* (9)

¼

*∂*2 *∂y*<sup>2</sup> þ

The symbol *δμν* is the Kronecker delta, and *I* represents the eight-by-eight identity matrix. The d'Alembertian (see [4], p. 290) and the Laplacian (see [4],

*:* (7)

*∂*2

*<sup>∂</sup>z*<sup>2</sup> *:* (8)

The imaginary quantity *i* represents the square root of minus one, and the physical quantity *c* corresponds to the speed of light in free space.

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

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

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

$$
\hat{M} = M\_1 \partial\_1 + M\_2 \partial\_2 + M\_3 \partial\_3 + M\_4 \partial\_4. \tag{3}
$$

The four eight-by-eight matrices *Mμ*, where *μ* ¼ 1*;* 2*;* 3*;* 4, are simply referred to as the spacetime matrices. These matrices have the following properties:

1. Each matrix *M<sup>μ</sup>* is equal to its own multiplicative inverse

$$\mathcal{M}\_{\mu} = \mathcal{M}\_{\mu}^{-1}. \tag{4}$$

2. These matrices satisfy the anti-commutation relation

$$\mathbf{M}\_{\mu}\mathbf{M}\_{\nu} + \mathbf{M}\_{\nu}\mathbf{M}\_{\mu} = 2\delta\_{\mu\nu}I.\tag{5}$$

3. Each matrix *M<sup>μ</sup>* is Hermitian

$$\mathcal{M}\_{\mu} = \mathcal{M}\_{\mu}^{\dagger}.\tag{6}$$

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

4.In addition

The partial derivative symbols are defined by the following:

*Compact matrix equations where the spacetime matrix operator M*^ *plays a central role*.

**Compact matrix equation Compact matrix equation description**

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup>*a*i ¼ <sup>∣</sup> *<sup>j</sup>*<sup>i</sup> Electromagnetic potential wave equations *<sup>M</sup>*^ <sup>∣</sup>*ϕ*i þ *<sup>κ</sup>*∣*ϕ*i ¼ <sup>∣</sup>*o*<sup>i</sup> Dirac spacetime matrix equation for free space

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup>*ϕ*i � *<sup>κ</sup>*<sup>2</sup>∣*ϕ*i ¼ <sup>∣</sup>*o*<sup>i</sup> Klein-Gordon spacetime matrix equation for free space *<sup>M</sup>*^ <sup>∣</sup>*ψ*i þ *<sup>κ</sup>*∣*ψ*i ¼ <sup>∣</sup>*o*<sup>i</sup> Generalized spacetime matrix equation for free space

*<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i ¼ <sup>∣</sup>*o*<sup>i</sup> Maxwell spacetime matrix equation for free space *<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i ¼ <sup>∣</sup> *<sup>j</sup>*<sup>i</sup> Maxwell matrix equation with charges and currents *<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i ¼ *<sup>M</sup>*^ <sup>∣</sup> *<sup>j</sup>*<sup>i</sup> Charge continuity and electromagnetic wave equations *<sup>M</sup>*^ <sup>∣</sup>*a*i ¼ <sup>∣</sup> *<sup>f</sup>*<sup>i</sup> Lorentz conditions and electromagnetic potentials

> *<sup>∂</sup>*<sup>2</sup> � *<sup>∂</sup> ∂y*

physical quantity *c* corresponds to the speed of light in free space.

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

*<sup>∂</sup>*<sup>3</sup> � *<sup>∂</sup> ∂z*

The imaginary quantity *i* represents the square root of minus one, and the

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

The spacetime matrix operator *M*^ , defined in Eq. (1), may also be expressed by

The four eight-by-eight matrices *Mμ*, where *μ* ¼ 1*;* 2*;* 3*;* 4, are simply referred to

*<sup>M</sup><sup>μ</sup>* <sup>¼</sup> *<sup>M</sup>*�<sup>1</sup>

*<sup>M</sup><sup>μ</sup>* <sup>¼</sup> *<sup>M</sup>*†

as the spacetime matrices. These matrices have the following properties:

1. Each matrix *M<sup>μ</sup>* is equal to its own multiplicative inverse

2. These matrices satisfy the anti-commutation relation

*<sup>M</sup>*^ <sup>¼</sup> *<sup>M</sup>*1*∂*<sup>1</sup> <sup>þ</sup> *<sup>M</sup>*2*∂*<sup>2</sup> <sup>þ</sup> *<sup>M</sup>*3*∂*<sup>3</sup> <sup>þ</sup> *<sup>M</sup>*4*∂*4*:* (3)

*MμM<sup>ν</sup>* þ *MνM<sup>μ</sup>* ¼ 2*δμνI:* (5)

*<sup>μ</sup> :* (4)

*<sup>μ</sup>:* (6)

*<sup>∂</sup>*<sup>4</sup> � <sup>1</sup> *ic ∂ ∂t*

*:* (2)

*<sup>∂</sup>*<sup>1</sup> � *<sup>∂</sup> ∂x*

employed throughout this chapter.

3. Each matrix *M<sup>μ</sup>* is Hermitian

**18**

the following equation:

**Table 1.**

*Progress in Relativity*

$$
\hat{\mathbf{M}}\hat{\mathbf{M}}=\hat{\mathbf{M}}^2 = I\,\Box^2.\tag{7}
$$

The symbol *δμν* is the Kronecker delta, and *I* represents the eight-by-eight identity matrix. The d'Alembertian (see [4], p. 290) and the Laplacian (see [4], p. 15) operators are defined by

$$
\Box^2 \equiv \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \qquad \text{and} \qquad \nabla^2 \equiv \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}. \tag{8}
$$

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.

#### **3.1 Maxwell spacetime matrix equation for free space**

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 *M*^ was published recently [1]. For free space, in the absence of charges and currents, this later version is given by

$$
\begin{bmatrix}
\mathbf{0} & -\partial\_{4} & \mathbf{0} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & -\partial\_{1} & -\partial\_{2} \\
\mathbf{0} & \mathbf{0} & -\partial\_{4} & \mathbf{0} & -\partial\_{2} & +\partial\_{1} & \mathbf{0} & -\partial\_{3} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -\partial\_{4} & +\partial\_{1} & +\partial\_{2} & +\partial\_{3} & \mathbf{0} \\
\mathbf{0} & +\partial\_{3} & -\partial\_{2} & +\partial\_{1} & +\partial\_{4} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
+\partial\_{2} & -\partial\_{1} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & \mathbf{0} & +\partial\_{4} & \mathbf{0} \\
\end{bmatrix}
\begin{bmatrix}
\dot{i}E\_{1} \\
iE\_{2} \\
iE\_{3} \\
\mathbf{0} \\
B\_{1} \\
B\_{2} \\
B\_{3} \\
\mathbf{0}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{bmatrix}.
\tag{9}
$$

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

$$
\hat{M}|f\rangle = |o\rangle. \tag{10}
$$

The wave function ∣ *f*i is an eight-by-one ket vector containing, in general, six nonzero scalar components associated with the electric field vector **E** ¼ ð Þ *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup> and the magnetic induction vector **B** ¼ ð Þ *B*<sup>1</sup> *B*<sup>2</sup> *B*<sup>3</sup> . The elements (4,1) and (8,1) in ∣ *f*i have 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 ∣*o*i represents the eight-by-one null vector.

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

$$
\nabla \cdot \mathbf{E} = \mathbf{0} \qquad \text{and} \qquad \nabla \cdot \mathbf{B} = \mathbf{0} \tag{11}
$$

$$
\nabla \times \mathbf{E} + \frac{1}{c} \frac{\partial}{\partial t} \mathbf{B} = \mathbf{0} \qquad \text{and} \qquad \nabla \times \mathbf{B} - \frac{1}{c} \frac{\partial}{\partial t} \mathbf{E} = \mathbf{0}.\tag{12}
$$

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

Eq. (19), when expanded, is equivalent to two divergences and two curl equations. The resulting four vector equations are referred to as the microscopic Max-

**<sup>J</sup>***<sup>m</sup>* and <sup>∇</sup> � **<sup>B</sup>** � <sup>1</sup>

The various scalar and vector quantities appearing in the microscopic Maxwell vector equations are the electric field vector **E** ¼ ð Þ *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup> , the magnetic induction vector **B** ¼ ð Þ *B*<sup>1</sup> *B*<sup>2</sup> *B*<sup>3</sup> , the electric current density vector **J***<sup>e</sup>* ¼ *Je*<sup>1</sup> *Je*<sup>2</sup> *Je*<sup>3</sup> ð Þ, the magnetic current density vector **J***<sup>m</sup>* ¼ *Jm*<sup>1</sup> *Jm*<sup>2</sup> *Jm*<sup>3</sup> ð Þ, the electric charge density *ρe*, the magnetic charge density *ρm*, and the speed of light *c* in 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

**3.3 Charge continuity and electromagnetic wave equations**

discovered in nature. The ket vector ∣ *f*i represents the eight-by-one column vector on the left-hand side of Eq. (19). The ket vector ∣ *j*i corresponds to the eight-by-one column vector on the right-hand side of Eq. (19) multiplied by the factor 4*π=*c.

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

Expanding this single matrix equation yields the charge continuity and electro-

*ρ<sup>e</sup>* ¼ 0 and ∇ � **J***<sup>m</sup>* þ

By using the spacetime matrix operator *M*^ , 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:

<sup>∇</sup> � **<sup>J</sup>***<sup>m</sup>* and □<sup>2</sup>

∇ � **E** ¼ þ4*πρ<sup>e</sup>* and ∇ � **B** ¼ þ4*πρ<sup>m</sup>* (21)

*c ∂ ∂t* **E** ¼ þ

*<sup>M</sup>*^ *<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i ¼ *<sup>M</sup>*^ <sup>∣</sup> *<sup>j</sup>*i*:* (23)

*∂ ∂t*

**<sup>B</sup>** <sup>¼</sup> <sup>4</sup>*<sup>π</sup> c*2 *∂ ∂t*

*ρ<sup>m</sup>* ¼ 0 (24)

*c*

∇ � **J***e:* (25)

**<sup>J</sup>***<sup>m</sup>* <sup>þ</sup> <sup>4</sup>*π*∇*ρ<sup>m</sup>* � <sup>4</sup>*<sup>π</sup>*

well field equations (see [8], pp. 283–290). They are given by

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

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

∇ � **E** þ

operator *M*^ . That is,

magnetic wave equations:

□2

**21**

**<sup>E</sup>** <sup>¼</sup> <sup>4</sup>*<sup>π</sup> c*2 *∂ ∂t* ∇ � **J***<sup>e</sup>* þ

**J***<sup>e</sup>* þ 4*π*∇*ρ<sup>e</sup>* þ

*∂ ∂t*

4*π c*

**3.4 Lorentz conditions and electromagnetic potentials**

1 *c ∂ ∂t*

**<sup>B</sup>** ¼ � <sup>4</sup>*<sup>π</sup> c*

*<sup>M</sup>*^ <sup>∣</sup> *<sup>f</sup>*i ¼ <sup>∣</sup> *<sup>j</sup>*i*:* (20)

4*π c*

**J***e:* (22)

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

$$\mathbf{E}(\mathbf{r},t) = \mathbf{E}\_\mathbf{o} \exp\left\{i(\mathbf{k}\cdot\mathbf{r}-\alpha t)\right\} \qquad \text{and} \qquad \mathbf{B}(\mathbf{r},t) = \mathbf{B}\_\mathbf{o} \exp\left\{i(\mathbf{k}\cdot\mathbf{r}-\alpha t)\right\} \tag{13}$$

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

$$
\mathbf{k} \cdot \mathbf{E\_0} = 0 \qquad \text{and} \qquad \mathbf{k} \cdot \mathbf{B\_0} = 0 \tag{14}
$$

$$\mathbf{k} \times \mathbf{E\_0} = +\frac{\alpha}{c} \mathbf{B\_0} \qquad \text{and} \qquad \mathbf{k} \times \mathbf{B\_0} = -\frac{\alpha}{c} \mathbf{E\_0}.\tag{15}$$

The quantities **k** and *ω* correspond to the wave vector and the angular frequency associated with the electromagnetic wave; **r** and *t* represent the position vector and the instantaneous time. From the preceding equations, we find the vectors **E***o*, **B***o*, and **k** are mutually perpendicular. That is,

$$\mathbf{k} \perp \mathbf{E}\_{\mathbf{o}} \qquad \mathbf{E}\_{\mathbf{o}} \perp \mathbf{B}\_{\mathbf{o}} \qquad \mathbf{k} \perp \mathbf{B}\_{\mathbf{o}}.\tag{16}$$

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

$$E\_o = B\_o \tag{17}$$

and

$$
\omega = k \mathcal{c} \qquad \lambda \mathcal{f} = \mathcal{c} \qquad \text{where} \qquad \mathcal{w} = 2 \pi \mathcal{f} \qquad k = 2 \pi/\lambda. \tag{18}
$$

The quantities *k*, *f*, and *λ* represent 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 **Eo** and **Bo** are 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

$$
\begin{bmatrix}
\mathbf{0} & -\partial\_{4} & \mathbf{0} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & -\partial\_{1} & -\partial\_{2} \\
\mathbf{0} & \mathbf{0} & -\partial\_{4} & \mathbf{0} & -\partial\_{2} & +\partial\_{1} & \mathbf{0} & -\partial\_{3} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -\partial\_{4} & +\partial\_{1} & +\partial\_{2} & +\partial\_{3} & \mathbf{0} \\
\mathbf{0} & +\partial\_{3} & -\partial\_{2} & +\partial\_{1} & +\partial\_{4} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
+\partial\_{2} & -\partial\_{1} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & \mathbf{0} & +\partial\_{4} & \mathbf{0} \\
\end{bmatrix}
\begin{bmatrix}
f\_{\ell 1} \\
i E\_{2} \\
0 \\
B\_{1} \\
B\_{2} \\
B\_{3} \\
0
\end{bmatrix} = \frac{4\pi}{c} \begin{bmatrix}
f\_{\ell 1} \\
f\_{\ell 2} \\
f\_{\ell 3} \\
\partial\_{1}
\end{bmatrix}.\tag{19}
$$

∇ � **E** þ

matter terms (see [8], pp. 362–368).

*Progress in Relativity*

the following set of equations:

important results

and

Gaussian units.

**20**

terms [1], is given by

1 *c ∂ ∂t*

**<sup>k</sup>** � **Eo** ¼ þ *<sup>ω</sup>*

and **k** are mutually perpendicular. That is,

*c*

**<sup>B</sup>** <sup>¼</sup> 0 and <sup>∇</sup> � **<sup>B</sup>** � <sup>1</sup>

We recognize these four equations as the traditional Maxwell field equations (Gaussian units) for free space in the absence of charges, currents, and ordinary

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

**E r**ð Þ¼ *; t* **Eo** exp f g *i*ð Þ **k** � **r** � *ωt* and **B r**ð Þ¼ *; t* **Bo** exp f g *i*ð Þ **k** � **r** � *ωt* (13)

will next be substituted back into the previous four vector equations. This yields

The quantities **k** and *ω* correspond to the wave vector and the angular frequency associated with the electromagnetic wave; **r** and *t* represent the position vector and the instantaneous time. From the preceding equations, we find the vectors **E***o*, **B***o*,

These properties represent transverse electromagnetic waves. We also obtain the

The quantities *k*, *f*, and *λ* represent 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 **Eo** and **Bo** are equal, a well-known result in electromagnetic wave propagation. Recall we are using

The Maxwell spacetime matrix equation, with the addition of charge and current

*iE*<sup>1</sup> *iE*<sup>2</sup> *iE*<sup>3</sup> 0 *B*1 *B*2 *B*3 0

<sup>¼</sup> <sup>4</sup>*<sup>π</sup> c*

*Je*1 *Je*2 *Je*3 *cρ<sup>m</sup> iJm*<sup>1</sup> *iJm*<sup>2</sup> *iJm*<sup>3</sup> �*icρ<sup>e</sup>*

*:* (19)

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

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

*ω* ¼ *kc λf* ¼ *c* where *ω* ¼ 2*πf k* ¼ 2*π=λ:* (18)

**Bo** and **<sup>k</sup>** � **Bo** ¼ � *<sup>ω</sup>*

*c ∂ ∂t*

**k** � **Eo** ¼ 0 and **k** � **Bo** ¼ 0 (14)

**k** ⊥**Eo Eo** ⊥ **Bo k** ⊥**Bo***:* (16)

*c*

*Eo* ¼ *Bo* (17)

**E** ¼ 0*:* (12)

**Eo***:* (15)

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

$$
\ddot{\mathcal{M}}|f\rangle = |j\rangle. \tag{20}
$$

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

$$
\nabla \cdot \mathbf{E} = +4\pi \rho\_{\varepsilon} \qquad \text{and} \qquad \nabla \cdot \mathbf{B} = +4\pi \rho\_{m} \tag{21}
$$

$$
\nabla \times \mathbf{E} + \frac{1}{c} \frac{\partial}{\partial t} \mathbf{B} = -\frac{4\pi}{c} \mathbf{J}\_m \qquad \text{and} \qquad \nabla \times \mathbf{B} - \frac{1}{c} \frac{\partial}{\partial t} \mathbf{E} = +\frac{4\pi}{c} \mathbf{J}\_c. \tag{22}
$$

The various scalar and vector quantities appearing in the microscopic Maxwell vector equations are the electric field vector **E** ¼ ð Þ *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup> , the magnetic induction vector **B** ¼ ð Þ *B*<sup>1</sup> *B*<sup>2</sup> *B*<sup>3</sup> , the electric current density vector **J***<sup>e</sup>* ¼ *Je*<sup>1</sup> *Je*<sup>2</sup> *Je*<sup>3</sup> ð Þ, the magnetic current density vector **J***<sup>m</sup>* ¼ *Jm*<sup>1</sup> *Jm*<sup>2</sup> *Jm*<sup>3</sup> ð Þ, the electric charge density *ρe*, the magnetic charge density *ρm*, and the speed of light *c* in 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 ∣ *f*i represents the eight-by-one column vector on the left-hand side of Eq. (19). The ket vector ∣ *j*i corresponds to the eight-by-one column vector on the right-hand side of Eq. (19) multiplied by the factor 4*π=*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 *M*^ . That is,

$$
\hat{\mathbf{M}}\hat{\mathbf{M}}|f\rangle = \hat{\mathbf{M}}|j\rangle. \tag{23}
$$

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

$$
\nabla \cdot \mathbf{J}\_{\epsilon} + \frac{\partial}{\partial t} \rho\_{\epsilon} = \mathbf{0} \qquad \text{and} \qquad \nabla \cdot \mathbf{J}\_{m} + \frac{\partial}{\partial t} \rho\_{m} = \mathbf{0} \tag{24}
$$

$$
\Box \Box^2 \mathbf{E} = \frac{4\pi}{c^2} \frac{\partial}{\partial t} \mathbf{J}\_\epsilon + 4\pi \nabla \rho\_\epsilon + \frac{4\pi}{c} \nabla \times \mathbf{J}\_m \qquad \text{and} \qquad \Box^2 \mathbf{B} = \frac{4\pi}{c^2} \frac{\partial}{\partial t} \mathbf{J}\_m + 4\pi \nabla \rho\_m - \frac{4\pi}{c} \nabla \times \mathbf{J}\_\epsilon. \tag{25}
$$

#### **3.4 Lorentz conditions and electromagnetic potentials**

By using the spacetime matrix operator *M*^ , 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:

$$
\begin{bmatrix}
\mathbf{0} & -\partial\_{4} & \mathbf{0} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & -\partial\_{1} & -\partial\_{2} \\
\mathbf{0} & \mathbf{0} & -\partial\_{4} & \mathbf{0} & -\partial\_{2} & +\partial\_{1} & \mathbf{0} & -\partial\_{3} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -\partial\_{4} & +\partial\_{1} & +\partial\_{2} & +\partial\_{3} & \mathbf{0} \\
\mathbf{0} & +\partial\_{3} & -\partial\_{2} & +\partial\_{1} & +\partial\_{4} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
+\partial\_{2} & -\partial\_{1} & \mathbf{0} & +\partial\_{3} & \mathbf{0} & \mathbf{0} & +\partial\_{4} & \mathbf{0} \\
\end{bmatrix}
\begin{bmatrix}
iE\_{1} \\
\end{bmatrix} = \begin{bmatrix}
iE\_{1} \\
iE\_{2} \\
iE\_{3} \\
0 \\
B\_{1} \\
B\_{2} \\
B\_{3} \\
0
\end{bmatrix}.
\tag{26}
$$

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

$$
\hat{M}|a\rangle = |f\rangle. \tag{27}
$$

**4. Dirac spacetime matrix equation**

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

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

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

�*∂*<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. (34) is given by

stant *κ* is defined by

*h* divided by 2*π*.

gence and two curl equations [1], namely,

*c ∂ ∂t*

<sup>∇</sup> � **<sup>U</sup>** ¼ � <sup>1</sup>

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 *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 poten-

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 con-

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

The Dirac spacetime matrix equation (34) when expanded is equivalent to eight partial differential equations. These eight equations can be rewritten as two diver-

**L** � *iκ***L** and ∇ � **L** ¼ þ

free space in the absence of charge, current, and ordinary matter terms.

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

*iU*<sup>1</sup> *iU*<sup>2</sup> *iU*<sup>3</sup> *L*1 *L*2 *L*3 

þ *κ*

*<sup>M</sup>*^ <sup>∣</sup>*ϕ*i þ *<sup>κ</sup>*∣*ϕ*i ¼ <sup>∣</sup>*o*i*:* (35)

*κ* � *moc=*ℏ*:* (36)

∇ � **U** ¼ 0 and ∇ � **L** ¼ 0 (37)

 *c ∂ ∂t*

**U** � *iκ***U***:* (38)

*iU*<sup>1</sup> *iU*<sup>2</sup> *iU*<sup>3</sup> *L*1 *L*2 *L*3 

*:* (34)

tials [11], the Dirac spacetime matrix equation for free space is given by

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:

$$
\nabla \cdot \mathbf{A}\_{\epsilon} + \frac{1}{c} \frac{\partial}{\partial t} \phi\_{\epsilon} = 0 \qquad \text{and} \qquad \nabla \cdot \mathbf{A}\_{m} + \frac{1}{c} \frac{\partial}{\partial t} \phi\_{m} = 0 \tag{28}
$$

$$\mathbf{E} = -\nabla\phi\_{\epsilon} - \frac{1}{c}\frac{\partial}{\partial t}\mathbf{A}\_{\epsilon} - \nabla \times \mathbf{A}\_{m} \qquad \text{and} \qquad \mathbf{B} = -\nabla\phi\_{m} - \frac{1}{c}\frac{\partial}{\partial t}\mathbf{A}\_{m} + \nabla \times \mathbf{A}\_{\epsilon}. \tag{29}$$

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

#### **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 *M*^ . This gives

$$
\hat{\mathbf{M}}\hat{\mathbf{M}}|a\rangle = \hat{\mathbf{M}}|f\rangle. \tag{30}
$$

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

$$
\hat{M}\hat{M}|\mathfrak{a}\rangle = |\mathfrak{j}\rangle.\tag{31}
$$

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

$$
\Box^2 \phi\_\varepsilon = -4\pi \rho\_\varepsilon \qquad \text{and} \qquad \Box^2 \phi\_m = -4\pi \rho\_m \tag{32}
$$

$$
\Box^2 \mathbf{A}\_\epsilon = -\frac{4\pi}{c} \mathbf{J}\_\epsilon \qquad \text{and} \qquad \Box^2 \mathbf{A}\_m = -\frac{4\pi}{c} \mathbf{J}\_m. \tag{33}
$$

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