**5.4 Wave propagation along the +***z* **direction for** *κ* ¼ *moc=***ℏ**

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

$$\mathbf{p} = p \begin{pmatrix} \mathbf{0} \ \mathbf{0} \ \mathbf{1} \end{pmatrix}. \tag{57}$$

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

$$
\begin{bmatrix}
0 & -E\_o & 0 & 0 & -ipc & 0 & 0 & 0 \\
0 & 0 & -E\_o & 0 & 0 & 0 & 0 & +ipc \\
0 & 0 & 0 & -E\_o & 0 & 0 & -ipc & 0 \\
0 & +ipc & 0 & 0 & +E\_o & 0 & 0 & 0 \\
0 & 0 & 0 & +ipc & 0 & 0 & +E\_o & 0 \\
0 & 0 & -ipc & 0 & 0 & 0 & 0 & +E\_o
\end{bmatrix}
\begin{bmatrix}
\Delta\_1 \\
\Delta\_2 \\
\Delta\_3 \\
\Delta\_4 \\
\Delta\_5 \\
\Delta\_6 \\
\Delta\_7 \\
\Delta\_8
\end{bmatrix} = E \begin{bmatrix}
\Delta\_1 \\
\Delta\_2 \\
\Delta\_3 \\
\Delta\_4 \\
\Delta\_5 \\
\Delta\_6 \\
\Delta\_7
\end{bmatrix} \tag{58}
$$

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

where

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

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

We will refer to Eq. (54) as the eigenvalue spacetime matrix equation. The

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

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 momen-

Without loss of generality, let us consider matter-wave propagation along the +*z*

�*μ* 0000 þ*iα*<sup>3</sup> �*iα*<sup>2</sup> þ*iα*<sup>1</sup> �*μ* 0 0 �*iα*<sup>3</sup> 0 þ*iα*<sup>1</sup> þ*iα*<sup>2</sup> 0 0 �*μ* 0 þ*iα*<sup>2</sup> �*iα*<sup>1</sup> 0 þ*iα*<sup>3</sup> �*μ* �*iα*<sup>1</sup> �*iα*<sup>2</sup> �*iα*<sup>3</sup> 0 þ*iα*<sup>3</sup> �*iα*<sup>2</sup> þ*iα*<sup>1</sup> þ*μ* 000 �*iα*<sup>3</sup> 0 þ*iα*<sup>1</sup> þ*iα*<sup>2</sup> 0 þ*μ* 0 0 þ*iα*<sup>2</sup> �*iα*<sup>1</sup> 0 þ*iα*<sup>3</sup> 0 0 þ*μ* 0 �*iα*<sup>1</sup> �*iα*<sup>2</sup> �*iα*<sup>3</sup> 0000 þ*μ*

compact matrix form of Eq. (54) is represented by

*μ* � *moc* 

**5.4 Wave propagation along the +***z* **direction for** *κ* ¼ *moc=***ℏ**

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

�*Eo* 0000 þ*ipc* 0 0 �*Eo* 0 0 �*ipc* 000 0 0 �*Eo* 0000 þ*ipc* �*Eo* 0 0 �*ipc* 0 þ*ipc* 0 0 þ*Eo* 000 �*ipc* 0000 þ*Eo* 0 0 þ*ipc* 0 0 þ*Eo* 0 0 0 �*ipc* 0000 þ*Eo*

following eigenvalue equation:

*pc*

in Eq. (54):

tum vector **p**.

direction, that is,

*Progress in Relativity*

∣*ψ*i ¼ ∣*ψo*i exp ½ � þ*i*ð Þ **p** � **r** � *Et =*ℏ *:* (53)

*H*∣*ψ*i ¼ *E*∣*ψ*i*:* (55)

*=pc* and **p** � *p* ð Þ *α*<sup>1</sup> *α*<sup>2</sup> *α*<sup>3</sup> *:* (56)

**p** ¼ *p* ð Þ 001 *:* (57)

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>

*E*

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>

(58)

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>

*E*

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>

*:* (54)

$$E\_o \equiv m\_o c^2.\tag{59}$$

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

$$H|\psi\_n\rangle = E\_n|\psi\_n\rangle \qquad n = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots \\ \text{8.} \tag{60}$$

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

$$E\_{+} = +E \quad \text{and} \quad E\_{-} = -E \qquad \text{where} \qquad E = \sqrt{E\_{o}^{2} + p^{2}c^{2}}.\tag{61}$$

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

$$E = \gamma E\_o \qquad p = \gamma m\_o \nu \qquad pc = \gamma \beta E\_o \qquad \gamma = \mathbf{1}/\sqrt{\mathbf{1} - \boldsymbol{\beta}^2} \qquad \beta = v/c. \tag{62}$$

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

$$
\langle \Psi\_m | \Psi\_n \rangle = \delta\_{mn}. \tag{63}
$$

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

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

$$a \equiv \frac{\sqrt{2}}{2} \sqrt{\frac{\chi + 1}{\chi}} \qquad \qquad \qquad a^2 + b^2 = 1 \qquad \qquad b \equiv \frac{\sqrt{2}}{2} \sqrt{\frac{\chi - 1}{\chi}}.\tag{64}$$

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


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 characteristic of a transverse wave in three dimensions.


on the left-hand side, and ∣*o*i is the four-by-one null ket vector appearing on the right-hand side. For time-harmonic plane-wave solutions, the ket vector ∣*σ*i may be

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

For the special case of wave propagation in the +*z* direction, the preceding

Again using the matrix software MATLAB, the four orthonormal eigenvectors

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

Note, the quantities *a* and *b* appearing in the traditional Dirac equation eigenvectors listed in **Table 3** are the same *a* and *b* quantities appearing in the generalized

and corresponding eigenvalues satisfying Eq. (69) are listed in the **Table 3**.

spacetime matrix equation eigenvectors listed in **Table 2** for *κ* ¼ *moc=*ℏ.

<sup>2</sup> *<sup>a</sup>* � ffiffi

<sup>2</sup> *<sup>a</sup>* <sup>þ</sup> ffiffi

<sup>2</sup> *<sup>b</sup>* � ffiffi

<sup>2</sup> *<sup>b</sup>* � ffiffi

*En E***<sup>1</sup>** *E***<sup>2</sup>** *E***<sup>3</sup>** *E***<sup>4</sup>** *E* þ*γEo* þ*γEo* �*γEo* �*γEo* ∣*σn*i ∣*σ*1i ∣*σ*2i ∣*σ*3i ∣*σ*4i

> 2 p

> 2 p

> 2 p

> 2 p

*Eigenvalues and orthonormal eigenvectors associated with the traditional Dirac equation for wave propagation*

<sup>2</sup> *<sup>a</sup>* <sup>þ</sup> ffiffi

<sup>2</sup> *<sup>a</sup>* <sup>þ</sup> ffiffi

<sup>2</sup> *<sup>b</sup>* � ffiffi

<sup>2</sup> *<sup>b</sup>* <sup>þ</sup> ffiffi

2 p

2 p

2 p

2 p

Σ1 Σ2 Σ3 Σ4

¼ *E*

þ*μ* 0 þ*α*<sup>3</sup> �*iα*<sup>2</sup> þ *α*<sup>1</sup> 0 þ*μ* þ*iα*<sup>2</sup> þ *α*<sup>1</sup> �*α*<sup>3</sup> þ*α*<sup>3</sup> �*iα*<sup>2</sup> þ *α*<sup>1</sup> �*μ* 0 þ*iα*<sup>2</sup> þ *α*<sup>1</sup> �*α*<sup>3</sup> 0 �*μ*

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

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

eigenvalue equation reduces to the following simplified form:

þ*Eo* 0 þ*pc* 0 0 þ*Eo* 0 �*pc* þ*pc* 0 �*Eo* 0 0 �*pc* 0 �*Eo*

The quantities *a* and *b* appearing in **Table 3** are defined by

∣*σ*i ¼ ∣*σo*i exp ½ � þ*i*ð Þ **p** � **r** � *Et =*ℏ *:* (67)

Σ1 Σ2 Σ3 Σ4

Σ1 Σ2 Σ3 Σ4

ffiffi 2 p 2

s

ffiffiffiffiffiffiffiffiffiffi *γ* � 1 *γ*

<sup>2</sup> *<sup>b</sup>* <sup>þ</sup> ffiffi

<sup>2</sup> *<sup>b</sup>* � ffiffi

<sup>2</sup> *<sup>a</sup>* � ffiffi

<sup>2</sup> *<sup>a</sup>* � ffiffi

¼ *E*

Σ1 Σ2 Σ3 Σ4

*:* (69)

*:* (70)

2 p <sup>2</sup> *b*

2 p <sup>2</sup> *b*

2 p <sup>2</sup> *a*

2 p <sup>2</sup> *a*

*:* (68)

expressed as

*pc*

*a* �

ffiffi 2 p 2

<sup>Σ</sup><sup>1</sup> � ffiffi

<sup>Σ</sup><sup>2</sup> � ffiffi

<sup>Σ</sup><sup>3</sup> � ffiffi

<sup>Σ</sup><sup>4</sup> <sup>þ</sup> ffiffi

*in the +z direction when κ* ¼ *moc=*ℏ.

**Table 3.**

**31**

2 p

2 p

2 p

2 p

s

ffiffiffiffiffiffiffiffiffiffi *γ* þ 1 *γ*

#### **Table 2.**

*Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave propagation in the +z direction when κ* ¼ *moc=*ℏ.

On the other hand, for wave propagation in the +*z* direction, the non-transverse waves have eigenvector solutions ∣*ψ*i where elements (1,1), (2,1), (5,1), and (6,1) are identically equal to zero. That is to say, Δ ¼ ð Þ 0 0 Δ<sup>3</sup> Δ<sup>4</sup> and Ω ¼ ð Þ 0 0 Ω<sup>3</sup> Ω<sup>4</sup> . This implies, Δ<sup>3</sup> and Ω<sup>3</sup> represent *z*-components. Δ<sup>4</sup> and Ω<sup>4</sup> represent the fourth components (unknown origin) in a four-dimensional space. Thus, for wave propagation in the +*z* direction, the non-transverse wave solutions have a *z* vector 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.

#### **5.5 Traditional Dirac equation**

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

$$
\begin{bmatrix}
+\partial\_{4} & \mathbf{0} & -i\partial\_{3} & -\partial\_{2} - i\partial\_{1} \\
\mathbf{0} & +\partial\_{4} & +\partial\_{2} - i\partial\_{1} & +i\partial\_{3} \\
+i\partial\_{3} & +\partial\_{2} + i\partial\_{1} & -\partial\_{4} & \mathbf{0} \\
\end{bmatrix}
\begin{bmatrix}
\Sigma\_{1} \\
\Sigma\_{2} \\
\Sigma\_{3} \\
\Sigma\_{4}
\end{bmatrix} + \kappa \begin{bmatrix}
\Sigma\_{1} \\
\Sigma\_{2} \\
\Sigma\_{3} \\
\Sigma\_{4}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{bmatrix} \tag{65}
$$

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

$$
\hat{D}|\sigma\rangle + \kappa|\sigma\rangle = |\sigma\rangle. \tag{66}
$$

The Dirac matrix operator *D*^ represents the four-by-four matrix operator on the left-hand side of Eq. (65), ∣*σ*i is the four-by-one ket vector appearing twice

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

on the left-hand side, and ∣*o*i is the four-by-one null ket vector appearing on the right-hand side. For time-harmonic plane-wave solutions, the ket vector ∣*σ*i may be expressed as

$$|\sigma\rangle = |\sigma\_o\rangle \exp\left[+i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar\right].\tag{67}$$

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

$$pc \begin{bmatrix} +\mu & 0 & +a\_3 & -ia\_2 + a\_1 \\ 0 & +\mu & +ia\_2 + a\_1 & -a\_3 \\ & +a\_3 & -ia\_2 + a\_1 & -\mu & 0 \\ +ia\_2 + a\_1 & -a\_3 & 0 & -\mu \end{bmatrix} \begin{bmatrix} \Sigma\_1 \\ \Sigma\_2 \\ \Sigma\_3 \\ \Sigma\_4 \end{bmatrix} = E \begin{bmatrix} \Sigma\_1 \\ \Sigma\_2 \\ \Sigma\_3 \\ \Sigma\_4 \end{bmatrix}.\tag{68}$$

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

$$
\begin{bmatrix} +E\_o & \mathbf{0} & +pc & \mathbf{0} \\ \mathbf{0} & +E\_o & \mathbf{0} & -pc \\ +pc & \mathbf{0} & -E\_o & \mathbf{0} \\ \mathbf{0} & -pc & \mathbf{0} & -E\_o \end{bmatrix} \begin{bmatrix} \Sigma\_1 \\ \Sigma\_2 \\ \Sigma\_3 \\ \Sigma\_4 \end{bmatrix} = E \begin{bmatrix} \Sigma\_1 \\ \Sigma\_2 \\ \Sigma\_3 \\ \Sigma\_4 \end{bmatrix} . \tag{69}
$$

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

The quantities *a* and *b* appearing in **Table 3** are defined by

$$a \equiv \frac{\sqrt{2}}{2} \sqrt{\frac{\chi + 1}{\chi}} \qquad \qquad \qquad a^2 + b^2 = 1 \qquad \qquad b \equiv \frac{\sqrt{2}}{2} \sqrt{\frac{\chi - 1}{\chi}}.\tag{70}$$

Note, the quantities *a* and *b* appearing in the traditional Dirac equation eigenvectors listed in **Table 3** are the same *a* and *b* quantities appearing in the generalized spacetime matrix equation eigenvectors listed in **Table 2** for *κ* ¼ *moc=*ℏ.


#### **Table 3.**

On the other hand, for wave propagation in the +*z* direction, the non-transverse waves have eigenvector solutions ∣*ψ*i where elements (1,1), (2,1), (5,1), and (6,1) are identically equal to zero. That is to say, Δ ¼ ð Þ 0 0 Δ<sup>3</sup> Δ<sup>4</sup> and Ω ¼ ð Þ 0 0 Ω<sup>3</sup> Ω<sup>4</sup> . This implies, Δ<sup>3</sup> and Ω<sup>3</sup> represent *z*-components. Δ<sup>4</sup> and Ω<sup>4</sup> represent the fourth components (unknown origin) in a four-dimensional space. Thus, for wave propagation in the +*z* direction, the non-transverse wave solutions have a *z* vector component (longitudinal in nature) and a fourth vector component (neither transverse nor longitudinal in nature, perhaps a "time" component) of a non-transverse wave

*Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave*

*En E***<sup>1</sup>** *E***<sup>2</sup>** *E***<sup>3</sup>** *E***<sup>4</sup>** *E***<sup>5</sup>** *E***<sup>6</sup>** *E***<sup>7</sup>** *E***<sup>8</sup>** *E* +*γEo* +*γEo* �*γEo* �*γEo* +*γEo* +*γEo* �*γEo* �*γEo* ∣*ψn*i ∣*ψ*1i ∣*ψ*2i ∣*ψ*3i ∣*ψ*4i ∣*ψ*5i ∣*ψ*6i ∣*ψ*<sup>7</sup> i ∣*ψ*8i Δ<sup>1</sup> 0 +*b* 0 +*a* 000 0 Δ<sup>2</sup> +*ib* 0 +*ia* 0000 0 Δ<sup>3</sup> 0 0 0 0+*b* 0 +*a* 0 Δ<sup>4</sup> 0 0 0 0 0+*ib* 0 +*ia* Ω<sup>1</sup> �*a* 0 +*b* 0000 0 Ω<sup>2</sup> 0 �*ia* 0 +*ib* 000 0 Ω<sup>3</sup> 000 00 �*a* 0 +*b* Ω<sup>4</sup> 000 0 �*ia* 0 +*ib* 0

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

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

The Dirac matrix operator *D*^ represents the four-by-four matrix operator on the left-hand side of Eq. (65), ∣*σ*i is the four-by-one ket vector appearing twice

Σ1 Σ2 Σ3 Σ4

*<sup>D</sup>*^ <sup>∣</sup>*σ*i þ *<sup>κ</sup>*∣*σ*i ¼ <sup>∣</sup>*o*i*:* (66)

Σ1 Σ2 Σ3 Σ4

(65)

in four dimensions.

terms, is given by

**Table 2.**

*Progress in Relativity*

given by

**30**

**5.5 Traditional Dirac equation**

*propagation in the +z direction when κ* ¼ *moc=*ℏ.

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

*Eigenvalues and orthonormal eigenvectors associated with the traditional Dirac equation for wave propagation in the +z direction when κ* ¼ *moc=*ℏ.

#### **5.6 Linear transformation equation**

For the special case of a matter wave traveling through free space in the + *z* direction, 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* ¼ *moc=*ℏ. These two sets of orthonormal eigenvectors are related [1] through the following linear transformation matrix equation:

$$
\begin{bmatrix}
\Sigma\_1\\\Sigma\_2\\\Sigma\_3\\\Sigma\_4
\end{bmatrix} = \frac{\sqrt{2}}{2} \begin{bmatrix}
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & +\mathbf{1} & -i & +\mathbf{1} & -i\\\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & +\mathbf{1} & +i & -\mathbf{1} & -i\\-\mathbf{1} & +i & -\mathbf{1} & +i & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0}\\-\mathbf{1} & -i & +\mathbf{1} & +i & \mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix} \begin{bmatrix}
\Delta\_1\\\Delta\_2\\\Delta\_3\\\Delta\_4\\\Delta\_5\\\Delta\_6\\\Delta\_7\\\Delta\_8
\end{bmatrix}.\tag{71}
$$

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

$$|\sigma\rangle = Z|\psi\rangle. \tag{72}$$

The constants *a* and *b* appearing in **Table 4** are now defined by

1. ∣*ψ*1i and ∣*ψ*2i represent transverse waves moving with speed þc.

2. ∣*ψ*3i and ∣*ψ*4i represent transverse waves moving with speed �c.

3. ∣*ψ*5i and ∣*ψ*6i represent non-transverse waves moving with speed þc.

4.∣*ψ*7i and ∣*ψ*8i represent non-transverse waves moving with speed �c.

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, characteristic of a transverse wave in three dimensions. Only those waves propagating at a speed in free space of +*c* represent real electromagnetic waves. On the other hand, for wave propagation in the +*z* direction, the non-transverse waves have eigenvector solutions ∣*ψ*i where elements (1,1), (2,1), (5,1), and (6,1) are identically equal to zero. That is to say, Δ ¼ ð Þ 0 0 Δ<sup>3</sup> Δ<sup>4</sup> and Ω ¼ ð Þ 0 0 Ω<sup>3</sup> Ω<sup>4</sup> . This implies, Δ<sup>3</sup> and Ω<sup>3</sup> represent *z*-components. Δ<sup>4</sup> and Ω<sup>4</sup> represent the fourth components in a four-dimensional space. Thus, for wave propagation in the +*z* direction, the non-transverse wave solutions have a *z* vector 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

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

þ*c* þ*c* �*c* �*c* þ*c* þ*c* �*c* �*c*

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

*Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave*

*En E***<sup>1</sup>** *E***<sup>2</sup>** *E***<sup>3</sup>** *E***<sup>4</sup>** *E***<sup>5</sup>** *E***<sup>6</sup>** *E***<sup>7</sup>** *E***<sup>8</sup>** *pnc* þ*pc* þ*pc* �*pc* �*pc* þ*pc* þ*pc* �*pc* �*pc* ∣*ψn*i ∣*ψ*1i ∣*ψ*2i ∣*ψ*3i ∣*ψ*4i ∣*ψ*5i ∣*ψ*6i ∣*ψ*<sup>7</sup> i ∣*ψ*8i Δ<sup>1</sup> 0 þ*b* 0 þ*a* 0000 Δ<sup>2</sup> þ*ib* 0 þ*ia* 00000 Δ<sup>3</sup> 0000 þ*b* 0 þ*a* 0 Δ<sup>4</sup> 00000 þ*ib* 0 þ*ia* Ω<sup>1</sup> �*a* 0 þ*b* 00000 Ω<sup>2</sup> 0 �*ia* 0 þ*ib* 0000 Ω<sup>3</sup> 00000 �*a* 0 þ*b* Ω<sup>4</sup> 0000 �*ia* 0 þ*ib* 0 *vn v*<sup>1</sup> *v*<sup>2</sup> *v*<sup>3</sup> *v*<sup>4</sup> *v*<sup>5</sup> *v*<sup>6</sup> *v*<sup>7</sup> *v*<sup>8</sup>

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

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

ffiffi 2 p

<sup>2</sup> (76)

*a* �

*propagation in the +z direction when κ* ¼ 0.

**Table 4.**

physics regarding these additional solutions.

**33**

ffiffi 2 p 2

When we substitute each the eight eigenvectors ∣*ψn*i from **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**:

$$|\sigma\_1\rangle = Z|\psi\_1\rangle \qquad |\sigma\_2\rangle = Z|\psi\_2\rangle \qquad |\sigma\_3\rangle = Z|\psi\_3\rangle \qquad |\sigma\_4\rangle = Z|\psi\_4\rangle. \tag{73}$$

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

$$|\sigma\_1\rangle = Z|\psi\_5\rangle \qquad |\sigma\_2\rangle = Z|\psi\_6\rangle \qquad |\sigma\_3\rangle = Z|\psi\_7\rangle \qquad |\sigma\_4\rangle = Z|\psi\_8\rangle. \tag{74}$$

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 +***z* **direction for** *κ*=0

For the special case of wave propagation in the +*z* direction, when *κ* ¼ 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 ∣*ψn*i also form an orthonormal set, that is,

$$
\langle \Psi\_m | \Psi\_n \rangle = \delta\_{mn}.\tag{75}
$$


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

**Table 4.**

**5.6 Linear transformation equation**

Σ1 Σ2 Σ3 Σ4

ffiffi 2 p 2

Eq. (71), we obtain the following results:

eigenvectors in **Table 3**:

equations (37) and (38).

**32**

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

**5.7 Wave propagation along the +***z* **direction for** *κ*=0

*Progress in Relativity*

**Table 3**:

For the special case of a matter wave traveling through free space in the + *z* direction, 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* ¼ *moc=*ℏ. These two sets of orthonormal eigenvectors are related [1] through the following linear transformation matrix equation:

> 0000 þ1 �*i* þ1 �*i* 0000 þ1 þ*i* �1 �*i* �1 þ*i* �1 þ*i* 0000 �1 �*i* þ1 þ*i* 0000

When we substitute each the eight eigenvectors ∣*ψn*i from **Table 2** back into

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

∣*σ*1i ¼ *Z*∣*ψ*1i ∣*σ*2i ¼ *Z*∣*ψ*2i ∣*σ*3i ¼ *Z*∣*ψ*3i ∣*σ*4i ¼ *Z*∣*ψ*4i*:* (73)

∣*σ*1i ¼ *Z*∣*ψ*5i ∣*σ*2i ¼ *Z*∣*ψ*6i ∣*σ*3i ¼ *Z*∣*ψ*7i ∣*σ*4i ¼ *Z*∣*ψ*8i*:* (74)

2. The four non-transverse eigenvectors in **Table 2** map into the same four

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

For the special case of wave propagation in the +*z* direction, when *κ* ¼ 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 ∣*ψn*i also form an orthonormal set, that is,

∣*σ*i ¼ *Z*∣*ψ*i*:* (72)

*ψ <sup>m</sup>*j*ψ<sup>n</sup>* h i ¼ *δmn:* (75)

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>

*:* (71)

> *Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave propagation in the +z direction when κ* ¼ 0.

The constants *a* and *b* appearing in **Table 4** are now defined by

$$a \equiv \frac{\sqrt{2}}{2} \qquad \qquad a^2 + b^2 = 1 \qquad \qquad b \equiv \frac{\sqrt{2}}{2} \tag{76}$$

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

1. ∣*ψ*1i and ∣*ψ*2i represent transverse waves moving with speed þc.

2. ∣*ψ*3i and ∣*ψ*4i represent transverse waves moving with speed �c.

3. ∣*ψ*5i and ∣*ψ*6i represent non-transverse waves moving with speed þc.

4.∣*ψ*7i and ∣*ψ*8i represent non-transverse waves moving with speed �c.

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, characteristic of a transverse wave in three dimensions. Only those waves propagating at a speed in free space of +*c* represent real electromagnetic waves.

On the other hand, for wave propagation in the +*z* direction, the non-transverse waves have eigenvector solutions ∣*ψ*i where elements (1,1), (2,1), (5,1), and (6,1) are identically equal to zero. That is to say, Δ ¼ ð Þ 0 0 Δ<sup>3</sup> Δ<sup>4</sup> and Ω ¼ ð Þ 0 0 Ω<sup>3</sup> Ω<sup>4</sup> . This implies, Δ<sup>3</sup> and Ω<sup>3</sup> represent *z*-components. Δ<sup>4</sup> and Ω<sup>4</sup> represent the fourth components in a four-dimensional space. Thus, for wave propagation in the +*z* direction, the non-transverse wave solutions have a *z* vector 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.
