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TITLE: An Eight-Dimensional Matrix Representation of Fundamental Physics Equations

The primary purpose of this article is to inform the reader that a number of the traditional scalar and vector partial differential equations may be re-expressed in a compact matrix form. This certainly has the advantage when numerical analysis or symbolic computations are involved. Off-the-shelf computer scientific software packages, like MATLAB and Wolfram Mathematica for example, are well suited for solving complicated problems when expressed in matrix form. Problems requiring solution of simultaneous equations, matrix multiplication, matrix inversion, eigenvalues and eigenvectors, and so on are just a few examples of the matrix operations which can be easily implemented using these software packages.

A number of examples illustrating these ideas are taken from the field of physics; in particular classical electromagnetism, relativistic quantum mechanics, and gravitoelectromagnetism. To be more specific, the summary table below lists ten matrix equations with a brief description of each. A more detailed discussion of each of these matrix equations will be presented in the sections to follow.

Notice that the ten equations listed in the table above have one thing in common and that is the matrix operator symbolized by $$\hat{M}$$. This matrix operator has 8 rows and 8 columns. For purposes of this discussion, it is referred to as the spacetime matrix operator. In the next section this operator is defined and several of its important properties are briefly described.

One other important point to be made and that is: students beginning their careers as either undergraduate or graduate students may find these matrix representations as a nice starting point or road map in their studies of matrices and their applications in physics, mathematics, and engineering. From each of these matrix equations the corresponding vector and scalar partial differential equations, traditionally found in introductory and advanced textbooks, are readily obtained. In the material to follow Gaussian units are employed.

The $8 × 8$ Spacetime Matrix Operator
The $8 × 8$ spacetime matrix operator $$\hat{M}$$ is defined by the following equation

where the partial derivative symbols appearing in this matrix operator are defined by


 * $$\begin{align}

\partial_1 \equiv \frac{\partial}{\partial x} && \partial_2 \equiv \frac{\partial}{\partial y} && \partial_3 \equiv \frac{\partial}{\partial z} && \partial_4 \equiv \frac{\partial}{ic\partial t}. \end{align}$$

The symbol $$c$$ represents the speed of light in vacuum and the imaginary unit $$i$$ represents the positive square-root of minus one.

The matrix operator $$\hat{M}$$ may also be expressed as


 * $$\begin{align}

\hat{M} = M_1\partial_1 + M_2\partial_2 + M_3\partial_3 + M_4\partial_4 \end{align}$$

where the following four $8 × 8$ matrices $$ M_\mu$$ for $$\mu$$ = 1, 2, 3, 4 are defined by


 * $$\begin{align}

\hat{M_1} \equiv \begin{bmatrix} O & O & O & -\sigma_1\\ O & O & +\sigma_1 & O\\ O & +\sigma_1 & O & O\\ -\sigma_1 & O & O & O\end{bmatrix} \quad \hat{M_2} \equiv \begin{bmatrix} O & O & O & +\sigma_3\\ O & O & -\sigma_3 & O\\ O & -\sigma_3 & O & O\\ +\sigma_3 & O & O & O\end{bmatrix} \quad \hat{M_3} \equiv \begin{bmatrix} O & O & -i\sigma_2 & O\\ O & O & O & -i\sigma_2\\ +i\sigma_2 & O & O & O\\ O & +i\sigma_2 & O & O\end{bmatrix} \quad \hat{M_4} \equiv \begin{bmatrix} -I & O & O & O\\ O & -I & O & O\\ O & O & +I & O\\ O & O & O & +I\end{bmatrix} \end{align}$$.

The $2 × 2$ matrices $$\sigma_1, \sigma_2, \sigma_3$$ are the Pauli matrices. The symbol $$O$$ represents the $2 × 2$ zero matrix and $$I$$ represents $2 × 2$ identity matrix.

Each matrix $$M_\mu$$ is a Hermitian matrix, that is


 * $$\begin{align} M_\mu = M^H_\mu \end{align}$$.

In addition, each matrix $$M_\mu$$ is equal to its own matrix inverse, namely


 * $$\begin{align} M_\mu = M^{-1}_\mu \end{align}$$.

These matrices also satisfy the anticommutator property


 * $$\begin{align}

M_\mu M_\nu + M_\nu M_\mu = 2 \delta_{\mu \nu} I_8 \end{align}$$

where $$\delta_{\mu \nu}$$ represents the Kronecker delta and $$ I_8 $$ corresponds to the $8 × 8$ identity matrix.

Not only does the $8 × 8$ spacetime operator $$\hat{M}$$ play a key role in the following sections, but so does the matrix product of $$\hat{M}$$ with itself. It can be easily shown that this matrix product is given by

where the d'Alembert operator is defined by


 * $$\begin{align}

\Box^2 \equiv {\partial_1^2} + {\partial_2^2} + {\partial_3^2} + {\partial_4^2} = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \end{align}$$

and $$\nabla^2$$ is the Laplace operator.

Gauss' law of electricity

 * $$\begin{align}

\nabla \cdot \mathbf{E} &= +4{\pi}{\rho_e} \end{align}$$

Gauss' law of magnetism

 * $$\begin{align}

\nabla \cdot \mathbf{B} &= +4{\pi}{\rho_m} \end{align}$$

Faraday's law of induction

 * $$\begin{align}

\nabla \times \mathbf{E} + \frac{1}{c} \frac{\partial\mathbf B}{\partial t} &= - \frac{4\pi}{c}\mathbf{J_m} \end{align}$$

Ampere-Maxwell law

 * $$\begin{align}

\nabla \times \mathbf{B} - \frac{1}{c} \frac{\partial\mathbf E}{\partial t} &= + \frac{4\pi}{c}\mathbf{J_e} \end{align}$$

Charge continuity equations

 * $$\begin{align}

\nabla \cdot \mathbf{J_e} + \frac{\partial\rho_e}{\partial t} = 0 \end{align}$$


 * $$\begin{align}

\nabla \cdot \mathbf{J_m} + \frac{\partial\rho_m}{\partial t} = 0 \end{align}$$

Electromagnetic field wave equations

 * $$\begin{align}

\Box^2 \mathbf{E} = \frac{4\pi}{c^2}\frac{\partial\mathbf{J_e}}{\partial t} + 4\pi\nabla \rho_e + \frac{4\pi}{c} \nabla \times    \mathbf{J_m} \end{align}$$


 * $$\begin{align}

\Box^2 \mathbf{B} = \frac{4\pi}{c^2}\frac{\partial\mathbf{J_m}}{\partial t} + 4\pi\nabla \rho_m - \frac{4\pi}{c} \nabla \times \mathbf{J_e} \end{align}$$

Lorentz conditions

 * $$\begin{align}

\nabla \cdot \mathbf{A_e} + \frac{1}{c} \frac{\partial\phi_e}{\partial t} = 0 \end{align}$$


 * $$\begin{align}

\nabla \cdot \mathbf{A_m} + \frac{1}{c} \frac{\partial\phi_m}{\partial t} = 0 \end{align}$$

Electromagnetic field and potential relations

 * $$\begin{align}

\mathbf{E} = -\nabla \phi_e - \frac{1}{c} \frac{\partial\mathbf{A_e}}{\partial t} - \nabla \times \mathbf{A_m} \end{align}$$


 * $$\begin{align}

\mathbf{B} = -\nabla \phi_m - \frac{1}{c} \frac{\partial\mathbf{A_m}}{\partial t} + \nabla \times \mathbf{A_e} \end{align}$$

Electromagnetic scalar wave equations

 * $$\begin{align}

\Box^2 \phi_e = - 4\pi \rho_e \end{align}$$


 * $$\begin{align}

\Box^2 \phi_m = - 4\pi \rho_m \end{align}$$

Electromagnetic vector wave equations

 * $$\begin{align}

\Box^2 \mathbf{A_e} = - \frac{4\pi}{c} \mathbf{J_e} \end{align}$$


 * $$\begin{align}

\Box^2 \mathbf{A_m} = - \frac{4\pi}{c} \mathbf{J_m} \end{align}$$

Scalar potentials and Hertz vector relations

 * $$\begin{align}

\phi_e = - \nabla\cdot\mathbf{\Pi_e} \end{align}$$


 * $$\begin{align}

\phi_m = + \nabla\cdot\mathbf{\Pi_m} \end{align}$$

Vector potentials and Hertz vector relations

 * $$\begin{align}

\mathbf{A_e} = +\frac{1}{c} \frac{\partial \mathbf{\Pi_e}}{\partial t} + \nabla \times \mathbf{\Pi_m} \end{align}$$


 * $$\begin{align}

\mathbf{A_m} = -\frac{1}{c} \frac{\partial \mathbf{\Pi_m}}{\partial t} + \nabla \times \mathbf{\Pi_e} \end{align}$$

Electric Hertz vector and electric field relation

 * $$\begin{align}

\Box^2 \mathbf{\Pi_e} = + \mathbf{E} \end{align}$$

Magnetic Hertz vector and magnetic field relation

 * $$\begin{align}

\Box^2 \mathbf{\Pi_m} = - \mathbf{B} \end{align}$$

Poynting's theorem

 * $$\begin{align}

\nabla \cdot \mathbf{S} + \frac{\partial u}{\partial t} + \mathbf{E} \cdot \mathbf{J_e} + \mathbf{B} \cdot \mathbf{J_m} = 0 \end{align}$$

Energy density

 * $$\begin{align}

u \equiv \frac{1}{8\pi}(\mathbf{E} \cdot \mathbf{E} + \mathbf{B} \cdot \mathbf{B}) \end{align}$$

Poynting vector

 * $$\begin{align}

\mathbf{S} \equiv \frac{c}{4 \pi} (\mathbf{E} \times \mathbf{B}) \end{align}$$

Quantoelectromagnetic Dirac Equations

 * $$\begin{align}

\kappa \equiv \frac{mc}{\hbar} \end{align}$$

Divergence equations

 * $$\begin{align}

\nabla \cdot \mathbf{U} = 0 \end{align}$$


 * $$\begin{align}

\nabla \cdot \mathbf{L} = 0 \end{align}$$

Curl equations

 * $$\begin{align}

\nabla \times \mathbf{U} + \frac{1}{c} \frac{\partial\mathbf L}{\partial t} = - i \kappa \mathbf{L} \end{align}$$


 * $$\begin{align}

\nabla \times \mathbf{L} - \frac{1}{c} \frac{\partial\mathbf U}{\partial t} &= - i \kappa \mathbf{U} \end{align}$$

Quantoelectromagnetic Klein-Gordon Equations

 * $$\begin{align}

\Box^2 \mathbf{U} = \kappa^2 \mathbf{u} \end{align}$$


 * $$\begin{align}

\Box^2 \mathbf{L} = \kappa^2 \mathbf{L} \end{align}$$

Tentative References
Baylis

Bocker 1

Bocker 2

Bocker 3

Bocker 4

Bohm

Corson

Davydov

Dirac

Fowles

Harmuth

Hylleraas 1

Hylleraas 2

Jackson 1

Jackson 2

Lorrain

Messiah

Musha

Ohanian

Powell

Roman

Schiff