On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Abstract

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

1. Introduction

The continuity equation (C.E.) is a first-order partial differential equation that describes the transport in time of a physical quantity through a region of space in terms of its density, flux, and an external source that continuously increases or decreases the quantity from the system. If the source is null, then the transported physical quantity is said to be conservative, and the C.E. represents the principle of local conservation of said quantity. In this form, the C.E. permeates virtually all branches of physics [1].

An iconic case where the C.E. played a fundamental role in the development of a physical theory occurred when J. C. Maxwell noticed the inconsistency between the primitive field equations of the electromagnetic theory and the C.E. for the electric charge. Maxwell is capable of solving said inconsistency by introducing the now renowned displacement ‘current’. Thanks to this addition, he was able to derive wave equations for the electric and magnetic fields, subsequently discovering that their speed of propagation is precisely the speed of light [2]. Another instance was in the development of the quantum-relativistic equation by P.A.M. Dirac following previous inconsistent attempts to reconcile both theories under a unified model. One such attempt is the Klein–Fock–Gordon equation, which yields negative energy eigenvalues and a non-positive definite probability density upon its reduction to a C.E. Dirac, using a matrix analysis and ensuring consistency with the C.E., constructed what we today know as Dirac’s gamma matrices, algebra, and equation, thus allowing for a description of spin-$1/2$ massive particles in the presence of external fields while taking into account relativistic effects [3].

Undoubtedly, the C.E. has served as a consistency criterion for the development of physical theories. A natural language to express and study the C.E. comprises W.K. Clifford’s geometric algebras (GAs). In physics, GAs have enjoyed a vast popularization since their ‘rediscovery’ by W.E. Pauli and Dirac in the 1920’s and the contributions of D. Hestenes in the 1960’s. By design, GA establishes itself as a holistic mathematical framework capable of algebraically manipulating multi-dimensional objects, operating on them with a clear geometric interpretation and nesting within itself several sub-algebras such as complex numbers, quaternions, four-vectors, and spinors [4,5]. As a result, GA finds numerous applications not only in physics, but also in mathematics, computer science, and engineering [6,7].

Axiomatically, a geometric algebra $\mathcal{G}(p,q)$ is a graded and associative algebra of dimension $2^{p+q}$ defined by a vector space ${\mathcal{V}}^{p+q}$ over the field of real numbers $\mathbb{R}$. Elements of a GA are called multivectors or Clifford numbers and operate through the geometric product, also called the Clifford product. The geometric product—denoted by juxtaposition—is ‘well behaved’ with scalars, distributive, and associative [8,9,10,11]. For vectors, the geometric product is linked to the inner product of the vector space and H.G. Grassmann’s outer product via the so-called fundamental identity:

$${\gamma }_{\mu }{\gamma }_{\nu }={\gamma }_{\mu }·{\gamma }_{\nu }+{\gamma }_{\mu }\wedge {\gamma }_{\nu }$$

(1)

In general, for two homogeneous multivectors ${\mathsf{\Psi }}_{\alpha }={\langle {\mathsf{\Psi }}_{\alpha }\rangle }_{\alpha }$ and ${\mathsf{\Phi }}_{\beta }={\langle {\mathsf{\Phi }}_{\beta }\rangle }_{\beta }$ of grades $\alpha$ and $\beta$, respectively, the inner and outer products are defined in terms of the geometric product as [12,13,14]:

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\alpha }·{\mathsf{\Phi }}_{\beta }& ={\langle {\mathsf{\Psi }}_{\alpha }{\mathsf{\Phi }}_{\beta }\rangle }_{|\alpha -\beta |}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\alpha }\wedge {\mathsf{\Phi }}_{\beta }& ={\langle {\mathsf{\Psi }}_{\alpha }{\mathsf{\Phi }}_{\beta }\rangle }_{\alpha +\beta }\hfill \end{array}$$

(3)

Consider the geometric algebra $\mathcal{G}(1,3)$, known as the space–time algebra (STA), generated by the orthonormal basis vectors $\left\{{\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3\right\}$, which satisfy [15,16]:

$${\gamma }_{\mu }·{\gamma }_{\nu }={\displaystyle \frac{1}{2}}\left({\gamma }_{\mu }{\gamma }_{\nu }+{\gamma }_{\nu }{\gamma }_{\mu }\right)={\eta }_{\mu \nu }$$

(4)

where ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$ is Minkowski’s metric tensor. A key corollary of STA inner product is that orthogonal vectors anticommute under the geometric product: ${\gamma }_{\mu }{\gamma }_{\nu }=-{\gamma }_{\nu }{\gamma }_{\mu }$, $\forall \mu \ne \nu$. The relations with the dual basis are:

$${\gamma }_{\mu }={\eta }_{\mu \nu }{\gamma }^{\nu },{\gamma }_{\mu }·{\gamma }^{\nu }={\delta }_{\mu }^{\nu }$$

(5)

where ${\delta }_{\mu }^{\nu }={\eta }^{\nu \alpha }{\eta }_{\alpha \mu }$ is the symmetric Kronecker delta tensor. Henceforth, we adopt Einstein’s summation convention, Greek indices run from zero, Latin indices from one, and the so-called ‘natural units’ ($c={ϵ}_0={\mu }_0=1$) are enforced. Let $I={\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3$ be the unit pseudo-scalar of $\mathcal{G}(1,3)$ with the properties

$$II=-1,I{\gamma }_{\mu }=-{\gamma }_{\mu }I$$

(6)

Two kinds of bivectors conform the bivector subspace: positive-squared ${({\gamma }_i\wedge {\gamma }_0)}^2=1$ and negative-squared ${({\gamma }_i\wedge {\gamma }_j)}^2=-1$. Since the even sub-algebra of STA is isomorphic to the algebra physical space (APS), ${\mathcal{G}}^{+}(1,3)\simeq \mathcal{G}(3,0)$, the bivector basis then splits into relative vectors and relative bivectors:

$$\begin{array}{cc}\hfill {\gamma }_i{\gamma }_0& ={\mathbf{\sigma }}_i\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\gamma }_i{\gamma }_j& =-{\epsilon }_{ijk}I{\mathbf{\sigma }}_k,i\ne j\hfill \end{array}$$

(8)

where ${\epsilon }_{ijk}$ is the antisymmetric Levi-Civita symbol. With this notation, the relative vectors possess the properties:

$$\begin{array}{cc}\hfill {\mathbf{\sigma }}_i·{\mathbf{\sigma }}_j& ={\delta }_{ij}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathbf{\sigma }}_i\wedge {\mathbf{\sigma }}_j& ={\epsilon }_{ijk}I{\mathbf{\sigma }}_k\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill I{\mathbf{\sigma }}_i& ={\mathbf{\sigma }}_{i}I\hfill \end{array}$$

(11)

Just like the relative bivectors can be expressed in terms of the relative vectors, the trivectors can be expressed as pseudo-vectors:

$${\gamma }_{\alpha }{\gamma }_{\beta }{\gamma }_{\mu }=-{\epsilon }_{\alpha \beta \mu \nu }I{\gamma }_{\nu }$$

(12)

In STA, the vector derivative operator with respect to space–time position $x=x^{\mu }{\gamma }_{\mu }=(t+\mathbf{x}){\gamma }_0$ is defined as

$$\nabla ={\gamma }^{\mu }{\partial }_{\mu }=\left({\partial }_t-\mathsf{\nabla }\right){\gamma }_0$$

(13)

where $x^0=t$ and $\mathbf{x}=x^i{\mathbf{\sigma }}_i$. For brevity, we will refer to Equation (13) as the nabla operator. Let us also define the structure of a generalized multivector of STA as:

$$\begin{array}{cc}\hfill \mathsf{\Psi }& ={\langle \mathsf{\Psi }\rangle }_0+{\langle \mathsf{\Psi }\rangle }_1+{\langle \mathsf{\Psi }\rangle }_2+{\langle \mathsf{\Psi }\rangle }_3+{\langle \mathsf{\Psi }\rangle }_4\hfill \\ \hfill & ={\psi }_2^0+{\psi }_1^{\mu }{\gamma }_{\mu }+{\psi }_2^i{\gamma }_i{\gamma }_0+{\psi }_3^{i}I{\gamma }_i{\gamma }_0+{\psi }_4^{\mu }I{\gamma }_{\mu }+{\psi }_3^{0}I\hfill \\ \hfill & ={\psi }_2^0+\left({\psi }_1^0+{\mathit{\psi }}_1\right){\gamma }_0+{\mathit{\psi }}_2+{\mathit{\psi }}_{3}I+\left({\psi }_4^0+{\mathit{\psi }}_4\right)I{\gamma }_0+{\psi }_3^{0}I\hfill \end{array}$$

(14)

where (throughout the document: $a,b=1,2,3,4$) ${\psi }_a^{\mu }={\psi }_a^{\mu }\left(x^{\mu }\right)$ are scalar functions of the coordinates of x. Respectively, they constitute the scalar ${\langle \mathsf{\Psi }\rangle }_0={\psi }_2^0$, vector ${\langle \mathsf{\Psi }\rangle }_1=\left({\psi }_1^0+{\psi }_1\right){\gamma }_0$, bivector ${\langle \mathsf{\Psi }\rangle }_2={\mathit{\psi }}_2+{\mathit{\psi }}_{3}I$, pseudovector ${\langle \mathsf{\Psi }\rangle }_3=\left({\psi }_4^0+{\mathit{\psi }}_4\right)I{\gamma }_0$, and pseudoscalar part ${\langle \mathsf{\Psi }\rangle }_4={\psi }_3^{0}I$ of the multivector $\mathsf{\Psi }$. Bold notation in Equation (13) and (14) denotes

$${\mathit{\psi }}_a={\psi }_a^i{\mathbf{\sigma }}_i,\mathsf{\nabla }={\mathbf{\sigma }}_i{\partial }_i$$

(15)

The reversal conjugation is defined by ${\widetilde{\langle \mathsf{\Psi }\rangle }}_{\alpha }={(-1)}^{\alpha (\alpha -1)/2}{\langle \mathsf{\Psi }\rangle }_{\alpha }$, so that, for the generalized multivector defined in Equation (14), we obtain:

$$\tilde{\mathsf{\Psi }}={\psi }_2^0+\left({\psi }_1^0+{\mathit{\psi }}_1\right){\gamma }_0-{\mathit{\psi }}_2-{\mathit{\psi }}_{3}I-\left({\psi }_4^0+{\mathit{\psi }}_4\right)I{\gamma }_0+{\psi }_3^{0}I$$

(16)

It is assumed that the scalar functions ${\psi }_a^{\mu }$ are invariant under the reversal, satisfy Clairaut–Schwarz’s theorem, and commute with the basis vectors, the pseudoscalar, and each other. Correspondingly, the reversal of the nabla Equation (13) indicates that the partial derivatives act to the left:

$$\tilde{\nabla }=\left(\overleftarrow{{\partial }_t}-\overleftarrow{\mathsf{\nabla }}\right){\gamma }_0$$

(17)

In STA’s treatment of classical electrodynamics, Maxwell’s field equations are condensed into the single equation

$$\nabla \mathit{F}=J$$

(18)

where $\mathit{F}=\mathit{E}+\mathit{B}I$ is Faraday’s electromagnetic field, $\mathit{E}$ is the electric field, $\mathit{B}$ is the magnetic field, $J=(\rho +\mathit{j}){\gamma }_0$ the four-current density, $\rho$ the electric charge density, and $\mathit{j}$ the electric current density. In macroscopic media, the Faraday field splits into $\mathit{F}={\mathit{F}}_f+{\mathit{F}}_b$, where ${\mathit{F}}_f=\mathit{D}+\mathit{H}I$ and ${\mathit{F}}_b=-\mathit{P}+\mathit{M}I$. Here, $\mathit{D}$ is the electric displacement field, $\mathit{H}$ is the macroscopic magnetic field, $\mathit{P}$ is the polarization field of the medium, and $\mathit{M}$ is the magnetization field of the medium. Similarly, the current density splits into $J=J_f+J_b$, where $J_f$ relates to the free charge currents and $J_b$ to the bound charge currents in the medium. In the absence of polarization and magnetization, $J_b=0$, $\mathit{F}={\mathit{F}}_f$, and $J=J_f$. For such a case, developing the geometric product and separating the components of Equation (18) yield (to express this in Gibbs–Heaviside’s vector calculus notation, $\mathsf{\nabla }·\left(I\mathit{F}\right)=I\mathsf{\nabla }\wedge \mathit{F}=-\mathsf{\nabla }\times \mathit{F}$. Throughout the document, $I\mathit{A}\wedge \mathit{B}$ means $I\left(\mathit{A}\wedge \mathit{B}\right)$):

$$\begin{array}{cc}\hfill \mathsf{\nabla }·\mathit{E}& =\rho \hfill \\ \hfill \mathsf{\nabla }·\mathit{B}& =0\hfill \\ \hfill {\partial }_t\mathit{E}+I\mathsf{\nabla }\wedge \mathit{B}& =-\mathit{j}\hfill \\ \hfill {\partial }_t\mathit{B}-I\mathsf{\nabla }\wedge \mathit{E}& =0\hfill \end{array}$$

(19)

The electromagnetic four-potential $A=(\phi +\mathit{A}){\gamma }_0$ is obtained from $\nabla A=\mathit{F}$, where $\phi$ is the electric potential and $\mathit{A}$ the magnetic potential. The wave equations for the fields are obtained by once again applying the nabla to Equation (18), and the continuity equation for the electric charge is obtained through the scalar $\nabla ·J=0$. A manipulation of Equation (18) and its reverse yields the energy–momentum vectors of the electromagnetic field, $T^{\mu }={\displaystyle \frac{1}{2}}\tilde{\mathit{F}}{\gamma }^{\mu }\mathit{F}$. The corresponding continuity equation gives the conservation of electromagnetic energy and linear momentum in terms of the Lorentz force density as the source, ${\partial }_{\mu }{T}^{\mu }=J·\mathit{F}$. In particular, $T^0=\left[{\displaystyle \frac{1}{2}}\left({\mathit{E}}^2+{\mathit{B}}^2\right)-I\mathit{E}\wedge \mathit{B}\right]{\gamma }_0$ gives the Poynting four-vector. It is then apparent that, from the simple structure of Equation (18), a multitude of important physical quantities and relations can be found [17,18].

With this preamble, we propose to study the continuity equation in the present paper by employing the mathematical framework of STA. Our objective is to show how common equations in mathematical physics can be identified and derived from the C.E.’s structure and how the ubiquitous C.E. may aid in the search and physical interpretation of further equations. Results will be formulated based on the generalized multivector structure of Equation (14) and later be reduced and simplified to the well-known expressions in the STA treatment.

Our starting point in Section 2 will consist of establishing an equation of structure $\nabla \mathsf{\Psi }=\mathsf{\Phi }$ and identifying the systems of C.E.s that emerge from each component. Then, we will construct a system of wave equations and the energy–momentum vectors analogs (along with their C.E.) associated with the continuity system of the generalized multivector structure. In Section 3, we will explore some decoupling cases for the different parts of $\mathsf{\Psi }$, which are coupled with each other through the continuity system. Subsequently, in Section 4, we shall determine transformation symmetries for the functions involved in the continuity system such that the system of continuity equations remains invariant in structure. From these symmetries, we plan to derive a system of equations with the same structure as Maxwell’s field equations. Finally, in Section 5, we will make use of the continuity system to study the diffusion equation and the conditions for decoupling in terms of external fields.

STA contains the APS, which further contains the quaternion algebra, $G^{+}(3,0)\simeq \mathbb{H}$. Thus, our approach distinguishes this work from similar explorations of the continuity equation using a quaternionic formalism [19,20]. Alternative derivations of Maxwell’s equations from the C.E. can be found in [21,22] by means of retarded potentials and the Poincaré lemma in GA, also in [23,24], using Jefimenko’s equations and retarded tensor fields without GA. For an exploration of the conservation principles in general relativity with a Clifford bundle formalism, see the work of Rodrigues et al. [25,26]. Further connections can be seen in [27], where an extensive discussion is provided on the relationship between the structure of the Maxwell field equations with the Dirac, Einstein, and Navier–Stokes equations.

2. Continuity Equations

Consider the equation

$$\nabla \mathsf{\Psi }=\mathsf{\Phi }$$

(20)

where $\mathsf{\Phi }=\mathsf{\Phi }\left(x^{\mu }\right)$ is another multivector with the structure of Equation (14). The geometric product between the nabla and the multivector $\mathsf{\Psi }$ yields

$$\begin{array}{cc}\hfill \nabla \mathsf{\Psi }=& \left({\partial }_t{\psi }_1^0+\mathsf{\nabla }·{\mathit{\psi }}_1\right)\hfill \\ \hfill & +\left({\partial }_t{\psi }_2^0+\mathsf{\nabla }·{\mathit{\psi }}_2-{\partial }_t{\mathit{\psi }}_2-\mathsf{\nabla }{\psi }_2^0-I\mathsf{\nabla }\wedge {\mathit{\psi }}_3\right){\gamma }_0\hfill \\ \hfill & -\left({\partial }_t{\mathit{\psi }}_1+\mathsf{\nabla }{\psi }_1^0+I\mathsf{\nabla }\wedge {\mathit{\psi }}_4\right)\hfill \\ \hfill & +\left({\partial }_t{\mathit{\psi }}_4+\mathsf{\nabla }{\psi }_4^0-I\mathsf{\nabla }\wedge {\mathit{\psi }}_1\right)I\hfill \\ \hfill & +\left(-{\partial }_t{\psi }_3^0-\mathsf{\nabla }·{\mathit{\psi }}_3+{\partial }_t{\mathit{\psi }}_3+\mathsf{\nabla }{\psi }_3^0-I\mathsf{\nabla }\wedge {\mathit{\psi }}_2\right)I{\gamma }_0\hfill \\ \hfill & -\left({\partial }_t{\psi }_4^0+\mathsf{\nabla }·{\mathit{\psi }}_4\right)I\hfill \end{array}$$

(21)

The terms in Equation (21) have been organized by parenthesis into the scalar, vector, relative vector, relative bivector, pseudovector, and pseudoscalar part, respectively. By hypothesis, $\mathsf{\Phi }$ has structure Equation (14), so matrix notation allows us to identify the following pair of systems of equations, which follow from the components of Equation (20). On the one hand, the scalar structure:

$${\partial }_t\left(\begin{array}{c}{\psi }_1^0\\ {\psi }_2^0\\ {\psi }_3^0\\ {\psi }_4^0\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\psi }}_1\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_3\\ {\mathit{\psi }}_4\end{array}\right)=\left(\begin{array}{c}{\varphi }_2^0\\ {\varphi }_1^0\\ -{\varphi }_4^0\\ -{\varphi }_3^0\end{array}\right)$$

(22)

On the other hand, the vector structure:

$${\partial }_t\left(\begin{array}{c}{\mathit{\psi }}_1\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_3\\ {\mathit{\psi }}_4\end{array}\right)+\mathsf{\nabla }\left(\begin{array}{c}{\psi }_1^0\\ {\psi }_2^0\\ {\psi }_3^0\\ {\psi }_4^0\end{array}\right)-I\mathsf{\nabla }\wedge \left(\begin{array}{c}-{\mathit{\psi }}_4\\ -{\mathit{\psi }}_3\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_1\end{array}\right)=\left(\begin{array}{c}-{\mathit{\varphi }}_2\\ -{\mathit{\varphi }}_1\\ {\mathit{\varphi }}_4\\ {\mathit{\varphi }}_3\end{array}\right)$$

(23)

The system of Equation (22) clearly possesses the canonical structure of the (scalar) C.E. with the existence of non-zero sources given by $\mathsf{\Phi }$. To observe that the system of Equation (23) has a continuity structure as well, but one of a vector character, we use the dual identity:

$$I\mathsf{\nabla }\wedge {\mathit{\psi }}_a=\mathsf{\nabla }·\left(I{\mathit{\psi }}_a\right)$$

(24)

and define the second-rank tensors ${\mathsf{T}}_a$ such that its components are connected with ${\psi }_a^0$ through the restriction

$${\partial }_i{T}_a^{ij}={\partial }_j{\psi }_a^0,\mathsf{\nabla }·{\mathsf{T}}_a=\mathsf{\nabla }{\psi }_a^0$$

(25)

With these results, Equations (22) and (23) compact into a visibly single continuity structure:

$${\partial }_t\left(\begin{array}{c}{\psi }_1^0\\ {\psi }_2^0\\ {\psi }_3^0\\ {\psi }_4^0\\ {\mathit{\psi }}_1\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_3\\ {\mathit{\psi }}_4\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\psi }}_1\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_3\\ {\mathit{\psi }}_4\\ {\mathsf{T}}_1+I{\mathit{\psi }}_4\\ {\mathsf{T}}_2+I{\mathit{\psi }}_3\\ {\mathsf{T}}_3-I{\mathit{\psi }}_2\\ {\mathsf{T}}_4-I{\mathit{\psi }}_1\end{array}\right)=\left(\begin{array}{c}{\varphi }_2^0\\ {\varphi }_1^0\\ -{\varphi }_4^0\\ -{\varphi }_3^0\\ -{\mathit{\varphi }}_2\\ -{\mathit{\varphi }}_1\\ {\mathit{\varphi }}_4\\ {\mathit{\varphi }}_3\end{array}\right)$$

(26)

${\psi }_a^0$ are then immediately identified with the densities, and ${\mathit{\psi }}_a$ with the fluxes of whatever physical quantity is being represented by $\mathsf{\Psi }$. We also see that Equation (20) is the compact form of the scalar and vector continuity system of equations Equation (26) and, thus, another form of the C.E. by itself.

2.1. Wave Equations

By applying from the left the nabla operator to Equation (20), we obtain the second-order differential equation

$$\square \mathsf{\Psi }=\nabla \mathsf{\Phi }$$

(27)

where $\square =\nabla \nabla =\nabla ·\nabla ={\eta }^{\mu \nu }{\partial }_{\mu \nu }^2$ is the d’Alembertian operator. The multivector $\nabla \mathsf{\Phi }$ will have by structure Equation (21), so by separating the components of Equation (27), we will obtain two systems of equations with the form of Equations (22) and (23). Defining the tensors ${\mathsf{N}}_a$ restricted to the analogous condition of Equation (25), but connected to the ${\varphi }_a^0$, the following system of wave equations is then obtained:

$${\partial }_t\left(\begin{array}{c}{\varphi }_1^0\\ {\varphi }_2^0\\ {\varphi }_3^0\\ {\varphi }_4^0\\ {\mathit{\varphi }}_1\\ {\mathit{\varphi }}_2\\ {\mathit{\varphi }}_3\\ {\mathit{\varphi }}_4\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\varphi }}_1\\ {\mathit{\varphi }}_2\\ {\mathit{\varphi }}_3\\ {\mathit{\varphi }}_4\\ {\mathsf{N}}_1+I{\mathit{\varphi }}_4\\ {\mathsf{N}}_2+I{\mathit{\varphi }}_3\\ {\mathsf{N}}_3-I{\mathit{\varphi }}_2\\ {\mathsf{N}}_4-I{\mathit{\varphi }}_1\end{array}\right)=\square \left(\begin{array}{c}{\psi }_2^0\\ {\psi }_1^0\\ -{\psi }_4^0\\ -{\psi }_3^0\\ -{\mathit{\psi }}_2\\ -{\mathit{\psi }}_1\\ {\mathit{\psi }}_4\\ {\mathit{\psi }}_3\end{array}\right)$$

(28)

The scalar continuity system of Equation (22) shows the direct coupling between the vector components $({\psi }_1^0,{\mathit{\psi }}_1)$ and the pseudovector components $({\psi }_4^0,{\mathit{\psi }}_4)$, between the scalar and relative vector $({\psi }_2^0,{\mathit{\psi }}_2)$ and between the pseudoscalar and relative bivector $({\psi }_3^0,{\mathit{\psi }}_3)$. We shall call these relations the ‘first couplings’. The first couplings remain in the vector continuity system of Equation (23), with the addition of the coupling between the space pseudovector with the vector $({\psi }_1^0,{\mathit{\psi }}_1,{\mathit{\psi }}_4)$, the space vector with the pseudovector $({\psi }_4^0,{\mathit{\psi }}_4,{\mathit{\psi }}_1)$, and the relative vector and bivector with each scalar $({\psi }_2^0,{\mathit{\psi }}_2,{\mathit{\psi }}_3)$, $({\psi }_3^0,{\mathit{\psi }}_3,{\mathit{\psi }}_2)$. For consistency, we call these the ‘second couplings’. An analogous argument applies to the components of the source $\mathsf{\Phi }$. The cost for decoupling—first or second—is a transition to a higher-order system of equations, as can be seen in Equation (28), where each component ${\psi }_a^{\mu }$ satisfies a non-homogeneous wave equation independently of any other component of $\mathsf{\Psi }$. In Section 3, we will explore some decoupling cases.

2.2. Energy–Momentum Vectors

Attending to the conjugations Equations (16) and (17), the reversal of Equation (20) is $\widetilde{\mathsf{\nabla }\mathsf{\Psi }}=\tilde{\mathsf{\Psi }}\tilde{\mathsf{\nabla }}=\tilde{\mathsf{\Phi }}$. The well-known method to derive the C.E. from the Schrödinger, Pauli, Klein–Fock–Gordon, and Dirac’s equations:

$$\begin{array}{cc}\hfill \left(\widetilde{\nabla \mathsf{\Psi }}\right)\mathsf{\Psi }+\tilde{\mathsf{\Psi }}\left(\nabla \mathsf{\Psi }\right)& =\tilde{\mathsf{\Phi }}\mathsf{\Psi }+\tilde{\mathsf{\Psi }}\mathsf{\Phi }\hfill \\ \hfill {\partial }_{\mu }\left({\displaystyle \frac{1}{2}}\tilde{\mathsf{\Psi }}{\gamma }^{\mu }\mathsf{\Psi }\right)& ={\displaystyle \frac{1}{2}}\left(\tilde{\mathsf{\Phi }}\mathsf{\Psi }+\tilde{\mathsf{\Psi }}\mathsf{\Phi }\right)\hfill \end{array}$$

(29)

allows us to define

$$\begin{array}{cc}\hfill J^{\mu }& ={\displaystyle \frac{1}{2}}\tilde{\mathsf{\Psi }}{\gamma }^{\mu }\mathsf{\Psi }\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill Л& ={\displaystyle \frac{1}{2}}\left(\tilde{\mathsf{\Phi }}\mathsf{\Psi }+\tilde{\mathsf{\Psi }}\mathsf{\Phi }\right)\hfill \end{array}$$

(31)

such that

$${\partial }_{\mu }{J}^{\mu }=Л$$

(32)

Equation (31) is a new source associated with the continuity system Equation (32). In terms of the components of $\mathsf{\Psi }$ and $\mathsf{\Phi }$, it has the explicit form:

$$\begin{array}{cc}\hfill Л=& {\eta }_{\mu \nu }\left({\varphi }_1^{\mu }{\psi }_1^{\nu }+{\varphi }_2^{\mu }{\psi }_2^{\nu }-{\varphi }_3^{\mu }{\psi }_3^{\nu }-{\varphi }_4^{\mu }{\psi }_4^{\nu }\right)\hfill \\ \hfill & +{\eta }_{\mu \nu }\left({\varphi }_1^{\mu }{\psi }_2^{\nu }+{\varphi }_2^{\mu }{\psi }_1^{\nu }-{\varphi }_3^{\mu }{\psi }_4^{\nu }-{\varphi }_4^{\mu }{\psi }_3^{\nu }\right){\gamma }_0\hfill \\ \hfill & +[{\psi }_2^0{\mathit{\varphi }}_1-{\psi }_1^0{\mathit{\varphi }}_2+{\psi }_4^0{\mathit{\varphi }}_3-{\psi }_3^0{\mathit{\varphi }}_4+{\varphi }_2^0{\mathit{\psi }}_1-{\varphi }_1^0{\mathit{\psi }}_2+{\varphi }_4^0{\mathit{\psi }}_3-{\varphi }_3^0{\mathit{\psi }}_4\hfill \\ \hfill & -I\left(-{\mathit{\varphi }}_1\wedge {\mathit{\psi }}_3+{\mathit{\varphi }}_3\wedge {\mathit{\psi }}_1-{\mathit{\varphi }}_2\wedge {\mathit{\psi }}_4+{\mathit{\varphi }}_4\wedge {\mathit{\psi }}_2\right)]{\gamma }_0\hfill \\ \hfill & +{\eta }_{\mu \nu }\left(-{\varphi }_1^{\mu }{\psi }_4^{\nu }+{\varphi }_2^{\mu }{\psi }_3^{\nu }+{\varphi }_3^{\mu }{\psi }_2^{\nu }-{\varphi }_4^{\mu }{\psi }_1^{\nu }\right)I\hfill \end{array}$$

(33)

Equation (30) represents four multivectors associated with the density of $\mathsf{\Psi }$. Unlike Equation (31), it is independent of the source $\mathsf{\Phi }$. In terms of the components of $\mathsf{\Psi }$, it has the explicit form:

$$\begin{array}{cc}\hfill J^0=& \left({\psi }_1^{\mu }{\psi }_2^{\mu }+{\psi }_3^{\mu }{\psi }_4^{\mu }\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\psi }_1^{\mu }{\psi }_1^{\mu }+{\psi }_2^{\mu }{\psi }_2^{\mu }+{\psi }_3^{\mu }{\psi }_3^{\mu }+{\psi }_4^{\mu }{\psi }_4^{\mu }+2{\psi }_1^{\mu }{\psi }_4^{\mu }\right){\gamma }_0\hfill \\ \hfill & +\left[({\psi }_1^0+{\psi }_4^0)({\mathit{\psi }}_1+{\mathit{\psi }}_4)-{\psi }_2^0{\mathit{\psi }}_2-{\psi }_3^0{\mathit{\psi }}_3-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right]{\gamma }_0\hfill \\ \hfill & +\left({\psi }_1^{\mu }{\psi }_3^{\mu }-{\psi }_2^{\mu }{\psi }_4^{\mu }\right)I\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill J^i=& \left[{\psi }_2^0{\psi }_1^i+{\psi }_1^0{\psi }_2^i+{\psi }_4^0{\psi }_3^i+{\psi }_3^0{\psi }_4^i+{\left(-I{\mathit{\psi }}_1\wedge {\mathit{\psi }}_3\right)}_i+{\left(-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_4\right)}_i\right]\hfill \\ \hfill & +\left[{\psi }_1^0{\psi }_1^i+{\psi }_2^0{\psi }_2^i+{\psi }_3^0{\psi }_3^i+{\psi }_4^0{\psi }_4^i+{\left(-I{\mathit{\psi }}_1\wedge {\mathit{\psi }}_4\right)}_i+{\left(-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right)}_i\right]{\gamma }_0\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\eta }_{\mu \nu }\left({\psi }_1^{\mu }{\psi }_1^{\nu }-{\psi }_2^{\mu }{\psi }_2^{\nu }-{\psi }_3^{\mu }{\psi }_3^{\nu }+{\psi }_4^{\mu }{\psi }_4^{\nu }\right){\gamma }_i\hfill \\ \hfill & +\left[{\psi }_1^i{\psi }_1^k-{\psi }_2^i{\psi }_2^k-{\psi }_3^i{\psi }_3^k+{\psi }_4^i{\psi }_4^k+\left(-{\psi }_1^0{\psi }_4^j+{\psi }_2^0{\psi }_3^j-{\psi }_3^0{\psi }_2^j+{\psi }_4^0{\psi }_1^j\right){\epsilon }_{ijk}\right]{\gamma }_k\hfill \\ \hfill & +\left[{\psi }_3^0{\psi }_1^i-{\psi }_4^0{\psi }_2^i+{\psi }_1^0{\psi }_3^i-{\psi }_2^0{\psi }_4^i+{\left(I{\mathit{\psi }}_1\wedge {\mathit{\psi }}_2\right)}_i+{\left(-I{\mathit{\psi }}_3\wedge {\mathit{\psi }}_4\right)}_i\right]I\hfill \end{array}$$

(35)

Notice ${\langle J^{\mu }\rangle }_2={\langle J^{\mu }\rangle }_3={\langle Л\rangle }_2={\langle Л\rangle }_3=0$. Equation (32), thus, yields a system of four C.E.s. From the scalar and pseudoscalar parts, we have:

$$\begin{array}{cc}\hfill & {\partial }_t\left(\begin{array}{c}{\psi }_1^0{\psi }_2^0+{\mathit{\psi }}_1·{\mathit{\psi }}_2+{\psi }_3^0{\psi }_4^0+{\mathit{\psi }}_3·{\mathit{\psi }}_4\\ {\psi }_1^0{\psi }_3^0+{\mathit{\psi }}_1·{\mathit{\psi }}_3-{\psi }_2^0{\psi }_4^0-{\mathit{\psi }}_2·{\mathit{\psi }}_4\end{array}\right)\hfill \\ \hfill & +\mathsf{\nabla }·\left(\begin{array}{c}{\psi }_2^0{\mathit{\psi }}_1+{\psi }_1^0{\mathit{\psi }}_2+{\psi }_4^0{\mathit{\psi }}_3+{\psi }_3^0{\mathit{\psi }}_4-I{\mathit{\psi }}_1\wedge {\mathit{\psi }}_3-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_4\\ {\psi }_3^0{\mathit{\psi }}_1-{\psi }_4^0{\mathit{\psi }}_2+{\psi }_1^0{\mathit{\psi }}_3-{\psi }_2^0{\mathit{\psi }}_4+I{\mathit{\psi }}_1\wedge {\mathit{\psi }}_2-I{\mathit{\psi }}_3\wedge {\mathit{\psi }}_4\end{array}\right)\hfill \\ \hfill & ={\eta }_{\mu \nu }\left(\begin{array}{c}{\varphi }_1^{\mu }{\psi }_1^{\nu }+{\varphi }_2^{\mu }{\psi }_2^{\nu }-{\varphi }_3^{\mu }{\psi }_3^{\nu }-{\varphi }_4^{\mu }{\psi }_4^{\nu }\\ -{\varphi }_1^{\mu }{\psi }_4^{\nu }+{\varphi }_2^{\mu }{\psi }_3^{\nu }+{\varphi }_3^{\mu }{\psi }_2^{\nu }-{\varphi }_4^{\mu }{\psi }_1^{\nu }\end{array}\right)\hfill \end{array}$$

(36)

The timelike vector part:

$$\begin{array}{cc}\hfill & {\partial }_t\left[{\displaystyle \frac{1}{2}}\left({\psi }_1^{\mu }{\psi }_1^{\mu }+{\psi }_2^{\mu }{\psi }_2^{\mu }+{\psi }_3^{\mu }{\psi }_3^{\mu }+{\psi }_4^{\mu }{\psi }_4^{\mu }\right)+{\psi }_1^{\mu }{\psi }_4^{\mu }\right]\hfill \\ \hfill & +\mathsf{\nabla }·\left({\psi }_1^0{\mathit{\psi }}_1+{\psi }_2^0{\mathit{\psi }}_2+{\psi }_3^0{\mathit{\psi }}_3+{\psi }_4^0{\mathit{\psi }}_4-I{\mathit{\psi }}_1\wedge {\mathit{\psi }}_4-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right)\hfill \\ \hfill & ={\eta }_{\mu \nu }\left({\varphi }_1^{\mu }{\psi }_2^{\nu }+{\varphi }_2^{\mu }{\psi }_1^{\nu }-{\varphi }_3^{\mu }{\psi }_4^{\nu }-{\varphi }_4^{\mu }{\psi }_3^{\nu }\right)\hfill \end{array}$$

(37)

The spacelike vector part:

$$\begin{array}{cc}\hfill & {\partial }_t\left[({\psi }_1^0+{\psi }_4^0)({\mathit{\psi }}_1+{\mathit{\psi }}_4)-{\psi }_2^0{\mathit{\psi }}_2-{\psi }_3^0{\mathit{\psi }}_3-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathsf{\nabla }\left({\psi }_1^{\mu }{\psi }_1^{\nu }-{\psi }_2^{\mu }{\psi }_2^{\nu }-{\psi }_3^{\mu }{\psi }_3^{\nu }+{\psi }_4^{\mu }{\psi }_4^{\nu }\right){\eta }_{\mu \nu }\hfill \\ \hfill & +\left(\mathsf{\nabla }·{\mathit{\psi }}_1+{\mathit{\psi }}_1·\mathsf{\nabla }\right){\mathit{\psi }}_1-\left(\mathsf{\nabla }·{\mathit{\psi }}_2+{\mathit{\psi }}_2·\mathsf{\nabla }\right){\mathit{\psi }}_2\hfill \\ \hfill & -\left(\mathsf{\nabla }·{\mathit{\psi }}_3+{\mathit{\psi }}_3·\mathsf{\nabla }\right){\mathit{\psi }}_3+\left(\mathsf{\nabla }·{\mathit{\psi }}_4+{\mathit{\psi }}_4·\mathsf{\nabla }\right){\mathit{\psi }}_4\hfill \\ \hfill & -I\mathsf{\nabla }\wedge \left({\psi }_4^0{\mathit{\psi }}_1-{\psi }_3^0{\mathit{\psi }}_2+{\psi }_2^0{\mathit{\psi }}_3-{\psi }_1^0{\mathit{\psi }}_4\right)\hfill \\ \hfill & ={\psi }_2^0{\mathit{\varphi }}_1-{\psi }_1^0{\mathit{\varphi }}_2+{\psi }_4^0{\mathit{\varphi }}_3-{\psi }_3^0{\mathit{\varphi }}_4+{\varphi }_2^0{\mathit{\psi }}_1-{\varphi }_1^0{\mathit{\psi }}_2+{\varphi }_4^0{\mathit{\psi }}_3-{\varphi }_3^0{\mathit{\psi }}_4\hfill \\ \hfill & -I\left(-{\mathit{\varphi }}_1\wedge {\mathit{\psi }}_3+{\mathit{\varphi }}_3\wedge {\mathit{\psi }}_1-{\mathit{\varphi }}_2\wedge {\mathit{\psi }}_4+{\mathit{\varphi }}_4\wedge {\mathit{\psi }}_2\right)\hfill \end{array}$$

(38)

As in Equation (25), we may identify the emergence of a tensor $\mathsf{R}$:

$$\begin{array}{cc}\hfill \mathsf{\nabla }·\mathsf{R}=& \mathsf{\nabla }{\displaystyle \frac{1}{2}}{\eta }_{\mu \nu }\left({\psi }_1^{\mu }{\psi }_1^{\nu }-{\psi }_2^{\mu }{\psi }_2^{\nu }-{\psi }_3^{\mu }{\psi }_3^{\nu }+{\psi }_4^{\mu }{\psi }_4^{\nu }\right)\hfill \\ \hfill & +\left(\mathsf{\nabla }·{\mathit{\psi }}_1+{\mathit{\psi }}_1·\mathsf{\nabla }\right){\mathit{\psi }}_1-\left(\mathsf{\nabla }·{\mathit{\psi }}_2+{\mathit{\psi }}_2·\mathsf{\nabla }\right){\mathit{\psi }}_2\hfill \\ \hfill & -\left(\mathsf{\nabla }·{\mathit{\psi }}_3+{\mathit{\psi }}_3·\mathsf{\nabla }\right){\mathit{\psi }}_3+\left(\mathsf{\nabla }·{\mathit{\psi }}_4+{\mathit{\psi }}_4·\mathsf{\nabla }\right){\mathit{\psi }}_4\hfill \end{array}$$

(39)

2.3. Cases

Let us look at some particular cases where $\mathsf{\Psi }$ takes a simpler form.

2.3.1. Bivector Field

For a pure bivector field,

$$\mathsf{\Psi }={\langle \mathsf{\Psi }\rangle }_2={\mathit{\psi }}_2+{\mathit{\psi }}_{3}I$$

Equations (26), (28) and (33)–(35) reduce to:

$$\begin{array}{cc}\hfill {\partial }_t\left(\begin{array}{c}0\\ 0\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_3\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\psi }}_2\\ {\mathit{\psi }}_3\\ I{\mathit{\psi }}_3\\ -I{\mathit{\psi }}_2\end{array}\right)& =\left(\begin{array}{c}{\varphi }_1^0\\ -{\varphi }_4^0\\ -{\mathit{\varphi }}_1\\ {\mathit{\varphi }}_4\end{array}\right)\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\partial }_t\left(\begin{array}{c}{\varphi }_1^0\\ {\varphi }_4^0\\ {\mathit{\varphi }}_1\\ {\mathit{\varphi }}_4\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\varphi }}_1\\ {\mathit{\varphi }}_4\\ {\mathsf{N}}_1+I{\mathit{\varphi }}_4\\ {\mathsf{N}}_4-I{\mathit{\varphi }}_1\end{array}\right)& =\square \left(\begin{array}{c}0\\ 0\\ -{\mathit{\psi }}_2\\ {\mathit{\psi }}_3\end{array}\right)\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill J^0& ={\displaystyle \frac{1}{2}}\left({\mathit{\psi }}_2·{\mathit{\psi }}_2+{\mathit{\psi }}_3·{\mathit{\psi }}_3\right){\gamma }_0-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3{\gamma }_0\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill J^i& ={\left(-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right)}_i{\gamma }_0+{\displaystyle \frac{1}{2}}\left({\mathit{\psi }}_2·{\mathit{\psi }}_2+{\mathit{\psi }}_3·{\mathit{\psi }}_3\right){\gamma }_i-\left({\psi }_2^i{\mathit{\psi }}_2+{\psi }_3^i{\mathit{\psi }}_3\right){\gamma }_0\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill Л& =-\left({\mathit{\varphi }}_1·{\mathit{\psi }}_2-{\mathit{\varphi }}_4·{\mathit{\psi }}_3\right){\gamma }_0+\left[-{\varphi }_1^0{\mathit{\psi }}_2+{\varphi }_4^0{\mathit{\psi }}_3-I\left(-{\mathit{\varphi }}_1\wedge {\mathit{\psi }}_3+{\mathit{\varphi }}_4\wedge {\mathit{\psi }}_2\right)\right]{\gamma }_0\hfill \end{array}$$

(44)

The scalar and pseudoscalar C.E.s of Equation (32) vanish:

$${\langle {\partial }_{\mu }{J}^{\mu }-Л\rangle }_{0,4}=0$$

(45)

while the vector parts yield the scalar and vectorial continuity systems:

$${\partial }_t\left[{\displaystyle \frac{1}{2}}\left({\mathit{\psi }}_2·{\mathit{\psi }}_2+{\mathit{\psi }}_3·{\mathit{\psi }}_3\right)\right]+\mathsf{\nabla }·\left(-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right)=-{\mathit{\varphi }}_1·{\mathit{\psi }}_2+{\mathit{\varphi }}_4·{\mathit{\psi }}_3$$

(46)

$$\begin{array}{cc}\hfill & {\partial }_t\left(-I{\mathit{\psi }}_2\wedge {\mathit{\psi }}_3\right)+\mathsf{\nabla }\left[{\displaystyle \frac{1}{2}}\left({\mathit{\psi }}_2·{\mathit{\psi }}_2+{\mathit{\psi }}_3·{\mathit{\psi }}_3\right)\right]-\left(\mathsf{\nabla }·{\mathit{\psi }}_2+{\mathit{\psi }}_2·\mathsf{\nabla }\right){\mathit{\psi }}_2\hfill \\ \hfill & -\left(\mathsf{\nabla }·{\mathit{\psi }}_3+{\mathit{\psi }}_3·\mathsf{\nabla }\right){\mathit{\psi }}_3=-\left({\varphi }_1^0{\mathit{\psi }}_2-I{\mathit{\varphi }}_1\wedge {\mathit{\psi }}_3\right)+{\varphi }_4^0{\mathit{\psi }}_3-I{\mathit{\varphi }}_4\wedge {\mathit{\psi }}_2\hfill \end{array}$$

(47)

2.3.2. Vector Potential

Let us consider the case where a pure vector field $P={\langle P\rangle }_1=\left(p_1^0+{\mathit{p}}_1\right){\gamma }_0$ relates to a pure bivector field $\mathsf{\Psi }={\langle \mathsf{\Psi }\rangle }_2$ in a potential-like manner:

$$\begin{array}{cc}\hfill \mathsf{\nabla }P& =\mathsf{\Psi }\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill \square P& =\mathsf{\nabla }\mathsf{\Psi }=\mathsf{\Phi }\hfill \end{array}$$

(49)

Then, Equations (26), (28) and (33)–(35) reduce to:

$$\begin{array}{cc}\hfill {\partial }_t\left(\begin{array}{c}{p}_1^0\\ {\mathit{p}}_1\\ 0\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{p}}_1\\ {\mathsf{T}}_1\\ -I{\mathit{p}}_1\end{array}\right)& =\left(\begin{array}{c}0\\ -{\mathit{\psi }}_2\\ {\mathit{\psi }}_3\end{array}\right)\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\partial }_t\left(\begin{array}{c}0\\ 0\\ {\mathit{\psi }}_2\\ {\mathit{\psi }}_3\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\psi }}_2\\ {\mathit{\psi }}_3\\ I{\mathit{\psi }}_3\\ -I{\mathit{\psi }}_2\end{array}\right)& =\square \left(\begin{array}{c}{p}_1^0\\ 0\\ -{\mathit{p}}_1\\ 0\end{array}\right)\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill J^0& =\left({\displaystyle \frac{1}{2}}{p}_1^0{p}_1^0+{\displaystyle \frac{1}{2}}{\mathit{p}}_1·{\mathit{p}}_1+p_1^0{\mathit{p}}_1\right){\gamma }_0\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill J^i& =p_1^0{p}_1^i{\gamma }_0+{\displaystyle \frac{1}{2}}\left(p_1^0{p}_1^0-{\mathit{p}}_1·{\mathit{p}}_1\right){\gamma }_i+p_1^i{\mathit{p}}_1{\gamma }_0\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill Л& =\left({\psi }_2^0{p}_1^0-{\mathit{\psi }}_2·{\mathit{p}}_1-p_1^0{\mathit{\psi }}_2+{\psi }_2^0{\mathit{p}}_1-I{\mathit{\psi }}_3\wedge {\mathit{p}}_1\right){\gamma }_0\hfill \end{array}$$

(54)

Once again, the scalar and pseudoscalar continuity of Equation (32) vanish, Equation (45), and the vector parts yield the scalar and vectorial continuity systems:

$${\partial }_t{\displaystyle \frac{1}{2}}\left(p_1^0{p}_1^0+{\mathit{p}}_1·{\mathit{p}}_1\right)+\mathsf{\nabla }·\left(p_1^0{\mathit{p}}_1\right)={\psi }_2^0{p}_1^0-{\mathit{\psi }}_2·{\mathit{p}}_1$$

(55)

$$\begin{array}{cc}\hfill & {\partial }_t\left(p_1^0{\mathit{p}}_1\right)+\mathsf{\nabla }{\displaystyle \frac{1}{2}}\left(p_1^0{p}_1^0-{\mathit{p}}_1·{\mathit{p}}_1\right)+\left(\mathsf{\nabla }·{\mathit{p}}_1+{\mathit{p}}_1·\mathsf{\nabla }\right){\mathit{p}}_1\hfill \\ \hfill & =-p_1^0{\mathit{\psi }}_2+{\psi }_2^0{\mathit{p}}_1-I{\mathit{\psi }}_3\wedge {\mathit{p}}_1\hfill \end{array}$$

(56)

2.3.3. Discussion

Upon direct comparison with [17,18], we see that the cases of the pure bivector fields and pure vector potentials of Section 2.3.1. and Section 2.3.2. are consistent with STA’s treatment of classical electrodynamics considering null bound charge currents. Specifically, if ${\mathit{\psi }}_2\to \mathit{E}$ is the electric field, ${\mathit{\psi }}_3\to \mathit{B}$ is the magnetic field, ${\varphi }_1^0\to {\rho }_e$ is the electric charge density, ${\mathit{\varphi }}_1\to {\mathit{j}}_e$ is the electric current density, ${\varphi }_4^0\to {\rho }_m$ is the magnetic charge density, ${\mathit{\varphi }}_4\to {\mathit{j}}_m$ is the magnetic current density, $p_1^0\to V$ is the electric potential, and ${\mathit{p}}_1\to \mathit{A}$ is the magnetic potential, then $\mathsf{\Psi }\to \mathit{F}$ is Faraday’s electromagnetic field, $\mathsf{\Phi }\to J_e+J_m$ are the electric and magnetic four-current densities, and $P\to A$ is the electromagnetic four-potential. Thus, reverting back the notation via Equations (24) and (25), Equation (40) are Maxwell’s field equations of Equation (19). Equation (41) are the conservation of electric and magnetic charge along with the wave equations for the electric and magnetic field. Equation (50) is Lorentz’s gauge and the relations between the potentials and the fields, while Equation (51) are the relations between the wave equations of the potentials and Maxwell’s equations. Correspondingly, the $J^{\mu }$ from Equations (42) and (43) are the energy–momentum vectors of the electromagnetic field—where, in particular, $J^0{\gamma }_0\to u+\mathit{S}$, u being the electromagnetic energy density and $\mathit{S}$ the Poynting vector. Equations (46) and (47) represent the conservation of electromagnetic energy and linear momentum, respectively, with $Л$ of Equation (44) the Lorentz force density that includes the magnetic charge. Lastly, the simplification of Equation (39) is identified with the divergence of the electromagnetic tensor.

Hence, we see that the nabla equation for a generalized STA multivector $\mathsf{\Psi }$, Equation (26), can be understood as a system of continuity equations for the different components of $\mathsf{\Psi }$, and upon reduction to simpler quantities (bivector fields), it encodes a system with the structure of Maxwell’s field equations. Repeated application of the nabla operator to $\mathsf{\Psi }$ yields the wave equation Equation (28), which relates the continuity system of the source $\mathsf{\Phi }$ with the wave equations for the different components of $\mathsf{\Psi }$. It encodes the wave equations of the fields involved in the Maxwell-like structure upon reduction to simpler quantities. Equations (33)–(35) extend the energy–momentum vectors and Lorentz force density structure beyond bivector fields up to generalized multivectors with multiple components. The corresponding continuity system Equation (32) encodes the analog of the conservation of four-momentum when reduced to the simpler case.

Because $\mathsf{\Psi }$ and $\mathsf{\Phi }$ are arbitrary, the results we have found and that are mentioned in the paragraph above need not be limited to electromagnetism, as we have seen that this is just a particular case where the multivector is a homogeneous bivector field or a potential-like vector field. In fact, an analog of Maxwell’s equations can be found in the theory of hydrodynamics, where if $\mathsf{\Psi }\to J{\gamma }_0=n+\mathit{j}$ is the turbulent or fluid–mechanical ‘charge’ density, ${\mathit{\psi }}_2\to \mathit{L}$ is Lamb’s vector, ${\mathit{\psi }}_3\to \mathit{w}$ is the vorticity, ${\mathit{p}}_1\to \mathit{v}$ is the fluid’s velocity, and $p_1^0\to h$ is the enthalpy per unit mass or equivalently $p_1^0\to \phi$ Bernoulli’s energy function, then Equation (40) are Maxwell’s field equations for a compressible fluid [28,29,30]. But, of course, the fact that Maxwell-like equations, energy–momentum vectors, and Lorentz force density analogs that are possible to be derived from an arbitrary current and source $(\mathsf{\Psi },\mathsf{\Phi }$) does not imply in the least a necessary physical correspondent. A simple counter-example is the energy and momentum conservation for the potentials, Equations (55) and (56), where no obvious physical interpretation seems to appear.

3. ‘Linear’ Decoupling

Because of the independence and symmetry between the C.E. regarding the second couplings, $\left({\psi }_1^{\mu },{\psi }_4^{\mu }\right)$ & $\left({\psi }_2^{\mu },{\psi }_3^{\mu }\right)$, consider the simplified system of Equation (26):

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0+\mathsf{\nabla }·{\mathit{\psi }}_a& ={\varphi }_a\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_b^0+\mathsf{\nabla }·{\mathit{\psi }}_b& =-{\varphi }_b\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\partial }_t{\mathit{\psi }}_a+\mathsf{\nabla }{\psi }_a^0+I\mathsf{\nabla }\wedge {\mathit{\psi }}_b& =-{\mathit{\varphi }}_a\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {\partial }_t{\mathit{\psi }}_b+\mathsf{\nabla }{\psi }_b^0-I\mathsf{\nabla }\wedge {\mathit{\psi }}_a& ={\mathit{\varphi }}_b\hfill \end{array}$$

(60)

where it is understood that, if $a=1$, then $b=4$, and correspondingly, if $a=2$, then $b=3$. The source terms ${\varphi }_a,{\mathit{\varphi }}_a$, etc., are given by Equation (26), although for now, we are not interested in their form. Let us proceed to determine some cases for ‘linear’ decoupling—for which we simply mean that we wish to decouple any one of the functions ${\psi }_a^{\mu }$ and ${\psi }_b^{\mu }$ from one another in the system Equation (57)–(60) without necessarily recurring to the wave system Equation (28) or the trivial case.

3.1. Mutual Decoupling

The simplest case of linear decoupling consists of imposing on both relative vectors the mutual restriction $\mathsf{\nabla }\wedge {\mathit{\psi }}_{a,b}=0$. Whence,

$${\mathit{\psi }}_{a,b}=\mathsf{\nabla }{\mathsf{\Gamma }}_{a,b}$$

(61)

where ${\mathsf{\Gamma }}_{a,b}={\mathsf{\Gamma }}_{a,b}\left(x^{\mu }\right)$ are scalar functions, and the index ($a,b$) denotes that the above equations apply for both functions simultaneously. With the mutual simultaneous decoupling of Equation (61), the system of Equation (57)–(60) reduces to:

$$\begin{array}{cc}\hfill {\partial }_t{\mathsf{\psi }}_{a,b}^0+{\mathsf{\nabla }}^2{\mathsf{\Gamma }}_{a,b}& =\pm {\varphi }_{a,b}\hfill \\ \hfill \mathsf{\nabla }\left({\psi }_{a,b}^0+{\partial }_t{\mathsf{\Gamma }}_{a,b}\right)& =\mp {\mathit{\varphi }}_{a,b}\hfill \end{array}$$

(62)

with ${\mathsf{\nabla }}^2=\mathsf{\nabla }·\mathsf{\nabla }={\delta }_{ij}{\partial }_{ij}^2$ the Laplacian operator of the APS. Thus, imposing the restriction of Equation (61) simultaneously to both relative vectors ${\mathit{\psi }}_a$ and ${\mathit{\psi }}_b$ decouples them in two pairs of equations with the same structure, but independent of each other.

3.2. Unilateral Decoupling

Suppose we now impose a restriction to just one of the terms, say ${\psi }_b^{\mu }$ with respect to ${\psi }_a^{\mu }$. Decoupling is achieved with the condition

$${\psi }_b^{\mu }={\partial }^{\mu }{\mathsf{\Gamma }}_b$$

(63)

or, explicitly, ${\psi }_b^0={\partial }_t{\mathsf{\Gamma }}_b,{\mathit{\psi }}_b=-\mathsf{\nabla }{\mathsf{\Gamma }}_b$, which ensures

$$\begin{array}{cc}\hfill \mathsf{\nabla }\wedge {\mathit{\psi }}_b=-\mathsf{\nabla }\wedge \mathsf{\nabla }{\mathsf{\Gamma }}_b& =0\hfill \\ \hfill {\partial }_t{\mathit{\psi }}_b+\mathsf{\nabla }{\mathsf{\psi }}_b^0=-{\partial }_t\left(\mathsf{\nabla }{\mathsf{\Gamma }}_b\right)+\mathsf{\nabla }\left({\partial }_t{\mathsf{\Gamma }}_b\right)& =0\hfill \end{array}$$

(64)

Then, the system of Equations (57)–(60) takes the form:

$${\partial }_t\left(\begin{array}{c}{\psi }_a^0\\ {\partial }_t{\mathsf{\Gamma }}_b\\ {\mathit{\psi }}_a\\ 0\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\psi }}_a\\ -\mathsf{\nabla }{\mathsf{\Gamma }}_b\\ {\mathsf{T}}_a\\ -I{\mathit{\psi }}_a\end{array}\right)=\left(\begin{array}{c}{\varphi }_a\\ -{\varphi }_b\\ -{\mathit{\varphi }}_a\\ {\mathit{\varphi }}_b\end{array}\right)$$

(65)

At the cost of restricting the structure of ${\psi }_b^{\mu }$, we obtain a wave equation for ${\mathsf{\Gamma }}_b$. However, we free ${\psi }_a^{\mu }$ in three equations independent of any other component of $\mathsf{\Psi }$.

3.3. Mixed Coupling

As a last notable case, consider instead a multivector for which its components are coupled in the form:

$$\mathsf{\Psi }=\left({\partial }_{t}f-\mathsf{\nabla }f-I\mathsf{\nabla }\wedge \mathit{F}\right){\gamma }_0+\left(-\mathsf{\nabla }·\mathit{F}+{\partial }_t\mathit{F}\right)I{\gamma }_0$$

(66)

where $f=f\left(x^{\mu }\right)$ is a scalar function and $\mathit{F}=\mathit{F}\left(x^{\mu }\right)$ is a vector function. With this structure, the system coupled system Equations (57)–(60) directly yields the two wave equations

$$\begin{array}{cc}\hfill \square f& ={\varphi }_a\hfill \\ \hfill \square \mathit{F}& ={\mathit{\varphi }}_b\hfill \end{array}$$

(67)

where we have used the identities

$$\begin{array}{cc}\hfill \mathsf{\nabla }·(I\mathsf{\nabla }\wedge \mathit{F})& =I\mathsf{\nabla }\wedge (\mathsf{\nabla }\wedge \mathit{F})=0\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill \mathsf{\nabla }·(\mathsf{\nabla }\wedge \mathit{F})& ={\mathsf{\nabla }}^2\mathit{F}-\mathsf{\nabla }(\mathsf{\nabla }·\mathit{F})\hfill \end{array}$$

(69)

Upon comparing with the general case, we see that Equations (66) and (67) can be put in terms of a potential multivector defined by f and $\mathit{F}$:

$$\begin{array}{cc}\hfill \mathsf{\Psi }& =\nabla (f+\mathit{F}I)\hfill \\ \hfill \mathsf{\Phi }& =\nabla \mathsf{\Psi }=\square (f+\mathit{F}I)\hfill \end{array}$$

(70)

4. Symmetries of the Continuity Equation

From the system of Equations (57)–(60), consider the proposal:

$${\partial }_t{\psi }_a^0+\mathsf{\nabla }·{\mathit{\psi }}_a={\partial }_t{\left({\psi }_a^0\right)}^{\prime }+\mathsf{\nabla }·{\left({\mathit{\psi }}_a\right)}^{\prime }$$

(71)

In other words, how can the functions ${\psi }_a^{\mu }$ transform in such a way that the structure of its scalar C.E. remains invariant? Making use of the identity of Equation (68), it is found that:

$$\begin{array}{cc}\hfill {\psi }_a^0& ={\left({\psi }_a^0\right)}^{\prime }+\mathsf{\nabla }·\mathsf{\Lambda }\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill {\mathit{\psi }}_a& ={\left({\mathit{\psi }}_a\right)}^{\prime }-{\partial }_t\mathsf{\Lambda }-I\mathsf{\nabla }\wedge \mathsf{\Theta }\hfill \end{array}$$

(73)

If these symmetry transformations for the scalar continuity are employed in Equation (59), we obtain

$${\partial }_t{\left({\mathit{\psi }}_a\right)}^{\prime }+\mathsf{\nabla }{\left({\psi }_a^0\right)}^{\prime }+I\mathsf{\nabla }\wedge {\mathit{\psi }}_b-\left[\square \mathsf{\Lambda }+\mathsf{\nabla }·(I{\partial }_t\mathsf{\Theta }+\mathsf{\nabla }\wedge \mathsf{\Lambda })\right]=-{\mathit{\varphi }}_a$$

(74)

where we have used identity Equation (69). Independent of decoupling or not, let us impose an invariance in the vector continuity Equation (59) upon the transformations Equations (72) and (73):

$${\partial }_t{\mathit{\psi }}_a+\mathsf{\nabla }{\psi }_a^0+I\mathsf{\nabla }\wedge {\mathit{\psi }}_b={\partial }_t{\left({\mathit{\psi }}_a\right)}^{\prime }+\mathsf{\nabla }{\left({\psi }_a^0\right)}^{\prime }+I\mathsf{\nabla }\wedge {\mathit{\psi }}_b$$

(75)

which then delivers the condition

$$\square \mathsf{\Lambda }=-I\mathsf{\nabla }\wedge ({\partial }_t\mathsf{\Theta }-I\mathsf{\nabla }\wedge \mathsf{\Lambda })$$

(76)

This allows us to define

$${\partial }_t\mathsf{\Theta }-I\mathsf{\nabla }\wedge \mathsf{\Lambda }=\mathsf{\Omega }$$

(77)

By applying the ${\partial }_t$ operator to Equation (77), the $I\mathsf{\nabla }\wedge$ operator to Equation (73), and combining the two expressions, we obtain

$$\square \mathsf{\Theta }={\partial }_t\mathsf{\Omega }-\mathsf{\nabla }(\mathsf{\nabla }·\mathsf{\Theta })-I\mathsf{\nabla }\wedge \left[{\mathit{\psi }}_a-{\left({\mathit{\psi }}_a\right)}^{\prime }\right]$$

(78)

Finally, it is convenient to define

$$\mathsf{\nabla }·\mathsf{\Theta }=-\omega$$

(79)

Thus, grouping all the results into matrix notation continuity systems:

$${\partial }_t\left(\begin{array}{c}0\\ 0\\ \mathsf{\Lambda }\\ \mathsf{\Theta }\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}\mathsf{\Lambda }\\ \mathsf{\Theta }\\ I\mathsf{\Theta }\\ -I\mathsf{\Lambda }\end{array}\right)=\left(\begin{array}{c}{\psi }_a^0-{\left({\psi }_a^0\right)}^{\prime }\\ -\omega \\ -\left[{\mathit{\psi }}_a-{\left({\mathit{\psi }}_a\right)}^{\prime }\right]\\ \mathsf{\Omega }\end{array}\right)$$

(80)

$${\partial }_t\left(\begin{array}{c}{\psi }_a^0-{\left({\psi }_a^0\right)}^{\prime }\\ \omega \end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{\psi }}_a-{\left({\mathit{\psi }}_a\right)}^{\prime }\\ \mathsf{\Omega }\end{array}\right)=\square \left(\begin{array}{c}0\\ 0\end{array}\right)$$

(81)

$${\partial }_t\left(\begin{array}{c}{\mathit{\psi }}_a-{\left({\mathit{\psi }}_a\right)}^{\prime }\\ \mathsf{\Omega }\end{array}\right)+\mathsf{\nabla }\left(\begin{array}{c}{\psi }_a^0-{\left({\psi }_a^0\right)}^{\prime }\\ \omega \end{array}\right)-I\mathsf{\nabla }\wedge \left(\begin{array}{c}-\mathsf{\Omega }\\ {\mathit{\psi }}_a-{\left({\mathit{\psi }}_a\right)}^{\prime }\end{array}\right)=\square \left(\begin{array}{c}-\mathsf{\Lambda }\\ \mathsf{\Theta }\end{array}\right)$$

(82)

Ergo, by comparing with the general structures Equations (26) and (28), it follows that, if

$$\begin{array}{cc}\hfill \mathit{Q}& =\mathsf{\Lambda }+\mathsf{\Theta }I\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill M& =\left[{\mathsf{\psi }}_a^0-{\left({\psi }_a^0\right)}^{\prime }+{\mathit{\psi }}_a-{\left({\mathit{\psi }}_a\right)}^{\prime }\right]{\gamma }_0+(\omega +\mathsf{\Omega })I{\gamma }_0\hfill \end{array}$$

(84)

then Equations (80)–(82) are nothing but

$$\begin{array}{cc}\hfill \nabla \mathit{Q}& =M\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill \square \mathit{Q}& =\mathsf{\nabla }M\hfill \end{array}$$

(86)

4.1. Free Waves

The condition for homogeneous wave equations for the fields $\mathsf{\Lambda }$ and $\mathsf{\Theta }$:

$$\square \mathsf{\Lambda }=\square \mathsf{\Theta }=0$$

(87)

on the one hand, imposes the invariance of Equation (60):

$$-I\mathsf{\nabla }\wedge {\mathit{\psi }}_a=-I\mathsf{\nabla }\wedge {\left({\mathit{\psi }}_a\right)}^{\prime }$$

(88)

and, on the other hand, the restrictions on ${\langle M\rangle }_3$:

$$\begin{array}{cc}\hfill \mathsf{\nabla }\wedge \mathsf{\Omega }& =0\hfill \\ \hfill {\partial }_t\mathsf{\Omega }+\mathsf{\nabla }\omega & =0\hfill \end{array}$$

(89)

But, from Section 3.2, these are just the conditions for the unilateral decoupling of ${\langle M\rangle }_3$ with respect to ${\langle M\rangle }_1$. Thus, the pseudovector of M takes the form:

$${\langle M\rangle }_3=({\partial }_t\zeta -\mathsf{\nabla }\zeta )I{\gamma }_0$$

(90)

4.2. Potentials

Section 2.3.2 and Section 3.3 and Equations (85) and (86) allow us to consider the equations:

$$\begin{array}{cc}\hfill \nabla P& =\mathit{Q}\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill \square P& =\nabla \mathit{Q}=M\hfill \end{array}$$

(92)

From structure Equation (26), we see that the bivector $\mathit{Q}$ is completely characterized by a potential multivector of the form

$$P=(p_1^0+{\mathit{p}}_1){\gamma }_0+(p_4^0+{\mathit{p}}_4)I{\gamma }_0$$

(93)

Therefore, Equation (91) and (92) for the potentials in explicit form are

$${\partial }_t\left(\begin{array}{c}{p}_1^0\\ p_4^0\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}{\mathit{p}}_1\\ {\mathit{p}}_4\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$

(94)

$${\partial }_t\left(\begin{array}{c}{\mathit{p}}_1\\ {\mathit{p}}_4\end{array}\right)+\mathsf{\nabla }\left(\begin{array}{c}{p}_1^0\\ p_4^0\end{array}\right)-I\mathsf{\nabla }\wedge \left(\begin{array}{c}-{\mathit{p}}_4\\ {\mathit{p}}_1\end{array}\right)=\left(\begin{array}{c}-\mathsf{\Lambda }\\ \mathsf{\Theta }\end{array}\right)$$

(95)

$$\square \left(\begin{array}{c}{p}_1^0\\ -p_4^0\\ -{\mathit{p}}_1\\ {\mathit{p}}_4\end{array}\right)={\partial }_t\left(\begin{array}{c}0\\ 0\\ \mathsf{\Lambda }\\ \mathsf{\Theta }\end{array}\right)+\mathsf{\nabla }·\left(\begin{array}{c}\mathsf{\Lambda }\\ \mathsf{\Theta }\\ I\mathsf{\Theta }\\ -I\mathsf{\Lambda }\end{array}\right)=\left(\begin{array}{c}{\psi }_1^0-{\left({\psi }_1^0\right)}^{\prime }\\ -\omega \\ -\left[{\mathit{\psi }}_1-{\left({\mathit{\psi }}_1\right)}^{\prime }\right]\\ \mathsf{\Omega }\end{array}\right)$$

(96)

Imposing the decoupling of ${\langle P\rangle }_3$ with respect to ${\langle P\rangle }_1$ through psi4 desacoplamiento permits expressing the fields in terms of the vector potential exclusively:

$$\begin{array}{cc}\hfill \mathsf{\Lambda }& =-\mathsf{\nabla }{p}_1^0-{\partial }_t{\mathit{p}}_1\hfill \\ \hfill \mathsf{\Theta }& =-I\mathsf{\nabla }\wedge {\mathit{p}}_1\hfill \end{array}$$

(97)

By the structure of Equation (83), the Lorentz force density and energy–momentum vectors (with their continuity) associated with the field $\mathit{Q}$ are completely described by Section 2.3.1. For the potential P, see Section 2.3.2.

4.3. Discussion

We have found that the invariance of the system of C.E.s for a multivector ${\langle \mathsf{\Psi }\rangle }_1$ due to symmetry transformations naturally allows for the construction of a system of eqs. with the structure of Maxwell’s field equations, Equation (80), for the bivector field $\mathit{Q}$ defined by Equation (83). Because of the symmetric nature in the system of Equations (57)–(60) for ${\langle \mathsf{\Psi }\rangle }_1+{\langle \mathsf{\Psi }\rangle }_3$ and ${\langle \mathsf{\Psi }\rangle }_0+{\langle \mathsf{\Psi }\rangle }_2+{\langle \mathsf{\Psi }\rangle }_4$, all of these relations hold if M has a bivector character and Q a vector character. The emergence of the new pseudovector term ${\langle M\rangle }_3=\left(\omega +\mathsf{\Omega }\right)I{\gamma }_0$ exists independently of the pseudovector part ${\langle \mathsf{\Psi }\rangle }_3=\left({\psi }_b^0+{\mathit{\psi }}_b\right)I{\gamma }_0$ of $\mathsf{\Psi }$:

$$M={\langle \mathsf{\Psi }\rangle }_1-{\langle {\mathsf{\Psi }}^{\prime }\rangle }_1+{\langle M\rangle }_3$$

(98)

The symmetries do admit ${\langle \mathsf{\Psi }\rangle }_3={\langle M\rangle }_3$, although this does not seem to be a requirement, but a special case. The whole system is obtained by only the symmetries of Equations (57) and (59), insofar as the invariance of Equation (60) emerges as a condition for the existence of free waves for the fields. The vector part ${\langle M\rangle }_1={\langle \mathsf{\Psi }\rangle }_1-{\langle {\mathsf{\Psi }}^{\prime }\rangle }_1$ may be regarded as a displacement in the transported physical quantity with a fixed coordinate system.

For the case in which ${\langle \mathsf{\Psi }\rangle }_1\to J{\gamma }_0=\rho +\mathit{j}$ in Equation (84) is the electric four-current density and $\mathit{Q}\to \mathit{F}=\mathit{E}+I\mathit{B}$ Faraday’s electromagnetic field, see the discussion in Section 2.3.3. Here, we will limit ourselves to say that the fact that the condition of free waves for the electric and magnetic fields implies the decoupling of the magnetic current density ${\langle M\rangle }_3$ with respect to the electric current density J (Section 4.1) may shed light on the nature of magnetic monopoles and their experimental elusiveness so far.

5. Diffusion Equation

To conclude, consider the case where one of the relative vectors satisfies Fick’s laws of diffusion, e.g.,

$${\mathit{\psi }}_a=-D\mathsf{\nabla }{\psi }_a^0$$

(99)

where $D=D\left({\psi }_a^0;x^{\mu }\right)$ is called the diffusion coefficient. With Equation (99), the continuity system of Equations (57)–(60) takes the form:

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0-\mathsf{\nabla }·\left(D\mathsf{\nabla }{\mathsf{\psi }}_a^0\right)& ={\varphi }_a\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_b^0+\mathsf{\nabla }·{\mathit{\psi }}_b& =-{\varphi }_b\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill \left(1-{\partial }_{t}D\right)\mathsf{\nabla }{\psi }_a^0-D\mathsf{\nabla }\left({\partial }_t{\psi }_a^0\right)+I\mathsf{\nabla }\wedge {\mathit{\psi }}_b& =-{\mathit{\varphi }}_a\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill {\partial }_t{\mathit{\psi }}_b+\mathsf{\nabla }{\psi }_b^0+I\mathsf{\nabla }D\wedge \mathsf{\nabla }{\psi }_a^0& ={\mathit{\varphi }}_b\hfill \end{array}$$

(103)

Equation (99) establishes a diffusion equation for ${\psi }_a^0$ in Equation (100). However, the solution and structure of this equation are conditioned by the rest of the equations in the system.

5.1. Constant Diffusion Coefficient

Consider the case $D=\mathrm{constant}$. For this condition, the system of Equations (100)–(103) reduces to:

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0-D{\mathsf{\nabla }}^2{\psi }_a^0& ={\varphi }_a\hfill \\ \hfill {\partial }_t{\psi }_b^0+\mathsf{\nabla }·{\mathit{\psi }}_b& =-{\varphi }_b\hfill \\ \hfill \mathsf{\nabla }\left({\psi }_a^0-D{\partial }_t{\psi }_a^0\right)+I\mathsf{\nabla }\wedge {\mathit{\psi }}_b& =-{\mathit{\varphi }}_a\hfill \\ \hfill {\partial }_t{\mathit{\psi }}_b+\mathsf{\nabla }{\psi }_b^0& ={\mathit{\varphi }}_b\hfill \end{array}$$

(104)

As expected, a constant diffusion coefficient yields a heat equation for ${\psi }_a^0$. The remaining equations are still coupled with it, so let us inspect the case for which ${\mathit{\psi }}_b$ is subjected to the imposition of Equation (61) such that Equation (104) takes the form:

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0-D{\mathsf{\nabla }}^2{\psi }_a^0& ={\varphi }_a\hfill \\ \hfill {\partial }_t{\psi }_b^0+{\mathsf{\nabla }}^2{\mathsf{\Gamma }}_b& =-{\varphi }_b\hfill \\ \hfill \mathsf{\nabla }\left({\psi }_a^0-D{\partial }_t{\psi }_a^0\right)& =-{\mathit{\varphi }}_a\hfill \\ \hfill \mathsf{\nabla }\left({\psi }_b^0+{\partial }_t{\mathsf{\Gamma }}_b\right)& ={\mathit{\varphi }}_b\hfill \end{array}$$

(105)

Decoupling of ${\psi }_a^{\mu }$ and ${\psi }_b^{\mu }$ is indeed achieved. However, it can be seen that the absence of external sources, $\nabla \mathsf{\Psi }=0$, represents a separation of space and time variables for ${\psi }_a^0$. That is, $\mathsf{\Phi }=0$ implies

$$\begin{array}{cc}\hfill {\psi }_a^0-D{\partial }_t{\psi }_a^0& =\mathrm{constant}\hfill \\ \hfill {\psi }_b^0+{\partial }_t{\mathsf{\Gamma }}_b& =\mathrm{constant}\hfill \end{array}$$

(106)

and, thus,

$$\begin{array}{cc}\hfill \left({\mathsf{\nabla }}^2-D^{-2}\right){\psi }_a^0& =\mathrm{const}.\hfill \end{array}$$

(107)

$$\begin{array}{cc}\hfill \square {\mathsf{\Gamma }}_b& =0\hfill \end{array}$$

(108)

For a pseudoscalar diffusion coefficient, $D={\left(D^{\prime }\right)}^{-1}I$, Equation (107) takes the structure of a Helmholtz equation: $\left({\mathsf{\nabla }}^2+D^{\prime 2}\right){\psi }_a^0=\mathrm{const}$. The separation of variables for ${\psi }_a^0$ constitutes by itself a transformation that decouples the system, as can be easily verified by imposing a priori Equation (106) on the system of Equation (104). Furthermore, the decoupling-by-gradient hypothesis for ${\mathit{\psi }}_b$ may also adopt a diffusion structure by considering a multivector $\mathsf{\Psi }=({\psi }_a-D_a\mathsf{\nabla }{\psi }_a)+\left({\psi }_b-D_b\mathsf{\nabla }{\psi }_b\right)I$, where $D_{a,b}$ are both constants. Equation (20) yields the ‘secondly decoupled’ system:

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_{a,b}-D_{a,b}{\mathsf{\nabla }}^2{\psi }_{a,b}& =\pm {\varphi }_{a,b}\hfill \\ \hfill \mathsf{\nabla }\left({\psi }_{a,b}-D_{a,b}{\partial }_t{\psi }_{a,b}\right)& =\mp {\mathit{\varphi }}_{a,b}\hfill \end{array}$$

(109)

Once again, the absence of external sources $\mathsf{\Phi }=0$—or, specifically, the absence of vector field sources ${\mathit{\varphi }}_{a,b}=0$—reduces the system into two Helmholtz-like equations, Equation (107).

5.2. Non-Constant Diffusion Coefficient

Let us decouple ${\psi }_b^{\mu }$ from the general diffusion system of Equations (100)–(103):

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0-\mathsf{\nabla }·\left(D\mathsf{\nabla }{\psi }_a^0\right)& ={\varphi }_a\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill \left(1-{\partial }_{t}D\right)\mathsf{\nabla }{\psi }_a^0-D\mathsf{\nabla }\left({\partial }_t{\psi }_a^0\right)& =-{\mathit{\varphi }}_a\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill I\mathsf{\nabla }D\wedge \mathsf{\nabla }{\psi }_a^0& ={\mathit{\varphi }}_b\hfill \end{array}$$

(112)

Consider the case where the diffusion coefficient is an explicit function of ${\psi }_a^0$, but not on the space–time coordinates, $D=D\left({\psi }_a^0\right)$:

$$\begin{array}{cc}\hfill {\partial }_{t}D& ={\displaystyle \frac{\partial D}{\partial {\psi }_a^0}}{\partial }_t{\psi }_a^0\hfill \\ \hfill \mathsf{\nabla }D& ={\displaystyle \frac{\partial D}{\partial {\psi }_a^0}}\mathsf{\nabla }{\psi }_a^0\hfill \end{array}$$

(113)

Then, from Equations (110)–(112), the following non-linear system is obtained:

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0-\mathsf{\nabla }·\left(D\mathsf{\nabla }{\psi }_a^0\right)& ={\varphi }_a\hfill \\ \hfill \mathsf{\nabla }{\psi }_a^0-{\displaystyle \frac{\partial D}{\partial {\psi }_a^0}}{\partial }_t{\psi }_a^0\mathsf{\nabla }{\psi }_a^0-D\mathsf{\nabla }\left({\partial }_t{\psi }_a^0\right)& =-{\mathit{\varphi }}_a\hfill \end{array}$$

(114)

Structure $D=D\left({\psi }_a^0\right)$ by virtue of Equation (113) implies that the diffusion flux is a conservative vector field, which further enables the introduction of a potential gradient:

$$\mathsf{\nabla }\vartheta =D\mathsf{\nabla }{\psi }_a^0$$

(115)

Ergo, system Equation (114) can also be expressed as

$$\begin{array}{cc}\hfill {\partial }_t{\psi }_a^0-{\mathsf{\nabla }}^2\vartheta & ={\varphi }_a\hfill \\ \hfill \mathsf{\nabla }\left({\psi }_a^0-{\partial }_t\vartheta \right)& =-{\mathit{\varphi }}_a\hfill \end{array}$$

(116)

Once again, in the absence of external sources, ${\psi }_a^0-{\partial }_t\vartheta =\mathrm{const}$, and therefore,

$$\square \vartheta =0$$

(117)

5.3. Interpretation

Consider the system of Equations (110)–(112) arising from $\mathsf{\nabla }\left({\psi }_a^0-D\mathsf{\nabla }{\psi }_a^0\right)$. Equations (111) and (112) condition the function ${\psi }_a^0$ in such a way that, for constant or explicitly dependent coefficients, the absence of external sources implies that the diffusion structure of Equation (110) is lost. This situation can be understood from the perspective of bi-dependence between the components of $\mathsf{\Psi }$ and $\mathsf{\Phi }$. From Equation (110), the density/concentration ${\psi }_a^0$ can be determined in terms of the scalar source ${\varphi }_a^0$ and the diffusive coefficient D. Then, $({\mathit{\varphi }}_a+I{\mathit{\varphi }}_b)$ may be conceived as an external vector-field source defined by Equations (111) and (112) in terms of ${\psi }_a^0$ and D. From the constant and explicitly dependent coefficient cases, we saw that, if this field is null, then the diffusive structure is transformed into Helmholtz-like wave structures. Therefore, the external field $({\mathit{\varphi }}_a+I{\mathit{\varphi }}_b)$ admits the interpretation of being the source that produces itself the diffusion phenomena.

6. Conclusions

In synthesis, we have shown that, for a generalized STA multivector, the nabla equation $\nabla \mathsf{\Psi }=\mathsf{\Phi }$ yields a system of eight continuity equations—four of a scalar character and four of a vector character. Thus, the nabla equation can be identified as a multivector C.E. in STA. From this, we constructed the system $\square \mathsf{\Psi }=\nabla \mathsf{\Phi }$, where each wave equation for the components of $\mathsf{\Psi }$ plays the role of the source for the continuity system of $\mathsf{\Phi }$. With the same method for which the C.E. is derived from the Schrödinger and Dirac eqs., we have constructed generalized analogous quantities to the energy–momentum vectors, the Lorentz force density, and their corresponding C.E.s, where the latter acts as the source. Upon reduction to well-known simpler quantities, the results found are consistent with STA’s treatment of classical electromagnetic theory and with hydrodynamics from fluid mechanics. The continuity systems allow for the identification of the coupled structures $\left({\langle \mathsf{\Psi }\rangle }_1,{\langle \mathsf{\Psi }\rangle }_3\right)$ and $\left({\langle \mathsf{\Psi }\rangle }_0,{\langle \mathsf{\Psi }\rangle }_2,{\langle \mathsf{\Psi }\rangle }_4\right)$. The cost of complete decoupling for either structure is a transition to a system of higher-order equations, such as the wave system. Faced with this, we have determined three cases of decoupling by imposing restrictions on the structure of at least one of the functions involved in the C.E.: if both functions are APS gradients, if one of them is a space–time gradient, and—considering instead a mixed coupling—a potential multivector structure was constructed for which the continuity system directly yields a wave system. Symmetry transformations that make the (scalar) C.E. invariant are found in terms of vector fields. By imposing that the system of C.E.s be invariant under these transformations, a system of field, wave, and potential equations with the structure of Maxwell’s equations was derived. Thus, we see that a Maxwellian system is not only possible to be derived from the nabla equation and the generalized continuity system as special cases (Section 2.3.1), but also from the symmetries of the C.E. structure as well. Finally, when one of the functions in the continuity system satisfies Fick’s laws, a diffusion equation arises coupled with the remaining C.E.s of the system. It was found that, for a decoupled system with a constant or explicitly dependent diffusion coefficient, the absence of external sources implies a loss in the diffusion equation structure being transformed to a Helmholtz-like structure and a wave system. For both cases considered, the external vector-field source can be defined by the complementary equations in the system in terms of the density/concentration and diffusion coefficient, and admit the interpretation of the diffusion phenomena producer.