Content-Type: multipart/mixed; boundary="-------------2107090024399" This is a multi-part message in MIME format. ---------------2107090024399 Content-Type: text/plain; name="21-36.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="21-36.comments" This is a revised, expanded, and hopefully improved version of mp arc 20-100 with the same title. This version is currently under review at Quantum Studies: Mathematical Foundations. ---------------2107090024399 Content-Type: text/plain; name="21-36.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="21-36.keywords" Quantum Theory, Dirac's Equation, Clifford Algebra, direct particle interaction, action-at-a-distance, time-symmetry, Majorana spinors, dark energy, super-symmetry ---------------2107090024399 Content-Type: application/x-tex; name="v14.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="v14.tex" \documentclass[twocolumn]{svjour3} % onecolumn (second format) \smartqed % flush right qed marks, e.g. at end of proof \usepackage{mathptmx} % use Times fonts if available on your TeX system \usepackage{mathtools} \usepackage{graphicx} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{hyperref} \usepackage{url} \usepackage{multirow} \usepackage{tabularx} \usepackage{color} \usepackage{textcomp} \usepackage{amssymb} \usepackage{amsxtra} \usepackage{wasysym} \usepackage{isomath} \usepackage{txfonts} \usepackage{upgreek} \usepackage{enumerate} \usepackage{tensor} \usepackage{pifont} \usepackage{soul} \usepackage{arydshln} \usepackage{cancel} \usepackage{nicefrac} \usepackage{leftidx} \usepackage{hyphenat} \usepackage[bb=boondox]{mathalfa} \usepackage[italian,english]{babel} \usepackage{bbm} \usepackage[multiple]{footmisc} \newcommand{\scalarproductdot}{\boldsymbol{\cdot}} \newcommand{\doublesum}{\sum_{k,l=1;\, k \neq l}^{N}} \newcommand{\GreensFunction}{G} \newcommand{\half}{\textstyle \frac{1}{2}} \newcommand{\dq}{\mathbf{\boldsymbol{\updelta}}} \hyphenpenalty=0 \journalname{Quantum Studies: Mathematics and Foundations} % \begin{document} \title{Electromagnetic Foundation of Dirac Theory } \titlerunning{EM Foundation of Quantum Theory} \author{Michael Ibison} \institute{M. Ibison \at Tel.: +1 512 669 6713 \\ \email{ngc5548@gmail.com} } \date{Received: date / Accepted: date} \maketitle \begin{abstract} The dynamics of classical charges subject to a particular variant of electromagnetic direct particle interaction are shown to derive from a homogeneous differential equation in a Clifford multivector. Under appropriate conditions the multivector can be factorized to give a Dirac Equation whose bi-spinor operands are eigenvectors of the multivector, thereby giving a classical electromagnetic basis for the Dirac Equation. The Clifford multivector is an ensemble of vector and bi-vector contributions from the potential and Faraday of the auxiliary (‘adjunct’) fields of direct particle interaction, each member generated by a unique current. The presumption of light-speed motion of the charge generates non-linear constraints on these fields. These conditions are shown to be responsible for the otherwise enigmatic eigenvalue selection / ‘wavefunction collapse’ behavior characteristic of Dirac bi-spinors. Though time-symmetric adjunct fields are intrinsic to the direct particle action paradigm, their elimination has been the main focus of previous work in this field in order to conform with Maxwell field theory. By contrast, this work presents the time-symmetric fields as the foundation of Dirac bi-spinors. Even so, accidentally we discover a novel explanation of the emergence of exclusively retarded radiation from the direct action paradigm that makes no appeal to special boundary conditions. % Insert your abstract here. Include keywords, PACS and mathematical % subject classification numbers as needed. \keywords{Quantum Theory \and Dirac’s Equation \and Clifford Algebra \and direct particle interaction \and action-at-a-distance \and time-symmetry \and Majorana spinors \and dark energy \and super-symmetry} \PACS{03.50.-z \and 03.50.De \and 03.50.Kk \and 03.65.-w \and 03.65.pm \and 03.65.Ta \and 11.10.-z \and 11.15.Kc \and 11.30.Pb \and 11.09.+t \and 12.90.+b \and 14.60.Cd \and 4.70.Bh \and 11.30.Pb \and 95.36.+x} \subclass{81P99 \and 81Q65} \end{abstract} \section{Introduction\label{mark-1}} \subsection{Historical Context\label{mark-1.1}} Direct Particle Interaction, henceforth DPI, is a version of electromagnetism distinct from the Maxwell theory that was first proposed by Schwarzschild \cite{ref-0079}, Tetrode \cite{ref-0080}, and Fokker \cite{ref-0081}, in which the EM fields and potentials are not independent dynamical variables, and the only electromagnetic contribution to the action comes from direct interaction between 4-currents. All electromagnetic energy and momentum is to be accounted for in the interaction between charges, so that any EM energy leaving a charge must be destined for absorption by another charge. Accordingly DPI does not admit strictly vacuum degrees of freedom, strictly on-shell photons, or radiation exactly as portrayed by field theory. Since its inception a challenge for DPI has been an explanation for the observational evidence apparently in favor of exclusively retarded radiation.\footnote{An outcome of this work is that the observational facts are compatible with a different interpretation.} Though Wheeler and Feynman \cite{ref-0082},\cite{ref-0083} showed that radiation-like behavior, including radiation reaction, could arise within DPI if the future is sufficiently absorbing, the subsequent discovery of accelerating cosmological expansion rendered their explanation untenable because the universe is nearly transparent on the forward light-cone \cite{ref-0084},\cite{ref-0085} (see also the works by Pegg \cite{ref-0086},\cite{ref-0087}). The books by Hoyle and Narlikar \cite{ref-0088},\cite{ref-0089} and Davies [\cite{ref-0084} are recommended for a comprehensive review of Direct Particle Interaction. \subsection{Relation to other work\label{mark-1.2}} \paragraph{Clifford Formalism} The focus of this work is on exposing the classical foundations of Quantum Theory, employing \linebreak[4] Clifford algebra primarily as an intermediate tool, eventually departing from the Clifford formalism to obtain Dirac bi-spinors that are strictly compliant with the traditional Dirac Theory. We share with Hestenes \cite{ref-0091},\cite{ref-0092} (see also \cite{ref-0093}) that the Dirac equation be founded on real (versus complex) quantities, though there are differences both in how that is implemented here, and in the outcome. As shown by Rodriguez \cite{ref-0094},\cite{ref-0095}, the Clifford object that is the operand of the Dirac-Hestenes equation operator is not a Dirac bi-spinor, and does not share the same Fierz Identities as those of the Dirac bi-spinor that is the focus of this work (see \cite{ref-0096}).\footnote{The Fierz Identities are bi-linear relationships between the different $n\in \left[0,4\right]$ blades in the outer product $\psi \overline{\psi }$ (i.e. between the $\left\langle \psi \overline{\psi }\right\rangle _{n}$) as a consequence of the reduced number of degrees of freedom in a multivector restricted to this form.} \paragraph{Pilot Wave Model} This work has in common with the pilot wave model of de Broglie \cite{ref-0097},\cite{ref-0098} and Bohm \cite{ref-0099}-\cite{ref-0099c} in both the non-relativistic (Schr\"{o}dinger) domain and its relativistic extension (for example \cite{ref-0100},\cite{ref-0101}) that the electron is a classical point charge following flow lines generated by a ‘field’. The book by Holland \cite{ref-0102} is recommended for a thorough exposition of the de Broglie\hyp{}Bohm theory. See \cite{ref-0103}-\cite{ref-0103b} for journal-paper reviews of the Broglie\hyp{}Bohm theory, including its extension to quantum field theory. In common with those extensions, and of relevance to this work, the original - Schr\"{o}dinger domain - model has since been re-cast by Hiley \cite{ref-0104}, \cite{ref-0105}, \cite{ref-0100} in terms of Clifford algebra. Though the psi-field and associated quantum potential of the pilot wave model are sufficient for the task of reproducing standard theory, the de Broglie\hyp{}Bohm model is silent on the origin of the field. Even so, that model is to be credited for its pioneering role in expanding the language employed to ‘explain’ QM to include a classical particle (in addition to the wavefunction) and by providing an example of a successful epistemological alternative to the Bohr / Copenhagen doctrine. \paragraph{Random Walks} There have been efforts to mimic the Schr\"{o}\-dinger and Dirac equations with classical diffusion processes by Nagasawa \cite{ref-0124}, Nelson \cite{ref-0125},\cite{ref-0126}, Ord \cite{ref-0127}-\cite{ref-0127b} and others, e.g. \cite{ref-0128}. Though Nelson in particular seems to have had some success in reproducing quantum behavior from diffusion processes, the rules governing the jump probabilities do not appear to have a strong physical motivation. It is important that all of these have in common that in order to establish a ‘classical’ probability distribution that matches those of QM the diffusion jump probability at \textbf{x} are not Markovian, but depend on the (probabilistic) history of visits to \textbf{x}. \paragraph{Time-Symmetric Presentations of QM} Though Cramer {\cite{ref-0132}-{\cite{ref-0132b} does not attempt to give an explicit electromagnetic foundation for the wavefunction, his ‘Transactional Interpretation’ of QM captures something of role of time\hyp{}symmetric exchanges in this work that are crucial to the emergence of Dirac dynamics from an entirely classical EM \linebreak[4] framework. Cramer’s casting in the non-relativistic domain of the Schr\"{o}dinger wavefunction and its charge conjugate as ‘offer’ and ‘accept’ waves approximately correspond, respectively, with the retarded and advanced components of time\hyp{}symmetric exchanges. The theory of weak\hyp{}value measurements due to Aharo\-nov, Albert, and Vaidman \cite{ref-0133} that grant equal status to the initial and final boundary conditions on the wavefunction has helped draw attention to the time\hyp{}symmetry already pre\-sent in traditional quantum theory, but which derives, according to this work, from the time-symmetry of the EM fields that underlie the wavefunction. Relatedly, Sutherland \cite{ref-0134}-\cite{ref-0134b} makes a case for retro\hyp{}causal influences underpinning QM, granting the final boundary condition employed to explain weak\hyp{}value measurement the same status as the initial boundary condition, with the effect that the wavefunction at all intermediate times depends symmetrically on both in all cases. Price, Wharton, Evans and Miller \cite{ref-0135}-\cite{ref-0135c}, have argued not only that the non-locality intrinsic to QM is suggestive of retro-causal influences, but have suggested (correctly, from the perspective of this work) this be taken as evidence of a direct particle interaction foundation of quantum dynamics. \paragraph{Barut Zanghi Paper} Barut and Zanghi \cite{ref-0136} showed how to reproduce the algebraic structure of the observables of the Dirac Theory with a classical theory of a point charge augmented with spinor degrees of freedom. The goal of that work was not to reproduce the Dirac \textit{equation}, however. Its achievement was in constructing a classical analog that was faithful to the Dirac equation so that ‘canonical quantization’ reproduces the algebra of the observables of the Dirac theory. By contrast, from a particular variant of classical EM theory, this work reproduces not only the Dirac equation \textit{ab initio} (and therefore the algebra of its observables) but also the attendant machinery of eigenvalue selection by observation, neither of which were the aim or focus of the Barut-Zanghi work. \subsection{This work\label{mark-1.3}} \paragraph{Outcome} This work derives the Dirac equation from classical electromagnetic direct particle interaction. The nominally local Dirac wavefunction is given in terms of time\hyp{}symmetric `adjunct' potential and Faraday mediating direct-particle interactions over cosmological scales. The derivation is founded on an ensemble of local \emph{possible} classical (null) currents that are mutually\hyp{}exclusive when there is just one local charge. This construction turns out to give the wavefunction its characteristic `quantum' property of eigenvalue selection / wavefunction collapse in any function\hyp{}space representation. \paragraph{Main ingredients} The direct particle interaction variant of classical EM theory replaces Maxwell field theory. The time\hyp{}symmetric foundation of DPI imposes a self\hyp{}consistency \linebreak[4] condition on the `adjunct fields' (see below for a definition) not otherwise present in the Maxwell theory. This is because the time-reflected response to a local current from distant charges can arrive contemporaneously with the local outgoing field of that current. Very roughly, one arrives at $\partial^{2} A_{out}^\mu\left(x\right) = j_{local}^\mu\left(x\right)$ where $j_{local}^\mu\left(x\right)=\hat{S}\left[A_{in}^\mu\left(x\right)\right]$ and where the contemporaneous component of time-like reflections is contained in $A_{in}^\mu\left(x\right)=\hat{R}\left[A_{out}^\mu\left(x\right)\right]$, where $\hat{S}$ and $\hat{R}$ are local operators. But classical currents are 1D worldlines, precluding a per-current relationship as might be implied by $j_{local}^\mu\left(x\right)$ $=$ $\hat{S}\left[A_{in}^\mu\left(x\right)\right]$. Hence we work instead with an appropriately constructed ensemble of possible currents $\{j_{local}^\mu\left(x\right)\}$ that approximate to a smooth function in $\mathbb{R}^4$. Together these considerations lead to \begin{equation} \label{eq-02} \partial ^{2} \{A_{out}^\mu\left(x\right)\} = \hat{R}\left[\hat{S}\left[\{A_{out}^\mu\left(x\right)\}\right]\right] \end{equation} within the DPI framework. $\hat{R}$ turns out to be a scalar, $\hat{R}\rightarrow R$ say, if the distant charges are uniformly distributed and there is no (retarded) radiation acting on the local current. $\hat{S}$ will be a scalar if the velocity of a charge is proportional to the incoming potential \begin{equation} \label{eq-03} v_{local}^\mu\left(t\right)=\lambda\left(x\right) A_{in}^\mu\left(x\right) \end{equation} where $x$ is evaluated on the worldline of the charge. Eq (\ref{eq-03}) is a particular solution of the relativistic Newton\hyp{}Lorentz equation when the current and incoming potential are both null, and is the foundation of the non-relativistic de Broglie\hyp{}Bohm rule that a charge follows the flow-lines of a `pilot wave'. Accordingly, here the DPI action is modified from that considered elsewhere so that currents are constrained to move at light speed, with the result that $\{j_{local}^\mu\left(x\right)\}$ in the above is ensemble of possible null currents. The incoming potential is not generally null however, which seemingly precludes (\ref{eq-03}) for arbitrary $A_{in}^\mu\left(x\right)$. A trick is to find a decomposition of $A_{in}^\mu\left(x\right)$ into null components so that a charge following the flows lines of one of the null potentials in that decomposition does not feel a force from the Faraday associated with any of the other null potentials in the same decomposition, at the same $x$. If this can be accomplished then $\hat{S}\rightarrow S$ and (\ref{eq-02}) becomes \begin{equation} \label{eq-03.5} \partial ^{2} \{A_{out}^\mu\left(x\right)\} + \kappa^2 \{A_{out}\left(x\right)\} = 0 \end{equation} for some $\kappa$ (the sign in (\ref{eq-03.5}) turns out to follow from (\ref{eq-03})). A decomposition with the required properties is found by expressing the dynamics (\ref{eq-02}) in terms of a Clifford multivector comprising the ensemble potential and Faraday \begin{equation} \label{eq-04} Q = \kappa\gamma_{\mu}A_{out}^\mu + i\gamma_{\mu}\gamma_{\nu} F_{out}^{\mu\nu} \end{equation} where $\kappa$ is a scalar and the ensemble notation has been suppressed. Given the considerations above one can re-write (\ref{eq-02}) as \begin{equation} \label{eq-05} \partial Q + i \kappa Q = 0. \end{equation} Singular-value decomposition of $Q$ represented in ${\mathbb{M}_4}\left( \mathbb{C} \right)$ gives the null components of the potential and associated Faraday such that a charge following the flow lines of any one of the null potentials does not feel a force from the Faraday of any of the other null potentials, with the effect that the null currents are dynamically independent. The eigenvectors of the singular-valued decomposition are the bi-spinor wavefunctions and satisfy the Dirac equation under appropriate conditions. These conditions derive from the requirement that nullity is preserved in whatever function-space $Q$ is expressed - Fourier space versus real space, say. \paragraph{Sequence in this document} Section \ref{sec-DPI} gives the mathematical framework of direct particle interaction. The adjustment required to accommodate light\hyp{}speed motion of classical \linebreak[4] charges is made in Section \ref{sec-0006}. Section \ref{sec-Ensembles} introduces ensembles of currents and associated adjunct fields that are required to simulate smooth fields in ${\mathbb{R}^4}$, and it discusses some consequences of mutually exclusivity of the ensemble members. An outcome of those sections is a system of homogeneous differential-difference equations in the ensemble of null adjunct potentials.\footnote{The difference aspect is missing from (\ref{eq-03.5}) because that equation is valid only in the absence of radiation.} Section \ref{ref-0025} finds two distinct modes in the differential-difference system corresponding to total (net) symmetric and anti-symmetric potentials relative to \linebreak[4] each local charge. The anti-symmetric modes correspond, approximately, to radiation in the framework of field theory. Their emergence in the DPI framework without reference to special boundary conditions and without constraints on cosmological expansion is a novel finding. In this work the analysis otherwise serves only to identify the conditions under which the free space Dirac equation is valid. Section \ref{ref-0038} gives the spectral decomposition of the Dirac multivector, and establishes the relationship between the associated eigenvectors and the bi-spinor solutions of the Dirac Equation. Section \ref{ref-0052} identifies the various Dirac currents that can be constructed from the eigenvector / bi-spinors with a focus on the relationship between the traditional Dirac current and the time\hyp{}symmetric fields from which - according to this work - it is derived. The constraints of nullity and mutual exclusion on ensemble members is discussed in Section \ref{ref-0057}. Though the focus of this work is primarily on the single particle theory there is a very brief discussion of the necessary emergence of delta-correlation between bi-spinors. %================================================================================================== \section{ Direct Particle Interaction\label{sec-DPI}} %================================================================================================== %----------------------------------------------------------------------------------------------------- \subsection{Action} %----------------------------------------------------------------------------------------------------- The electromagnetic direct particle interaction is \begin{equation} \label{eq-1} I_{DPI} =-\int \mathrm{d}^{4}x\int \mathrm{d}^{4}x'\GreensFunction\left(x-x'\right)\mathbbm{j}\left(x\right)\circ \mathbbm{j}\left(x'\right) \end{equation} where \begin{equation} \label{eq-2.5} \GreensFunction\left(x\right) =\delta \left(x^{2}\right)\big/\left({4\pi}\right)\Rightarrow \partial ^{2}\GreensFunction\left(x\right)=\delta ^{4}\left(x\right) \end{equation} and \begin{equation} \label{eq-2} \mathbbm{j}\left(x\right) =\left| e\right| \int \mathrm{d}\lambda \,\mathbbm{v}\left(\lambda \right)\delta ^{4}\left(x-q\left(\lambda \right)\right);\quad \mathbbm{v}\left(\lambda \right)=\mathrm{d}\mathbbm{q}\left(\lambda \right)/\mathrm{d}\lambda. \end{equation} $\mathbbm{j}$ is a Lorentz vector, $q\left(\lambda \right)$ is a Lorentz vector, $\mathbbm{v}$ is a 4-vector and also a Lorentz vector when $\lambda $ is a Lorentz scalar. A double strike font signifies the object is to be considered an element of $\mathrm{C}\mathrm{l}_{1,3}\left(\mathrm{\mathbb{R}}\right)$ rendered in $\mathrm{M}_{4}\left(\mathrm{\mathbb{C}}\right)$, where appropriate. (An exception introduced later is the EM multivector, which is in $\mathrm{C}\mathrm{l}_{1,3}\left(\mathrm{\mathbb{C}}\right)$). $x$ and $q\left(\lambda \right)$ are also Lorentz vectors, but so-written are considered to be represented more conventionally, i.e. in $\mathrm{\mathbb{R}}^{4}$ with Minkowski norm, and $x^{2}=x^{\mu }x_{\mu }$ etc. Where necessary we refer to components in 3+1 D, e.g. $x=\left(t,\mathbf{x}\right)$. Hence, since they all refer to the same object, $\mathbbm{q}\left(\lambda \right)\cong q\left(\lambda \right)\cong \left(q^{0}\left(\lambda \right),\mathbf{q}\left(\lambda \right)\right)$. Due to the structure of \eqref{eq-1} an anti-symmetric component of $\GreensFunction\left(x\right)$ makes no contribution to the action. Consequently DPI effectively mandates a Green’s function that is invariant under negation of any of the coordinates, and is thereby distinguished from traditional theory by its restriction to time\hyp{}symmetric interactions relative to the sources. Let the currents be broken into segments parameterized by laboratory time $t$: $q\left(\lambda \right)\rightarrow \left\{q_{1}\left(t \right),q_{2}\left(t \right),\ldots ,q_{N}\left(t \right)\right\}$ each with constant sign of $\mathrm{d}q_{l}\left(t\right)/\mathrm{d}t$, and where $q_{l}^0\left(t \right)=t$. The current in \eqref{eq-2} is then \begin{equation}\label{eq-3} \mathbbm{j}\left(x\right) =\sum _{l=1}^{N}\mathbbm{j}_{l}\left(x\right);\;\; \mathbbm{j}_{l}\left(x\right) =e_{l}\mathbbm{v}_{l}\left(t\right)\delta ^{3}\left(\mathbf{x}-\mathbf{q}_{l}\left(t\right)\right) \end{equation} where $\mathbbm{v}_{l}\left(t\right) =\left(1,\mathbf{v}_{l}\left(t\right)\right)$ and $\mathbf{v}_{l}\left(t\right)=\mathrm{d}\mathbf{q}_{l}\left(t\right)/\mathrm{d}t$. Using \eqref{eq-3} in \eqref{eq-1} and denying self-action leads to \begin{equation} \label{eq-4} \begin{aligned} I_{DPI} \rightarrow &-\doublesum \int \mathrm{d}^{4}x\int \mathrm{d}^{4}x'\GreensFunction\left(x-x'\right)\mathbbm{j}_{k}\left(x\right)\circ \mathbbm{j}_{l}\left(x'\right)\\ =&-\doublesum\frac{e_{k}e_{l}}{4\pi }\int \mathrm{d}t\int \mathrm{d}t'\delta \left(s_{k,l}^{2}\left(t,t'\right)\right)\mathbbm{v}_{k}\left(t\right)\circ \mathbbm{v}_{l}\left(t'\right) \end{aligned} \end{equation} where $s_{k,l}\left(t,t'\right)=q_{k}\left(t\right)-q_{l}\left(t'\right)$.\footnote{Eq. \eqref{eq-4} is time-reparameterization invariant wherein $t$ plays the role of a ‘speed parameter’ for the space\hyp{}time curve $x=q\left(t\right)$ in $\mathrm{\mathbb{R}}^{4}$. Accordingly the worldlines in \eqref{eq-4} can be parameterized with any monotonic function of $t$. Alternatively the action can be written without any reference to $t$, for example as \begin{equation*} I_{DPI} =-\frac{e^{2}}{4\pi }\doublesum \int \int \mathrm{d}q_{k}\circ \mathrm{d}q_{l}\delta \left(\left(q_{k}-q_{l}\right)^{2}\right). \end{equation*} } %----------------------------------------------------------------------------------------------------- \subsection{Adjunct fields\label{sec-2.2}} %----------------------------------------------------------------------------------------------------- \paragraph{Classical Current} With reference to the second part of \eqref{eq-4} the subsequent introduction of $x$ to denote pre-existing $\mathrm{\mathbb{R}}^{4}$ spacetime is an intermediate computational tool. This applies to the current \eqref{eq-2}, which in DPI therefore has a derivative status relative to $q\left(t\right)$. To be consistent with the adjunct potential as coined by Wheeler and Feynman (see below) the subjects of \eqref{eq-2} and \eqref{eq-3} should be called \emph{adjunct} currents. %..................................................................................................... \paragraph{Adjunct Potential} The adjunct potential \cite{ref-0082},\cite{ref-0083} generated by the \textit{l}$^{\mathrm{th}}$ charge is \begin{equation}\label{eq-5} \begin{aligned} \mathbb{A}_{l}\left(x\right) &=\int \mathrm{d}^{4}x'\GreensFunction\left(x-x'\right)\mathbbm{j}_{l}\left(x'\right)\\ &=\frac{e_{l}}{4\pi }\int \mathrm{d}t'\mathbbm{v}_{l}\left(t'\right)\delta \left(\left(x-q_{l}\left(t'\right)\right)^{2}\right) \end{aligned} \end{equation} a consequence of which is \begin{equation}\label{eq-6} \partial^{2}\mathbb{A}_{l}\left(x\right)=\mathbbm{j}_{l}\left(x\right). \end{equation} The total adjunct potential from \textit{N} charges is \begin{equation}\label{eq-7} \mathbb{A}\left(x\right) =\sum _{l=1}^{N}\mathbb{A}_{l}\left(x\right). \end{equation} We will also need to refer to the potential of all but the \textit{l}$^{\mathrm{th}}$ current \begin{equation}\label{eq-8} \mathbb{A}_{\overline{l}}\left(x\right) =\mathbb{A}\left(x\right)-\mathbb{A}_{l}\left(x\right). \end{equation} The technique of distinguishing between fields according to their origin is due to Leiter \cite{ref-0137}. Note that $x$ in $\mathbb{A}\left(x\right)$ should not be taken to imply a pre-existing $\mathrm{\mathbb{R}}^{4}$ spacetime `canvas'; direct particle interaction grants the adjunct potential a physically meaningful role only on the worldline of a charge. The Lorenz gauge is mandated by the structure of \eqref{eq-5}, in particular because the Green’s function $\GreensFunction\left(x,x'\right)$ $\rightarrow$ \linebreak[4] $\GreensFunction\left(x-x'\right)$ depends only on the coordinate difference $x-x'$:\footnote{$\cancel{\partial }=\upgamma ^{\mu }\partial _{\mu }$ has the usual meaning. $\mathrm{a}\circ \mathrm{b}=\left(\mathrm{ab}+\mathrm{ba}\right)/2$ is the scalar product of two Clifford vectors. Likewise $a\circ b=a^{\mu }b_{\mu }$.} \begin{equation}\label{eq-10} \begin{aligned} \cancel{\partial }\circ \mathbb{A}_{l}\left(x\right) &=\int \mathrm{d}^{4}x'\cancel{\partial }\GreensFunction\left(x-x'\right)\circ \mathbbm{j}_{l}\left(x'\right)\\ &=\int \mathrm{d}^{4}x'\GreensFunction\left(x-x'\right)\cancel{\partial }^{'}\circ \mathbbm{j}_{l}\left(x'\right)\\ &=0. \end{aligned} \end{equation} Clearly \eqref{eq-10} implies $\cancel{\partial }\circ \mathbb{A}\left(x\right) =0$. Applying \eqref{eq-5} and \eqref{eq-8} to \eqref{eq-4} gives \begin{equation} \label{eq-DPI-basic} I_{DPI} =-\sum _{l=1}^{N}\int \mathrm{d}^{4}x\,\mathbbm{j}_{l}\left(x\right)\circ \mathbb{A}_{\overline{l}}\left(x\right) =-\int \mathrm{d}^{4}x\,\mathbbm{j}\left(x\right)\circ \mathbb{A}\left(x\right)-I_{self} \end{equation} where \begin{equation} \label{eq-12} I_{self} =-\sum _{l=1}^{N}\int \mathrm{d}^{4}x\,\mathbbm{j}_{l}\left(x\right)\circ \mathbb{A}_{l}\left(x\right). \end{equation} Note that the $\mathrm{q}_{l}\left(t\right)$ are the only dynamical degrees of freedom - the action is not extremized by variation of the $\mathbb{A}_{l}\left(x\right)$. %..................................................................................................... \paragraph{Properties} The adjunct potential of direct particle interaction differs from a potential of traditional field theory in that the adjunct potential:\begin{enumerate}[i)] \item[i)] Is always sourced. \item[ii)] Necessarily satisfies the Lorenz gauge condition. \item[iii)] Is time\hyp{}symmetric relative to the source. \item[iv)] Is physically consequential only where it originates and where it is terminated.\footnote{The Wheeler and Feynman adjunct potential satisfies i), ii) and iii) only. The termination requirement iv) is understood but not built in to the structure. Their adjunct potential is mathematically indistinguishable from a field-theory potential satisfying the same conditions (i.e. just i), ii) and iii) ) because it is non-zero on all future and past oriented null rays passing through the worldline of the source. On that basis Hoyle and Narlikar have argued (incorrectly from the point of view of this work) that the electromagnetic direct action stress-energy is essentially no different from that of the Maxwell theory.}\footnote{Feynman subsequently changed his position on the role of self-action, and so by implication on the status of the adjunct potential at its source.} \end{enumerate} Consequent to iv) is that the solutions of $\partial ^{2}\mathbb{A}=\mathbb{0}$ are everywhere physically inconsequential. %..................................................................................................... \paragraph{Adjunct Faraday} The adjunct Faraday bi-vector is \footnote{$\mathrm{a}\wedge \mathrm{b}=\left(\mathrm{ab}-\mathrm{ba}\right)/2$ is the anti-symmetric product of two vectors.} \begin{equation} \label{eq-12.5} \mathrm{\mathbb{F}}_{l}=\mathrm{\mathbb{F}}_{l}\left(x\right)=\cancel{\partial }\wedge \mathbb{A}_{l}=\cancel{\partial }\mathbb{A}_{l}-\cancel{\partial }\circ \mathbb{A}_{l}=\cancel{\partial }\mathbb{A}_{l}. \end{equation} We will need also \begin{equation} \label{eq-13} \mathrm{\mathbb{F}}\left(x\right) =\sum _{l=1}^{N}\mathrm{\mathbb{F}}_{l}\left(x\right),\quad \mathrm{\mathbb{F}}_{\overline{l}}\left(x\right) =\mathrm{\mathbb{F}}\left(x\right)-\mathrm{\mathbb{F}}_{l}\left(x\right). \end{equation} Taking into account \eqref{eq-6} (using $\cancel{\partial }^{2}=\partial ^{2}$) the ‘field equations’ appear to be those of the Maxwell electrodynamics in the Lorenz gauge \begin{equation} \label{eq-14} \cancel{\partial }\mathbb{A}_{l}=\mathrm{\mathbb{F}}_{l},\quad \cancel{\partial }\mathrm{\mathbb{F}}_{l}=\mathbbm{j}_{l} \end{equation} though $\mathbb{A}_{l}$ and $\mathbb{F}_{l}$ are under-constrained by \eqref{eq-14} because they admit an unphysical complementary function solution to \linebreak[4]$\partial ^{2}\mathbb{A}_{l}=\mathbb{0}$. %----------------------------------------------------------------------------------------------------- \subsection{Time-Symmetry\label{mark-2.3}} %----------------------------------------------------------------------------------------------------- The DPI action employs a Green’s function that is time\hyp{}symmetric. Accordingly the adjunct potential and Faraday are time\hyp{}symmetric relative to their source. The physical content of DPI however is restricted to the interactions at both ends of a light-like connection. These null ray line segments extend along the forward and backward light cone from a nominally local charge. Their angular distribution and their distribution in time (forwards versus backwards) depend on the distribution of other charges in space and time. Taking into account Cosmological evolution this distribution will not generally be time\hyp{}symmetric - except perhaps at the future cosmological conformal singularity. Further, due to superposition the total incoming response potential might in extreme cases vanish, even though it is the result of any number of other, distant charges.\footnote{This is the foundation of the Wheeler-Feynman absorber theory, wherein the presumed complete future absorption results in complete cancellation of the advanced component of the response.} Broadly then, though DPI is a time\hyp{}symmetric theory, the manifestation of that property depends on the actual distribution of matter. In contrast with earlier attempts to reconcile DPI with observation, in this work we allow for the possibility that the advance component of the DPI adjunct potential is not, in general, canceled at its source by the response of other charges. An outcome is that the universal system of charges can remain tightly coupled by whatever symmetric component remains, post recombination. In Section \ref{ref-0025} the totality of DPI modes are shown to correspond to those of an elastic lattice with optical and acoustic branches. The modes of the optical branch correspond very closely to the vacuum modes of field theory, thereby explaining the emergence of retarded radiation without appeal to a thermodynamic arrow of time. The acoustic modes are subsequently shown to underpin the Dirac wavefunction. Effectively, this work resolves the difficulties attributed to DPI with a re-interpretation of the supposed deficiency as the foundation of QM. %================================================================================================== \section{ Light-speed charge in a given potential\label{sec-0006}} %================================================================================================== %----------------------------------------------------------------------------------------------------- \subsection{ Light-speed motion} %----------------------------------------------------------------------------------------------------- We depart from classical traditional by asserting light\hyp{}speed motion of the electron \begin{equation} \label{eq-16} \mathbbm{v}_{l}^{2}\left(t\right)=0\;\forall\; l\in \left[1,N\right]. \end{equation} Justifications for this assertion are: \begin{enumerate}[i)] % \item[i)] The time\hyp{}symmetric interaction appears to demand that the mass be dynamically determined,\footnote{To be submitted.} which \eqref{eq-16} \linebreak[5] achieves, though not uniquely so.\footnote{A property of the time\hyp{}symmetric interaction is that the incoming adjunct potential response of other nominally distant charges to the light\hyp{}speed motion of the local charge `arrives' contemporaneously with that motion. The causal loop is closed with the requirement that the local electron motion in the presence of the incoming response potential is consistent with the motion that brought about that response. In the work cited here electron mass appears as an eigenvalue of the fields of that exchange.} % \item[ii)] The self-energy of a classical charge is ill-defined by traditional classical theory, which ambiguity can be removed in favor of a (definite) finite energy in that limit - without affecting the predictions of classical theory at subluminal speeds \cite{ref-0139}. % \item[iii)] The eigenvalues of the velocity operator for the Dirac electron are $\pm 1$. % \item[iv)] The Dirac Equation is an outcome of this (classical) analysis. % \end{enumerate} Eq. \eqref{eq-16} can be enforced via a semi-holonomic constraint in an action \begin{equation} \label{eq-17} I_{LS} =-\frac{1}{2}\sum _{l=1}^{N}\int \mathrm{d}t\,\mu _{l}\left(t\right)\mathbbm{v}_{l}^{2}\left(t\right) \end{equation} extremized by variation of $\mu_{l}\left(t\right)$. For $I_{LS}$ to be a Lorentz scalar the $\mu _{l}\left(t\right)$ must transform as $\mathrm{d}t$. Alternatively each path can be parameterized with a monotonic\-ally increasing \linebreak[4] Lorentz scalar, including an appropriately defined frame-independent time. It turns out however that the Euler equations will be such as to grant $\mu _{l}\left(t\right)$ the appropriate transformation property automatically.\footnote{This outcome is an automatic consequence of the relationship \eqref{eq-velocity-potential} established with the potential, which is a true Lorentz vector.} With \eqref{eq-DPI-basic} the full action is \begin{equation} \label{eq-DPI-total} \begin{aligned} I=&I_{LS}+I_{DPI}\\ =&-\sum _{l=1}^{N}\int \mathrm{d}t\,\left(\frac{1}{2}\mu _{l}\left(t\right)\mathbbm{v}_{l}^{2}\left(t\right)+e_{l}\mathbbm{v}_{l}\left(t\right)\circ \mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)\right) \end{aligned} \end{equation} $\mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)$ is the adjunct potential of all but the \textit{l}$^{\mathrm{th}}$ charge evaluated on the path of the \textit{l}$^{\mathrm{th}}$ charge. It is the ‘incoming’ adjunct potential relative to the current with label \textit{l}. The Euler equations are the corresponding Newton\hyp{}Lorentz equations \footnote{$\left\langle \right\rangle _{1}$ extracts the vector part of its Clifford operand, $\left\langle \right\rangle _{2}$ extracts the bi-vector part, etc.}\footnote{The space part of (\ref{eq-19}) is \newline \indent\indent\indent{$\frac{\operatorname{d} }{{\operatorname{d} t}}\bigg( {\mu_l \left( t \right){\mathbf{v}_l}\left( t \right) }\bigg) = e\big( {{\mathbf{E}_{\overline{l}}}\left( {t,{\mathbf{q}_l}\left( t \right)} \right) + {\mathbf{v}_l}\left( t \right) \times {\mathbf{B}_{\overline{l}}}\left( {t,{\mathbf{q}_l}\left( t \right)} \right)} \big)$}.} \begin{equation} \label{eq-19} \frac{\mathrm{d}}{\mathrm{d}t}\left[\mu _{l}\left(t\right)\mathbbm{v}_{l}\left(t\right)\right]=e_{l}\left\langle \mathrm{\mathbb{F}}_{\overline{l}}\left(q_{l}\left(t\right)\right)\mathbbm{v}_{l}\left(t\right)\right\rangle _{1} \end{equation} where, using an over-dot to identify the target of $\cancel{\partial }$, \begin{equation} \label{eq-20} \left\langle \mathrm{\mathbb{F}}_{\overline{l}}\left(q_{l}\left(t\right)\right)\mathbbm{v}_{l}\left(t\right)\right\rangle _{1}=\left[\cancel{\partial }\left[\mathbbm{v}_{l}\left(t\right)\circ \overset{\boldsymbol{\cdot} }{\mathbb{A}}_{\overline{l}}\left(x\right)\right]\right]_{x={q_{l}}\left(t\right)}-\frac{\mathrm{d}\mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)}{\mathrm{d}t} \end{equation} in which terms \eqref{eq-19} can be written \begin{equation} \label{eq-Newton-Lorentz} \frac{\mathrm{d}}{\mathrm{d}t}\left[\mu _{l}\left(t\right)\mathbbm{v}_{l}\left(t\right)+e_{l}\mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)\right] =e_{l}\left[\cancel{\partial }\left[\mathbbm{v}_{l}\left(t\right)\circ \overset{\boldsymbol{\cdot} }{\mathbb{A}}_{\overline{l}}\left(x\right)\right]\right]_{x={q_{l}}\left(t\right)}. \end{equation} The left\textendash{}hand side is the time rate of change of the total (mechanical plus electromagnetic) 4-momentum of the local charge. The electromagnetic part of the momentum is specific to the charge subject to the potential $\mathbb{A}_{\overline{l}}$ at $q_{l}\left(t\right)$. %----------------------------------------------------------------------------------------------------- \subsection{First integral of Newton-Lorentz equation\label{sec-Newton-Lorentz}} %----------------------------------------------------------------------------------------------------- \paragraph{Null Incoming Potential} Suppose initially that the incoming potential is null. Then a particular solution of \eqref{eq-Newton-Lorentz} is \footnote{The space part of (\ref{eq-velocity-potential}) is \newline \indent\indent\indent{$\mu _{l}\left(t\right)\mathbf{v}_{l}\left(t\right)+e_{l}\mathbf{A}_{\overline{l}}\left(t, \mathbf{q}_{l}\left(t\right)\right)=\mathrm{0}$}.}\footnote{This technique can be extended to give the solution of the traditional proper-time form of the Newton-Lorentz equation, i.e. when the charge is not constrained to move at light-speed. Neither solution seems to have been previously recognized.} \begin{equation} \label{eq-velocity-potential} \mu _{l}\left(t\right)\mathbbm{v}_{l}\left(t\right)+e_{l}\mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)=\mathrm{0} \end{equation} and the total momentum is zero. The time-component of \eqref{eq-velocity-potential} gives that \begin{equation} \label{eq-mu-phi} \mu_{l}\left(t\right) =-e_{l}\phi _{\overline{l}}\left(q_{l}\left(t\right)\right) \end{equation} and therefore \begin{equation} \label{eq-flow-line} \mathbbm{v}_{l}\left(t\right) =\mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)/\phi _{\overline{l}}\left(q_{l}\left(t\right)\right)\Rightarrow \mathbf{v}_{l}\left(t\right) =\mathbf{A}_{\overline{l}}\left(q_{l}\left(t\right)\right)/\phi _{\overline{l}}\left(q_{l}\left(t\right)\right). \end{equation} Hence the null current follows the flow lines of an incoming null adjunct potential.\footnote{Since only derivatives of the incoming potential appear in \eqref{eq-Newton-Lorentz} it follows that a more general solution is $\mu _{l}\left(t\right)\mathbbm{v}_{l}\left(t\right)+e_{l}\mathbb{A}_{\overline{l}}\left(q_{l}\left(t\right)\right) =e_{l}\mathrm{u}_{\overline{l}}$ for any constant vector $\mathrm{u}_{\overline{l}}$.} %----------------------------------------------------------------------------------------------------- \paragraph{General Case\label{sec-general-case}} %----------------------------------------------------------------------------------------------------- Any non-null potential can be decomposed into null components. It will turn out to be useful to decompose the incoming Faraday in an analogous way, which in combination will give rise to 4 distinguishably different null potentials. Initially we presume that the charge follows just one of those null potentials, accepting the possibility of subsequent revision to account for the presence of the other potentials. It turns out that however that it is always possible to find a decomposition in which the 4 null paths are independent of each other, provided the potentials are modes of the acoustic branch (see below). Optical branch mode potentials (Section \ref{ref-0026}) require separate treatment however. To implement this strategy let an arbitrary incoming potential be decomposed into \textit{r} null potentials \begin{equation} \label{eq-potential-over-r} \mathbb{A}_{\overline{l}}\left(x\right)=\sum _{n=1}^{r}\mathbb{A}_{\overline{l},n}\left(x\right);\quad \mathbb{A}_{\overline{l},n}^{2}\left(x\right)=0 \end{equation} where for now the number of terms \textit{r} in the decomposition is left undetermined. Each $\mathbb{A}_{\overline{l},n}\left(x\right)$ generates a set of flow-lines, the possible occupancy of each member of which by a charge will initially be considered independently, in accord with the above. Then the solution \eqref{eq-flow-line} can be applied to each of these: \begin{equation} \label{eq-flow-line-over-r} \mathbbm{v}_{l,n}\left(t\right)=\mathbb{A}_{\overline{l},n}\left(q_{l,n}\left(t\right)\right)/\phi _{\overline{l},n}\left(q_{l,n}\left(t\right)\right);\quad \left[\mathbb{A}_{\overline{l},n}\left(q_{l,n}\left(t\right)\right)\right]^{2}=0 \end{equation} Here $q_{l,n}\left(t\right)$ is the worldline of the \textit{l}$^{\mathrm{th}}$ charge following the flow-line of the \textit{n}$^{\mathrm{th}}$ null potential in an \textit{r}-fold decomposition of the potential of all other charges. Provided the $\mathbb{A}_{\overline{l},n}\left(x\right)$ independently satisfy the Lorenz gauge then it is straight-forward to show that \begin{equation} \label{eq-32} \mathrm{d}\phi_{\overline{l},n} \left(q_{l,n}\left(t\right)\right) / {\mathrm{d}t} =\left[\cancel{\partial }\circ \mathbb{A}_{\overline{l},n}\left(x\right)\right]_{x={q_{l,n}}\left(t\right)} = 0. \end{equation} It follows that the paths $q_{l,n}\left(t\right)$ that satisfy \eqref{eq-flow-line-over-r} are the characteristics of $\phi_{\overline{l},n}$, i.e. on which $\phi_{\overline{l},n}$ is constant. In particular \begin{equation} \label{eq-32b} \phi_{\overline{l},n}\left(q_{l,n}\left(t\right)\right)=\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right). \end{equation} %----------------------------------------------------------------------------------------------------- \paragraph{Connection with de Broglie-Bohm Model} %----------------------------------------------------------------------------------------------------- Eq. \eqref{eq-flow-line-over-r} with \eqref{eq-32b} is the classical electromagnetic foundation of the de Broglie\hyp{}Bohm pilot-wave mechanism, consistent with which is the absence of a role in the dynamics for magnitude of the 4-potential. When however the description is subsequently extended to cover an ensemble of charges the time-component of the null potential (which in this context can be equated with the ‘magnitude’) will be ‘re-purposed’ to carry information about the occupation probabilities of the flow lines. %----------------------------------------------------------------------------------------------------- \subsection{Signs of mass and charge} %----------------------------------------------------------------------------------------------------- The stipulation that the time component of $v_{l}\left(t\right)$ is equal to 1 forces the parameterization of all particles to be in the same direction along the time dimension. Informally this means that all charges proceed forwards in time, regardless of the sign of the charge. To align with convention we also arrange for the sign of the dynamic mass to be positive. Taking into account \eqref{eq-32b}, the time component of the $n^{th}$ potential in an $r$-fold decomposition of \eqref{eq-velocity-potential} is \begin{equation} \label{eq-28} \mu_{l,n}\left(t\right) = \mu_{l,n}\left(0\right)=-e_{l,n}\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right). \end{equation} A positive constant mass therefore requires \begin{equation} \label{eq-29} \mu_{l,n}\left(0\right)=\left| e\right| \left| \phi _{\overline{l},n}\left(q_{l,n}\left(0\right)\right)\right| \Rightarrow e_{l,n} =-\left| e\right| \mathrm{sgn}\left(\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)\right). \end{equation} The sign of the charge is the negative of the sign of \linebreak[4] $\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)$, which implies a restriction of elec\-tron / posi\-tron flow-lines to the positive / negative 'half\hyp{}cycles' of a null time-varying potential and a corresponding restriction on the current vector (Section \ref{mark-7.4}).\footnote{See however the discussion in Section \ref{mark-7.1}, and also the remarks in footnote \ref{footnote_33}.} %================================================================================================== \section{Ensembles\label{sec-Ensembles}} %================================================================================================== %----------------------------------------------------------------------------------------------------- \subsection{Sum over mutually exclusive possibilities\label{ref-0021}} %----------------------------------------------------------------------------------------------------- Eq. \eqref{eq-flow-line-over-r} with \eqref{eq-32b} is the first order differential equation \footnote{Here we revert to a component representation of the Lorentz vectors to avoid discussion of functions of Clifford vectors necessitated by writing $\mathrm{d}\mathbbm{q}_{l,n}\left(t\right)/\mathrm{d}t=\mathbb{A}_{\overline{l},n}\left(\mathbbm{q}_{l,n}\left(t\right)\right)/\phi _{\overline{l},n}\left(\mathbbm{q}_{l,n}\left(t\right)\right)$.} \begin{equation} \label{eq-38} \frac{\mathrm{d}q_{l,n}\left(t\right)}{\mathrm{d}t}=\frac{1}{\phi _{\overline{l},n}\left(q_{l,n}\left(0\right)\right)}A_{\overline{l},n}\left(q_{l,n}\left(t\right)\right);\quad \left[\mathbb{A}_{\overline{l},n}\left(q_{l,n}\left(t\right)\right)\right]^{2}=0. \end{equation} If $A_{\overline{l},n}\left(x\right)$ is given then in principle $q_{l,n}\left(t\right)$ can be found by solving \eqref{eq-38}. Taking into account \eqref{eq-29} the 4-current $\mathbbm{j}_{l}\left(x\right)$ in \eqref{eq-3} must be distinguished accordingly as one of $\mathbbm{j}_{l,n}\left(x\right)$ for $n\in \left[1,r\right]$ \begin{equation} \label{eq-36} \begin{aligned} \mathbbm{j}_{l,n}\left(x\right) &=-\frac{\left|e\right|\delta^{3}\left(\mathbf{x}-\mathbf{q}_{l,n}\left(t\right)\right)}{\left|\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)\right|}\mathbb{A}_{\overline{l},n}\left(q_{l,n}\left(t\right)\right)\\ &=-\frac{\left|e\right|\delta^{3}\left(\mathbf{x}-\mathbf{q}_{l,n}\left(t\right)\right)}{\left|\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)\right|}\mathbb{A}_{\overline{l},n}\left(x\right) \end{aligned} \end{equation} where $q_{l,n}\left(t\right)$ is a solution of \eqref{eq-38}. Let us write \begin{equation} \label{eq-40} \mathbf{q}_{l,n}\left(t\right)=\dq_{l,n}\left(t\right)+\mathbf{q}_{l,n}\left(0\right) \end{equation} where $\dq_{l,n}\left(0\right)=\mathbf{0}$, ie $\mathbf{q}_{l,n}\left(0\right)$ is the particular solution of \eqref{eq-38} that passes through the origin at $t=0$. We now form a statis\-tically-weighted ensemble, summing the currents \eqref{eq-36} over the initial conditions. Let $p_{l,n}\left(\mathbf{q}_{l,n}\left(0\right)\right)$ be the weight of the \textit{n}$^{\mathrm{th}}$ null current passing through $\mathbf{x}=\mathbf{q}_{l,n}\left(0\right)$ at $t=0$. Then \begin{equation} \label{eq-42} \left\{\mathbbm{j}_{l,n}\left(x\right)\right\} =\int \mathrm{d}^{3}q_{l,n}\left(0\right)p_{l,n}\left(\mathbf{q}_{l,n}\left(0\right)\right)\mathbbm{j}_{l,n}\left(x\right) \end{equation} is an ensemble current, the members of which are mutually exclusive when there is just one local charge. Consider the particular weights \begin{equation} \label{eq-43} p_{l,n}\left(\mathbf{q}_{l,n}\left(0\right)\right)= \left|\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)\right|\left.\beta_{l,n}^2\middle/\left|e\right|\right. = \left|\phi_{\overline{l},n}\left(0,\mathbf{q}_{l,n}\left(0\right)\right)\right| \left.\beta_{l,n}^2\middle/\left|e\right|\right. \end{equation} where $\beta_{l,n}$ is a constant with dimensions $L^{-1}$. Substitution of \eqref{eq-43} and \eqref{eq-36} into \eqref{eq-42} and using \eqref{eq-40} gives \begin{equation} \label{eq-47} \begin{aligned} \left\{\mathbbm{j}_{l,n}\left(x\right)\right\} &=-\beta_{l,n}^2 \int \mathrm{d}^{3}q_{l,n}\left(0\right) \delta^{3}\left(\mathbf{x}-\mathbf{q}_{l,n}\left(t\right)\right)\mathbb{A}_{\overline{l},n}\left(x\right)\\ &=-\beta_{l,n}^2 \mathbb{A}_{\overline{l},n}\left(x\right). \end{aligned} \end{equation} The ensemble current $\left\{\mathbbm{j}_{l,n}\left(x\right)\right\}$ is conserved iff each of $\mathbb{A}_{\overline{l},n}\left(x\right)$ satisfy the Lorenz gauge, and vice-versa. Despite appearances, Eq. \eqref{eq-43} is not a restriction on the weights because $\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)$ is not \emph{given}. A consequence of \eqref{eq-47} is that $\phi_{\overline{l},n}\left(x\right)$ (now) satisfies a homogeneous coupled differential equation (see Section \ref{ref-0025}), for which $\phi_{\overline{l},n}\left(q_{l,n}\left(0\right)\right)$ can be cast as a boundary condition - with no constraint on its functional form. %----------------------------------------------------------------------------------------------------- \subsection{Coupling strength\label{ref-0021.5}} %----------------------------------------------------------------------------------------------------- We will find that $\beta$ sets the strength of the coupling of the local current to the time-symmetric response potential in the absence of radiation. And we will find that $n$ has 4 possible values corresponding to the sign of the charge and its two possible polarizations (or spin - depending on how one chooses to allocate the quantum numbers to a pair of Fermions). Coupling that depends on the sign of the charge or on spin orientation is not discounted, but is ignored here on the grounds that it can be presumed to arise only when the distribution of distant charges is somehow exceptional. The dependence of $\beta$ on the index $l$ connotes, effectively, a dependency of the coupling constant on the location of the local charge. Hence, to a good approximation one expects to be able to write $\beta_{l,n} \rightarrow \beta\left(x\right)$ where $x$ dependency is generally very slow compared to that of the incoming potential, and therefore the local current. Its appearance in the Dirac Equation in that form is equivalent to propagation in conformal spacetime \cite{ref-0140}. The possible connection with GR is ignored here however; we will take $\beta$ to be constant with the understanding that further analysis is restricted to Minkowski spacetime. With the above understood we re-write (\ref{eq-47}) as \begin{equation} \label{eq-47.} \left\{\mathbbm{j}_{l,n}\left(x\right)\right\} =-\beta^2 \mathbb{A}_{\overline{l},n}\left(x\right). \end{equation} and form an $r$-fold ensemble of the null ensemble currents: \begin{equation} \label{eq-48} \left\{\mathbbm{j}_{l}\left(x\right)\right\}=\sum _{n=1}^{r}\left\{\mathbbm{j}_{l,n}\left(x\right)\right\} =-\beta^{2}\mathbb{A}_{\overline{l}}\left(x\right) \end{equation} where we used \eqref{eq-potential-over-r}. Introducing the \textit{ensemble} adjunct potential \begin{equation} \label{eq-51} \left\{\mathbb{A}_{l}\left(x\right)\right\}=\int \mathrm{d}^{4}x'\GreensFunction\left(x-x'\right)\left\{\mathbbm{j}_{l}\left(x'\right)\right\} \end{equation} the ensemble version of \eqref{eq-14} is \begin{equation} \label{eq-52} \cancel{\partial }\left\{\mathbb{A}_{l}\left(x\right)\right\}=\left\{\mathrm{\mathbb{F}}_{l}\left(x\right)\right\},\quad \cancel{\partial }\left\{\mathrm{\mathbb{F}}_{l}\left(x\right)\right\}=-\beta^{2}\mathbb{A}_{\overline{l}}\left(x\right). \end{equation} These ensembles simulate smooth fields satisfying differential equations. They hide the constraint that the adjunct potential is physically consequential only at the point of contact with the charge, and the non-linear constraint that the generators of flow lines for the current are null. %----------------------------------------------------------------------------------------------------- \subsection{ Post hoc enforcement of mutual exclusion\label{sec-mutex}} %----------------------------------------------------------------------------------------------------- Nothing in the above enforces mutual exclusion; the \linebreak[4] $p_{l,n}\left(\mathbf{q}_{l,n}\left(0\right)\right)$ above appear to be independent. By contrast, if it is known that there is just one particle then mutual exclusivity of flow-line occupancy requires \footnote{If \textit{a} and \textit{b} are discrete and mutually exclusive then $p\left(a|b\right)=\delta _{a,b}$, and Bayes Theorem $p\left(a,b\right)=p\left(a|b\right)p\left(b\right)$ becomes $p\left(a,b\right)=\delta _{a,b}p\left(b\right)=\delta _{a,b}p\left(a\right)$.} \begin{equation} \label{eq-53} p_{l,n}\left(\mathbf{q}_{l,n}\left(0\right),\mathbf{q}'_{l,n}\left(0\right)\right)=\delta ^{3}\left(\mathbf{q}_{l,n}\left(0\right)-\mathbf{q}'_{l,n}\left(0\right)\right)p_{l,n}\left(\mathbf{q}_{l,n}\left(0\right)\right) \end{equation} and suitably extended to cover higher orders of correlation. An implementation, viable at least in a single particle theory, is to compute the dynamics at first ignoring mutual exclusion, treating \eqref{eq-42} as an ordinary integral and \eqref{eq-48} as an ordinary sum - i.e. both in the sense of a superposition - and enforce mutual exclusion only on \emph{products} of mutually exclusive possibilities. For example, squaring $\left\{\mathbbm{j}_{l}\left(x\right)\right\}$ in \eqref{eq-48}, and supposing for simplicity that $n\in \left\{1,2\right\}$, one has \begin{equation} \label{eq-54} \left\{\mathbbm{j}_{l}\left(x\right)\right\}^{2}=\left\{\mathbbm{j}_{l,1}\left(x\right)\right\}^{2}+\left\{\mathbbm{j}_{l,2}\left(x\right)\right\}^{2}+2\left\{\mathbbm{j}_{l,1}\left(x\right)\right\}\circ \left\{\mathbbm{j}_{l,2}\left(x\right)\right\} \end{equation} The first two terms on the right are null, the third term vanishes because it is the product of two-mutually exclusive possibilities, and therefore $\left\{\mathbbm{j}_{l}\left(x\right)\right\}$ is effectively null. This property extends to the $\left\{\mathbbm{j}_{l,n}\left(x\right)\right\}$ given by \eqref{eq-47} and \eqref{eq-48}: squaring \eqref{eq-48}, one has \begin{equation} \label{eq-55} \left\{\mathbbm{j}_{l,n}\left(x\right)\right\}^{2}=\int \mathrm{d}^{3}a\int \mathrm{d}^{3}bp_{l,n}\left(\mathbf{a}\right)p_{l,n}\left(\mathbf{b}\right)\mathbbm{j}_{l,n}\left(x;\mathbf{a}\right)\circ \mathbbm{j}_{l,n}\left(x;\mathbf{b}\right) \end{equation} To remove the mutually-exclusive terms one can make the replacement \begin{equation} \label{eq-56} p_{l,n}\left(\mathbf{a}\right)p_{l,n}\left(\mathbf{b}\right)\rightarrow p_{l,n}\left(\mathbf{a},\mathbf{b}\right)=p_{l,n}\left(\mathbf{a}\right)\delta ^{3}\left(\mathbf{a}-\mathbf{b}\right) \end{equation} whereupon \eqref{eq-55} becomes \begin{equation} \label{eq-57} \left\{\mathbbm{j}_{l,n}\left(x\right)\right\}^{2}=\int \mathrm{d}^{3}ap_{l,n}\left(\mathbf{a}\right)\left[\mathbbm{j}_{l,n}\left(x;\mathbf{a}\right)\right]^{2}=0. \end{equation} as required. The $\mathrm{SU}\left(2\right)$ representation of a null vector can be `factorized' as an outer-product of Weyl spinors. A null Faraday, which turns out also to play a prominent role in the dynamics, can be similarly factorized. Subsequently we show that nullity is automatically preserved when the dynamics is expressed in terms of Weyl spinors rather than Lorentz vectors and bi-vectors. Further, and of relevance to the above, mutual exclusion can be then enforced by requiring that products of Weyl spinors that are factors of mutually exclusive null currents do not contribute to expectation of observables. This issue is briefly revisited in Section \ref{ref-0057}. The suggestive connection with the anti-commutators of QFT is not discussed in this document however, which is primarily focused on the single particle theory. %----------------------------------------------------------------------------------------------------- \subsection{Back reaction} %----------------------------------------------------------------------------------------------------- $ \mathbb{A}_{\overline{l}}\left(x\right)$ is the total incoming potential, including the distant response to the particular (`actual') motion of the local \linebreak[4] charge associated with just one member of the ensemble $\left\{\mathbbm{j}_{l,n}\left(x\right)\right\}$, i.e. for some particular $n \in [1,r]$, and non-zero only at some singular $\mathbf{x}$ at $t=0$. Yet the ensemble average in Section \ref{ref-0021} was performed assuming the incoming potential was \emph{given}, and therefore independent of which of the ensemble members was `actual'. (`Actuality' here corresponds to pilot wave flow-line occupancy in the de Broglie\hyp{}Bohm model.) A contradiction is avoided because the incoming potential is physically consequential only on the worldline of a charge - there is no requirement that the potential elsewhere have any particular value. From this perspective $ \mathbb{A}_{\overline{l},n}\left(x\right)$ is already a composite object of mutually exclusive possibilities. From the field theory perspective it is assembled from flowlines each taken from a different Maxwell-type potential, each of \emph{these} being the field-theoretical response to a single local current. For all practical purposes therefore in Eq. (\ref{eq-52}) we can make the replacement \begin{equation} \label{eq-58} \mathbb{A}_{\overline{l},n}\left(x\right)\rightarrow\left\{\mathbb{A}_{\overline{l},n}\left(x\right)\right\}\Rightarrow \mathbb{A}_{\overline{l}}\left(x\right)\rightarrow\left\{\mathbb{A}_{\overline{l}}\left(x\right)\right\}. \end{equation} where $\left\{\mathbb{A}_{\overline{l}}\left(x\right)\right\}$ is unaffected by which of the local flow-lines is actually occupied. Using \eqref{eq-58} in \eqref{eq-52} gives \begin{equation} \label{eq-59} \cancel{\partial }\left\{\mathbb{A}_{l}\left(x\right)\right\}=\left\{\mathrm{\mathbb{F}}_{l}\left(x\right)\right\},\quad \cancel{\partial }\left\{\mathrm{\mathbb{F}}_{l}\left(x\right)\right\}=-\beta^{2}\left\{\mathbb{A}_{\overline{l}}\left(x\right)\right\} \end{equation} Note that the relationship \eqref{eq-59} is exclusively between ensembles. There is no corresponding direct relationship between particular members $\mathbbm{j}_{l}\left(x\right)$ and $\mathbb{A}_{\overline{l}}\left(x\right)$ of their respective ensembles.\footnote{Each member current is delta-valued on the worldline of the charge, whereas every incoming potential - every member of $\left\{\mathbb{A}_{\overline{l}}\left(x\right)\right\}$ - is a smooth function of co-dimension 1 in $\mathrm{\mathbb{R}}^{4}$ on the double light-cone of its source.} %----------------------------------------------------------------------------------------------------- \section{ Normal Modes\label{ref-0025}} %----------------------------------------------------------------------------------------------------- \subsection{ Acoustic and optical branches\label{ref-0026}} %----------------------------------------------------------------------------------------------------- Eqs. \eqref{eq-59} are equivalent to \begin{equation} \label{eq-61} \partial ^{2}\left\{\mathbb{A}_{l}\left(x\right)\right\}=-\beta^{2}\left\{\mathbb{A}_{\overline{l}}\left(x\right)\right\} \end{equation} subject to the constraints \begin{equation} \label{eq-62} \cancel{\partial }\circ \left\{\mathbb{A}_l\left(x\right)\right\}=\cancel{\partial }\circ \left\{\mathbb{A}_{\overline{l}}\left(x\right)\right\}=0. \end{equation} Eq. \eqref{eq-61} is a differential difference equation. The same information can be represented in a pair of homogeneous differential equations, which can be constructed with the help of an equation ‘adjoint’ to \eqref{eq-61}. Suppressing arguments \begin{equation} \label{eq-63} \begin{aligned} \partial ^{2}\left\{\mathbb{A}_{\overline{l}}\right\} &=-\beta ^{2}\sum_{k=1;\,k \neq l}^N\left\{\mathbb{A}_{\overline{k}}\right\}\\ &=-\beta ^{2}\sum_{k=1;\,k \neq l}^N\left[\left\{\mathbb{A}\right\}-\left\{\mathbb{A}_{k}\right\}\right]\\ &=-\beta ^{2}\left[\left(N-1\right)\left\{\mathbb{A}\right\}-\left\{\mathbb{A}_{\overline{l}}\right\}\right]. \end{aligned} \end{equation} Here $\left\{\mathbb{A}\right\}$ is the total ensemble potential \begin{equation} \label{eq-64} \left\{\mathbb{A}\right\} =\sum _{l=1}^{N}\left\{\mathbb{A}_{l}\right\}=\left\{\mathbb{A}_{l}\right\}+\left\{\mathbb{A}_{\overline{l}}\right\} \end{equation} using which \eqref{eq-63} can be written just in terms of $\left\{\mathbb{A}_{l}\right\}$ and $\left\{\mathbb{A}_{\overline{l}}\right\}$ \begin{equation} \label{eq-65} \partial ^{2}\left\{\mathbb{A}_{\overline{l}}\right\}=-\beta^{2}\left[\left(N-1\right)\left\{\mathbb{A}_{l}\right\}+\left(N-2\right)\left\{\mathbb{A}_{\overline{l}}\right\}\right]. \end{equation} Eqs. \eqref{eq-61} and \eqref{eq-65} form the coupled system \begin{equation} \label{eq-66} \begin{array}{ll} \left[\begin{array}{ll} \partial ^{2},\,\, & \beta^{2}\\ \left(N-1\right)\beta^{2},\,\, & \partial ^{2}+\left(N-2\right)\beta^{2} \end{array}\right] & \left[\begin{array}{l} \left\{\mathbb{A}_{l}\right\}\\ \left\{\mathbb{A}_{\overline{l}}\right\} \end{array}\right] \end{array}=0. \end{equation} Adding the two rows gives an equation for the total adjunct potential \begin{equation} \label{eq-67} \left[\partial ^{2}+\kappa^{2}\right]\left\{\mathbb{A}\right\}=\mathrm{0} \end{equation} where $\kappa=\beta\sqrt{N-1}$. Subtracting the second row from $N-1$ times the first row gives \begin{equation} \label{eq-69} \left[\partial^{2}-\beta^{2}\right]\left\{\overset{\sim}{\mathbb{A}}_{l}\right\}=0 \end{equation} where \begin{equation} \label{eq-70} \left\{\overset{\sim}{\mathbb{A}}_{l}\right\}=\left\{\mathbb{A}_{l}\right\}-\left\{\mathbb{A}_{\overline{l}}\right\}/\left(N-1\right) \end{equation} is an anti-symmetric combination of the potential of the local charge and the potential of all other distant charges, as it acts on the local charge. The relative weights are such that the potentials of distant charges contribute coherently. The anti-symmetry is suggestive of an analogy with the optical modes of an elastic lattice, whereas $\left\{\mathbb{A}\right\}$ represents the symmetric modes of a coupled \textit{N}-particle system, analogous to the acoustic modes of an elastic lattice. Eq. \eqref{eq-67} is a Klein-Gordon equation for the total adjunct ensemble potential $\left\{\mathbb{A}\right\}$ with mass-frequency $\kappa $. Given $N\sim 10^{80}$ say, this is of order $10^{40}$ times the magnitude of $\beta $ in \eqref{eq-69}. If $\kappa $ corresponds to a known elementary particle then $\beta$ must be tiny. If for example $\kappa $ is the Compton frequency of the electron with wavelength $2.4\times 10^{-12}$m, then the wavelength associated with $\beta$ is of order of the present Hubble radius, and the frequency has a period of order of the Cosmological age. At frequencies much greater than this $\left\{\overset{\sim}{\mathbb{A}}_{l}\right\}$ behaves like a free (vacuum) potential: \footnote{Note that, however small, $\beta $ forces agreement between field theory and direct particle interaction on the necessity that the (free) potential satisfies the Lorenz gauge. }\footnote{Due to the sign of $\beta ^{2}$ in \eqref{eq-69} the radiation modes are exponentially unstable over the Hubble time-frame, suggestive of a DPI basis for dark energy.} \begin{equation} \label{eq-71} \partial ^{2}\left\{\overset{\sim}{\mathbb{A}}_{l}\right\} \approx 0. \end{equation} Eq. \eqref{eq-71} is a novel demonstration of the existence of endogenous quasi-vacuum modes in a DPI theory, without recourse to special boundary conditions presumed in earlier work. Examination of the connection with retarded EM radiation is outside the scope of this report, which is focused on the origin of the Dirac equation. %----------------------------------------------------------------------------------------------------- \subsection{ Acoustic branch with no radiation\label{ref-0031}} %----------------------------------------------------------------------------------------------------- If it is known that no radiation is present, i.e. $\left\{\overset{\sim}{\mathbb{A}}_{l}\right\}=0$, then \eqref{eq-70} gives that the incoming and locally-generated adjunct ensemble potentials are proportional, \begin{equation} \label{eq-72} \left\{\mathbb{A}_{l}\right\}=\left\{\mathbb{A}_{\overline{l}}\right\}/\left(N-1\right) \end{equation} in which case the total potential is \begin{equation} \label{eq-73} \left\{\mathbb{A}\right\} =\left\{\mathbb{A}_{l}\right\}+\left\{\mathbb{A}_{\overline{l}}\right\}=N\left\{\mathbb{A}_{l}\right\} \end{equation} and the local potential $\left\{\mathbb{A}_{l}\right\}$ satisfies the Klein-Gordon equation \eqref{eq-67}. It follows from \eqref{eq-72} that under these conditions (of no radiation), the local current is proportional to its own potential as \begin{equation} \label{eq-74x} \left\{\mathbbm{j}_{l}\right\}=-\kappa^{2}\left\{\mathbb{A}_{l}\right\} \end{equation} and Eq. \eqref{eq-61} now reads \begin{equation} \label{eq-75} \left[\partial ^{2}+\kappa^{2}\right]\left\{\mathbb{A}_{l}\right\}=0 \end{equation} with the subsidiary condition \begin{equation} \label{eq-76} \cancel{\partial }\circ \left\{\mathbb{A}_{l}\right\}=0. \end{equation} %----------------------------------------------------------------------------------------------------- \subsection{ EM multivector} %----------------------------------------------------------------------------------------------------- The Lorenz gauge constraint can be incorporated into the dynamics via the ensemble multivector \begin{equation} \label{eq-Q} \left\{\mathrm{\mathbb{Q}}_{l}\right\}=\kappa\left\{\mathbb{A}_{l}\right\}+i\left\{\mathrm{\mathbb{F}}_{l}\right\} \end{equation} (where $\left\{\mathrm{\mathbb{F}}_{l}\right\}=\cancel{\partial }\left\{\mathbb{A}_{l}\right\}$), in which terms \eqref{eq-75} and \eqref{eq-76} can be combined into the `multi-vector Dirac equation' \begin{equation} \label{eq-dQ} \left[\cancel{\partial }+i\kappa\right]\left\{\mathrm{\mathbb{Q}}_{l}\right\}=\mathrm{0}. \end{equation} Hereafter we refer to any linear combination of the potential and Faraday as an ‘EM multivector’ (to distinguish it from any other multivector containing other non-zero blades). We note in passing that in the Majorana representation \eqref{eq-dQ} can be expressed entirely in terms of real quantities. Suppressing the particle label and re-writing as \eqref{eq-dQ} \begin{equation*} \left[\left[\cancel{\partial }+i\kappa \right]/i\right]\left[\left\{\mathrm{\mathbb{Q}}\right\}/i\right]=\mathrm{0} \end{equation*} then \begin{equation} \label{eq-81} \left[\cancel{\partial }+i\kappa \right]/i=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \frac{\partial }{\partial x}+\kappa & -\frac{\partial }{\partial z} & 0 & \frac{\partial }{\partial y}-\frac{\partial }{\partial t}\\ -\frac{\partial }{\partial z} & -\frac{\partial }{\partial x}+\kappa & -\frac{\partial }{\partial y}+\frac{\partial }{\partial t} & 0\\ 0 & -\frac{\partial }{\partial y}-\frac{\partial }{\partial t} & \frac{\partial }{\partial x}+\kappa & -\frac{\partial }{\partial z}\\ \frac{\partial }{\partial y}+\frac{\partial }{\partial t} & 0 & -\frac{\partial }{\partial z} & -\frac{\partial }{\partial x}+\kappa \end{array}\right] \end{equation} and, expressed in terms of contra-variant components, \begin{equation} \label{eq-82} \begin{aligned} &\left\{\mathrm{\mathbb{Q}}\right\}/i=\\ &\resizebox{\hsize}{!}{$\left\{\left[\begin{array}{c@{\,}c@{\,}c@{\,}c} \kappa \,A_{x}-E_{y} & -\kappa \,A_{z}+B_{y} & E_{z}-B_{x} & -\kappa \,\phi +\kappa \,A_{y}+B_{z}+E_{x}\\ -\kappa \,A_{z}-B_{y} & -\kappa \,A_{x}-E_{y} & \kappa \,\phi -\kappa \,A_{y}+B_{z}+E_{x} & -E_{z}+B_{x}\\ E_{z}+B_{x} & -\kappa \,\phi -\kappa \,A_{y}-B_{z}+E_{x} & \kappa \,A_{x}+E_{y} & -\kappa \,A_{z}+B_{y}\\ \kappa \,\phi +\kappa \,A_{y}-B_{z}+E_{x} & -E_{z}-B_{x} & -\kappa \,A_{z}-B_{y} & -\kappa \,A_{x}+E_{y} \end{array}\right]\right\}.$} \end{aligned} \end{equation} It is established in Section \ref{ref-0038} that $\mathrm{\mathbb{Q}}$ transforms like the outer product of a Dirac bi-spinor with its adjoint. By contrast, each of the individual columns in $\mathrm{\mathbb{Q}}$ in \eqref{eq-82} do not transform like a Dirac bi-spinor, even though in any frame each of those columns satisfies a Dirac-like equation. %----------------------------------------------------------------------------------------------------- \section{ Dirac Equation\label{ref-0038}} %----------------------------------------------------------------------------------------------------- \subsection{Multivector projections\label{mark-6.1}} %----------------------------------------------------------------------------------------------------- Eq. \eqref{eq-dQ} is a coupled first order system in the components of the potential and Faraday. The focus of this work is on the ensemble current, which can found from solutions of \eqref{eq-dQ} provided the conditions described in Section \ref{sec-Newton-Lorentz} are met. If so then each flow line of each of the null components of that current is a possible - mutually exclusive - path for the local electron. One could form a Dirac equation of sorts by right multiplication of \eqref{eq-dQ} with a constant 4-vector to project onto $\mathrm{\mathbb{C}}^{4}$. But $\left\{\mathrm{\mathbb{Q}}_{l}\left(x\right)\right\}$ times a constant 4-vector does not transform as a bi-spinor (see below). By contrast a Lorentz invariant 4-vector (bi-spinor) description of the dynamics can be obtained from a projection of the phase-space representation of the Clifford Multivector, because in that representation there is no constraint that the projection 4-vector be constant. The dimensionality of $\left\{\mathrm{\mathbb{Q}}_{l}\left(x\right)\right\}$ mandates there are 4 such independent projections that generate 4 Dirac equations, each associated with a different conserved current. %----------------------------------------------------------------------------------------------------- \subsection{ Multivector eigenvectors (Dirac bi-spinors)} %----------------------------------------------------------------------------------------------------- \paragraph{Phase-space representation} %----------------------------------------------------------------------------------------------------- We suppress the particle index \textit{l}, and distinguish between real-space, phase-space, and \linebreak[4] Fourier domain functions by their arguments. Using the trans\-form convention \begin{equation} \label{eq-89} f\left(k\right)=\int \mathrm{d}^{4}xe^{ik\circ x}f\left(x\right)\Rightarrow f\left(x\right)=\left(2\pi \right)^{-4}\int \mathrm{d}^{4}ke^{-ik\circ x}f\left(k\right) \end{equation} let \begin{equation} \label{eq-90} f\left(x,k\right)=e^{-ik\circ x}f\left(k\right) \end{equation} for any function $f\left(k\right)$ so that \begin{equation} \label{eq-91} f\left(x\right)=\left(2\pi \right)^{-4}\int \mathrm{d}^{4}kf\left(x,k\right). \end{equation} In these terms the multivector \eqref{eq-Q} is \begin{equation} \label{eq-92} \left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=\kappa \left\{\mathbb{A}\left(x,k\right)\right\}+i\left\{\mathrm{\mathbb{F}}\left(x,k\right)\right\}=\left[\kappa +\mathbbm{k}\right]\left\{\mathbb{A}\left(x,k\right)\right\} \end{equation} and \eqref{eq-dQ} can be written as either of \footnote{$\mathbb{k}$ is $\cancel{k}$ of the traditional Feynman slash notation. } \begin{equation} \label{eq-93} \left[\cancel{\partial }+i\kappa \right]\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=i\left[\kappa -\mathbbm{k}\right]\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=0. \end{equation} The second of \eqref{eq-93} can be written \begin{equation} \label{eq-94} \mathrm{\mathbb{P}}_{-}\left(k\right)\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=0 \end{equation} where $\mathrm{\mathbb{P}}_{\sigma }\left(k\right)=\left(\kappa +\sigma \mathrm{k}\right)/2\kappa $, $\sigma =\pm 1$, are a complementary pair of projection matrixes each with rank 2. Consequently, $\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}$ has rank 2, and can be decomposed, therefore, as the sum of two outer-products of 4-component vectors in $\mathrm{\mathbb{C}}^{4}$, though the form of that decomposition is constrained by conditions on $\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}$ due to the reality of the underlying potential and Faraday, and \textendash{} relatedly - the symmetries of their matrix representations. Substitution of \eqref{eq-92} into \eqref{eq-93} gives the Klein-Gordon type condition $k^{2}=\kappa ^{2}$. The two roots can be accommodated by reduction of the dimensionality of the k-space integrations, replacing \eqref{eq-91} with \begin{equation} \label{eq-95} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\left(2\pi \right)^{-3}\int \mathrm{d}^{3}k\left(\left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}e^{i k\circ x}+\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}e^{-i k\circ x}\right) \end{equation} where \begin{equation} \label{eq-96} k =\left(k^{0},\mathbf{k}\right);\quad k^{0} = {}_+\sqrt{\kappa^{2}+\mathbf{k}^{2}}. \end{equation} One infers from \eqref{eq-91} that \begin{equation} \label{eq-97} \begin{aligned} \left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}&=2\pi \left(\left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}e^{ik\circ x}+\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}e^{-ik\circ x}\right)\\ &\times\delta \left(k^{0}-\tensor*[_{+}^{}]{\sqrt{\kappa ^{2}+\mathbf{k}^{2}}}{}\right) \end{aligned} \end{equation} Since $\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}$ has rank 2, $\left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}$ and $\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}$ must each have at least rank 1. (Because $\left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}e^{ik\circ x}$ and \linebreak[4] $\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}e^{-ik\circ x}$ are functionally independent from the point of view of a Fourier decomposition of solutions of \eqref{eq-93} it cannot be the case that one of these has rank 2, and the other rank 0.) Taking $\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}$ to be the more fundamental physical quantity, we now seek rank 1 representations of $\left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}$ and $\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}$ that have sufficient degrees of freedom to satisfy symmetry constraints on $\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}$. %----------------------------------------------------------------------------------------------------- \paragraph{Relativistic Covariance} %----------------------------------------------------------------------------------------------------- Corresponding to a Lorentz transformation \begin{equation} \label{eq-98} x^{\mu }\rightarrow x'^{\mu }={L^{\mu }}_{\nu }x^{\nu };\quad L^{T}L=1 \end{equation} where $L=\left\{{L^{\mu }}_{\nu }\right\}$, the transformation rule for $\cancel{\partial }$ is \begin{equation} \label{eq-99} \cancel{\partial }\rightarrow \cancel{\partial }^{'}=\mathbb{S}\cancel{\partial }\mathbb{S}^{-1} \end{equation} for a constant matrix $\mathbb{S}\left(L\right)$. One finds \begin{equation} \label{eq-100} \cancel{\partial }=\upgamma ^{\mu }\frac{\partial }{\partial x^{\mu }}\rightarrow \upgamma ^{\nu }\frac{\partial }{\partial x'^{\mu }}=\upgamma ^{\nu }\frac{\partial x^{\mu }}{\partial x'^{\nu }}\frac{\partial }{\partial x^{\mu }}=\upgamma ^{\nu }{\left(L^{-1}\right)^{\mu }}_{\nu }\frac{\partial }{\partial x^{\mu }} \end{equation} and therefore $\mathbb{S}\left(L\right)$ is the solution of \begin{equation} \label{eq-101} \mathbb{S}\left(L\right)\upgamma ^{\mu }\mathbb{S}^{-1}\left(L\right)=\upgamma ^{\nu }{\left(L^{-1}\right)^{\mu }}_{\nu } \end{equation} which up to an overall scalar factor is \cite{ref-0141} \begin{equation} \label{eq-102} \mathbb{S}\left(L\right)=e^{\left[\upgamma _{b},\upgamma _{a}\right]{\omega ^{ab}}};\quad \omega ^{ab}=\left(g^{ab}-L^{ab}\right)/8. \end{equation} Since $\cancel{\partial }$ is a proto-typical vector it follows that the potential must transform likewise \begin{equation} \label{eq-103} \left\{\mathbb{A}\left(x\right)\right\}\rightarrow \left\{\mathbb{A}'\left(x'\right)\right\}=\mathbb{S}\left\{\mathbb{A}\left(x\right)\right\}\mathbb{S}^{-1} \end{equation} and therefore \begin{equation} \label{eq-104} \left\{\mathrm{\mathbb{F}}\left(x\right)\right\}=\cancel{\partial }\left\{\mathbb{A}\left(x\right)\right\}\rightarrow \left\{\mathrm{\mathbb{F}}'\left(x'\right)\right\}=\cancel{\partial }^{'}\left\{\mathbb{A}'\left(x'\right)\right\}=\mathbb{S}\left\{\mathrm{\mathbb{F}}\left(x\right)\right\}\mathbb{S}^{-1}. \end{equation} Consequently \begin{equation} \label{eq-105} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}\rightarrow \left\{\mathrm{\mathbb{Q}}'\left(x'\right)\right\}=\mathbb{S}\left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}\mathbb{S}^{-1} \end{equation} and \eqref{eq-93} is invariant under Lorentz transformations: \begin{equation} \label{eq-106} \begin{aligned} \left[\cancel{\partial }+i\kappa \right]\left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}&\rightarrow \left[\cancel{\partial }^{'}+i\kappa \right]\left\{\mathrm{\mathbb{Q}}'\left(x'\right)\right\}\\ &=\mathbb{S}\left[\cancel{\partial }+i\kappa \right]\left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}\mathbb{S}^{-1}\\ &=0. \end{aligned} \end{equation} It follows from \eqref{eq-105} that $\left\{\mathbb{Q}\left(x\right)\right\}$ transforms as an outer\hyp{}product of Dirac-theory bi\hyp{}spinors $\psi \left(x\right)$, the transformation rule for which (see for example \cite{ref-0141}) is $\psi '\left(x'\right)=\mathbb{S}\psi \left(x\right)$. Since $\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}$ has rank two it must be decomposable in $\mathrm{M}_{4}\left(\mathrm{\mathbb{C}}\right)$ as \begin{equation} \label{eq-108} \left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=r\left(x,k\right)\overline{s}\left(x,k\right)+u\left(x,k\right)\overline{v}\left(x,k\right) \end{equation} where $r,s$ transform as Dirac bi-spinors, $\overline{s},\overline{v}$ transform as adjoint bi-spinors, and the overbar has the traditional meaning for a bi-spinor $\psi $ that $\overline{\psi }=\psi ^{\dagger }\upgamma ^{0}$. Consistent with \eqref{eq-108}, and taking into account the discussion above, we now seek a sufficient decomposition of $\left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}$ and $\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}$ in \eqref{eq-95} as \begin{equation} \label{eq-109} \left\{\mathrm{\mathbb{Q}}_{+}\left(\mathbf{k}\right)\right\}=r\left(\mathbf{k}\right)\overline{s}\left(\mathbf{k}\right),\quad \left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}=u\left(\mathbf{k}\right)\overline{v}\left(\mathbf{k}\right) \end{equation} whereupon \eqref{eq-95} becomes \begin{equation} \label{eq-110} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\left(2\pi \right)^{-3}\int \mathrm{d}^{3}k\left(r\left(\mathbf{k}\right)\overline{s}\left(\mathbf{k}\right)e^{i{k}\circ x}+u\left(\mathbf{k}\right)\overline{v}\left(\mathbf{k}\right)e^{-i{k}\circ x}\right). \end{equation} We have not used the ensemble notation for the bi-spinors because $r\left(\mathbf{k}\right)$ and $\overline{s}\left(\mathbf{k}\right)$ in \eqref{eq-109} for example are outer-product vector \textit{factors} of an ensemble - they do not each represent an ensemble of bi-spinors. %----------------------------------------------------------------------------------------------------- \paragraph{Restriction to $\mathrm{C}\mathrm{l}_{1,3}\left(\mathrm{\mathbb{R}}\right)$} %----------------------------------------------------------------------------------------------------- The degrees of freedom in $r,s$ $\overline{s},\overline{v}$ must be restricted to conform with intrinsic symmetries of the gamma matrixes \begin{equation} \label{eq-111} \upgamma ^{\mu }=\upgamma ^{0}\upgamma ^{\mu \dagger }\upgamma ^{0}=\upgamma ^{0}\mathrm{C\upgamma }^{\mu *}\mathrm{C\upgamma }^{0}. \end{equation} The first of \eqref{eq-111} applied to a real-space potential and Faraday yield \begin{equation} \label{eq-112} \upgamma ^{0}\left\{\mathbb{A}^{\dagger }\left(x\right)\right\}\upgamma ^{0}=\left\{\mathbb{A}\left(x\right)\right\} \end{equation} and \begin{equation} \label{eq-113} \begin{aligned} \upgamma ^{0}\left\{\mathrm{\mathbb{F}}^{\dagger }\left(x\right)\right\}\upgamma ^{0}&=\upgamma ^{0}\left[\cancel{\partial }\left\{\mathbb{A}\left(x\right)\right\}\right]^{\dagger }\upgamma ^{0}\\ &=\upgamma ^{0}\left\{\mathbb{A}^{\dagger }\left(x\right)\right\}\overset{\leftarrow }{\cancel{\partial }}^{\dagger }\upgamma ^{0}\\ &=\left\{\mathbb{A}\left(x\right)\right\}\overset{\leftarrow }{\cancel{\partial }}\\ &=-\left\{\mathrm{\mathbb{F}}\left(x\right)\right\} \end{aligned} \end{equation} and therefore \begin{equation} \label{eq-114} \upgamma ^{0}\left[i\left\{\mathrm{\mathbb{F}}\left(x\right)\right\}\right]^{\dagger }\upgamma ^{0}=i\left\{\mathrm{\mathbb{F}}\left(x\right)\right\}. \end{equation} Applied to \eqref{eq-Q} these give \begin{equation} \label{eq-115} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\upgamma ^{0}\left\{\mathrm{\mathbb{Q}}^{\dagger }\left(x\right)\right\}\upgamma ^{0}. \end{equation} A similar application of the second of \eqref{eq-111} leads to \begin{equation} \label{eq-116} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\upgamma ^{0}\mathrm{C}\left\{\mathrm{\mathbb{Q}}^{*}\left(x\right)\right\}\mathrm{C\upgamma }^{0}. \end{equation} Recalling \eqref{eq-89}, \eqref{eq-90} and \eqref{eq-91}, the phase-space representations of $\left\{\mathbb{A}\right\}$, $\left\{\mathrm{\mathbb{F}}\right\}$, and $\left\{\mathrm{\mathbb{Q}}\right\}$ must have the same symmetries, and therefore \begin{equation} \label{eq-117} \left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=\upgamma ^{0}\left\{\mathrm{\mathbb{Q}}^{\dagger }\left(x,k\right)\right\}\upgamma ^{0}=\upgamma ^{0}\mathrm{C}\left\{\mathrm{\mathbb{Q}}^{*}\left(x,k\right)\right\}\mathrm{C\upgamma }^{0}. \end{equation} The first of these implies that $\mathrm{\mathbb{Q}}\left(x\right)\upgamma ^{0}$ and $\mathrm{\mathbb{Q}}\left(x,k\right)\upgamma ^{0}$ are Hermitian. Applied to \eqref{eq-110} the decomposition is restricted to \begin{equation} \label{eq-118} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\left(2\pi \right)^{-3}\int \mathrm{d}^{3}k\left(r\left(\mathbf{k}\right)\overline{s}\left(\mathbf{k}\right)e^{i{k}\circ x}+s\left(\mathbf{k}\right)\overline{r}\left(\mathbf{k}\right)e^{-i{k}\circ x}\right). \end{equation} Denoting the charge-conjugate of a bi-spinor by $\psi ^{c}$ $=$ \linebreak[4] $\upgamma ^{0}\mathrm{C}\psi ^{*}$, the second of \eqref{eq-117} connotes charge conjugation invariance of the whole matrix, which requires $r^{c}\left(\mathbf{k}\right)=s\left(\mathbf{k}\right)\Leftrightarrow r\left(\mathbf{k}\right)=s^{c}\left(\mathbf{k}\right)$, and therefore \begin{equation} \label{eq-120} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\left(2\pi \right)^{-3}\int \mathrm{d}^{3}k\left[ \begin{array}{l} \psi \left(\half\mathbf{k}\right)\overline{\psi^c}\left(\half\mathbf{k}\right)e^{i{k}\circ x}\\ +\;\psi ^{c}\left(\half\mathbf{k}\right)\overline{\psi }\left(\half\mathbf{k}\right)e^{-i{k}\circ x} \end{array} \right] \end{equation} for some bi-spinor $\psi \left(\mathrm{½}\mathbf{k}\right)$. Through a change of scale of the integration \eqref{eq-120} can be written \begin{equation} \label{eq-121} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\pi ^{-3}\int \mathrm{d}^{3}k\left[ \psi \left(x,\mathbf{k}\right)\overline{\psi^c}\left(x,\mathbf{k}\right) +\psi ^{c}\left(x,\mathbf{k}\right)\overline{\psi }\left(x,\mathbf{k}\right)\right] \end{equation} where \begin{equation} \label{eq-122} \psi \left(x,\mathbf{k}\right)=\psi \left(\mathbf{k}\right)e^{i{k_{c}}\circ x} \end{equation} and where the wave-vector is now \begin{equation} \label{eq-123} k_{c}=\left(k_{c}^{0},\mathbf{k}\right);\quad k_{c}^{0} =\tensor*[_{+}^{}]{\sqrt{\kappa _{c}^{2}+\mathbf{k}^{2}}}{},\quad \kappa _{c}=\kappa /2. \end{equation} The subscript \textit{c} alludes to the Compton frequency, which is half the frequency of the rest-frame adjunct potential. Eq. \eqref{eq-121} implies that $\left\{\mathrm{\mathbb{Q}}_{-}\left(\mathbf{k}\right)\right\}=\left\{\mathrm{\mathbb{Q}}_{+}^{c}\left(\mathbf{k}\right)\right\}$, and also that \eqref{eq-95} could be written more efficiently as \begin{equation} \label{eq-124} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\pi ^{-3}\int \mathrm{d}^{3}k\left(\left\{\mathrm{\mathbb{Q}}\left(\mathbf{k}\right)\right\}e^{i{k}\circ x}+\left\{\mathrm{\mathbb{Q}}^{c}\left(\mathbf{k}\right)\right\}e^{-i{k}\circ x}\right) \end{equation} where \begin{equation} \label{eq-125} \left\{\mathrm{\mathbb{Q}}^{c}\left(\mathbf{k}\right)\right\} =\upgamma ^{0}\mathrm{C}\left\{\mathrm{\mathbb{Q}}^{*}\left(\mathbf{k}\right)\right\}\mathrm{C\upgamma }^{0} \end{equation} where \begin{equation} \label{eq-126} \left\{\mathrm{\mathbb{Q}}\left(\mathbf{k}\right)\right\}=\psi \left(\mathrm{½}\mathbf{k}\right)\overline{\psi^c}\left(\mathrm{½}\mathbf{k}\right). \end{equation} %----------------------------------------------------------------------------------------------------- \subsection{Dirac Equation\label{mark-6.3}} %----------------------------------------------------------------------------------------------------- Applying \eqref{eq-91} to \eqref{eq-97}, a Fourier phase factor form of the multivector differential equation \eqref{eq-dQ} is \begin{equation} \label{eq-127} \left[\cancel{\partial }+i\kappa \right]\left\{\mathrm{\mathbb{Q}}\left(x,k\right)\right\}=0. \end{equation} With the substitution \eqref{eq-124} Eq. \eqref{eq-dQ} can also be expressed in the form \begin{equation} \label{eq-128} \left[\cancel{\partial }+i\kappa \right]\left\{\mathrm{\mathbb{Q}}\left(\mathbf{k}\right)\right\}e^{i{k}\circ x}=0 \end{equation} with the component form of $k$ given in \eqref{eq-96}. This is sufficient because the charge-conjugate of \eqref{eq-128} takes care of the second term in \eqref{eq-124}. Expressed instead in terms of the eigenvector decomposition \eqref{eq-126}, Eq. \eqref{eq-128} is \begin{equation} \label{eq-130} \left[\cancel{\partial }+i\kappa \right]\psi \left(\mathrm{½}\mathbf{k}\right)\overline{\psi^c}\left(\mathrm{½}\mathbf{k}\right)e^{i{k}\circ x}=0\Rightarrow \left[\cancel{\partial }+i\kappa _{c}\right]\psi \left(\mathbf{k}\right)\overline{\psi^c}\left(\mathbf{k}\right)e^{i{k_{c}}\circ x}=0. \end{equation} Using \eqref{eq-122} it follows that a sufficient condition for the satisfaction of \eqref{eq-dQ} is that each phase-space component $\psi \left(x,\mathbf{k}\right)$, $\forall x,\mathbf{k}$ satisfies the Dirac equation \begin{equation} \label{eq-131} \left[\cancel{\partial }+i\kappa _{c}\right]\psi \left(x,\mathbf{k}\right)=i\left[\kappa _{c}+\mathbbm{k}_{c}\right]\psi \left(x,\mathbf{k}\right)=0. \end{equation} %----------------------------------------------------------------------------------------------------- \section{ Dirac Currents\label{ref-0052}} %----------------------------------------------------------------------------------------------------- \subsection{Electron-positron current\label{mark-7.1}} %----------------------------------------------------------------------------------------------------- Solutions $\left\{\mathbb{Q}_{l}\left(x\right)\right\}$ of \eqref{eq-dQ} can be assembled from solutions $\psi \left(x,\mathbf{k}\right)$ of \eqref{eq-131} using \eqref{eq-121}, the bi-vector and vector parts of which are the ensemble Faraday and ensemble potential, respectively. The latter is \footnote{The Minkowski components of a Clifford vector $\mathbbm{V}=\left\langle \psi \overline{\psi^c}\right\rangle _{1}=V^{\mu }\upgamma _{\mu }$ are $V^{\mu }=\overline{\psi^c}\upgamma _{\mu }\psi $.} \begin{equation} \label{eq-133} \left\{\mathbb{A}\left(x\right)\right\}=\frac{1}{\pi ^{3}\kappa }\int \mathrm{d}^{3}k\left\langle \psi \left(x,\mathbf{k}\right)\overline{\psi^c}\left(x,\mathbf{k}\right)+ \psi ^{c}\left(x,\mathbf{k}\right)\overline{\psi }\left(x,\mathbf{k}\right)\right\rangle_{1}. \end{equation} The ensemble potential is proportional to an ensemble of local currents through \eqref{eq-74x}. Specifically \begin{equation} \label{eq-134} \left\{\mathbbm{j}\left(x\right)\right\}=-\frac{2\kappa _{c}}{\pi ^{3}}\int \mathrm{d}^{3}k\left\langle \psi \left(x,\mathbf{k}\right)\overline{\psi^c}\left(x,\mathbf{k}\right) +\psi^{c}\left(x,\mathbf{k}\right)\overline{\psi }\left(x,\mathbf{k}\right)\right\rangle _{1} \end{equation} where we used $\kappa =2\kappa _{c}$. Let us confirm that $\left\{\mathbb{A}\left(x\right)\right\}$ and therefore $\left\{\mathbbm{j}\left(x\right)\right\}$ satisfy the Lorenz gauge condition. Suppressing arguments \begin{equation} \label{eq-135} \begin{aligned} \partial \circ \left\{\mathbb{A}\right\} &=\frac{1}{\pi ^{3}\kappa }\int \mathrm{d}^{3}k \left\langle \cancel{\partial} \left[\left\langle \psi \overline{\psi^c}\right\rangle_{1} +\left\langle \psi ^{c}\overline{\psi }\right\rangle _{1}\right]\right\rangle_{0}\\ &=\frac{1}{\pi ^{3}\kappa }\int \mathrm{d}^{3}k\left[\overline{\psi^c}\overset{\leftrightarrow }{\cancel{\partial}}\psi +\overline{\psi }\overset{\leftrightarrow }{\cancel{\partial}}\psi ^{c}\right]. \end{aligned} \end{equation} This vanishes because $\cancel{\partial} \psi =-i\kappa_{c}\psi$, $\cancel{\partial} \psi ^{c}=-i\kappa_{c}\psi^{c}$, $\overline{\psi}\overset{\leftarrow }{\cancel{\partial}}=i\kappa_{c}\overline{\psi}$, and $\overline{\psi^c}\overset{\leftarrow }{\cancel{\partial}}=i\kappa_{c}\overline{\psi}^{c}$. Hence $\left\{\mathbbm{j}\left(x\right)\right\}$ is a conserved current. Due to \eqref{eq-43} we will refer to the $\left\{\mathbbm{j}\left(x\right)\right\}$ given by \eqref{eq-134} as the electron-positron ensemble current. The overall factor $2{\kappa}_c /\pi ^{3}$ can be replaced to comply with a normalization condition on the charge. To facilitate a physical interpretation of the current we express the $\psi $ in terms of the eigenvectors of charge conjugation. These are Majorana bi-spinors, which will be denoted here by a change of font to $\boldsymbol{\uppsi}$. They can be projected out of an arbitrary $\psi$ using \begin{equation} \label{eq-136} \boldsymbol{\uppsi}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)=\hat{\mathrm{\mathbb{P}}}_{{\sigma _{e}}}\left[\psi \left(x,\mathbf{k}\right)\right]=\frac{1}{2}\left[\psi \left(x,\mathbf{k}\right)+\sigma _{e}\psi ^{c}\left(x,\mathbf{k}\right)\right]. \end{equation} The inverse relations are \begin{equation} \label{eq-137} \begin{aligned} \psi \left(x,\mathbf{k}\right)&=\boldsymbol{\uppsi}_{+}\left(x,\mathbf{k}\right)+\boldsymbol{\uppsi}_{-}\left(x,\mathbf{k}\right)\\ \psi^{c}\left(x,\mathbf{k}\right)&=\boldsymbol{\uppsi}_{+}\left(x,\mathbf{k}\right)-\boldsymbol{\uppsi}_{-}\left(x,\mathbf{k}\right). \end{aligned} \end{equation} Substitution of \eqref{eq-137} into \eqref{eq-121} gives \begin{equation} \label{eq-138} \left\{\mathrm{\mathbb{Q}}\left(x\right)\right\}=\frac{2}{\pi ^{3}}\int \mathrm{d}^{3}k\left[\boldsymbol{\uppsi}_{+}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{+}\left(x,\mathbf{k}\right)-\boldsymbol{\uppsi}_{-}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{-}\left(x,\mathbf{k}\right)\right] \end{equation} in which terms \eqref{eq-134} is \begin{equation} \label{eq-139} \left\{\mathbbm{j}\left(x\right)\right\}=-\frac{4\kappa _{c}}{\pi ^{3}}\int \mathrm{d}^{3}k\left\langle \begin{array}{c} \boldsymbol{\uppsi}_{+}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{+}\left(x,\mathbf{k}\right)\\ - \boldsymbol{\uppsi}_{-}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{-}\left(x,\mathbf{k}\right) \end{array} \right\rangle _{1}. \end{equation} From the definition \eqref{eq-136} (see also \eqref{C18.5}) it can be shown that outer-products of Majorana bi-spinors have vector and bi-vector parts only - all other Clifford blades vanish - as do all bi-linear combinations of the vector and bi-vector parts (see Eq. \eqref{C19}). These constraints can be represented by expressing the individual outer-products of the Majorana bi-spinors above in terms of electron and positron multivectors $\mathrm{\mathbb{Q}}_{{\sigma}_e}$ where, suppressing arguments \begin{equation} \label{eq-138.3} \left\{\mathrm{\mathbb{Q}}_{{\sigma}_e}\right\}=\boldsymbol{\uppsi}_{{\sigma}_e}\overline{\boldsymbol{\uppsi}}_{{\sigma}_e} = \kappa \left\{\mathbb{A}_{{\sigma}_e}\right\}+i\left\{\mathbb{F}_{{\sigma}_e}\right\}, \end{equation} and where, suppressing braces, the component forms of the potential and Faraday satisfy \footnote{The electric and magnetic fields emulate the radiation zone fields of an oscillating dipole. A charge following the flow-lines of the potential moves in the direction of the momentum of those fields.}\footnote{Eq. \eqref{eq-138.3} follows from \eqref{eq-139} and the property of the Majorana bi-spinors \eqref{C19}. In Lorentz notation, in total these are ${\tilde F^{\mu \nu }}{F_{\mu \nu }} = {\tilde F^{\mu \nu }}{\tilde F_{\mu \nu }} = {F^{\mu \nu }}{F_{\mu \nu }} = {\tilde F^{\mu \nu }}{A_\nu } = {F^{\mu \nu }}{A_\nu } = {A^\nu }{A_\nu } = 0$ where $F^{\mu \nu }$, $\tilde F^{\mu \nu }$, and $A^{\mu}$ respectively are the bi-vector, its dual, and the vector parts of $\boldsymbol{\uppsi}_{{\sigma}_e}\overline{\boldsymbol{\uppsi}}_{{\sigma}_e}$.} \begin{equation} \label{eq-138.5} A_{{\sigma}_e}=\phi_{{\sigma}_e} \left(1,\hat{\mathbf{E}}_{{\sigma}_e}\times \hat{\mathbf{B}}_{{\sigma}_e}\right),\quad \mathbf{E}_{{\sigma}_e}\scalarproductdot\mathbf{B}_{{\sigma}_e}=\mathbf{B}^{2}_{{\sigma}_e}-\mathbf{E}^{2}_{{\sigma}_e}=0. \end{equation} $\mathbf{E}_{{\sigma}_e}$, $\mathbf{B}_{{\sigma}_e}$, and $\mathbf{A}_{{\sigma}_e}$ are mutually orthogonal therefore. Note that the individual $\left\{\mathrm{\mathbb{Q}}_{{\sigma}_e}\right\}$ generally do not satisfy the multi-vector Dirac equation \eqref{eq-dQ}, since the requirement is only that \begin{equation} \label{eq-138.6} \left[ \cancel{\partial } + i\kappa \right]\left[ {\left\{ {{\mathbb{Q}_ + }} \right\} + \left\{ {{\mathbb{Q}_ - }} \right\}} \right] = 0. \end{equation} Accordingly \begin{equation} \label{eq-138.7} \begin{array}{rl} \{\mathbb{F}\}&=\cancel{\partial} \{\mathbb{A}\}\\ {\Rightarrow} \;\; \{\mathbb{F}_{+}\}+ \{\mathbb{F}_{-}\}&=\cancel{\partial} \{\mathbb{A}_{+}\}+\cancel{\partial} \{\mathbb{A}_{-}\}\\ \cancel{\Rightarrow} \;\;\{\mathbb{F}_{{\sigma}_e}\}&=\cancel{\partial}\{\mathbb{A}_{{\sigma}_e}\}. \end{array} \end{equation} Of particular relevance here is that \eqref{eq-138.5} implies that $A_{{\sigma}_e}$ is null, and therefore that \eqref{eq-137} is a decomposition of the total current into 2 null currents. Their further decomposition into polarized null currents is discussed in Appendix \ref{mark-7.6}. At fixed $t$ the ensemble $\left\{\mathbbm{j}\left(x\right)\right\}$ therefore comprises 4 null currents - 2 null currents of each of the two charge species - passing through every $\mathbf{x}$.\footnote{And therefore $r=4$ in \eqref{eq-48}.} Upon substitution of \eqref{eq-122} into \eqref{eq-136} the outer-products in \eqref{eq-138} and \eqref{eq-139} become \begin{equation} \label{eq-140} \begin{array}{ll} \boldsymbol{\uppsi}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)&\overline{\boldsymbol{\uppsi}}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)=\frac{1}{4}\left[\psi \left(\mathbf{k}\right)\overline{\psi }\left(\mathbf{k}\right)+\psi ^{c}\left(\mathbf{k}\right)\overline{\psi^c}\left(\mathbf{k}\right)\right]\\ &+\frac{\sigma _{e}}{4}\left[\psi ^{c}\left(\mathbf{k}\right)\overline{\psi }\left(\mathbf{k}\right)e^{-2i{k_{c}}\circ x}+\psi \left(\mathbf{k}\right)\overline{\psi^c}\left(\mathbf{k}\right)e^{2i{k_{c}}\circ x}\right] \end{array}. \end{equation} Hence at fixed $\mathbf{k}$ the two terms in \eqref{eq-139} each comprise an oscillatory component offset by a constant mean. Moreover, the magnitude of the mean is the same for both species. Consequently the static terms cancel upon substitution of \eqref{eq-140} into \eqref{eq-139}, leaving \begin{equation} \label{eq-141} \left\{\mathbbm{j}\left(x\right)\right\}=-\frac{\kappa _{c}}{\pi ^{3}}\int \mathrm{d}^{3}k\left[ \begin{array}{r} \left\langle \psi ^{c}\left(\mathbf{k}\right)\overline{\psi }\left(\mathbf{k}\right)\right\rangle _{1}e^{-2i{k_{c}}\circ x}\\ +\left\langle \psi \left(\mathbf{k}\right)\overline{\psi^c}\left(\mathbf{k}\right)\right\rangle _{1}e^{2i{k_{c}}\circ x} \end{array} \right]. \end{equation} The electron-positron current \eqref{eq-139} is purely sinusoidal therefore, as would be expected of a solution of the Klein-Gordon equation. %----------------------------------------------------------------------------------------------------- \subsection{Traditional Dirac current \label{mark-7.4}} %----------------------------------------------------------------------------------------------------- The electron and positron bi-spinors $\boldsymbol{\uppsi}_{-}$ and $\boldsymbol{\uppsi}_{+}$ independently solve the Dirac equation, and contribute independ\-ently to the electron and positron currents in \eqref{eq-139} (the cross terms vanish). In the Majorana representation they are respectively purely real and purely imaginary - or vice-versa - up to an overall phase factor. A input to this presentation of DPI is that the individual members of the current ensemble are always null. By contrast the ensemble current is generally non-null. It will be non-null in the case of \eqref{eq-139} due to interference between the electron and positron currents.\footnote{I.E.: not as a result of interference between bi-spinors.} A purely electron ensemble current is constrained to move at light speed, therefore. The time component of each null current is proportional to \begin{equation} \label{eq-142} \begin{aligned} \overline{\boldsymbol{\uppsi}}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)\upgamma ^{0}\boldsymbol{\uppsi}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)=&\overline{\psi }\left(\mathbf{k}\right)\upgamma ^{0}\psi \left(\mathbf{k}\right)\\ +&\sigma _{e}\mathrm{Re}\left\{\overline{\psi }\left(\mathbf{k}\right)\upgamma ^{0}\psi ^{c}\left(\mathbf{k}\right)e^{-2i{k_{c}}\circ x}\right\} \end{aligned} \end{equation} Since \begin{equation} \label{eq-143} \overline{\boldsymbol{\uppsi}}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)\upgamma ^{0}\boldsymbol{\uppsi}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)=\boldsymbol{\uppsi}_{\sigma _{e}}^{\dagger }\left(x,\mathbf{k}\right)\boldsymbol{\uppsi}_{{\sigma _{e}}}\left(x,\mathbf{k}\right)>0 \end{equation} and \begin{equation} \label{eq-144} \overline{\psi }\left(\mathbf{k}\right)\upgamma ^{0}\psi \left(\mathbf{k}\right)=\psi ^{\dagger }\left(\mathbf{k}\right)\psi \left(\mathbf{k}\right)>0. \end{equation} It follows the magnitude of the oscillating part of \eqref{eq-139} is never greater than the magnitude of the static part. As we have said, the magnitudes of the static part of the two terms are equal. The oscillating parts also have equal magnitude but have different (intrinsic) signs, with the result that their subtraction re-enforces the sinusoidal part. Hence the subscripts $+$ and $-$ here refer to the relative phase of the oscillating parts of two null currents, whereas the sign of the charge is determined solely by the sign of extrinsic multiplier. In other words, the terms on the right of \eqref{eq-139} correspond to positron and electron respectively because they appear with different signs, not because they have different subscripts. Though we are considering only one Fourier component, it is clear from the above that the two terms in \eqref{eq-139} each cover half of the `space' of possible amplitudes and phases spanned by each Fourier mode (split here according to relative phase) and therefore, considered over all $\mathbf{k}$, they cover half of the Fourier function space. It follows that an arbitrary distribution (over $\mathbf{x}$) of a charge of \emph{one} species in ${\mathbb{R}^3}$ must comprise a superposition of both terms. In the Majorana representation this can be accomplished with the composition \footnote{The \{\} are retained to remind the reader of the ensemble origins.} \begin{equation} \label{eq-154} \psi_{\text{Dirac}} \left(x,\mathbf{k}\right) := \boldsymbol{\uppsi}_{-} \left(x,\mathbf{k}\right) + i \boldsymbol{\uppsi}_{+} \left(x,\mathbf{k}\right) \end{equation} provided $\boldsymbol{\uppsi}_{-}$ and $\boldsymbol{\uppsi}_{+}$ are linearly independent. For definiteness both can be taken as real. We define a current \begin{equation} \label{eq-154.7} \left\{\mathbbm{j}_{\text{Dirac}}\left(x\right)\right\}:=\int \mathrm{d}^{3}k\left\langle \psi_{\text{Dirac}} \left(x,\mathbf{k}\right)\overline{\psi }_{\text{Dirac}}\left(x,\mathbf{k}\right)\right\rangle _{1} \end{equation} and then find using \eqref{eq-154} that \begin{equation} \label{eq-154.5} \left\{\mathbbm{j}_{ \text{Dirac}}\left(x\right)\right\}=\int \mathrm{d}^{3}k\left[ \begin{array}{r} \left\langle \boldsymbol{\uppsi}_{-}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{-}\left(x,\mathbf{k}\right)\right\rangle _{1}\\ +\left\langle \boldsymbol{\uppsi}_{+}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{+}\left(x,\mathbf{k}\right)\right\rangle _{1} \end{array} \right] \end{equation} where, suppressing the Dirac subscripts, we used $\langle\overline{\psi}^c\psi^c\rangle_1=\langle\overline{\psi}\psi\rangle_1$. Hence even though the composition \eqref{eq-144} is of Majorana spinors rather than null currents, it succeeds in generating the two required null currents due to vanishing of the cross terms when extracting the vector part. Eq. \eqref{eq-154.7} is the traditional Dirac current, to be compared with \eqref{eq-134}. Unlike \eqref{eq-141}, Eq. \eqref{eq-154.7} is not a solution of the Klein-Gordon equation. It remains conserved because the currents of the two charge species are independently conserved. %----------------------------------------------------------------------------------------------------- \subsection{Double cover of ${\mathbb{R}^3}$\label{sec-double-cover}} %----------------------------------------------------------------------------------------------------- The electron\hyp{}positron current \eqref{eq-134} is proportional to a time-symmetric potential $\left\{\mathbb{A}\left(x\right)\right\}$. A single k-space component of the positron-only part $\left\langle\boldsymbol{\uppsi}_{+}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{+}\left(x,\mathbf{k}\right)\right\rangle _{1}$ can be interpre\-ted as due to occupancy of the flow-lines of the positron null half of the originating time-symmetric potential, with the other null half $-\left\langle\boldsymbol{\uppsi}_{-}\left(x,\mathbf{k}\right)\overline{\boldsymbol{\uppsi}}_{-}\left(x,\mathbf{k}\right)\right\rangle _{1}$ still generated but left unoccupied (by an electron). From that point of view the second term in \eqref{eq-154.5} is due to occupancy of the flow-lines of the positron null half of a different potential $\left\{\mathbb{A}'\left(x\right)\right\}$ say, again with the other null half generated but unoccupied, and where \begin{equation} \label{eq-154.8} \left\{\mathbb{A}'\left(x\right)\right\} = -\left\{\mathbb{A}\left(x\right)\right\}. \end{equation} From the point of view of flow-line occupancy this interpretation requires maintenance of a distinction between the potential and it negation, even though the potential and Faraday (irrespective of occupancy) are already functionally `complete' - in the sense that the functional freedom permitted by the dynamics allows for arbitrary specification of $\left\{\mathbb{A}\left(0,\mathbf{x}\right)\right\}$ and $\left\{\mathbb{F}\left(0,\mathbf{x}\right)\right\}$, say. Hence if it is admitted that $\left\{\mathbbm{j}_{\text{Dirac}}\left(0,\mathbf{x}\right)\right\}$ can be specified arbitrarily for each charge \linebreak[4] species, independently, then together these electron and posi\-tron Dirac current vectors will cover the space of the time-symmetric potential twice. From the perspective of the electron\hyp{}positron current \linebreak[4] \eqref{eq-134} the Dirac current \eqref{eq-154} contains contributions from electrons following both electron and positron flow lines, which undermines the association between the sign of $\sigma_e$ and that of the charge.\footnote{Recall from \eqref{eq-flow-line} that the direction of the (velocity of the) charge is decided entirely by the incoming vector potential; it is independent of the sign of the charge. Hence positrons and electrons can share the same flow lines.} Since in this work we will not need to refer to Dirac currents of both species we can avoid introducing another label to distinguish between these cases, instead letting the meaning of $\sigma_e$ be decided by the context. Applied to the Dirac current (by default, of a single species) it will index the two members of the double cover of the incoming potential, rather than the two charge species in a single cover, as it does for the electron-positron current \eqref{eq-134}. %----------------------------------------------------------------------------------------------------- \subsection{External Coupling\label{sec-external-coupling}} %----------------------------------------------------------------------------------------------------- Coupling of the null ensemble currents to external potentials involves an inhomgeneous differential system in the potential and Faraday that results when $\left\{\overset{\sim}{\mathbb{A}}_{l}\right\}$ is not zero. And it involves the projection of that system onto a homogeneous system in a Dirac bi-spinor. In the end one expects to be able to identify an `interaction' of the kind traditionally added to a `free-space Dirac action', with the outcome that the Euler equations are those of the Dirac equation coupled to an external potential. We do not go into this in detail here, though remark in passing that the traditional interaction \begin{equation} \label{eq-156.2} L_{int}=-\int \mathrm{d}^{4}x\left\{\mathbbm{j}_{\text{ Dirac}}\left(x\right)\right\}\circ \mathbb{A}_{\text{ ext}}\left(x\right) \end{equation} is problematic from the perspective of this work. $\mathbb{A}_{\text{ ext}}\left(x\right)$ in \eqref{eq-156.2} is a vacuum potential.\footnote{In the context of this work it is more accurately the anti-symmetric $\left\{\overset{\sim}{\mathbb{A}}_{l}\right\}$ that solves \eqref{eq-70}.} Variation of $\overline{\psi }_{\text{Dirac}}$ in this and the free-space Dirac action will result in a Dirac equation that now includes a term proportional to $\mathbb{A}_{\text{ ext}}\psi_{\text{Dirac}}$. From the perspective of the electron-positron current \eqref{eq-139}, Eq. \eqref{eq-156.2} couples an EM potential to a current without regard for the sign of the charge species. The total current will still be conserved however, because the two species are independently conserved. By contrast, the appropriate coupling to the Dirac current must account its origin, effectively, in \emph{two} copies of the time-symmetric potential, required to host the full set of flow-lines implied by \eqref{eq-154.5} - even though a single species can occupy no more than 50\% of the flow-lines of each copy. Though use of \eqref{eq-156.2} is the source of well-known problems with the traditional presentation of the single particle Dirac theory, discussion of alternative forms of interaction consistent with the framework developed here is outside the scope of this paper. %----------------------------------------------------------------------------------------------------- \subsection{Dynamic independence of the currents\label{ref-dof0}} %----------------------------------------------------------------------------------------------------- It is shown in Appendix \ref{mark-7.6} that the conserved charge currents can be further separated into independently conserved charged spin currents via an appropriately chosen spin projection $\mathbb{P}_s$, ${\boldsymbol{\uppsi}_{s,{\sigma _e}}} = {\mathbb{P}_s}{\boldsymbol{\uppsi}_{{\sigma _e}}}$, giving a total of 4 conserved null currents that can be derived from a general solution of the Dirac equation. Specifically, the Dirac current \eqref{eq-154.7} is \begin{equation}\label{eq-4currents} \begin{aligned} \left\{ {\mathbb{j}_{Dirac}\left( x \right)} \right\} = & \sum\limits_{\begin{subarray}{l} s \in \uparrow \downarrow \\ {\sigma _e} = \pm 1 \end{subarray} } {\int {{\operatorname{d} ^3}k} {{\left\langle {{\boldsymbol{\uppsi} _{s,{\sigma _e}}}\left( {x,{\mathbf{k}}} \right){{\bar {\boldsymbol{\uppsi}} }_{s,{\sigma _e}}}\left( {x,{\mathbf{k}}} \right)} \right\rangle }_1}} \\ = & \sum\limits_{\begin{subarray}{l} s \in \uparrow \downarrow \\ {\sigma _e} = \pm 1 \end{subarray} } {\int {{\operatorname{d} ^3}k} \left\{ {{\mathbb{j}_{s,{\sigma _e}}}\left( {x,{\mathbf{k}}} \right)} \right\}} . \end{aligned} \end{equation} The absence in the current of cross terms of the form \linebreak[4] ${{\boldsymbol{\uppsi} _{s',{\sigma'_e}}}\left( {x,{\mathbf{k}}} \right){{\bar {\boldsymbol{\uppsi}} }_{s,{\sigma _e}}}\left( {x,{\mathbf{k}}} \right)}$ where $s',{\sigma'_e}$ differ from $s,{\sigma_e}$ is an outcome of the properties of the projectors. The 4 currents $\left\{ {{\mathbb{j}_{s,{\sigma _e}}}\left( {x,{\mathbf{k}}} \right)} \right\}$ are coinciding but mutually exclusive ensembles of possible paths of a light-speed charge following the flow lines of a null potential. Crucial to the applicability of the method of Section \ref{sec-0006} to the general case that the incoming potential and Faraday are non-null is that these currents are dynamically independent, the demonstration of which is given Appendix \ref{mark-7.6}. %----------------------------------------------------------------------------------------------------- \section{Superposition, anti-commutation, and wavefunction collapse\label{ref-0057}} %----------------------------------------------------------------------------------------------------- Eq. \eqref{eq-4currents} is an integral superposition of an outer-product of phase-space bi-spinors, each term corresponding to a single Fourier $\mathbf{k}$-space component of the current. The constraint that the current is null will be satisfied if each of the \linebreak[4] $\left\{\mathbbm{j}_{\sigma_{p},{\sigma _{e}}}\left(x,\mathbf{k}\right)\right\}$ - i.e. for each possible $\sigma_{p},\sigma_{e},\mathbf{k}$ over all $x$ - is mutually exclusive. Under these conditions each term in the superposition (i.e. the integrand) is a candidate for the role of \textit{sole} contributor to a single instance current, whose relative magnitude therefore corresponds to the probability of that being the case in any single instance. Due to the co-occurrence in the integrand in \eqref{eq-4currents} of $\boldsymbol{\uppsi}$ and $\bar{\boldsymbol{\uppsi}}$ with the same $\mathbf{k}$ the current can be said to be explicitly diagonal in that representation. It is implicitly diagonal also in the sub-space indexed by $s,\sigma_e$ due to the automatic vanishing of cross terms ${{\boldsymbol{\uppsi} _{s',{\sigma'_e}}}\left( {x,{\mathbf{k}}} \right){{\bar {\boldsymbol{\uppsi}} }_{s,{\sigma _e}}}\left( {x,{\mathbf{k}}} \right)}$. In the $\mathbf{k}$-space representation the amplitude of a mode is the classical probability for the occupation of \emph{any} flowline within that mode. More generally, nullity can be preserved in an ensemble cast as a superposition through mutual exclusion in the joint distribution of all such mode probabilities. The ensemble potential can be regarded equally as the generator of flow-lines for both $\mathbf{x}$-space and $\mathbf{k}$-space representations - subject to the constraint that its ‘parent’ multivector satisfy \eqref{eq-dQ}.\footnote{The joint distribution to enforce nullity through mutual exclusion of flow-line occupancy in the real-space representation is not simply related to the joint distribution required to enforce nullity through mutual exclusion of $\mathbf{k}$-space mode occupancy because mutual exclusion is a non-linear constraint on the joint occupancy probability in each representation.} Nullity will be preserved in the latter case (i.e. for individual flow-lines rather than $\mathbf{k}$-space modes) by assigning mutually-exclusive classical occupation probabilities to each $\mathbf{x}$ at some fixed $t$. From the perspective of this work the freedom to choose the function space originates in the multi-vector Dirac equation, not in the Dirac equation. The distinction is important because singular value decomposition of the multivector does not generally commute with transformation of the function space. The Fourier space bi-spinors appearing in \eqref{eq-131} differ from the Fourier transform of real-space solutions of the real space Dirac equation, for example. These are different because the multivector is quadratic in the bi-spinor, and its factorization in the Fourier domain gives rise to bi-spinors that are \emph{not} the Fourier transform of bi-spinor factors of the real-space multivector. The necessity of expressing the dynamics in terms of bi-spinors rather than vector currents can be traced to the constraint that the charge moves at light speed; the light-speed constraint is \emph{intrinsic} to the Majorana bi-spinor when employed as the generator of the current, whereas it is not similarly mathematically intrinsic to a vector representation of that current. These considerations do not apply to the time-symmetric fields, which are adequately expressed in terms of a vector potential and bi-vector Faraday. Hence the `use' of those fields as generators of flow lines for the local current has the effect of granting the Dirac equation a derivative status relative to the multi-vector Dirac equation. Specifically, the Dirac equation is valid for the bi-spinor factors of the multivector in whatever function-space the dynamics of the latter is expressed. As a consequence, a current expressed in terms of bi-spinors is valid only when it is diagonal in that representation. The supervening role of the multi-vector Dirac equation can be overcome, with the effect of granting at least equal status to the (traditional) Dirac equation, by attributing the property of mutual exclusion to the \emph{bi-spinors} - rather than to the null currents they generate. To implement this strategy will require treating the bi-spinors, rather than the currents, as delta-correlated in whatever function space the current is expressed.\footnote{Investigation of the connection with the anti-commutators of quantum field theory is beyond the scope of this work, though the overlap is already clear.} In a discrete function space however delta-correlation simply involves striking out off-diagonal terms in the density matrix. In that case, the amplitude of the remaining diagonal terms will retain their role as mutually exclusive probabilities, \emph{in that representation}. This of course is the (traditional) path that has been taken. %----------------------------------------------------------------------------------------------------- \section{ Summary\label{ref-0062}} %----------------------------------------------------------------------------------------------------- The Dirac Equation is shown to derive from an equation for the Clifford multivector of the time\hyp{}symmetric potential and Faraday of classical direct particle electrodynamics. The probabilistic aspect is seen to be a consequence of embedding the dynamics of a single current in an ensemble of hypothetical currents. Wavefunction collapse / representation-independent eigenvalue selection is shown to be a consequence of non-linear constraints on the solutions of a linear differential equation. \appendix \numberwithin{equation}{section} %----------------------------------------------------------------------------------------------------- \section{Dynamic independence of the currents\label{mark-7.6}} \subsection{Spin projection\label{mark-7.60}} %----------------------------------------------------------------------------------------------------- The ${\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}\left( x,\mathbf{k} \right)$ can be further decomposed into two independent components each of which satisfies the Dirac equation, and each of which generates its own null current. It follows that though $\left\{ {{j}_{+}} \right\}$ and $\left\{ {{j}_{-}}\right\}$ are null, they cannot be the generators of the unique flow-lines assumed in Section 3. Bearing in mind \eqref{eq-131}, the components of any decomposition of a given solution of the Dirac equation will be independent solutions of the Dirac equation, and their associated currents will be conserved, if the eigenvectors of the associated projector commute with ${\mathbb{k}_{c}}$. A candidate projection with this property (in addition to and different from charge conjugation) is \begin{equation}\label{C11} {{\mathbb{P}}_{s}}:=\frac{1}{2}\left[ 1+{{\sigma }_{s}}{{\upgamma}^{5}}\mathbb{n} \right];\quad s\in \left\{ \uparrow ,\downarrow \right\},\quad {{\sigma }_{\uparrow }}=1,\quad {{\sigma }_{\downarrow }}=-1 \end{equation} where $n$ is any vector satisfying $n\circ {{k}_{c}}=0$ and ${{n}^{2}}=-1$. Writing the projections as \begin{equation}\label{C12} {{\psi }_{s}}\left( x;\mathbf{k} \right):={{\mathbb{P}}_{s}}\psi \left( x;\mathbf{k} \right) \end{equation} the ${{\psi }_{s}}\left( x;\mathbf{k} \right)$ solve the Dirac equation \eqref{eq-131}. ${{\mathbb{P}}_{s}}$ projects out the two possible spin orientations. The associated spin currents are \begin{equation}\label{C14} \left\{ {{j}_{s}}\left( x \right) \right\}=\lambda \int{{{\operatorname{d}}^{3}}k{{\left\langle {{\psi }_{s}}\left( x,\mathbf{k} \right){{{\bar{\psi }}}_{s}}\left( x,\mathbf{k} \right) \right\rangle }_{1}}} \end{equation} for some constant $\lambda$. Conservation of $\left\{ {{j}_{s}}\left( x \right) \right\}$ follows from \eqref{eq-131} with $\psi \left( x,\mathbf{k} \right)$ replaced by ${{\psi }_{s}}\left( x,\mathbf{k} \right)$. The charge and spin projectors commute \begin{equation}\label{C15} {\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}\left( x;\mathbf{k} \right):={{\mathbb{P}}_{s}}{{\hat{\mathbb{P}}}_{{{\sigma }_{e}}}}\left[ \psi \left( x;\mathbf{k} \right) \right]={{\hat{\mathbb{P}}}_{{{\sigma }_{e}}}}\left[ {{\mathbb{P}}_{s}}\psi \left( x;\mathbf{k} \right) \right] \end{equation} where ${\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}\left( x;\mathbf{k} \right)$ are the Majorana bi-spinor components of a decomposition of a general $\psi \left( x,\mathbf{k} \right)$ into charge and spin components: $\psi ={\boldsymbol{\uppsi}_{\uparrow ,+}}+{\boldsymbol{\uppsi}_{\uparrow ,-}}+{\boldsymbol{\uppsi}_{\downarrow ,+}}+{\boldsymbol{\uppsi}_{\downarrow ,-}}$. The current can be decomposed likewise \begin{equation}\label{C16} \begin{aligned} \left\{ j\left( x \right) \right\}=& \sum\limits_{ \begin{smallmatrix} s\in \uparrow \downarrow \\ {{\sigma }_{e}}=\pm 1 \end{smallmatrix}} {\left\{ {{j}_{s,{{\sigma }_{e}}}}\left( x \right) \right\}}\\ \left\{ {{j}_{s,{{\sigma }_{e}}}}\left( x \right) \right\}=&{{\sigma }_{e}}\lambda \int{{{\operatorname{d}}^{3}}k{{\left\langle {\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right){{{\bar{\boldsymbol{\uppsi}}}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\rangle }_{1}}} \end{aligned} \end{equation} for some $\lambda$, where each of the four currents is conserved and null: \begin{equation}\label{C17} \cancel\partial\circ \left\{ {{j}_{s,{{\sigma }_{e}}}}\left( x \right) \right\}={{\left\{ {{j}_{s,{{\sigma }_{e}}}}\left( x \right) \right\}}^{2}}=0;\quad s\in \left\{ \uparrow \downarrow \right\},\quad {{\sigma }_{e}}\in \left\{ \pm 1 \right\}. \end{equation} Hence there are four possible null-current charge paths passing through every space-time point - two for each sign of charge. \subsection{Condition on EM fields for independence\label{mark-7.61}} Let $s\in \left\{ \uparrow \downarrow \right\},\,{{\sigma }_{e}}\in \left\{ \pm 1 \right\}$ be collected into superscript labels $j,k$ $\in$\linebreak[4] $\left\{ 1,2,3,4 \right\}$, with no commitment to a particular relationship between the two sets of labels. In terms of the second set of labels, dynamic independence demands that current 1 does not feel a force from the Faraday of currents 2, 3, 4, etc., cyclically. Taking into account that $\left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}} \right\}$ and $\left\{ {{j}_{s,{{\sigma }_{e}}}} \right\}$ can be used interchangeably in this context (because a constant of proportionality is inconsequential to the determination of dynamical independence) the necessary condition for dynamic independence can be written \begin{equation}\label{C18} {{\left\langle \left\{ {{\mathbb{A}}^{\left( k \right)}} \right\}\cancel\partial\left\{ {{\mathbb{A}}^{\left( j \right)}} \right\} \right\rangle }_{1}}=0\quad\forall j,k \in [1,4] \wedge j\ne k\;. \end{equation} The case $j\text{=}k$ is excluded because, even though explicitly electromagnetic self-interaction was excluded by construction in \eqref{eq-4}, it re-enters the action through the light-speed constraint via \eqref{eq-17}. %----------------------------------------------------------------------------------------------------- \subsection{Independence expressed in terms of Majorana bi-spinors\label{mark-7.62}} %----------------------------------------------------------------------------------------------------- It can readily be shown from the defining property of a Majorana bi-spinor, which for our purposes can be taken to be \begin{equation}\label{C18.5} \left[\boldsymbol{\uppsi}_{{\sigma _{e}}}\right]^c=\sigma _{e}\boldsymbol{\uppsi}_{{\sigma _{e}}}, \end{equation} that the outer-product ${\boldsymbol{\uppsi}}\bar{\boldsymbol{\uppsi}}$ of Majorana bi-spinors is a multivector sum of a vector and bi-vector, every bi-linear combination of which vanishes: \begin{equation}\label{C19} {{\left\langle {\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{\sigma }_{e}}}} \right\rangle }_{g}}{{\left\langle {\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{\sigma }_{e}}}} \right\rangle }_{{{g}'}}}=0\quad \forall g,{g}'\in \left[ 0,4 \right],\quad {{\sigma }_{e}}\in \left\{ +1,-1 \right\}. \end{equation} Let us identify the potential and Faraday in $\left\langle {\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{\sigma }_{e}}}} \right\rangle $ explicitly via \begin{equation}\label{C20} \left\langle {\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{\sigma }_{e}}}} \right\rangle =2\kappa_c \left\{ {{\mathbb{A}}_{{{\sigma }_{e}}}} \right\}+i\left\{ {{\mathbb{F}}_{{{\sigma }_{e}}}} \right\} \end{equation} in which terms the total potential and Faraday are \begin{equation}\label{C21} \left\{ \mathbb{A} \right\}=\left\{ {{\mathbb{A}}_{+}} \right\}+\left\{ {{\mathbb{A}}_{-}} \right\},\ \ \left\{ \mathbb{F} \right\}=\left\{ {{\mathbb{F}}_{+}} \right\}+\left\{ {{\mathbb{F}}_{-}} \right\} \end{equation} where \begin{equation}\label{C22} \cancel\partial\left\{ \mathbb{A} \right\}=\left\{ \mathbb{F} \right\},\quad \cancel\partial\left\{ \mathbb{F} \right\}=-4\kappa_c^2\left\{ \mathbb{A} \right\}. \end{equation} Note that \eqref{C20}, \eqref{C21} and \eqref{C22} do not mandate the particular relation $\left\{ {{\mathbb{F}}_{{{\sigma }_{e}}}} \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{{{\sigma }_{e}}}} \right\}$. Inverting \eqref{C20} \begin{equation}\label{C23} i\left\{ {{\mathbb{F}}_{{{\sigma }_{e}}}} \right\}:={{\left\langle {\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{\sigma }_{e}}}} \right\rangle }_{2}},\quad 2\kappa_c\left\{{{\mathbb{A}}_{{{\sigma }_{e}}}} \right\}:={{\left\langle {\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{\sigma }_{e}}}} \right\rangle }_{1}} \end{equation} and comparing with \eqref{C19} we infer that \eqref{C18} will be satisfied provided \begin{equation}\label{C24} \left\{ {{\mathbb{F}}_{+}} \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{-}} \right\},\quad \left\{ {{\mathbb{F}}_{-}} \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{+}} \right\}. \end{equation} We notice however that the relations \eqref{C21} and \eqref{C22} permit \begin{equation}\label{C25} \left\{ {{\mathbb{F}}_{+}} \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{-}} \right\}+\mathbb{G},\quad \left\{ {{\mathbb{F}}_{-}} \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{+}} \right\}-\mathbb{G} \end{equation} and therefore \begin{equation}\label{C26} \left\{ {{\mathbb{F}}_{{{\sigma }_{e}}}} \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{{{{\bar{\sigma }}}_{e}}}} \right\}+{{\sigma }_{e}}\mathbb{G} \end{equation} where $\mathbb{G}$ can have scalar and bi-vector parts. Due to \eqref{C26} $\mathbb{G}$ will contribute equal and opposite forces on $\left\{ {{\mathbb{A}}_{{{\sigma }_{e}}}} \right\}$ via its presence in $\left\{ {{\mathbb{A}}_{{{{\bar{\sigma }}}_{e}}}} \right\}$ and in $\left\{ {{\mathbb{A}}_{{{\sigma }_{e}}}} \right\}$, resulting in a net zero contribution from $\mathbb{G}$ to the total force on the current. Hence its presence does not affect the dynamic independence of the currents, where this is now interpreted more specifically as the absence of a force on any of the null currents except (at most) from the \emph{intrinsic} part of its own Faraday (thereby excluding $\mathbb{G}$ from consideration).\footnote{There is corresponding indeterminacy in the decomposition of $\cancel\partial \{\mathbb{F}\} = -\kappa^2 \{\mathbb{A}\}$ into null components $\cancel\partial \{\mathbb{F}_{\sigma_e}\} = -\kappa^2 \{\mathbb{A}_{\sigma_e}\} + {\sigma_e}\mathbb{H}$, where $\mathbb{H}$ can have vector and pseudo-vector parts. For a monochromatic `radiation-zone' time-symmetric field one finds that $\mathbb{G}$ is a static magnetic dipole, and $\mathbb{H}$ is a static charge density, which admits the interpretation that these static components are not \emph{directly} conveyed by the time-symmetric fields, but are `reconstituted' locally, when the potential and Faraday are analyzed into their null components.\label{footnote_33}} But \eqref{C24} is not sufficient for the satisfaction of \eqref{C18}. The latter demands, in addition, that ${{\left\langle \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}} \right\}\left\{ {{\mathbb{F}}_{{s}',{{{\bar{\sigma }}}_{e}}}} \right\} \right\rangle }_{1}}=\mathbb{0}$ for one of the two possibilities $s={s}'$ or $s\ne {s}'$. Extracting the vector and bi-vector parts of ${\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{\bar{\boldsymbol{\uppsi}}}_{{{\sigma }_{e}}}}$ using \begin{equation}\label{C27} \begin{aligned} & i\left\{ {{\mathbb{F}}_{{s}',{{{\bar{\sigma }}}_{e}}}} \right\}={{\left\langle {\boldsymbol{\uppsi}_{{s}',{{{\bar{\sigma }}}_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{s}',{{{\bar{\sigma }}}_{e}}}} \right\rangle }_{2}}=\frac{1}{2}\left[ {\boldsymbol{\uppsi}_{{s}',{{{\bar{\sigma }}}_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{s}',{{{\bar{\sigma }}}_{e}}}}+{{\gamma }^{5}}{\boldsymbol{\uppsi}_{{s}',{{{\bar{\sigma }}}_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{s}',{{{\bar{\sigma }}}_{e}}}}{{\gamma }^{5}} \right] \\ & 2\kappa_c \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}} \right\}={{\left\langle {\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{s,{{\sigma }_{e}}}} \right\rangle }_{1}}=\frac{1}{2}\left[ {\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{s,{{\sigma }_{e}}}}-{{\gamma }^{5}}{\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{s,{{\sigma }_{e}}}}{{\gamma }^{5}} \right] \\ \end{aligned} \end{equation} we have \begin{equation}\label{C28} {\left\langle {\left\{ {{\mathbb{A}_{s,{\sigma _e}}}} \right\}\left\{ {{\mathbb{F}_{s',{{\bar \sigma }_e}}}} \right\}} \right\rangle _1} = \frac{-i}{{8\kappa_c }}{\left\langle {\begin{array}{*{20}{c}} {}&{\left[{\mathbb{P}_s}\boldsymbol{\uppsi}_{\sigma_e}\bar{\boldsymbol{\uppsi}}_{\sigma_e}{\mathbb{P}_s} - \upgamma^5{\mathbb{P}_s}\boldsymbol{\uppsi}_{\sigma_e}\bar{\boldsymbol{\uppsi}}_{\sigma_e}{\mathbb{P}_s}\upgamma^5\right]} \\ {\times} &{\left[{\mathbb{P}_{s'}}\boldsymbol{\uppsi}_{\bar{\sigma_e}}\bar{\boldsymbol{\uppsi}}_{\bar{\sigma_e}}{\mathbb{P}_{s'}} + \upgamma^5{\mathbb{P}_{s'}}\boldsymbol{\uppsi}_{\bar{\sigma_e}}\bar{\boldsymbol{\uppsi}}_{\bar{\sigma_e}}{\mathbb{P}_{s'}}\upgamma^5\right]} \end{array}} \right\rangle _1}. \end{equation} Multiplying out the terms and using that ${{\left\langle \mathbb{Q}{{\gamma }^{5}} \right\rangle }_{1}}=-{{\left\langle {{\gamma }^{5}}\mathbb{Q} \right\rangle }_{1}}$ for any $\mathbb{Q}$ this reduces to \begin{equation}\label{C29} {{\left\langle \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}} \right\}\left\{ {{\mathbb{F}}_{{s}',{{{\bar{\sigma }}}_{e}}}} \right\} \right\rangle }_{1}}= a{{\left\langle {{\mathbb{P}}_{s}}{\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{{\bar{\sigma }}}_{e}}}}{{\mathbb{P}}_{{{s}'}}} \right\rangle }_{1}}+b{{\left\langle {{\mathbb{P}}_{s}}{\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}{{{\bar{\boldsymbol{\uppsi}}}}_{{{{\bar{\sigma }}}_{e}}}}{{\mathbb{P}}_{{{s}'}}}{{\gamma }^{5}} \right\rangle }_{1}} \end{equation} where $a,b$ are the scalars \begin{equation}\label{C30} a=-\frac{i}{4\kappa_c }{{\bar{\boldsymbol{\uppsi}}}_{{{\sigma }_{e}}}}{{\mathbb{P}}_{s}}{{\mathbb{P}}_{{{s}'}}}{\boldsymbol{\uppsi}_{{{{\bar{\sigma }}}_{e}}}},\quad b=-\frac{i}{4\kappa_c }{{\bar{\boldsymbol{\uppsi}}}_{{{\sigma }_{e}}}}{{\mathbb{P}}_{s}}{{\gamma }^{5}}{{\mathbb{P}}_{{{s}'}}}{\boldsymbol{\uppsi}_{{{{\bar{\sigma }}}_{e}}}}. \end{equation} A sufficient condition that ${{\left\langle \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}} \right\}\left\{ {{\mathbb{F}}_{{s}',{{{\bar{\sigma }}}_{e}}}} \right\} \right\rangle }_{1}}$ vanish is that $a=b=0$. Using that the ${\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}={\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}\left( x,\mathbf{k} \right)$ satisfy the Dirac equation $\left[ {{\kappa }_{c}}-{\mathbb{k}_{c}} \right]{\boldsymbol{\uppsi}_{{{\sigma }_{e}}}}$ $=0$ we can write \begin{equation}\label{C31} \begin{aligned} 4i\kappa_c b=&{{\bar{\boldsymbol{\uppsi}}}_{{{\sigma }_{e}}}}{\mathbb{k}_{c}}{{\mathbb{P}}_{s}}{{\gamma }^{5}}{{\mathbb{P}}_{{{s}'}}}{\mathbb{k}_{c}}{\boldsymbol{\uppsi}_{{{{\bar{\sigma }}}_{e}}}}/\kappa _{c}^{2}\\ &=-{{\bar{\boldsymbol{\uppsi}}}_{{{\sigma }_{e}}}}k_{c}^{2}{{\mathbb{P}}_{s}}{{\gamma }^{5}}{{\mathbb{P}}_{{{s}'}}}{\boldsymbol{\uppsi}_{{{{\bar{\sigma }}}_{e}}}}/\kappa _{c}^{2}\\ &=-{{\bar{\boldsymbol{\uppsi}}}_{{{\sigma }_{e}}}}{{\mathbb{P}}_{s}}{{\gamma }^{5}}{{\mathbb{P}}_{{{s}'}}}{\boldsymbol{\uppsi}_{{{{\bar{\sigma }}}_{e}}}}\\ &=-4i\kappa_c b. \end{aligned} \end{equation} $b$ vanishes automatically therefore, regardless of the relative values of $s$ and ${s}'$. Consequently it is sufficient for ${{\left\langle \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}} \right\}\left\{ {{\mathbb{F}}_{{s}',{{{\bar{\sigma }}}_{e}}}} \right\} \right\rangle }_{1}}$ to vanish that ${s}'\ne s$, because then ${{\mathbb{P}}_{s}}{{\mathbb{P}}_{{{s}'}}}=0$ and therefore $a=0$, also. %----------------------------------------------------------------------------------------------------- \subsection{Summary\label{mark-7.63}} %----------------------------------------------------------------------------------------------------- We have shown that the 4 null currents $\left\{ {{j}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}\propto \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}$, where ${{\sigma }_{e}}\in \left\{ \pm 1 \right\}$ and $s,{s}'\in \left\{ \uparrow ,\downarrow \right\}$, are dynamically independent. Specifically, they satisfy \begin{equation}\label{C32} {{\left\langle \left\{ {{\mathbb{F}}_{{s}',{{{{\sigma }'}}_{e}}}}\left( x,\mathbf{k} \right) \right\}\left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\} \right\rangle }_{1}}=0 \end{equation} unless ${{\sigma }'_{e}}={{\bar{\sigma }}_{e}}$ and $s'={s}$, where \begin{equation}\label{C33} \begin{aligned} i\left\{ {{\mathbb{F}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}=&{{\left\langle {\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right){{{\bar{\boldsymbol{\uppsi}}}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\rangle }_{2}}\\ 2\kappa_c \left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}=&{{\left\langle {\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right){{{\bar{\boldsymbol{\uppsi}}}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\rangle }_{1}} \end{aligned} \end{equation} and where the ${\boldsymbol{\uppsi}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right)$ are projections according to \eqref{C15} of a general solution of the Dirac equation \eqref{eq-131}. The demonstration of independence is predicated on the particular association \begin{equation}\label{C34} \left\{ {{\mathbb{F}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}=\cancel\partial\left\{ {{\mathbb{A}}_{s,{{{\bar{\sigma }}}_{e}}}}\left( x,\mathbf{k} \right) \right\}+{{\sigma }_{e}}\mathbb{G}, \end{equation} which is accommodated within the supervening relations \begin{equation}\label{C35} \begin{aligned} \left\{ \mathbb{A}\left( x,\mathbf{k} \right) \right\}= & \sum\limits_{s\in \uparrow \downarrow }{\sum\limits_{{{\sigma }_{e}}=\pm 1}{\left\{ {{\mathbb{A}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}}} \\ \left\{ \mathbb{F}\left( x,\mathbf{k} \right) \right\}= & \sum\limits_{s\in \uparrow \downarrow }{\sum\limits_{{{\sigma }_{e}}=\pm 1}{\left\{ {{\mathbb{F}}_{s,{{\sigma }_{e}}}}\left( x,\mathbf{k} \right) \right\}}} \\ \left\{ \mathbb{F}\left( x,\mathbf{k} \right) \right\} = &\cancel\partial\left\{ \mathbb{A}\left( x,\mathbf{k} \right) \right\}. \end{aligned} \end{equation} Hence the Dirac equation describes the evolution of 4 independent null ensemble currents. 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