1 Introduction↩︎

Quantum mechanics and quantum electrodynamics are among the most successful physical theories ever constructed. Their predictions for atomic and molecular structure, condensed matter, scattering amplitudes and radiative processes have been verified with extraordinary precision. Nevertheless, these theories remain conceptually and mathematically incomplete, being accompanied by foundational paradoxes as well as the occurrence of divergences requiring renormalization. The purpose of the present paper is therefore to ask whether the equations and effective structures of relativistic quantum theory admit a different foundational interpretation in terms of persistent stochastic processes.

The approach developed here is inspired by two distinct lines of thought. The first is Nelson’s stochastic mechanics, in which the nonrelativistic Schrödinger equation is derived from an underlying conservative Brownian motion [1], [2]. In Nelson’s framework the wavefunction is not a fundamental postulate, but encodes the statistical structure of a deeper stochastic process. The second is the relativistic extension due to Gaveau, Jacobson, Kac and Schulman [3], who showed that a Poisson process with finite propagation speed [4], after analytic continuation, gives rise to the Dirac equation. In this construction the particle propagates at speed \(c\), while random Poisson-distributed reversals switch its propagation sector. The reversal rate is directly related to the mass parameter after analytic continuation.

The essential conceptual shift proposed in the present paper is to take this persistent stochastic structure seriously as a process theory. In such a theory, the primary objects are not particles or fields with fixed intrinsic properties, but dynamical stochastic processes with finite propagation speed, internal sector structure, and finite correlation scales. Relativistic wave equations then arise as effective descriptions of these processes. Physical parameters such as mass and charge are not regarded as primitive bare constants, but as emergent characteristics of stochastic persistence and stochastic coupling. In this respect the programme has some affinity with Schwinger’s source theory, which sought to formulate electrodynamics in terms of physical sources and finite response amplitudes rather than bare operator fields [5]–[7].

This viewpoint is especially natural for the Dirac equation and electrodynamics. Maxwell’s equations in vacuum can be written in Dirac-like form by means of the Riemann–Silberstein vector \[\mathbf{F}_{\pm}=\mathbf{E}\pm i\mathbf{B}.\] The resulting photon wave equation has the same first-order structure as a massless relativistic wave equation [8], with the spin-\(1\) matrices replacing the spin-\(1/2\) Pauli matrices. Thus the Dirac and Maxwell equations may be viewed as arising from a common persistent stochastic mechanism, differing only in the spin representation carried by the process [9].

This observation suggests that the distinction between matter and radiation may not lie in fundamentally different ontological entities, but in different representation structures of a common persistent stochastic dynamics. The similarity is therefore not merely mathematical, but points toward a unified process-theoretic interpretation of relativistic quantum dynamics, a spin-\(1/2\) persistent process giving rise to the Dirac equation [3] and a spin-\(1\) persistent process giving rise to the photon equation [9]. The formal similarity is not merely mathematical: it points toward a unified process-theoretic interpretation of relativistic quantum dynamics.

A further important feature of the Kac process [4] is its multi-sector probability structure. If \(P_+\) and \(P_-\) denote the probabilities associated with the two propagation sectors, they are not separately conserved. Probability is continually transferred between them by the stochastic switching terms. Only the total probability \[P=P_++P_-\] satisfies a continuity equation. This feature distinguishes persistent stochastic mechanics from ordinary nonrelativistic Schrödinger mechanics, where the probability density behaves as a single conserved fluid. It also makes the persistent stochastic framework structurally closer to relativistic quantum field theory, in which individual particle-number sectors are not in general conserved, while appropriate total charges remain conserved.

The paper also revisits radiative processes from this standpoint. In standard quantum electrodynamics, spontaneous and stimulated emission are described by coupling atomic states to quantized radiation modes [10], [11]. In the process-theoretic picture, an excited stationary state is interpreted as a metastable persistent stochastic mode. In the nonrelativistic diffusive limit, the persistent stochastic framework reduces to Nelson-type stochastic mechanics. For a nondegenerate bound state, the current velocity then vanishes while the osmotic structure remains nontrivial, giving rise to the quantum potential.

From the more fundamental persistent-process viewpoint, however, the stationary state should be interpreted not merely as a diffusive stochastic equilibrium, but as a metastable persistent propagation mode possessing internal sector dynamics and finite correlation structure. Spontaneous emission corresponds to stochastic destabilization of such a metastable persistent mode, while stimulated emission corresponds to resonant synchronization of the persistent stochastic transition currents by an incident radiation field.

The same viewpoint suggests a different interpretation of radiative dressing. Instead of viewing radiative corrections as ultraviolet-divergent corrections to fixed bare parameters, one may regard observed masses, charges and magnetic moments as effective response parameters of coupled persistent stochastic matter–radiation processes. The anomalous magnetic moment of the electron provides a particularly sharp test case: the Schwinger term \(a_e=\alpha/2\pi\) [12] would appear as the leading weak-coupling approximation to a more general emergent stochastic spin response.

This structural resemblance strengthens the possibility that persistent stochastic process theory and quantum field theory may be observationally equivalent descriptions of the same physical phenomena, while assigning very different ontological meaning to the underlying structures. The point is not that the Kac process literally reproduces particle creation and annihilation at the outset, but that it already contains an internal sector-transfer architecture absent from ordinary diffusion-based stochastic mechanics.

The structure of the paper is as follows. Section 2 reviews the transition from Nelson’s Brownian stochastic mechanics to Kac-type persistent stochastic processes. It then discusses the emergence of the Dirac equation from the Kac process and emphasizes the interpretation of mass as persistence. Section 3 reformulates Maxwell’s equations using the Riemann–Silberstein vector and shows how the photon equation fits into the same process-theoretic structure. It also discusses the multi-sector probability structure of the Kac process and its relation to relativistic quantum field theory. Section 4 develops the interpretation of stationary states, osmotic motion and the quantum potential, and applies the framework to spontaneous and stimulated emission. Section 5 discusses radiative corrections, the Pauli term and the anomalous magnetic moment of the electron. Section 6 comments on gauge symmetry, particle multiplets and the Standard Model. Section 7 concludes.

2 From Nelson Stochastic Mechanics to Persistent Processes↩︎

Nelson’s Stochastic Mechanics

Nelson’s stochastic mechanics begins with the hypothesis that the motion of a microscopic particle is described not by a differentiable classical trajectory, but by a continuous stochastic process [1], [2]. The underlying process is taken to be a conservative Brownian motion with diffusion coefficient \[\nu=\frac{\hbar}{2m}.\] Because Brownian paths are everywhere continuous but nowhere differentiable, one must distinguish between forward and backward stochastic derivatives. The forward stochastic process is written \[d\mathbf{x}(t) = \mathbf{b}(\mathbf{x},t)\,dt + d\mathbf{W}(t),\] where \(\mathbf{b}\) is the forward drift velocity and \(d\mathbf{W}\) is the Wiener increment satisfying \[\langle dW_i\,dW_j\rangle = 2\nu\,\delta_{ij}\,dt.\] Similarly, the backward process possesses a backward drift velocity \(\mathbf{b}_*\).

The current and osmotic velocities are then defined by \[\mathbf{v} = \frac{1}{2}(\mathbf{b}+\mathbf{b}_*), \qquad \mathbf{u} = \frac{1}{2}(\mathbf{b}-\mathbf{b}_*).\] The probability density \(\rho(\mathbf{x},t)\) satisfies the continuity equation \[\partial_t\rho+\nabla\cdot(\rho\mathbf{v})=0,\] while the osmotic velocity is related to the density by \[\mathbf{u} = \nu\nabla\ln\rho.\] Introducing the phase function \(S(\mathbf{x},t)\) through \[m\mathbf{v}=\nabla S,\] one may define the complex wavefunction \[\psi = \sqrt{\rho}\, e^{iS/\hbar}.\] Nelson showed that the stochastic dynamical equations are then equivalent to the Schrödinger equation \[i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi.\] A central feature of Nelson’s construction is that the Born probability rule is not imposed independently. Rather, it emerges naturally from the probability density of the underlying stochastic process. The wavefunction therefore acquires a statistical interpretation without requiring an additional measurement postulate.

However, the Wiener process underlying Nelson’s theory possesses infinite propagation speed. The trajectories are continuous but nowhere differentiable, and disturbances propagate instantaneously through the stochastic medium. This feature is acceptable in the nonrelativistic domain, but it becomes problematic for relativistic theories where finite propagation speed is essential.

Persistent Kac Processes

A natural relativistic generalization of Brownian motion is provided by the persistent stochastic process introduced by Kac. Unlike Wiener diffusion, the Kac process possesses finite propagation speed. In one spatial dimension, the particle moves with speed \(c\), but randomly reverses direction according to a Poisson process with rate \(\lambda\).

Let \(P_+(x,t)\) and \(P_-(x,t)\) denote the probabilities for right-moving and left-moving propagation respectively. The stochastic evolution equations are \[\partial_t P_+ = -c\,\partial_x P_+ -\lambda P_+ +\lambda P_-,\] \[\partial_t P_- = +c\,\partial_x P_- +\lambda P_+ -\lambda P_-.\] The stochastic switching terms continuously transfer probability between the two propagation sectors.

Defining the total probability density \[P=P_++P_-,\] and the current \[J=c(P_+-P_-),\] one obtains the continuity equation \[\partial_t P+\partial_x J=0.\] Thus total probability remains conserved even though the individual sector probabilities are not.

Eliminating the current yields the Telegrapher equation \[\partial_t^2 P + 2\lambda\,\partial_t P = c^2\partial_x^2 P.\] The damping term does not represent loss of total probability. Rather, it reflects relaxation of directional persistence and continual stochastic exchange between the propagation sectors.

The Kac process therefore differs fundamentally from ordinary diffusion. The system possesses finite propagation speed, finite correlation time, and intrinsic memory structure. The process is not simply diffusive, but persistent.

Analytic Continuation and Relativistic Wave Equations

A remarkable result obtained by Gaveau, Jacobson, Kac and Schulman [3] is that the Kac process with the addition of helicity becomes directly related to the Dirac equation after analytic continuation of the reversal rate, \[\lambda \rightarrow \frac{i m c^2}{\hbar}.\label{mlambda}\tag{1}\] Under this continuation, the persistent stochastic switching between the two propagation sectors acquires oscillatory phase structure.

The resulting equations in 3-dimensions may be rewritten in spinorial form as \[i\hbar\frac{\partial\psi}{\partial t} = m c^2 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \psi + c \begin{pmatrix} \sigma_x & 0 \\ 0 & -\sigma_x \end{pmatrix} \frac{\hbar}{i}\partial_x\psi,\] which is the Dirac equation in the Weyl (chiral) respresentation.

The important conceptual point is that the mass parameter arises directly from the persistence scale of the underlying stochastic process (1 ). In this picture, mass is not introduced as a primitive intrinsic particle property. Rather, it emerges from the stochastic reversal dynamics itself.

This observation suggests a fundamentally process-theoretic interpretation of relativistic quantum mechanics. The relativistic wavefunction is not viewed as a fundamental object, but as an effective description of persistent stochastic propagation possessing finite velocity, internal sector structure and finite correlation time.

Persistent Stochasticity and Relativistic Structure

The persistent stochastic process already possesses several structural features normally associated with relativistic quantum field theory.

First, the propagation speed is finite from the outset. Second, the process possesses an intrinsically multi-sector structure. Third, the individual sector probabilities are not separately conserved. Only the total probability obeys a continuity equation.

This differs fundamentally from ordinary Schrödinger mechanics, where the probability density behaves as a single conserved fluid. Persistent stochastic mechanics therefore possesses a richer dynamical structure even before analytic continuation.

After analytic continuation, the sector-transfer dynamics acquires Dirac-like form. The resulting framework shares not only the relativistic wave equations of quantum field theory, but also part of its deeper dynamical architecture: sector conversion, finite propagation speed and oscillatory internal dynamics.

These observations strengthen the possibility that relativistic quantum field theory and persistent stochastic mechanics may be observationally equivalent descriptions of the same physical phenomena, while differing fundamentally in their ontological interpretation.

3 Electrodynamics and the Riemann–Silberstein Representation↩︎

Maxwell Equations in Dirac-like Form

A particularly important feature of electrodynamics is that Maxwell’s equations in vacuum may be written in a first-order form closely resembling relativistic wave mechanics. This becomes especially transparent in the Riemann–Silberstein representation [8], in which the electric and magnetic fields are combined into the complex vectors \[\mathbf{F}_{\pm} = \mathbf{E} \pm i\mathbf{B}.\] In vacuum, Maxwell’s equations are \[\nabla\cdot\mathbf{E}=0, \qquad \nabla\cdot\mathbf{B}=0,\] \[\partial_t\mathbf{E} = c\,\nabla\times\mathbf{B}, \qquad \partial_t\mathbf{B} = -c\,\nabla\times\mathbf{E}.\] Combining these equations yields \[i\partial_t\mathbf{F}_{\pm} = \pm c\,\nabla\times\mathbf{F}_{\pm}.\] The divergence-free condition becomes \[\nabla\cdot\mathbf{F}_{\pm}=0.\] Introducing the spin-\(1\) matrices \[(s_i)_{jk} = -i\epsilon_{ijk},\] the curl operator may be written \[\nabla\times\mathbf{F} = -i\,(\mathbf{s}\cdot\nabla)\mathbf{F}.\] If one combines the two helicity components into a single wave function \[\psi(\mathbf{r},t)= \begin{pmatrix} F_{+}(\mathbf{r},t) \\ F_{-}(\mathbf{r},t) \end{pmatrix}\] with six components, the equation for \(\psi\) in free space takes the form \[i\hbar\,\partial_t \psi(\mathbf{r},t) =-i\hbar c\,({\boldsymbol{\Sigma}}\cdot {\mathbf{\nabla} })\,\psi(\mathbf{r},t),\] where \[{\boldsymbol{\Sigma}}= \begin{pmatrix} {\boldsymbol{s}} & 0 \\ 0 & -{\boldsymbol{s}} \end{pmatrix}.\] This is of the same form as the Dirac equation except for the spin structure and the absence of the mass term.

One may regard this equation also as arising from a persistent Kac-type spin-\(1\) process. The persistent stochastic evolution for the helicity components has the form \[\partial_t F_{+} = -c\,(\mathbf{s}\cdot\nabla)F_{+} -\lambda_\gamma F_{+} +\lambda_\gamma F_{-},\] \[\partial_t F_{-} = +c\,(\mathbf{s}\cdot\nabla) F_{-} +\lambda_\gamma F_{-} -\lambda_\gamma F_{+} .\] In matrix form, \[\partial_t \psi = -c\,\Sigma\cdot\nabla\,\psi -\lambda_\gamma(1-\sigma_1)\psi,\] where \[\psi = \begin{pmatrix} F_{+} \\ F_{-} \end{pmatrix},\] and \(\sigma_1\) acts on the helicity-sector index while the matrices \(s_i\) act on the spin-\(1\) vector index. The physical photon equation is recovered in the limit \[\lambda_\gamma\rightarrow 0,\] leaving only the helicity-preserving Maxwell dynamics [9].

The distinction between matter and radiation therefore appears not as a difference between fundamentally different physical substances, but as a difference in the internal representation structure of the underlying persistent stochastic process. This viewpoint suggests a common process-theoretic origin for relativistic wave equations. Persistent stochastic propagation together with internal sector structure generates relativistic dynamics, while the representation carried by the process determines the observed spin-statistics character.

Multi-sector probability structure of the Kac process The analogy becomes even more suggestive when one recalls that the Kac process itself involves continual stochastic transfer between propagation sectors. Although Maxwell’s equations in vacuum preserve helicity in free propagation, matter–radiation interaction may couple the helicity sectors through stochastic interaction processes.

This suggests that the distinction between stochastic matter propagation and electromagnetic propagation may be less fundamental than ordinarily assumed.

Charge and Minimal Coupling

In relativistic quantum theory, electromagnetic interaction is introduced through minimal coupling. The canonical momentum is replaced according to \[\mathbf{p} \rightarrow \mathbf{p} - \frac{e}{c}\mathbf{A}.\] The corresponding relativistic wave equation becomes \[i\hbar\partial_t\psi = c\,\boldsymbol{\alpha}\cdot \left( \hat{\mathbf{p}} - \frac{e}{c}\mathbf{A} \right)\psi + \beta mc^2\psi + e\phi\,\psi.\] Within the present framework, this interaction is interpreted not as coupling between fundamentally distinct fields and particles, but as stochastic coupling between different persistent propagation processes.

The electric charge therefore acquires a different conceptual meaning. Rather than representing a primitive intrinsic property of a particle, it becomes an effective coupling parameter governing stochastic interaction between matter and radiation processes.

This viewpoint suggests that charge, like mass, may ultimately arise from deeper stochastic interaction structure rather than existing as a fundamental bare attribute.

4 Stationary States in Stochastic Mechanics↩︎

In ordinary classical mechanics, a stationary bound state corresponds to the absence of motion. In stochastic mechanics, however, the situation is fundamentally different. Even a stationary quantum state possesses underlying stochastic dynamics.

Consider a stationary state \[\psi_n(\mathbf{r},t) = \phi_n(\mathbf{r}) e^{-iE_n t/\hbar}.\] Writing \[\psi = e^{R+iS/\hbar},\] the probability density is \[\rho = |\psi|^2 = e^{2R}.\] The current and osmotic velocities are \[\mathbf{v} = \frac{1}{m}\nabla S, \qquad \mathbf{u} = \frac{\hbar}{m}\nabla R.\] For a nondegenerate bound state, the spatial wavefunction may be chosen real. The spatial phase is therefore constant, implying \[\mathbf{v}=0.\] However, unless the probability density is spatially uniform, \[\mathbf{u}\neq 0.\] Thus a stationary bound state is not a state of classical rest. The state possesses nontrivial osmotic structure even though the current velocity vanishes.

Quantum Potential and Osmotic Motion

The quantum potential is \[Q = -\frac{\hbar^2}{2m} \frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.\] Since \[\sqrt{\rho}=e^R,\] one has \[\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}} = \nabla^2R + (\nabla R)^2.\] Using \[\mathbf{u} = \frac{\hbar}{m}\nabla R,\] the quantum potential becomes \[Q = -\frac{m}{2}\mathbf{u}^2 - \frac{\hbar}{2}\nabla\cdot\mathbf{u}.\]

The quantum potential is therefore determined entirely by the osmotic structure of the stochastic process.

For a stationary state with vanishing current velocity, the stochastic Hamilton–Jacobi equation reduces to \[-E_n+V+Q=0,\] or \[V+Q=E_n.\] The external potential and the osmotic quantum potential therefore balance exactly to produce the stationary bound state.

This interpretation differs sharply from the ordinary Copenhagen viewpoint. The stationary state is not a static abstract wavefunction, but a dynamical stochastic equilibrium maintained by nonvanishing osmotic motion.

Persistent Stochastic Interpretation

The preceding discussion refers primarily to the nonrelativistic diffusive limit associated with Nelson stochastic mechanics. The persistent stochastic framework permits a deeper relativistic reinterpretation of stationary states. In Nelson’s Wiener process the stochastic dynamics is diffusive and Markovian. By contrast, the Kac process possesses (i) finite propagation speed, (ii) finite correlation time, (iii) internal propagation-sector structure, and (iv) intrinsic temporal persistence.

The relation between stationary states in Nelson stochastic mechanics and in persistent stochastic mechanics may be clarified through the Gordon decomposition of the conserved Dirac current \[j^\mu = \bar{\psi}\gamma^\mu\psi.\] The Gordon decomposition gives \[j^\mu = \frac{i\hbar}{2m} \left( \bar{\psi}\partial^\mu\psi - (\partial^\mu\bar{\psi})\psi \right) + \frac{\hbar}{2m} \partial_\nu (\bar{\psi}\sigma^{\mu\nu}\psi).\] The first term represents the convective part of the current, while the second term represents the intrinsic spin-current contribution.

This decomposition is closely analogous to the decomposition of the stochastic velocity in Nelson mechanics into current and osmotic parts, \[\mathbf{v} = \frac{1}{2}(\mathbf{b}+\mathbf{b}_*), \qquad \mathbf{u} = \frac{1}{2}(\mathbf{b}-\mathbf{b}_*).\] Within the persistent stochastic framework, the propagation-sector structure of the Kac process naturally provides a relativistic analogue of this decomposition. The combinations \[P_++P_-\] and \[P_+-P_-\] respectively describe total convective transport and internal sector-transfer structure associated with stochastic switching dynamics.

The internal sector-transfer contribution therefore plays a role analogous to the osmotic component in Nelson’s stochastic mechanics. A stationary persistent stochastic state may thus possess vanishing net convective transport while retaining nontrivial internal sector dynamics.

From this viewpoint, relativistic stationary states are interpreted not as static wavefunctions, but as metastable persistent stochastic structures maintained by continual internal sector-transfer dynamics. This gives stationary quantum states a dynamical interpretation. The state resembles a self-maintained stochastic resonance structure rather than a static configuration.

Metastability of Excited States

Excited atomic states are particularly important from this viewpoint. In standard quantum mechanics, an excited state is represented by an energy eigenfunction. Its instability is introduced phenomenologically through interaction with the radiation field.

In the present framework, the excited state itself is interpreted as a metastable persistent stochastic structure. The persistence of the state is maintained by the internal stochastic dynamics, while its finite lifetime reflects instability of the underlying persistent mode.

This interpretation gives spontaneous decay a natural dynamical meaning. Spontaneous emission corresponds to stochastic destabilization of a metastable persistent structure rather than merely probabilistic collapse of an abstract wavefunction.

Spontaneous Emission as Persistent Mode Destabilization

Consider an excited state \[\psi_e(\mathbf{r},t) = \phi_e(\mathbf{r})e^{-iE_e t/\hbar},\] together with a lower state \[\psi_g(\mathbf{r},t) = \phi_g(\mathbf{r})e^{-iE_g t/\hbar}.\] The transition frequency is \[\omega_0 = \frac{E_e-E_g}{\hbar}.\] In the persistent stochastic picture, the excited state is not perfectly stable because the underlying propagation-sector dynamics continuously generates stochastic fluctuations in the persistent mode structure. Spontaneous emission then corresponds to stochastic destabilization of the excited persistent mode together with transfer of energy into the radiation sector. Unlike ordinary classical radiation theory, the instability is intrinsic to the persistent stochastic structure itself.

Stimulated Emission as Resonant Synchronization

Now consider the presence of incident radiation with average occupation number \(\bar n\). The external electromagnetic field modifies the stochastic current structure through minimal coupling, \[\mathbf{p} \rightarrow \mathbf{p}-\frac{e}{c}\mathbf{A}.\] The current velocity becomes \[\mathbf{v} = \frac{1}{m} \left( \nabla S - \frac{e}{c}\mathbf{A} \right).\] Near resonance, \[\omega \simeq \omega_0,\] the incident field couples strongly to the persistent stochastic transition currents associated with the excited-state decay process.

The essential point is that the incident field can synchronize the phase structure of the stochastic transition dynamics. From this viewpoint, stimulated emission is interpreted as resonant synchronization of persistent stochastic transition currents. This immediately explains the difference between spontaneous and stimulated emission: (a) spontaneous emission is incoherent because the stochastic transition phases are random, whereas (b) stimulated emission is coherent because the incident radiation field phase-locks the stochastic transition process. The coherence of stimulated emission therefore does not arise from coherence of the spontaneous process itself. Rather, it arises from synchronization of the underlying persistent stochastic dynamics by the incident radiation mode.

This interpretation is conceptually close to Bose’s original insight that probabilistic matter–radiation interaction naturally generates stimulated radiation in thermal equilibrium [13]. The present framework provides a possible dynamical realization of that idea in terms of persistent stochastic processes.

5 Radiative Structure and the Anomalous Magnetic Moment↩︎

Radiative Corrections in Quantum Electrodynamics

One of the central achievements of quantum electrodynamics is the successful description of radiative corrections. In standard perturbative QED, observable quantities such as the electron mass, charge and magnetic moment receive corrections arising from interaction with the quantized electromagnetic field [10], [11].

Gauge-Covariant Persistent Stochastic Dynamics↩︎

While the connection between Kac-type stochastic processes [4] and the Dirac equation has been explored by Gaveau, Jacobson, Kac and Schulman [3], and related stochastic path approaches have been extended to external electromagnetic fields, a fully coupled gauge-covariant persistent stochastic process theory in which both matter and the electromagnetic field are represented as interacting Kac-type processes does not appear to exist in the current literature.

The real Kac equations discussed in the preceding sections describe a genuine persistent stochastic process for directional probabilities. However, the introduction of gauge coupling requires a further conceptual step. Gauge covariance acts naturally on complex amplitudes rather than directly on real probabilities. We therefore generalize the persistent Kac structure from a probability-level process to a sector-amplitude dynamics retaining the same persistent switching architecture. The probabilities are subsequently introduced as modulus squares.

Consider two particle species labelled \(a\) and \(b\). Introduce for each species a two-component persistent sector amplitude \[\Phi^{(s)}(x,t) = \begin{pmatrix} \phi^{(s)}_+(x,t)\\ \phi^{(s)}_-(x,t) \end{pmatrix}, \qquad s=a,b.\] The free persistent stochastic evolution may then be written compactly as \[\partial_t \Phi^{(s)} = -c\,\sigma_3\,\partial_x\Phi^{(s)} - \lambda_s (1-\sigma_1)\Phi^{(s)},\] where \[\sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.\] The electromagnetic interaction is introduced through the gauge-covariant derivative \[D_x^{(s)} = \partial_x-i e_s A_x,\] where \(e_s\) is the charge associated with species \(s\). The interacting persistent stochastic equations become \[\partial_t \Phi^{(s)} = -c\,\sigma_3 D_x^{(s)}\Phi^{(s)} - \lambda_s (1-\sigma_1)\Phi^{(s)}.\] Explicitly, \[\partial_t \phi^{(s)}_+ = -c D_x^{(s)}\phi^{(s)}_+ - \lambda_s\phi^{(s)}_+ + \lambda_s\phi^{(s)}_-,\] \[\partial_t \phi^{(s)}_- = +c D_x^{(s)}\phi^{(s)}_- + \lambda_s\phi^{(s)}_+ - \lambda_s\phi^{(s)}_-.\] For two species one has \[D_x^{(a)} = \partial_x-i e_a A_x, \qquad D_x^{(b)} = \partial_x-i e_b A_x.\] In the physically important case of opposite charges, \[e_a=e, \qquad e_b=-e,\] so that \[D_x^{(a)} = \partial_x-i e A_x, \qquad D_x^{(b)} = \partial_x+i e A_x.\] The physical sector probabilities are then defined by \[P^{(s)}_\pm = |\phi^{(s)}_\pm|^2,\] while the total probability density for species \(s\) is \[P^{(s)} = |\phi^{(s)}_+|^2 + |\phi^{(s)}_-|^2.\] Gauge covariance is automatic provided the amplitudes transform as \[\Phi^{(s)} \rightarrow e^{i e_s\chi(x,t)} \Phi^{(s)},\] together with \[A_x \rightarrow A_x+\partial_x\chi.\] Including a scalar potential \(A_0\), one introduces the full covariant derivatives \[D_t^{(s)} = \partial_t+i e_s A_0, \qquad D_x^{(s)} = \partial_x-i e_s A_x.\] The gauge-covariant persistent stochastic equations then take the form \[D_t^{(s)}\Phi^{(s)} = -c\,\sigma_3 D_x^{(s)}\Phi^{(s)} - \lambda_s (1-\sigma_1)\Phi^{(s)}.\] To see explicitly how the Dirac form emerges, first rewrite the above equation as \[D_t^{(s)}\Phi^{(s)} = -c\,\sigma_3D_x^{(s)}\Phi^{(s)} -\lambda_s\Phi^{(s)} +\lambda_s\sigma_1\Phi^{(s)} .\] The scalar term \(-\lambda_s\Phi^{(s)}\) represents the overall Poisson survival factor. It may be removed by the transformation \[\Phi^{(s)}(x,t)=e^{-\lambda_s t}\,\widetilde{\Phi}^{(s)}(x,t),\] which gives \[D_t^{(s)}\widetilde{\Phi}^{(s)} = -c\,\sigma_3D_x^{(s)}\widetilde{\Phi}^{(s)} +\lambda_s\sigma_1\widetilde{\Phi}^{(s)} .\] The analytic continuation [3] \[\lambda_s \longrightarrow -\frac{i m_s c^2}{\hbar},\] then yields the Dirac equation \[i\hbar D_t^{(s)}\widetilde{\Phi}^{(s)} = -i\hbar c\,\sigma_3D_x^{(s)}\widetilde{\Phi}^{(s)} + m_s c^2\sigma_1\widetilde{\Phi}^{(s)}. \label{A}\tag{2}\] Thus the stochastic switching term becomes the Dirac mass term after analytic continuation. The exponential decay factor of the real Poisson process is thereby transformed into the oscillatory relativistic phase associated with the rest energy.

This structure is strongly suggestive. The propagation sectors of the Kac process behave analogously to internal spinorial degrees of freedom, while the gauge field couples naturally through the covariant derivative prior to the formation of probabilities. The resulting framework therefore provides a natural route toward a gauge-covariant persistent stochastic process theory in which (i) propagation-sector amplitudes evolve stochastically, (ii) gauge interactions enter through covariant transport, (iii) probabilities emerge only after formation of modulus squares, (iv) and relativistic quantum dynamics appears as an emergent collective structure of the underlying persistent stochastic process.

Three-Dimensional Charged Matter Sector

For a physical charged spin-\(\frac{1}{2}\) particle, the persistent stochastic sector amplitude should be generalized from the one-dimensional two-sector form above to a four-component spinor structure [3]. Let \[\Psi^{(s)}(\mathbf{r},t) = \begin{pmatrix} \psi^{(s)}_L(\mathbf{r},t)\\ \psi^{(s)}_R(\mathbf{r},t) \end{pmatrix},\] where \(\psi_L\) and \(\psi_R\) are two-component Weyl spinors representing the two chiral propagation sectors of species \(s\). The gauge-covariant derivatives are \[D_t^{(s)} = \partial_t+\frac{i e_s}{\hbar}A_0, \qquad \mathbf{D}^{(s)} = \nabla-\frac{i e_s}{\hbar c}\mathbf{A} .\] The one-dimensional Dirac equation (2 ) then takes the form \[i\hbar D_t^{(s)}\Psi^{(s)} = -\,i\hbar c\,\boldsymbol{\alpha}\cdot\mathbf{D}^{(s)}\Psi^{(s)} + m_s c^2\beta \Psi^{(s)} ,\] where \[\boldsymbol{\alpha} = \begin{pmatrix} -\boldsymbol{\sigma} & 0\\ 0 & \boldsymbol{\sigma} \end{pmatrix}, \qquad \beta = \begin{pmatrix} 0&I_2\\ I_2&0 \end{pmatrix}.\] Equivalently, in terms of the two Weyl sectors, \[i\hbar D_t^{(s)}\psi_L^{(s)} = +i\hbar c\,\boldsymbol{\sigma}\cdot\mathbf{D}^{(s)} \psi_L^{(s)} + m_s c^2 \psi_R^{(s)},\] \[i\hbar D_t^{(s)}\psi_R^{(s)} = -i\hbar c\,\boldsymbol{\sigma}\cdot\mathbf{D}^{(s)} \psi_R^{(s)} + m_s c^2 \psi_L^{(s)} .\] In the process-theoretic interpretation, the two Weyl components represent persistent propagation sectors. The mass term couples these sectors and is identified with the analytically continued Poisson switching rate, \[\lambda_s \longrightarrow \frac{i m_s c^2}{\hbar}.\] Thus the inertial mass is interpreted as the persistence scale associated with sector switching.

For two charged species \(a\) and \(b\), one writes \[D_\mu^{(a)} = \partial_\mu+\frac{i e_a}{\hbar c}A_\mu, \qquad D_\mu^{(b)} = \partial_\mu+\frac{i e_b}{\hbar c}A_\mu,\] or, for opposite charges, \[e_a=e, \qquad e_b=-e.\] The physical matter densities are then \[\rho^{(s)} = \Psi^{(s)\dagger}\Psi^{(s)} = \psi_L^{(s)\dagger}\psi_L^{(s)} + \psi_R^{(s)\dagger}\psi_R^{(s)} .\] The conserved matter current is \[j_s^\mu = \bar\Psi^{(s)}\gamma^\mu\Psi^{(s)} ,\] and the interaction with the electromagnetic process is represented at the effective level by the usual current coupling \[j_s^\mu A_\mu .\] Thus, in three dimensions, the charged matter sector is a gauge-covariant persistent stochastic spin-\(\frac{1}{2}\) process, while the electromagnetic sector is a precribed gauge field. Their coupling appears, at the emergent wave-equation level, as the standard minimal electromagnetic coupling.

Including the Electromagnetic Process

The preceding equations treat \(A_\mu\) as an externally prescribed gauge field. For a genuine process-theoretic formulation, the electromagnetic field must also be represented as a persistent stochastic process. This can be done as in Section 3, but here we do so by writing Kac-type equations for a spin-\(1\) process whose sector amplitudes correspond to the helicity components of the Riemann–Silberstein field. Analogous to Section 3, define the six-component spin-\(1\) sector amplitude \[\Psi_{\rm em} = \begin{pmatrix} \mathbf{F}_+\\ \mathbf{F}_- \end{pmatrix}.\] As we have seen, in the absence of sources, Maxwell’s equations may be written in the Dirac-like form [8] \[i\hbar\,\partial_t\Psi_{\rm em} = c\,(\Sigma\cdot\hat{\mathbf{p}})\Psi_{\rm em},\label{photon}\tag{3}\] where \[\Sigma_i = \begin{pmatrix} s_i&0\\ 0&-s_i \end{pmatrix}, \qquad (s_i)_{jk}=-i\epsilon_{ijk}.\] Schematically, introduce helicity sector amplitudes \(\Phi^{(\gamma)}_+\) and \(\Phi^{(\gamma)}_-\), each taking values in the spin-\(1\) representation space. The persistent stochastic evolution has the form \[\partial_t\Phi^{(\gamma)}_+ = -c\,(\mathbf{s}\cdot\nabla)\Phi^{(\gamma)}_+ -\lambda_\gamma \Phi^{(\gamma)}_+ +\lambda_\gamma \Phi^{(\gamma)}_-,\] \[\partial_t\Phi^{(\gamma)}_- = +c\,(\mathbf{s}\cdot\nabla)\Phi^{(\gamma)}_- +\lambda_\gamma \Phi^{(\gamma)}_+ -\lambda_\gamma \Phi^{(\gamma)}_- .\] In matrix form, \[\partial_t\Phi^{(\gamma)} = -c\,\Sigma\cdot\nabla\,\Phi^{(\gamma)} -\lambda_\gamma(1-\sigma_1)\Phi^{(\gamma)},\] where \[\Phi^{(\gamma)} = \begin{pmatrix} \Phi^{(\gamma)}_+\\ \Phi^{(\gamma)}_- \end{pmatrix},\] and \(\sigma_1\) acts on the helicity-sector index while the matrices \(s_i\) act on the spin-\(1\) vector index. The photon equation (3 ) is recovered in the limit \[\lambda_\gamma\rightarrow 0,\] leaving only the helicity-preserving Maxwell dynamics [9]. In this sense the electromagnetic field is not merely an externally imposed gauge potential, but a spin-\(1\) persistent stochastic process whose emergent wave equation is Maxwell’s equation.

Matter–radiation interaction is then described by coupling the spin-\(1/2\) matter process to this spin-\(1\) electromagnetic process. At the effective level this coupling appears as the usual gauge-covariant replacement \[\partial_\mu\rightarrow D_\mu=\partial_\mu-i e A_\mu,\] but in the process-theoretic interpretation \(A_\mu\) is itself an emergent collective variable associated with the spin-\(1\) persistent stochastic process.

The electromagnetic interaction is introduced at the level of persistent propagation-sector amplitudes rather than directly at the level of observable probabilities. The gauge-covariant stochastic dynamics therefore modifies the sector amplitudes themselves, and the physical probabilities emerge only after formation of modulus squares. At the emergent level, the resulting collective dynamics reproduces the standard Dirac and Maxwell equations together with their familiar electromagnetic couplings.

From this viewpoint, quantities such as the electron mass and charge are not fundamental microscopic parameters inserted into the theory from the outset. Rather, they are effective collective characteristics of the coupled matter–radiation persistent stochastic process. The observed relativistic particle and field structures therefore emerge from the stabilization of the underlying stochastic sector dynamics.

Anomalous Magnetic Moment as Stochastic Spin Dressing

Within the persistent stochastic framework, the free spin-\(\frac{1}{2}\) Kac process naturally reproduces the Dirac magnetic moment with gyromagnetic ratio \[g=2.\] This value reflects the intrinsic spinorial transport structure of the free persistent process itself.

The anomalous magnetic moment arises only after coupling the spin-\(\frac{1}{2}\) matter process to the spin-\(1\) electromagnetic persistent process. The interaction modifies the internal propagation-sector dynamics and the associated spin-current structure appearing in the Gordon decomposition of the Dirac current. The resulting collective stochastic dressing produces a small correction to the effective magnetic response.

From this viewpoint, the anomaly \[a_e=\frac{g-2}{2}\] does not represent the self-energy correction of a bare point particle, as in quantum electrodynamics. Rather, it measures the finite modification of the effective spin transport generated by repeated stochastic matter–radiation interaction.

The Schwinger value \[a_e=\frac{\alpha}{2\pi}\] is then interpreted as the leading weak-coupling approximation to this collective stochastic spin dressing. The full observed anomalous magnetic moment reflects the complete structure of the coupled persistent stochastic matter–radiation process.

At the present stage, it would be premature to claim that the stochastic framework derives the observed anomaly without any analogue of parameter renormalization. What can be said is that the anomaly may be interpreted not as a shift of the defining process parameters \(m\) and \(e\), but as an emergent finite response coefficient generated by the coupled stochastic matter–radiation dynamics.

This viewpoint changes the conceptual meaning of radiative corrections. Instead of: \[\text{bare particle} + \text{divergent vacuum self-energy},\] one has: \[\text{persistent stochastic process} \rightarrow \text{effective finite dynamical response}.\] The observable magnetic moment is therefore interpreted not as the property of an isolated particle modified by vacuum fluctuations, but as an emergent effective characteristic of coupled stochastic propagation and interaction.

Relation to Schwinger’s Source Theory

The present programme bears some conceptual similarity to Schwinger’s source theory [5]–[7]. Schwinger sought to formulate electrodynamics directly in terms of physically observable sources and finite transition amplitudes, avoiding the ontological primacy of operator-valued quantum fields and the associated emphasis on bare particles and vacuum divergences.

Similarly, the present framework places primary emphasis not on quantized operator fields, but on persistent stochastic propagation processes and their effective observable responses.

The analogy should not be overstated. Persistent stochastic mechanics is not a reformulation of source theory. Nevertheless, both approaches suggest that the successful empirical structure of quantum electrodynamics may admit a deeper interpretation in terms of finite physical processes rather than divergent bare entities. .

6 Gauge Symmetry, Particle Multiplets and the Standard Model↩︎

Symmetry in Quantum Theory

Symmetry principles play a central role in modern quantum theory. The Standard Model is constructed around local gauge symmetries together with internal group representations that organize the observed particle multiplets [10], [11].

Within conventional quantum field theory, these symmetries are introduced at the level of operator-valued fields and local gauge invariance. Renormalizable interactions are then constructed by requiring invariance under the relevant symmetry groups. The present process-theoretic framework does not deny the empirical success of this structure. Rather, it raises the possibility that the observed symmetry structure may itself emerge from deeper properties of persistent stochastic propagation.

Symmetry as Structure of Persistent Processes

In the persistent stochastic picture, the fundamental entities are not particles or quantum fields, but propagation processes possessing internal sector structure. The emergence of relativistic wave equations from persistent stochastic dynamics already suggests that representation theory enters naturally through the internal structure of the process itself. Different particle types may therefore correspond to different persistent stochastic representations: spin-\(1/2\) representations generate Dirac-type processes, spin-\(1\) representations generate Maxwell-type processes, and more complicated internal symmetry structures may generate multiplet structure analogous to that appearing in the Standard Model.

From this viewpoint, particle multiplets need not be regarded as fundamentally distinct collections of particles. Instead, they may correspond to different collective modes or representations of underlying persistent stochastic dynamics.

Gauge Symmetry and Stochastic Coupling

Gauge symmetry occupies a special place in conventional quantum field theory. In standard formulations, local gauge invariance determines the interaction structure and plays an essential role in renormalizability. In the present framework, however, renormalizability is not the primary guiding principle. The emphasis shifts toward finite stochastic interaction structure and effective dynamical response.

This raises the possibility that gauge symmetry itself may acquire a different interpretation. Rather than being a fundamental principle imposed at the level of operator fields, gauge structure may emerge as an effective large-scale invariance of the underlying persistent stochastic dynamics.

The appearance of minimal coupling in electrodynamics, an Abelian gauge theory, is interpreted in the present framework not merely as a formal replacement rule, but as the effective large-scale manifestation of stochastic coupling between persistent matter and radiation processes. In this sense, the Abelian gauge interaction is already incorporated process-theoretically through coupled propagation-sector amplitude dynamics. This naturally raises the question of whether non-Abelian gauge interactions may likewise emerge from more general internal stochastic sector structures and collective coupling symmetries of persistent processes. Gauge invariance would then reflect redundancy in the effective large-scale description of these coupled stochastic dynamics rather than a fundamental property of bare microscopic fields.

At present this idea remains speculative. Nevertheless, the process-theoretic framework naturally invites such reinterpretation because the fundamental ontology differs so radically from that of ordinary local quantum field theory.

Spontaneous Symmetry Breaking and the Higgs Boson

A similar reinterpretation may apply to spontaneous symmetry breaking. In the Standard Model, the Higgs field acquires a nonzero vacuum expectation value, generating particle masses through spontaneous breaking of electroweak symmetry. Within the present framework, however, mass already acquires a direct dynamical interpretation through persistent stochastic sector-switching, and renormalization is not an issue.

This suggests that the Higgs mechanism itself may represent an effective collective description of deeper stochastic persistence structure. The Higgs boson would then be interpreted not necessarily as a fundamental elementary scalar field, but as a collective excitation associated with symmetry-breaking properties of an underlying stochastic medium.

Such a viewpoint would resemble the appearance of collective excitations in condensed matter systems, where effective quasiparticles emerge from deeper many-body dynamics.

The present paper does not attempt to construct such a theory explicitly. Rather, the point is conceptual: once relativistic quantum dynamics is reinterpreted process-theoretically, the Standard Model symmetry structure may itself admit a deeper dynamical interpretation.

Observational Equivalence and Ontology

The central claim of the present programme is not that the Standard Model or quantum electrodynamics are empirically incorrect. On the contrary, the remarkable success of relativistic quantum field theory strongly suggests that any viable process-theoretic description must reproduce its effective large-scale predictions to extremely high precision.

The proposal is instead that quantum field theory may represent an effective emergent description of deeper persistent stochastic propagation processes. From this viewpoint (i) relativistic wave equations emerge from persistent stochastic dynamics, (ii) mass reflects persistence scale, (iii) charge reflects stochastic coupling strength, (iv) radiative structure reflects effective stochastic dressing, and (v) symmetry structure reflects collective properties of persistent stochastic propagation. The observational content of relativistic quantum field theory may therefore be retained while the underlying ontology is radically transformed.

This possibility is especially intriguing because the persistent stochastic framework already reproduces several structural features ordinarily associated with relativistic quantum field theory: (i) finite relativistic propagation speed, (ii) multi-sector dynamics, (iii) sector-transfer structure, (iv) Dirac-type relativistic evolution, and (v) coupled matter–radiation interaction.

These similarities strengthen the possibility that persistent stochastic mechanics and relativistic quantum field theory may be observationally equivalent descriptions of the same physical phenomena while differing fundamentally in their interpretation of physical reality.

7 Conclusion↩︎

The framework proposed reproduces several deep structural features of relativistic quantum theory while providing a radically different physical interpretation. Relativistic wave equations emerge from persistent stochastic propagation; the Dirac and Maxwell equations arise from closely related spin-\(\frac{1}{2}\) and spin-\(1\) persistent processes; and gauge interactions are introduced at the level of propagation-sector amplitudes prior to the emergence of observable probabilities.

Within this viewpoint, the parameters \(m\) and \(e\) appearing in the stochastic equations play the role of defining process parameters associated respectively with persistence scale and stochastic coupling strength. The observable relativistic particle and field structures then emerge as stable collective modes of the coupled stochastic dynamics. Radiative structure, including the anomalous magnetic moment, is interpreted not as divergent renormalization of bare point-particle entities, but as effective stochastic dressing of the internal propagation-sector dynamics generated by repeated matter–radiation interaction.

The present work therefore suggests the possibility that relativistic quantum field theory may ultimately represent an emergent effective description of deeper persistent stochastic process dynamics.

8 Acknowledgement↩︎

I acknowledge use of ChatGPT for language polishing.

References↩︎