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Modern Physics Letters A

TIME REVERSAL IN STOCHASTIC PROCESSES AND THE DIRAC EQUATION II: COUPLING TO AN EXTERNAL ELECTROMAGNETIC FIELD

D. G. C. McKeon, Garnet N. Ord

Published Mar 21, 2000• 3 citations
Publisher

Abstract

We extend a stochastic model underlying the Dirac equation in (1+1) dimensions to include the electromagnetic field. The mechanism which produces quantum behavior is the allowed Brownian motion in both time and space. The necessary complexification of the spinor field is achieved by allowing the movement of a particle from one sublattice in (1+1) dimensions to another orthogonal sublattice in a manner dictated by the external "electromagnetic" field. The Dirac equation is obtained without recourse to a formal analytic continuation in either the time variable or the probabilities involved.

DOI
10.1142/s0217732300000608

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We extend a stochastic model underlying the Dirac equation in (1+1) dimensions to include the electromagnetic field. The mechanism which produces quantum behavior is the allowed Brownian motion in both time and space. The necessary complexification of the spinor field is achieved by allowing the movement of a particle from one sublattice in (1+1) dimensions to another orthogonal sublattice in a manner dictated by the external "electromagnetic" field. The Dirac equation is obtained without recourse to a formal analytic continuation in either the time variable or the probabilities involved.
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The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
This paper extends the stochastic electrodynamic framework of Papers 1, 2, and 3 [1–3] to relativistic quantum mechanics, deriving the Dirac equation from the relativistic Coulomb field. The non-relativistic Brownian motion of Paper 1 is generalised to a four-dimensional stochastic process in Minkowski spacetime, parameterised by proper time τ . The relativistic diffusion coefficient isD4D = ℏ/m—exactly twice the non-relativistic value ℏ/2m—arising from the additional contribution of the temporal component of the four-dimensional Wiener process. The covariant zero-point field spectral tensor Sμν(ω) = (ℏω^3/6π^2ϵ0c^3)gμν is derived from the relativistic Coulomb field through the Lorentz-covariant Einstein–Hopf detailed balance and the ergodic theorem, whose applicability to the SED electron-vacuum system is proved in Paper 3 [3]. Relativistic stochastic optimal control gives the Hamilton–Jacobi–Bellman (HJB) equation in four-dimensional configuration space, which yields the Klein–Gordon equation for spin-0 particles through the relativistic Itô correction (ℏ/2m)□. Linearisation of the relativistic Lagrangian −mc√uμuμ via Dirac matrices γμ satisfying {γμ, γν} = 2gμν converts the second-order HJB equation into a first-order equation—the Dirac equation: (iℏγμ∂μ−eγμAμ/c−mc)ψ = 0. The central new result is that spin- 1/2 is not introduced by the linearisation: it is already present in the geometry of the four-dimensional stochastic path. The mass-shell constraint uμuμ = −c^2 forces the relativistic Brownian path to be helical, with radius ℏ/mc and angular frequency 2mc^2/ℏ (Zitterbewegung). The angular momentum of this helix is ℏ/2 (spin- 1/2 ) and the magnetic moment gives g = 2—both following from the Coulomb field without additional assumptions. The linearisation then finds the unique first-order wave equation consistent with this pre-existing helical spin structure. The four-component Dirac spinor encodes four degrees of freedom already determined by the stochastic path: two spin states (helix handedness) and two time directions (particle/antiparticle).