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Journal of Physics A Mathematical and General

Stochastic mechanics of a Dirac particle in two spacetime dimensions

Gian Fabrizio De Angelis, Giovanni Jona-Lasinio, Maurizio Serva, Nino Zanghı̀

Published Apr 21, 1986• 35 citations
Publisher

Abstract

The authors develop the stochastic mechanics of a Dirac particle interacting with an arbitrary external electromagnetic field in two-dimensional Minkowski space. Their construction provides a consistent stochastic interpretation of solutions of the true (real time) Dirac equation and, in particular, gives a stochastic description of the zitterbewegung.

DOI
10.1088/0305-4470/19/6/017

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The authors develop the stochastic mechanics of a Dirac particle interacting with an arbitrary external electromagnetic field in two-dimensional Minkowski space. Their construction provides a consistent stochastic interpretation of solutions of the true (real time) Dirac equation and, in particular, gives a stochastic description of the zitterbewegung.
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).
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We extend a stochastic model underlying the Dirac equation in 1+1 dimensions to include the electromagnetic field. The model obtains the Dirac equation without requiring a formal analytic continuation. The mechanism, which produces quantum behaviour, is the allowed Brownian motion in time. The microscopic time reversals in the Brownian motion are the source of quantum interference. The electromagnetic field is also obtained from a stochastic model and the interaction with Dirac particles is treated as a scattering process.PACS Nos.: 03.65.-w, 03.20.+i, 05.40.+j
Abstract Quantum mechanics describes multi-electronic systems by using probabilistic theories. All of these theories provide a probabilistic conception of quantum mechanics, with an astonishing accuracy (for instance the Helium atom ground state is computed with more than 20 digits of accuracy) but unfortunately seems to be a bit “disconnected” with macro-scale physics. Indeed, the real observable world is for us a 3D+time Minkowski space but quantum mechanics is expressed in multi-dimensional Hilbert space: which is not intuitively physical but seems to be purely mathematical. In this article we try to construct a relativistic equation that gives an approximately correct description of atomic physics in the spacetime (i.e the “real space”) for 2 electrons systems: i.e a 2-electrons Dirac equation. The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a matter-wave. For this we start from the Hartree equation, which gives already a spacetime description of each electron wave function. Then, we switch to Hartree-Fock representation, which gives birth to the exchange potential, that can be also expressed in spacetime. By extending the concept of exchange potential to the relativistic 4-exchange: we obtain a relativistic 2-electrons spacetime Dirac-Hartree-Fock (STDHF) equation (i.e in Minkowski space). That equation takes into account relativistic effects but not correlation effects, which is the difficulty of this study. Then we will show the equivalence of the 2-electrons STDHF equation with the Hartree-Fock equation in non-relativistic, stationary and scalar regime. Electron correlation issues will be discussed as well.