Dirac equation and dynamics of Brownian motion

R.F Roose, R. Dagonnier

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

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

L. Favella

Roumen Tsekov

where m is the particle mass at rest, c is the speed of light, p is the particle momentum and U is a potential. The root in Eq. (1) complicates substantially the way of quantizing the relativistic motion. In the literature there are two important solutions of this problem: the Klein-Gordon and Dirac equations. Their complex mathematical structures and some problems with definition of the probability density make it difficult to extend these equations for description of relativistic quantum Brownian motion. Moreover, it is not clear if the non-relativistic energy ˆ t E i   and momentum p i   operators, used in quantizing of Eq. (1), are valid in the relativistic quantum mechanics as well. The present paper proposes an alternative approach via a relativistic Wigner-Klein-Kramers equation. The quantum Brownian motion is described by the Madelung quantum hydrodynamics [4] coupled to the Breit-Fermi Hamiltonian. According to the relativistic dynamics the stochastic Langevin equation for a classical relativistic particle moving in a classical non-relativistic environment acquires the form

V. I. Mel'Nikov, S. V. Meshkov

B. S. Rajput

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.