Nonlinearity

Mapping of the relativistic kinetic balance equations onto the Klein–Gordon and second-order Dirac equations

Adriana I. Pesci, Raymond E. Goldstein, Hermann Uys

Published Feb 24, 2005• 5 citations

Abstract

In previous work we have shown that the quantum potential can be derived from the classical kinetic equations both for particles with and without spin. Here, we extend these mappings to the relativistic case. The essence of the analysis consists of Fourier transforming the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. In the non-relativistic case the ansatz of either separability, or separability and additivity, imposed on the probability distribution function produces mappings onto the Schrödinger equation and the Pauli equation, respectively. The former corresponds to an irrotational flow, the latter to a fluid with non-zero vorticity. In this work we show that the relativistic mappings lead to the Klein–Gordon equation in the irrotational case and to the second-order Dirac equation in the rotational case. These mappings are irreversible; an approximate inverse is the Wigner function. Taken together, these results provide a statistical interpretation of quantum mechanics.

DOI: 10.1088/0951-7715/18/3/018

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