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Journal of Mathematical Physics

Stochastic fields from stochastic mechanics

Steven M. Moore

Published Aug 1, 1980• 19 citations
Publisher

Abstract

Stochastic field theory for a real scalar field, considering both zero and positive temperatures, is developed from complements to Nelson’s stochastic mechanics. These complements include path integral formulas for the moments of the stochastic process, a functional differential equation for the generating functional, and a virial theorem. Using these and Yasue’s nonstandard analysis formulation of stochastic field theory, a rigorous meaning is given to the path integral formulas for the field moments and to the functional differential equation of the field’s generating functional.

DOI
10.1063/1.524705

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Journal of Mathematical Physics•Apr 1, 1981•3 citations
Stochastic field theory for a real scalar field, considering both zero and positive temperatures, is developed from complements to Nelson’s stochastic mechanics. These complements include path integral formulas for the moments of the stochastic process, a functional differential equation for the generating functional, and a virial theorem. Using these and Yasue’s nonstandard analysis formulation of stochastic field theory, a rigorous meaning is given to the path integral formulas for the field moments and to the functional differential equation of the field’s generating functional.
The finite-temperature behavior of supersymmetry is considered from the viewpoint of stochastic field theory. To this end, it is considered that Nelson’s stochastic mechanics may be generalized to the quantization of a Fermi field when the classical analog of such a field is taken to be a scalar nonlocal field where the internal space is anisotropic in nature such that when quantized this gives rise to two internal helicities corresponding to fermion and antifermion. Stochastic field theory at finite temperature is then formulated from stochastic mechanics which incorporates Brownian motion in the external space as well as in the internal space of a particle. It is shown that when the anisotropy of the internal space is suppressed so that the internal time ξ0 vanishes and the internal space variables are integrated out one has supersymmetry at finite temperature. This result is true for T=0, also. However, at this phase equilibrium will be destroyed. Thus for a random process van Hove’s result involving quantum mechanical operators, i.e., that when supersymmetry remains unbroken at T=0 it will also remain unbroken at T≠0, occurs. However, this formalism indicates that when at T=0 broken supersymmetry results, supersymmetry may be restored at a critical temperature Tc.
By using path integrals, the stochastic process associated to the time evolution of the quantum probability density is formally rewritten in terms of a stochastic differential equation, given by Newton's equation of motion with an additional multiplicative stochastic force. However, the term playing the role of the stochastic force is defined by a non-positive-definite probability functional, providing a clear example of the negative (or "extended") probabilities characteristic of quantum mechanics.