Time evolution with symmetric stochastic action

Abstract

The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation. Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases. For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions. The solution corresponds to a c-number field equilibrating in an additional dimension. Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space. We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics. This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.