Can Quantum Field Theory be Recovered from Time-Symmetric Stochastic Mechanics? Part I: Generalizing the Liouville Equation

TL;DR

This paper generalizes the classical Liouville equation to include stochastic effects, aiming to connect quantum field theory with a time-symmetric stochastic framework that could resolve the measurement problem.

Contribution

It derives a generalized Liouville equation with specific constraints and shows its form matches the Schr"odinger equation in certain bosonic quantum field theories.

Findings

Derived a Fokker-Planck type equation with a symmetric, traceless diffusion matrix.

Connected the generalized Liouville equation to the phase-space formulation of bosonic QFT.

Provided a foundation for interpreting quantum mechanics as an objective stochastic process.

Abstract

We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter $\hbar\mapsto0$, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schr\"odinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.