Can Quantum Field Theory be Recovered from Time-Symmetric Stochastic Mechanics? Part II: Prospects for a Trajectory Interpretation

TL;DR

This paper explores the potential for interpreting quantum field theory through time-symmetric stochastic trajectories, highlighting both promising aspects and fundamental limitations related to non-Markovian dynamics.

Contribution

It demonstrates that time-symmetric stochastic dynamics can model quantum evolution but faces challenges in representing all quantum states as trajectory averages.

Findings

Trajectory interpretation applies to fixed boundary conditions.

The dynamics are inherently non-Markovian due to time symmetry.

Major no-go theorems do not exclude this stochastic approach.

Abstract

In a companion paper we derived a unique time-reversal-invariant stochastic generalization of the Liouville equation and showed that it coincides with the evolution equation for the Husimi $Q$-function in a broad class of bosonic quantum field theories. Here we investigate the prospects for interpreting that evolution equation in terms of underlying stochastic trajectories. Drawing on Drummond's time-symmetric stochastic action formalism, we show that the traceless diffusion Fokker-Planck equation defines a natural measure over stochastic trajectories conditional on mixed-time boundary conditions. However, we identify a significant gap: it has not been established that every $Q$-function can be represented as a weighted average of these conditional probabilities over boundary values. The trajectory interpretation holds for ensembles with fixed boundary conditions but does not straightforwardly extend to arbitrary quantum states. Despite this limitation, we show that Drummond's trajectory dynamics are fundamentally non-Markovian -- a natural consequence of combining stochasticity with time-reversal invariance. This non-Markovianity places the dynamics outside the scope of the ontological models framework and thereby explains why the major no-go theorems for hidden-variable theories do not rule out the approach. These results clarify both the achievements and the remaining challenges in the project of understanding quantum field theory as the statistical mechanics of time-symmetric stochastic processes.