Simulating Nelsonian Quantum Field Theory

TL;DR

This paper explores Nelson's stochastic mechanics applied to lattice quantum field theory, demonstrating how it generates typical field configurations and offers an intuitive ontological picture of quantum states and particles.

Contribution

It extends Nelson's stochastic mechanics to quantum field theory, providing numerical simulations and visualizations of field configurations and particle-like features.

Findings

Nelson's theory can generate typical field configurations for quantum states.

The beable picture for the Fock vacuum is intuitively visualized.

The framework appears qualitatively similar for interacting scalar fields.

Abstract

We describe the picture of physical processes suggested by Edward Nelson's stochastic mechanics when generalized to quantum field theory regularized on a lattice, after an introductory review of his theory applied to the hydrogen atom. By performing numerical simulations of the relevant stochastic processes, we observe that Nelson's theory provides a means of generating typical field configurations for any given quantum state. In particular, an intuitive picture is given of the field ``beable'' -- to use a phrase of John Stewart Bell -- corresponding to the Fock vacuum, and an explanation is suggested for how particle-like features can be exhibited by excited states. We then argue that the picture looks qualitatively similar when generalized to interacting scalar field theory. Lastly, we compare the Nelsonian framework to various other proposed ontologies for QFT, and remark upon their relative merits in light of the effective field theory paradigm. Links to animations of the corresponding beables are provided throughout.