From stochastic quantization to bulk quantization: Schwinger-Dyson equations and S-matrix

TL;DR

This paper extends stochastic quantization to a broader bulk quantization framework, providing a direct proof of equivalence to 4D theories via Schwinger-Dyson equations, and expresses the S-matrix in terms of 5D correlation functions.

Contribution

It introduces a generalized bulk quantization approach with a topological 5D action, proving equivalence to 4D theories without relying on stochastic relaxation assumptions.

Findings

Proves 4D theory equivalence using Schwinger-Dyson equations.

Derives 5D Landau-Cutkosky rules including unitarity.

Expresses the S-matrix through 5D correlation functions.

Abstract

In stochastic quantization, ordinary 4-dimensional Euclidean quantum field theory is expressed as a functional integral over fields in 5 dimensions with a fictitious 5th time. This is advantageous, in particular for gauge theories, because it allows a different type of gauge fixing that avoids the Gribov problem. Traditionally, in this approach, the fictitious 5th time is the analog of computer time in a Monte Carlo simulation of 4-dimensional Euclidean fields. A Euclidean probability distribution which depends on the 5th time relaxes to an equilibrium distribution. However a broader framework, which we call ``bulk quantization", is required for extension to fermions, and for the increased power afforded by the higher symmetry of the 5-dimensional action that is topological when expressed in terms of auxiliary fields. Within the broader framework, we give a direct proof by means of Schwinger-Dyson equations that a time-slice of the 5-dimensional theory is equivalent to the usual 4-dimensional theory. The proof does not rely on the conjecture that the relevant stochastic process relaxes to an equilibrium distribution. Rather, it depends on the higher symmetry of the 5-dimensional action which includes a BRST-type topological invariance, and invariance under translation and inversion in the 5-th time. We express the physical S-matrix directly in terms of the truncated 5-dimensional correlation functions, for which ``going off the mass-shell'' means going from the 3 physical degrees of freedom to 5 independent variables. We derive the Landau-Cutokosky rules of the 5-dimensional theory which include the physical unitarity relation.