From path integral quantization to stochastic quantization: a pedestrian's journey

TL;DR

This paper presents two new proofs demonstrating the equivalence between path integral and stochastic quantizations for scalar Euclidean quantum field theories, using forest-indexed Taylor interpolations.

Contribution

It introduces novel proofs of equivalence between path integral and stochastic quantization, employing forest-based Taylor interpolations at different levels.

Findings

Proofs establish equivalence at the level of Feynman expansion terms.

Proofs establish equivalence at the path integral level without full perturbation expansion.

Abstract

We give two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. Our proofs rely on Taylor interpolations indexed by forests, in the fashion of constructive field theory. The first proof works at the level of individual terms in the Feynman expansion, with the forests appearing as spanning forests in Feynman graphs. The second one works at the level of the path integral and avoids the full expansion of the Feynman perturbation series.