How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism

TL;DR

This paper develops a method to derive Feynman diagrams directly from the BV formalism by explicitly calculating the homology of BV algebras, bridging the gap between algebraic structures and diagrammatic techniques.

Contribution

It introduces a novel approach to obtain Feynman rules from BV algebra homology, bypassing traditional diagrammatic derivations in quantum field theory.

Findings

Explicit calculation of BV algebra homology

Derivation of Wick's Theorem from algebraic principles

New algebraic foundation for Feynman diagram rules

Abstract

The Batalin-Vilkovisky formalism in quantum field theory was originally invented to address the difficult problem of finding diagrammatic descriptions of oscillating integrals with degenerate critical points. But since then, BV algebras have become interesting objects of study in their own right, and mathematicians sometimes have good understanding of the homological aspects of the story without any access to the diagrammatics. In this note we reverse the usual direction of argument: we begin by asking for an explicit calculation of the homology of a BV algebra, and from it derive Wick's Theorem and the other Feynman rules for finite-dimensional integrals.