Nonperturbative correlation functions from homotopy algebras

TL;DR

This paper introduces a new nonperturbative formula for quantum correlation functions using homotopy algebras, which does not require dividing the action into free and interaction parts, and is validated through numerical experiments.

Contribution

It presents a novel formula for correlation functions that works nonperturbatively and handles multiple Lefschetz thimbles, advancing the algebraic approach to quantum field theory.

Findings

The new formula reproduces perturbative correlation functions accurately.

Numerical evidence shows the formula's validity for scalar theories in zero dimensions.

Applicable to both Euclidean and Lorentzian cases with finite coupling constants.

Abstract

The formula for correlation functions based on quantum $A_\infty$ algebras in arXiv:2203.05366, arXiv:2305.11634, and arXiv:2305.13103 requires us to divide the action into the free part and the interaction part. We present a new form of the formula which does not involve such division. The new formula requires us to choose a solution to the equations of motion which does not have to be real, and we claim that the formula gives correlation functions evaluated on the Lefschetz thimble associated with the solution we chose. Our formula correctly reproduces correlation functions in perturbation theory, but it can be valid nonperturbatively, and we present numerical evidence for scalar field theories in zero dimensions both in the Euclidean case and the Lorentzian case that correlation functions for finite coupling constants can be reproduced. When the theory consists of a single Lefschetz thimble, our formula gives correlation functions of the theory by choosing the solution corresponding to the thimble. When the theory consists of multiple Lefschetz thimbles, we need to evaluate the ratios of the partition functions for those thimbles and we describe a method of such evaluations based on quantum $A_\infty$ algebras in a forthcoming paper.