Loop calculations in quantum-mechanical non-linear sigma models

TL;DR

This paper derives correct Feynman rules for one-dimensional path integrals in curved spacetime by analyzing operator and path integral methods, confirming results through explicit two-loop calculations.

Contribution

It establishes a clear prescription for handling distribution products in path integrals, unifying discretized and continuum approaches in quantum-mechanical sigma models.

Findings

Derived unambiguous Feynman rules for curved spacetime path integrals

Validated rules with explicit two-loop calculations

Clarified the treatment of distribution products in quantum path integrals

Abstract

By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved spacetime. Although the prescription how to deal with the products of distributions that appear in the computation of Feynman diagrams in configuration space is surprising, this prescription follows unambiguously from the discretized path integral. We check our results by an explicit two-loop calculation.