Self-normalizing Path Integrals

TL;DR

This paper investigates how the normalization of path integrals in quantum field theory is determined by the inner product on field configurations and introduces the concept of self-normalizing path integrals that are independent of scale.

Contribution

It demonstrates that the inner product on field configurations dictates path integral normalization and introduces self-normalizing path integrals, providing a new perspective on scale dependence and anomalies.

Findings

Inner product determines path integral normalization.

Self-normalizing path integrals are scale-independent.

Scale dependence encodes physical anomalies like the chiral anomaly.

Abstract

The normalization in the path integral approach to quantum field theory, in contrast with statistical field theory, can contain physical information. The main claim of this paper is that the inner product on the space of field configurations, one of the fundamental pieces of data required to be added to quantize a classical field theory, determines the normalization of the path integral. In fact, dimensional analysis shows that the introduction of this structure necessarily introduces a scale that is left unfixed by the classical theory. We study the dependence of the theory on this scale. This allows us to explore mechanisms that can be used to fix the normalization based on cutting and gluing different integrals. "Self-normalizing" path integrals, those independent of the scale, play an important role in this process. Furthermore, we show that the scale dependence encodes other important physical data: we use it to give a conceptually clear derivation of the chiral anomaly. Several explicit examples, including the scalar and compact bosons in different geometries, supplement our discussion.