Feynman path integrals and Lebesgue-Feynman measures

TL;DR

This paper explores Lebesgue-Feynman measures on locally convex spaces, analyzing their transformations and connections to quantum anomalies, and clarifies differing perspectives in quantum field theory literature.

Contribution

It introduces a detailed study of Lebesgue-Feynman measures, their transformations, and their relation to quantum anomalies, improving upon previous results.

Findings

Established properties of Lebesgue-Feynman measures under transformations

Connected measure transformations to quantum anomalies

Clarified contrasting views on quantum anomalies in literature

Abstract

We call a Lebesgue-Feynman measure (LFM) any generalized measure (distribution in the sense of Sobolev and Schwartz) on a locally convex topological vector space E which is translation invariant. In the present paper, we investigate transformations of the LFM generated by transformations of the domain and also discuss the connections of these transformations of the LFM with so-called quantum anomalies, improving some recent results of teh authors and co-workers. We revisit the contradiction between the points of view on quantum anomalies presented in the books of Fujikawa and Suzuki on the one hand, and of Cartier and DeWitt-Morette on the other.