$\eta$ regularisation and the functional measure

TL;DR

This paper develops a generalized framework for regularizing the functional measure in quantum field theory, connecting spectral asymmetry, number theory, and topological invariants to deepen understanding of the chiral anomaly.

Contribution

It extends $ ext{eta}$ regularisation to operator language, unifies analytic and topological perspectives, and explores measure transformations and divergences in anomaly calculations.

Findings

Unified regularisation framework linking spectral asymmetry and topology.

Derived a function encoding measure dependence on regularisation scale.

Connected $ ext{eta}$ regularisation with Schwinger proper-time formalism.

Abstract

In this paper, we revisit Fujikawa's path integral formulation of the chiral anomaly and develop a generalised framework for systematically defining a regularised functional measure. This construction extends the $\eta$ regularisation scheme to operator language, making the connection between spectral asymmetry and measure transformation fully explicit. Before recovering Fujikawa's expression for the chiral anomaly from the regularised measure, we explore the deeper number-theoretic structure underlying the ill-defined spectral sum associated with the anomaly, interpreting it through the lens of smoothed asymptotics. Our approach unifies two complementary perspectives: the analytic regularisation of Fujikawa and the topological characterisation given by the Atiyah-Singer index theorem. We further investigate how the measure transforms under changes to the regularisation scale and derive a function $\iota_E(\Lambda)$ that encodes this dependence, showing how its Mellin moments govern the appearance of divergences. Finally, we comment on the conceptual relationship between the regularised measure, $\eta$ regularisation, and the generalised Schwinger proper-time formalism, with a particular focus on the two-dimensional Schwinger model.