Effective QED Actions: Representations, Gauge Invariance, Anomalies and Mass Expansions

TL;DR

This paper develops explicit gauge-invariant representations of effective abelian gauge actions generated by charged fermions, analyzing their properties at finite temperature, including anomalies, large gauge transformations, and mass expansions.

Contribution

It introduces a Fourier representation of the effective action based on holonomies, clarifies the role of Chern-Simons terms, and provides rigorous gauge-invariant expansions for small and large fermion masses in arbitrary dimensions.

Findings

Gauge invariance is preserved at any temperature via ζ-function regularization.

Large gauge transformations are non-perturbative and influence Ward identities.

Explicit computations in 3D illustrate finite temperature effects on the effective action.

Abstract

We analyze and give explicit representations for the effective abelian vector gauge field actions generated by charged fermions with particular attention to the thermal regime in odd dimensions, where spectral asymmetry can be present. We show, through $\zeta-$function regularization, that both small and large gauge invariances are preserved at any temperature and for any number of fermions at the usual price of anomalies: helicity/parity invariance will be lost in even/odd dimensions, and in the latter even at zero mass. Gauge invariance dictates a very general ``Fourier'' representation of the action in terms of the holonomies that carry the novel, large gauge invariant, information. We show that large (unlike small) transformations and hence their Ward identities, are not perturbative order-preserving, and clarify the role of (properly redefined) Chern-Simons terms in this context. From a powerful representation of the action in terms of massless heat kernels, we are able to obtain rigorous gauge invariant expansions, for both small and large fermion masses, of its separate parity even and odd parts in arbitrary dimension. The representation also displays both the nonperturbative origin of a finite renormalization ambiguity, and its physical resolution by requiring decoupling at infinite mass. Finally, we illustrate these general results by explicit computation of the effective action for some physical examples of field configurations in the three dimensional case, where our conclusions on finite temperature effects may have physical relevance. Nonabelian results will be presented separately.