QED Fermi-Fields as Operator Valued Distributions and Anomalies

TL;DR

This paper explores the operator valued distribution framework in quantum electrodynamics, emphasizing causality, gauge invariance, and anomalies, and extends existing methods to higher dimensions with implications for non-perturbative approaches.

Contribution

It introduces a generalized formulation of QED using operator valued distributions that incorporates causality and gauge invariance, extending the heat kernel method for anomalies.

Findings

Preserves solvability of the Schwinger model at D=2

Provides a natural extension of the heat kernel method for anomalies in D=4

Highlights the role of causality in gauge invariance restoration

Abstract

The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on the importance of causality following the work of Epstein and Glaser. Gauge invariant theories demand the extension of the usual translation operation on OPVD, thereby leading to a generalized $QED$ formulation. At D=2 the solvability of the Schwinger model is totally preserved. At D=4 the paracompactness property of the Euclidean manifold permits using test functions which are decomposition of unity and thereby provides a natural justification and extension of the non perturbative heat kernel method (Fujikawa) for abelian anomalies. On the Minkowski manifold the specific role of causality in the restauration of gauge invariance (and mass generation for $QED_{2}$) is examplified in a simple way.