Generalized Abel-Plana formula as a renormalization tool in quantum field theory

TL;DR

This paper generalizes the Abel-Plana formula to handle more complex series in quantum field theory, aiding in the regularization and renormalization of divergent vacuum expectation values in systems with boundaries or nontrivial topology.

Contribution

The paper introduces a generalized Abel-Plana formula applicable to series over zeros of functions, extending its use in quantum field theory for diverse boundary conditions.

Findings

Provides a new mathematical tool for regularization in quantum field theory.

Enables finite integral representations for more complex series.

Facilitates analysis of quantum fields with curved boundaries or nontrivial topology.

Abstract

In quantum field theory the vacuum expectation values of physical observables, bilinear in the field operator, diverge. Among the most important points in the investigations of those expectation values is the regularization of divergent expressions, separation of divergences and the renormalization. In problems with boundaries the expectation values are expressed in the form of the difference of the divergent series and the corresponding integral. In problems with planar boundaries a finite integral representation for that difference is provided by the Abel-Plana summation formula. In the present contribution we consider the generalization of the Abel-Plana formula that allows to obtain similar representations for more general classes of series where the summation goes over the zeros of a given function. Applications are discussed in quantum field theoretical problems with nontrivial spatial topology and curved boundaries.