On the issue of imposing boundary conditions on quantum fields

TL;DR

This paper examines the complex relationship between mathematical boundary conditions and physical quantum fields, analyzing different regularization methods to address infinities in quantum field theory results.

Contribution

It provides a critical analysis of recent and past approaches to imposing boundary conditions on quantum fields, clarifying the mathematical and physical implications.

Findings

Highlights the interplay between physics and mathematics in boundary conditions

Compares zeta-function regularization with Hadamard regularization

Offers insights into handling infinities in quantum field theory

Abstract

An interesting example of the deep interrelation between Physics and Mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields. This procedure has recently been re-examined with care. Comments on that and previous analysis are here provided, together with considerations on the results of the purely mathematical zeta-function method, in an attempt at clarifying the issue. Hadamard regularization is invoked in order to fill the gap between the infinities appearing in the QFT renormalized results and the finite values obtained in the literature with other procedures.