Towards quantum field theory on spacetimes with boundaries

TL;DR

This paper develops an explicit algebraic quantization method for a massless scalar field on spacetimes with boundaries, using the method of images to relate boundary problems to Minkowski solutions, highlighting its limitations.

Contribution

It introduces a concrete construction for quantizing fields on bounded spacetimes, extending algebraic quantum field theory to new geometries with boundaries.

Findings

The method of images effectively relates boundary value problems to Minkowski solutions.

The approach provides explicit algebraic structures for fields on bounded spacetimes.

Limitations of the method are identified and discussed.

Abstract

Algebraic quantization has been applied on the class of globally hyperbolic spacetime for many decades, leading to remarkable results. Nonetheless, the presence of a boundary calls for a separate treatment, since, in general, it breaks global hyperbolicity and it requires the introduction of boundary conditions to the field equation. Although a general theory for such situations is lacking, we develop an explicit construction of the algebraic quantization of a massless scalar field on two prototypical geometries, by exploiting the method of images. This well known technique gives a way to relate solutions of the boundary value problem to solution of the field equation on Minkowski spacetime, filling the gap with the standard theory. At the same time, we point out some limits of reliability of this scheme.