The Casimir effect from the point of view of algebraic quantum field theory

TL;DR

This paper develops an algebraic quantum field theory framework for the Casimir effect, constructing states and observables for scalar fields between plates, and confirms results consistent with traditional calculations.

Contribution

It introduces a novel algebraic approach to quantize scalar fields with boundary conditions, explicitly constructs Hadamard states, and connects algebraic states with standard Minkowski spacetime states.

Findings

Constructed a suitable algebra of observables for bounded regions.

Established the correspondence of algebraic states with Minkowski spacetime states via the method of images.

Derived the two-point function and energy density consistent with the Casimir effect.

Abstract

We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital *-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincar\'e vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.