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

December 03, 2014

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 labeling space which is at the same time optimal and separating.
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
PoincarĂ© vacuum and KMS states. Eventually we use our results in both systems
to introduce the notion of Wick polynomials, showing that a global extended
algebra does not exist. Furthermore 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.

Keywords:

algebraic quantum field theory, Casimir effect