A FUNCTIONAL ANALYTIC FRAMEWORK FOR LOCAL ZETA REGULARIZATION AND THE SCALAR CASIMIR EFFECT

It is developed a functional analytic framework allowing to formulate a rigorous implementation of zeta regularization for a canonically quantized scalar field, living on an arbitrary spatial domain and interacting with a classical background potential. This framework relies on the construction of an infinite scale of graded Hilbert spaces associated to the real powers of some given, positive self-adjoint operator. When the latter is a Schr\"odinger-type differential operator, this formulation provides a natural language to study the integral kernels related to a large class of operators, fulfilling minimal regularity requirements; particular attention is devoted to the regularity of these kernels and to the construction of their analytic continuations with respect to some parameters. Within this framework, complex powers of the elliptic operator giving rise to the Klein-Gordon equation are used to define a zeta-regularized version of the Wightman field whose pointwise evaluation is well-posed. This regularized field determines regularized local observables (such as the stress-energy tensor), whose vacuum expectation values can be expressed in terms of the above mentioned integral kernels. This allows to make contact with the theory of the Casimir effect. Renormalization is achieved by analytic continuation, which is proved to give finite results for the previously mentioned expectation values in most cases of interest. Finally, to exhibit the computational efficiency of the above methods, some explicit examples are discussed.