Vacuum fluctuations with zero-range potentials

The Casimir effect is a manifestation of van der Waals polarization forces acting between atoms and molecules, in a regime where relativistic retardation effects cannot be neglected. An effective description can be given in terms of quantum fields influenced by classical objects,
such as boundaries or external potentials. This leads to the emergence of ultraviolet divergences which can be treated by means of the so-called zeta-function regularization. In this seminar I first present a rigorous implementation of the zeta-function technique, using the language of canonical quantization for a scalar field. To this purpose, complex powers of the elliptic operator describing the dynamics are used to define smeared versions of the Wightman field and renormalization of expectation values is ultimately attained by analytic continuation. Next, I address the issue of boundary anomalies, discussing some explicitly solvable models with zero-range potentials modeling semi-transparent walls.
Based on joint works with Livio Pizzocchero (University of Milano).