Zeta Function Regularization in Casimir Effect Calculations and J. S. Dowker's Contribution

Abstract

A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so called operator regularization procedure are presented.

The multiplicative anomaly of three or more commuting elliptic operators

“…Even with Wodzicki's formula, the multiplicative anomaly M 2 (A, B) attached to pairs of commuting operators is in general difficult to compute. It has been explicitly computed in terms of special functions only for a handful of cases (see [El2,§2.3] and the references there). Perhaps for this reason the joint multiplicative anomaly…”