Five Physical Applications of the Inhomogeneous Generalized Epstein–Hurwitz Zeta Functions

Abstract

In this chapter some explicit applications to the regularization, by means of Hurwitz zeta-functions considered in previous chapters, of different problems which have appeared recently in the physical literature, are considered. This kind of zeta functions show up profusely in many applications of quantum physics where regularization techniques are needed; in particular, when one deals with a massive quantum field theory in a (totally or partially) compactified spacetime (spherical or toroidal compactification, for instance). Aside from the interest that a detailed mathematical study of these functions may have on its own (e.g., in number theory), what is actually needed for most physical applications is always the numerical value of these functions, and of their derivatives with respect to the main variable and some of the accompanying parameters, before proceeding with the analytical continuation of the functions and also with the treatment of the (possible) poles. This procedure has already been checked in several situations, where the result is usually given in terms of sums of Hurwitz zeta functions and, generically, under the form of asymptotic expansions. Actually, this situation has been described in some detail in Chap. 4 already. In the present one, five additional, completely different applications are dealt with. Namely, in the first section, the calculation of the Casimir energy corresponding to compact universes without boundary. In the second, the calculation of the sum over one-loop integrals which yields the cross section of a scattering process in a Kaluza–Klein model with spherical compactification. Another application is the study of the critical behavior of a field theory at non-zero temperature. As the fourth example of this chapter, the quantization of two-dimensional gravity by means of the Wheeler–De Witt equation is discussed. Finally, the last case considered is the use of the spectral zeta function for both scalar and vector fields on a spacetime with a noncommutative toroidal part.