Explicit zeta functions for bosonic and fermionic fields on a non-commutative toroidal spacetime†

Abstract

Explicit formulae for the zeta functions ζα(s) corresponding to bosonic (α = 2) and to fermionic (α = 3) quantum fields living on a non-commutative, partially toroidal spacetime are derived. Formulae for the most general case of the zeta function associated with a quadratic + linear + constant form (in Z) are obtained. They provide the analytical continuation of the zeta functions in relation to the whole complex s plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulae. As is well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function; in particular, the residua of the poles and their finite parts are explicitly given. An important novelty is the fact that simple poles show up at s = 0, as well as in other places (simple or double, depending on the number of compactified, non-compactified and non-commutative dimensions of the spacetime) where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.