Explicit Zeta Functions for Bosonic and Fermionic Fields on a Noncommutative Toroidal Spacetime

TL;DR

This paper derives explicit formulas for zeta functions of bosonic and fermionic fields on noncommutative toroidal spacetimes, providing analytical continuations and revealing novel pole structures at s=0 that impact regularization methods.

Contribution

It introduces new explicit formulas for zeta functions on noncommutative spacetimes, including cases with quadratic, linear, and constant forms, and uncovers novel pole behaviors at s=0.

Findings

Explicit formulas for zeta functions with exponential convergence series.

Analytical continuation of zeta functions to the whole complex plane.

Discovery of simple poles at s=0 affecting regularization techniques.

Abstract

Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are obtained. They provide the analytical continuation of the zeta functions in question to the whole complex $s-$plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulas. As 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 there. 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, noncompactified, and noncommutative dimensions of the spacetime), where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.