Spectral Zeta Functions in Non-Commutative Spacetimes

TL;DR

This paper derives formulas for spectral zeta functions in non-commutative spacetimes, analyzing their poles and implications for regularization in quantum field theories on such backgrounds.

Contribution

It provides the most general formulas for zeta functions associated with quadratic+linear+constant forms and applies them to quantum fields on non-commutative spacetimes.

Findings

Spectral zeta functions exhibit poles at s=0 and other points.

Poles can be simple or double depending on spacetime dimensions.

Regularization procedures face challenges due to these poles.

Abstract

Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are given. As examples, the spectral 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 investigated. 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). This poses a challenge to the zeta-function regularization procedure.