Spectral Zeta Functions in Non-Commutative Spacetimes

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 $ζ_α(s)$ corresponding to bosonic ($α=2$) and to fermionic ($α=3$) 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.