Effective Finite Temperature Partition Function for Fields on Non-Commutative Flat Manifolds

TL;DR

This paper calculates the first quantum correction to the finite temperature partition function for a scalar field on a non-commutative flat manifold, revealing potential non-regularity in the associated zeta function.

Contribution

It introduces a method to evaluate quantum corrections on non-commutative manifolds using dimensional regularization and zeta-function techniques, addressing regularity issues.

Findings

Zeta function may be nonregular at the origin.

Regularized vacuum energy determination is discussed.

Quantum corrections depend on non-commutative geometry.

Abstract

The first quantum correction to the finite temperature partition function for a self-interacting massless scalar field on a $D-$dimensional flat manifold with $p$ non-commutative extra dimensions is evaluated by means of dimensional regularization, suplemented with zeta-function techniques. It is found that the zeta function associated with the effective one-loop operator may be nonregular at the origin. The important issue of the determination of the regularized vacuum energy, namely the first quantum correction to the energy in such case is discussed.