Non-commutative heat kernel

TL;DR

This paper extends heat kernel analysis to non-commutative spaces with Moyal star products, deriving an asymptotic expansion, explicit coefficients, and identifying UV/IR mixing phenomena.

Contribution

It introduces a non-commutative generalization of heat kernel expansion, explicitly computes the first four coefficients, and explores UV/IR mixing effects.

Findings

Established existence of a heat kernel asymptotic expansion for non-commutative operators.

Explicitly calculated the first four coefficients of the expansion.

Identified an analog of UV/IR mixing in the non-commutative setting.

Abstract

We consider a natural generalisation of the Laplace type operators for the case of non-commutative (Moyal star) product. We demonstrate existence of a power law asymptotic expansion for the heat kernel of such operators on T^n. First four coefficients of this expansion are calculated explicitly. We also find an analog of the UV/IR mixing phenomenon when analysing the localised heat kernel.