On the ultraviolet behaviour of quantum fields over noncommutative manifolds

TL;DR

This paper develops a Hamiltonian framework for fermion quantum fields on noncommutative manifolds, analyzing their ultraviolet behavior, especially over noncommutative 3-tori, and discusses implications for other noncommutative geometries.

Contribution

It introduces a novel Hamiltonian approach for fermion fields in noncommutative geometry based on Connes' axioms and Fredholm modules, extending quantum field analysis to noncommutative spaces.

Findings

Ultraviolet behavior of fields over noncommutative 3-tori analyzed

Framework connects Fredholm modules with canonical quantization

Discussion on expected behaviors on other noncommutative manifolds

Abstract

By exploiting the relation between Fredholm modules and the Segal-Shale-Stinespring version of canonical quantization, and taking as starting point the first-quantized fields described by Connes' axioms for noncommutative spin geometries, a Hamiltonian framework for fermion quantum fields over noncommutative manifolds is introduced. We analyze the ultraviolet behaviour of second-quantized fields over noncommutative 3-tori, and discuss what behaviour should be expected on other noncommutative spin manifolds.