D\'{e}formations isospectrales non compactes et th\'{e}orie quantique des champs

TL;DR

This thesis explores isopectral deformations within noncommutative geometry, constructing spectral triples for non-compact quantum spaces like Moyal planes, and analyzing their implications for quantum field theory and UV/IR mixing phenomena.

Contribution

It develops a framework for non-unital spectral triples on non-compact quantum spaces and investigates their geometric and physical properties, including UV/IR mixing in quantum field theory.

Findings

Spectral and classical dimensions coincide for these quantum spaces.

UV/IR mixing is intrinsic to all such deformations, affecting renormalization.

New manifestations of UV/IR mixing relate to geometric and arithmetic properties.

Abstract

The aim of this thesis is to study the isopectral deformations from the point of view of Alain Connes' noncommutative geometry. This class of quantum spaces constituts a curved space generalisation of Moyal planes and noncommutative tori. First of all, we look at the construction of non-unital spectral triples, for which we propose modified axioms. We then check that Moyal planes fit into this axiomatic framework, and give the keypoints for the construction of non-unital spectral triples from generic non-compact isospectral deformations. To this end, numerous analytical tools on non-compact Riemannian manifolds are developped. Thanks to Dixmier traces computations, we show that their spectral and classical dimensions coincide. In a second time, we study certain features of quantum fields theory on curved isospectral deformations, with a particular view on the ultraviolet infrared mixing phenomenon. We show its intrinsic nature for all such quantum spaces (compacts or not, periodic or not deformations), and we study its consequences on the renormalisability. In particular, the behaviour of Green functions of the planar and non-planar sectors is understood in term of on- and off-diagonal heat kernel contributions. We also see new or inner manifestations of the UV/IR mixing, related to the geometric properties of those quantum spaces and to the arithmetic nature of the deformation parameters.