[Paper Review] Déformations isospectrales non compactes et théorie quantique des champs

Victor Gayral|Jul 21, 2005

Advanced Operator Algebra Research3 citations

TL;DR

This thesis develops non-compact isospectral deformations in noncommutative geometry, extending Connes' spectral triple framework to non-unital algebras and applying it to Moyal planes. It establishes Dixmier trace computations for non-compact settings, proves UV/IR mixing in noncommutative field theory, and demonstrates that spectral invariants remain unchanged under deformation, providing a rigorous non-perturbative framework for quantum field theory on noncommutative spacetimes.

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.

Motivation & Objective

• To extend Connes' spectral triple formalism to non-unital, non-compact geometries relevant for quantum field theory.
• To analyze the behavior of the heat kernel and Dixmier trace in non-compact, non-unital Moyal planes.
• To investigate UV/IR mixing in noncommutative field theories on isospectral deformations.
• To establish the invariance of spectral invariants under non-compact isospectral deformations.
• To provide a rigorous mathematical foundation for noncommutative quantum field theory using spectral geometry.

Proposed method

• Adapts the Connes-Landi and Connes-Dubois-Violette constructions of isospectral deformations to non-compact, non-unital settings.
• Applies Hilbert space techniques to analyze the deformed Moyal product and its spectral properties.
• Uses Dixmier trace and weak Schatten class theory to compute spectral invariants in non-compact geometries.
• Applies the action spectral functional to non-unital spectral triples on Moyal planes.
• Analyzes the effective action and loop contributions in noncommutative φ⁴ theory on isospectral deformations.
• Applies Diophantine conditions to control divergences in non-planar Feynman diagrams in the non-compact case.

Experimental results

Research questions

• RQ1 How can spectral triples be generalized to non-unital, non-compact noncommutative geometries?
• RQ2 What is the behavior of the heat kernel and Dixmier trace in non-compact Moyal planes?
• RQ3 Does UV/IR mixing persist in non-compact isospectral deformations of Moyal spaces?
• RQ4 How do spectral invariants transform under non-compact isospectral deformations?
• RQ5 Can the action spectral functional be consistently defined and computed in non-unital, non-compact settings?

Key findings

• The Dixmier trace is well-defined and computable for non-compact Moyal planes, extending the spectral invariance of isospectral deformations beyond compact cases.
• UV/IR mixing is shown to occur in noncommutative φ⁴ theory on non-compact isospectral deformations, confirming earlier results in a broader geometric setting.
• The non-planar contributions to the effective action in the non-compact case are controlled by Diophantine conditions on the deformation parameter θ.
• The spectral triple framework is successfully generalized to non-unital algebras, allowing the description of non-compact noncommutative spaces via spectral data.
• The action spectral functional yields finite results for the Moyal plane in the non-compact case, supporting its use in quantum field theory.
• The heat kernel trace class properties in the weak Schatten class are established, enabling rigorous trace computations in non-compact geometries.