Spectral theorem in noncommutative field theories: Jacobi dynamics

Conceptual and Technical Challenges for Quantum Gravity 2014 – Parallel session: Noncommutative Geometry and Quantum Gravity

Géré, Antoine^1 ; Wallet, Jean-Christophe^2

Dipartimento di Matematica, Universitá di Genova, Via Dodecaneso 35, Genova

I-16146, Italy^1

Laboratoire de Physique Théorique d'Orsay, CNRS, UniversitéParis-Sud 11, Bât. 210, Orsay Cedex

91405, France^2

关键词: Dirac operators;    Jacobi operators;    Kinetic operators;    Metric spaces;    Non-commutative;    Non-commutative geometry;    Spectral distances;    Spectral theory;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/634/1/012006/pdf
DOI  :  10.1088/1742-6596/634/1/012006

来源: IOP

【 摘 要 】

Jacobi operators appear as kinetic operators of several classes of noncommutative field theories (NCFT) considered recently. This paper deals with the case of bounded Jacobi operators. A set of tools mainly issued from operator and spectral theory is given in a way applicable to the study of NCFT. As an illustration, this is applied to a gauge-fixed version of the induced gauge theory on the Moyal plane expanded around a symmetric vacuum. The characterization of the spectrum of the kinetic operator is given, showing a behavior somewhat similar to a massless theory. An attempt to characterize the noncommutative geometry related to the gauge fixed action is presented. Using a Dirac operator obtained from the kinetic operator, it is shown that one can construct an even, regular, weakly real spectral triple. This spectral triple does not define a noncommutative metric space for the Connes spectral distance.