Spectral theorem in noncommutative field theories: Jacobi dynamics

Abstract

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.

31 pages. Improved version to be published. Section 4 modified. Various misprints corrected

References in corpus (14)

• Algebras of distributions suitable for phase-space quantum mechanics. I
• Noncommutative Induced Gauge Theory
• Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
• Induced Gauge Theory on a Noncommutative Space
• Vacuum configurations for renormalizable non-commutative scalar models
• Emergent Gravity, Matrix Models and UV/IR Mixing
• On the vacuum states for noncommutative gauge theory
• Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom
• Renormalization of the Orientable Non-commutative Gross-Neveu Model
• Quantum gauge theories on noncommutative 3-d space
• One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model
• Noncommutative Induced Gauge Theories on Moyal Spaces
• Localization for Yang-Mills Theory on the Fuzzy Sphere
• Heat kernel expansion and induced action for the matrix model Dirac operator

Cited by in corpus (3)

• Gauge theories on quantum spaces
• Exact Partition Functions for Gauge Theories on ${\mathbb{R}}_{\lambda }^3$
• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics