Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

Abstract

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator ${\square }_g$ is known to be essentially self-adjoint. We define complex powers $({\square }_g-i\epsilon {)}^{-\alpha }$ by functional calculus, and show that the trace density exists as a meromorphic function of $\alpha$. We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine-Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments operate in Lorentzian signature directly and do not rely on a transition from the Euclidean setting. The results hold also true in the case of ultrastatic spacetimes.

65 pages; v4: Prop. 3.16 fixed and other minor corrections, accepted in J. Eur. Math. Soc. (JEMS)

References in corpus (7)

• The noncommutative residue and canonical trace in the light of Stokes' and continuity properties
• Limiting absorption principle and equivalence of Feynman propagators on asymptotically Minkowski spacetimes
• Spectral Action in Noncommutative Geometry
• Essential self-adjointness of real principal type operators
• Dynamical residues of Lorentzian spectral zeta functions
• Local Index Theory for Lorentzian Manifolds
• Strichartz estimates for non-degenerate Schrödinger equations

Cited by in corpus (5)

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• Particle models from special Jordan backgrounds and spectral triples
• Spectral action and the electroweak $\theta$-terms for the Standard Model without fermion doubling