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

TL;DR

This paper develops a framework for defining and analyzing complex powers of the wave operator on Lorentzian spacetimes, relating spectral properties to geometric quantities and proposing a Lorentzian spectral action model.

Contribution

It introduces a method to define complex powers of the wave operator on Lorentzian manifolds and relates their trace densities to geometric invariants, forming a Lorentzian spectral action.

Findings

Trace density as a meromorphic function of alpha

Poles related to scalar curvature

Spectral action principle for Lorentzian spacetimes

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 $(\square_g-i\varepsilon)^{-\alpha}$ 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.