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

Nguyen Viet Dang, Michał Wrochna
December 01, 2020
We consider the class of non-trapping Lorentzian scattering spaces, 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, proving therefore a Lorentzian analogue of the Kastler–Kalau–Walze theorem. We interpret this result in terms of 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.