Abstract
We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator □g is known to be essentially self-adjoint. We define complex powers .□g – i"/–α by functional calculus, and show that the trace density exists as a meromorphic function of α. 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 work in Lorentzian signature directly and do not rely on transition from the Euclidean setting.
| Original language | English |
|---|---|
| Pages (from-to) | 971-1054 |
| Number of pages | 84 |
| Journal | Journal of the European Mathematical Society |
| Volume | 27 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 16 Dec 2023 |
| Externally published | Yes |
Bibliographical note
Funding Information:The authors are particularly grateful to Fabien Besnard, Christian Brouder, Jan Derezin\u0301ski, Peter Hintz, Rapha\u00EBl Ponge, Kouichi Taira, Gunther Uhlmann and Andr\u00E1s Vasy for helpful discussions. We thank the MSRI in Berkeley and the Mittag\u2013Leffler Institute in Djursholm for their kind hospitality during thematic programs and workshops in 2019\u201320. Support from the grant ANR-16-CE40-0012-01 is gratefully acknowledged. N.V.D. acknowledges the support of the Institut Universitaire de France.
Publisher Copyright:
© 2023 European Mathematical Society.