Exact Results in Quantum Theory

The spectral theory of the Laplace–Beltrami operator on Riemannian manifolds is known to be intimately related to geometric invariants such as the Einstein-Hilbert action. These relationships have inspired many developments in physics including the Chamseddine–Connes action principle in the non-commutative geometry programme. However, a priori they do only apply to the case of Euclidean signature. The physical setting of Lorentzian manifolds has in fact remained largely problematic: elliptic theory no longer applies and something different is needed.In this talk I will report on joint work on this problem with Nguyen Viet Dang. We consider perturbations of Minkowski space and more general spacetimes on which the d’Alembertian P is essentially self-adjoint (thanks to recent results by Dereziński–Siemssen, Vasy and Nakamura–Taira). It is then possible to define functions of P, and we demonstrate that their Schwartz kernels have geometric content largely analogous to the Riemannian setting. In particular, we define a Lorentzian spectral zeta function and relate one of its poles to the Einstein–Hilbert action, paralleling thus a result in Euclidian signature attributed to Connes, Kastler and Kalau–Walze.The primary consequence is that gravity can be obtained from a spectral action directly in Lorentzian signature. The proofs involve mathematical ingredients from Quantum Field Theory on curved spacetime, in particular the Feynman propagator.Zoom Meeting ID: 830 2558 9543 Passcode: 263370)