An index theorem on asymptotically static spacetimes with compact Cauchy surface

Abstract

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat-Hörmander. The proof combines methods from time-dependent scattering theory with a variant of Egorov's theorem for pseudo-differential hyperbolic systems.

Diagonalization of elliptic systems via pseudodifferential projections

“…Being adjusted to pseudodifferential projections, our construction is particularly suitable to study the relation between the spectrum of the original operator and that of its diagonalized version. We believe that the self-contained explicit construction presented here will prove useful in applications -for example, in the context of quantum field theory in curved spacetime, compare [4,16,5] and [25,26,27,49] -and be more accessible for the wider mathematical physics community. From a theoretical perspective, this paper complements the analysis of invariant subspaces of elliptic systems carried out in [13,14], and the spectral results therein.…”

Diagonalization of elliptic systems via pseudodifferential projections

“…Being adjusted to pseudodifferential projections, our construction is particularly suitable to study the relation between the spectrum of the original operator and that of its diagonalized version. We believe that the self-contained explicit construction presented here will prove useful in applications -see, for example, [4,15,5] and [24,25,26,43] -and be more accessible for the wider mathematical physics community. From a theoretical perspective, this paper complements the analysis of invariant subspaces of elliptic systems carried out in [12,13], and the spectral results therein.…”

Dynamical residues of Lorentzian spectral zeta functions

“…The Feynman wavefront set condition plays an important role in various developments connecting the global theory of hyperbolic operators with local geometry, in particular in works on index theory by Bär-Strohmaier and other authors [3,4,59], and on trace formulas and Weyl laws by Strohmaier-Zelditch [63,65,64] (including a spectral-theoretical formula for the scalar curvature).…”