Feynman Propagators on Static Spacetimes

Abstract

We consider the Klein-Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on $C_{\mathrm{c}}^{\infty }$. We discuss various distinguished inverses and bisolutions of the Klein-Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein-Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schrödinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein-Gordon operator.

19 pages

References in corpus (5)

• Quantum out-states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property
• On a Recent Construction of "Vacuum-like" Quantum Field States in Curved Spacetime
• Wave Equations on Lorentzian Manifolds and Quantization
• On the Spectral Decomposition of Dichotomous and Bisectorial Operators
• Feynman propagators and Hadamard states from scattering data for the Klein-Gordon equation on asymptotically Minkowski spacetimes

Cited by in corpus (15)

• Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
• Limiting absorption principle and equivalence of Feynman propagators on asymptotically Minkowski spacetimes
• A remark on the essential self-adjointness for Klein-Gordon type operators
• Dynamical residues of Lorentzian spectral zeta functions
• Essential self-adjointness for the Klein-Gordon type operators on asymptotically static spacetime
• Wick rotation of the time variables for two-point functions on analytic backgrounds
• Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups
• An index theorem on asymptotically static spacetimes with compact Cauchy surface
• Green Operators in Low Regularity Spacetimes and Quantum Field Theory
• Spatially flat FLRW spacetimes with a Big Bang from matrix geometry
• Propagators in curved spacetimes from operator theory
• Analytic semigroups approaching a Schrödinger group on real foliated metric manifolds
• Equality of the Hilbert Hamiltonian and the canonical Hamiltonian for gauge theories in a static spacetime
• Adiabatic Ground States in Non-Smooth Spacetimes
• Remarks on the geodesically completeness and the smoothing effect on asymptotically Minkowski spacetimes