Propagators in Curved Spacetimes from Operator Theory

Abstract We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein–Gordon field on a Lorentzian manifold M . Typically, we assume that M is globally hyperbolic. The term propagator here refers to any Green function or bisolution of the Klein–Gordon equation pertinent to Quantum Field Theory. The off-shell setting is based on the Hilbert space $$L^2(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called in-out Feynman and out-in anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well. The on-shell setting is based on the Krein space $$\mathcal {W}_\textrm{KG}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mtext>KG</mml:mtext> </mml:msub> </mml:math> of solutions of the Klein–Gordon equation. It allows us to define 2-point functions associated with two, possibly distinct, Fock states as the Klein–Gordon kernels of projectors onto maximal uniformly positive subspaces of $$\mathcal {W}_\textrm{KG}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mtext>KG</mml:mtext> </mml:msub> </mml:math> . After a general discussion, we review a number of examples. We start with static and asymptotically static spacetimes, which are especially well suited for Quantum Field Theory. Then we discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schrödinger operators. We compare various approaches to de Sitter space where, curiously, the off-shell approach gives non-physical propagators. Finally, we discuss the universal cover of anti-de Sitter spaces, where the on-shell approach may require boundary conditions, unlike the off-shell approach.