An evolution equation approach to the Klein–Gordon operator on curved spacetime

Abstract

We develop a theory of the Klein-Gordon equation on curved spacetimes. Our main tool is the method of (non-autonomous) evolution equations on Hilbert spaces. This approach allows us to treat low regularity of the metric, of the electromagnetic potential and of the scalar potential. Our main goal is a construction of various kinds of propagators needed in quantum field theory.2010 Mathematics Subject Classification: 35L05, 47D06, 58J45, 81Q10, 81T20. loc (Σ)) with ∂ i u ∈ C(R; L 2 loc (Σ)) and Ku ∈ L 2 loc (M), thenProof. Note that, as a subset of Σ, we have suppẼ[u](t) supp u(t) ∪ supp u(t). We show that u(x) 0 for any) by an application of Prop. E.2 for all smoothˆ ≻ . For any such x, J − (x) does not intersect (supp Ku ∩ M + ) ∪ {t}× suppẼ[u](t). Prop. E.2 now shows that u vanishes in J − (x) ∩ M + and thus also at x. We have thus shown that supp u ∩ M ± ⊂ J ± (supp Ku ∩ M ± ) ∪ {t}× supp u(t) ∪ supp u(t) for all smoothˆ ≻ . It follows that (E.5) holds, because a vector is -causal if and only if it isˆ -timelike for all smoothˆ ≻ by Prop. 1.5 of [7] and thereforeThe embedding for J − follows by time reversal and remaining embedding by the union of the embeddings for J + and J − .

Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary

“…Remark 34: We expect that a result similar in spirit to Theorem 30 can be derived also in the framework of [DS17], up to a suitable modification of the geometrical setting considered therein. This would allow to consider the wave operator with the insertion of an electromagnetic potential and with possibly low regularity of both the metric and the electromagnetic potential in the sense of [DS17]. We are currently investigating this topic.…”

Feynman propagators on static spacetimes

“…There exists large literature about the Klein-Gordon equation on curved spacetimes, see e.g. [1,8,21]. However, we think that our paper offers some novel conceptual points on this subject.…”

“…However, the questions that we pose (the self-adjointness of the Klein-Gordon operator, the existence of the boundary values of the resolvent and its relationship to the Feynman propagator) can be formulated for non-static spacetimes. Thus, our paper points towards non-trivial further questions, of physical relevance, which we plan to investigate [8,9]. Note in particular, that the question of the self-adjointness of a non-static Klein-Gordon operator is much more difficult from the static case.…”

The Massive Feynman Propagator on Asymptotically Minkowski Spacetimes II

“…Here Φ s is the bicharacteristic flow acting on the left component of diag T * M (i.e., the diagonal in (T * M × T * M ) \o), and π : N × N → N is the projection to the left component. This leaves open the question of the existence of a canonical Feynman inverse G F satisfying (1.1) on a globally hyperbolic spacetime (M, g), see [DS2] for a discussion.…”

“…Recently, Vasy [Va2] considered the same problem by working directly on the scalar operator P using microlocal methods, motivated by the issue of essential self-adjointness of P (see [DS1]). Solving in this setting a conjecture by Dereziński and Siemssen [DS2,D], he constructed the Feynman inverse G F between microlocal Sobolev spaces as the boundary value (P − i0) −1 of the resolvent of P .…”