Proof of Local Lorentz Invariance of Feynman Propagator in Curved Spaces

[Thor90]

I am looking for a proof that the Feynman propagator is locally a lorentz invariant (at least for scalar fields) also in curved space-times if the background geometry is smooth enough.
I mean, since it is of course a lorentz invariant on flat spaces, this should also be true if a choose a sufficiently small portion of the space and the background geometry is sufficiently smooth, even if it should contain non-local terms when evaluated between well distanced spacetime points.
Some advice on where I can find something like that?

 

[Ambitwistor]

Hello,

Thank you for your question. The proof that the Feynman propagator is locally Lorentz invariant for scalar fields in curved space-times can be found in several sources.

One of the most commonly cited sources is the book "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics" by Robert M. Wald. In Chapter 2, he discusses the invariance of the Feynman propagator under local Lorentz transformations in curved space-time.

Another source is the paper "Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations" by Christian Gerard. In Section 3.3, he discusses the local Lorentz invariance of the Feynman propagator in curved space-time.

In general, the proof relies on the fact that the Feynman propagator is a Green's function, which is a solution to the Klein-Gordon equation. The Klein-Gordon equation is a relativistic wave equation that is invariant under local Lorentz transformations.

I hope this helps. Good luck with your research!