I am reading QFT in curved spacetime. In this book Wald writes that if the spacetime is stationary, there is a natural "particle interpretation" even if we have no positive frequency wave planes. As we have stationarity when we have a Killing field he uses the Lie derivative along it to get positive eigenvalues and to build a Fock space.
What happens if we take another Killing field? Would we get another equivalent Fock space?