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### Definition of vacua in QFT in generic spacetimes

I have been learning QFT in curved spaces from various sources (Birrell/Davies, Tom/Parker, some papers), and one thing that confuses me the most is the choice of vacua in various spacetimes, and the ...
Define a single-particle Green's function as $$i\hbar G(xt;x't') = \langle x| e^{-iH(t-t')/\hbar} | x'\rangle.$$ By inserting the completeness relation, we have \begin{...