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Feb 4 at 0:46 comment added Igor Khavkine I only have one comment. If you can think of a way to compute spatially separated correlations, you can think of similar reasons to compute temporally separated correlations.
Feb 2 at 22:41 comment added MathMath Schwinger functions are evaluated at different times, and this is what makes the whole difference in my opinion. In other words, you cannot simply motivate these functions by arguing that one wants to compute averages, because the fields are taken at different times. And this led me to this question. If one is interested in Schwinger functions in the first place, one must have some physical reason other than "computing averages".
Feb 2 at 22:39 comment added MathMath Igor, thanks for your answer. I like your comments because this is what made me think about these questions in the first case. In fact, I was trying to use the analogy with statistical mechanics, where one wants to compute expectation of fields (which are pretty natural objects to study, of course). However, I came to the conclusion that the natural extension to quantum mechanics would be either expectation of time-independent creation and annihilation operators (if one is not interested in time evolution) or expectations of time dependent operators but evaluated at the same time. (cont)
Feb 2 at 17:03 history answered Igor Khavkine CC BY-SA 4.0