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A correlation function is a statistical correlation between random variables at two different points in space, time, or other parameter space, usually as a function of the variable distance between these points. In QFT, field autocorrelation functions are propagators, so use the "propagator" tag, instead.
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Proof for a time-ordering equation in Negele & Orland (1998)
I accepted lionelbrits' answer because I've already figured it out with his/her hint to "slide the operator past". I will write now a more detailed version for others who are interested in the identy …
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Proof for a time-ordering equation in Negele & Orland (1998)
Let $T$ be the time-ordering operator which orders operators $A_1(t_1), A_2(t_2), \ldots$ such that the time parameter decreases from left to right:
$$T[A_1(t_1) A_2(t_2)] = A_2(t_2) A_1(t_1) \text{ …