Retarded and advanced Green's function and Feynman propagator 
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*Is there a use of advanced Green's functions? If yes then when or in which context?


*In quantum field theory, why do we always use Feynman's prescription for finding the propagator and not the retarded one?


*Why does Feynman's prescription for the propagator not make sense in a classical field theory?
 A: David Tong notes in his lecture notes that the advanced propagator "is useful if we know the end point of a field configuration and want to figure out where it came from".
A: *

*The full Green's function of an equation like the Klein-Gordon equation is the difference of the retarded and advanced Green's functions. It is only when the equation in question is an equation involving time that we often discard the advanced anti-causal part to get physically sensible solutions. In QFT, the full Green's function appears for example as the expectation value of the commutator of a scalar field with itself.

*The Feynman propagator is what corresponds to the expectation value of the time ordered product of a field with itself, and it is this time ordered expectation value that appears in the LSZ reduction formula for scattering amplitudes.
A: I recommend you read how these division of the Fenyman propagator appear for scalar fields, in advanced and delayed components, through the expected value, in the vacuum state for the temporally ordered product, of a scalar field with itself.
Read in QFT Ryder pages 193-194
