What are Feynman boundary conditions?

Question

For a hyperbolic second order linear PDE such as the Klein-Gordon equation, to get a unique solution we must specify the value of the field at spatial infinity, as well the value and normal derivative on some space-like surface. The retarded Green's function satisfies the BC of zero at spatial infinity, as well as zero field and normal derivative at some spatial slice in the far past. The advanced Green's function puts that spatial slice in the far future. What BC's does the Feynman Green's function satisfy?

For shimrod; How do you uniquely split a function into negative and positive frequency parts? My understanding is that this requires Fourier transforming the function in time, however a function that does not go to zero at t=+ and - infinity cannot be Fourier transformed, and the Feynman Green's function (and the retarted or advanced Green's functions, for that matter) doesn't go to zero at t=+ and - infinity. Thanks for the answer by the way.

Answer 1

I don't remember exactly, but as I recall, the Feynman Green's function is a linear combination of the retarded and advanced (because Feynman diagrams are time-order insensitive). That is, the FGF has a particle part and an antiparticle part, defined (in wavenumber representation) by different signs on the ie in the denominator. Since these are what determine the boundary conditions in time, that info should help.

Answer 2

Any spacetime function f(x,t) can be written uniquely as the sum of a positive frequency component and a negative frequency component by restricting the integral in its Fourier representation to the positive (exp(-iEt)) or negative (exp(+iEt)) frequencies respectively.

The boundary condition for the Feynman propagator is that for t--> +oo the negative frequency component is zero, and for t--> -oo, the positive frequency component is zero.

This boundary condition is similar to and has the same strength as the ones for the retarded and advanced Green's function except that the two parts (+ and - frequencies) are differentiated from each other, the former being the same as the boundary condition for the retarded Green's function and the latter being the same as the one for the advanced Green's function.