Michael E. Peskin & Daniel V. Schroeder, An Introduction to Quantum Field Theory, (Perseus Books, 1995) – Chapter 2.
Although we’ve already gone through a derivation of the Feynman propagator for the Klein-Gordon field (see here and subsequent posts, listed as pingbacks at the bottom of that post), it’s worth revisiting it here to review the derivation in P&S, which is quite a bit shorter now that we have a few tools ready to deal with it.
In our original derivation of the Green’s function for the Klein-Gordon equation we defined the function
The integral is done in the complex plane as a contour integral where the contour runs along the real axis from to , skirting the poles at with little semicircular arcs that go above the poles. For , the contour is closed with a large semicircle in the lower half plane so that the contour encloses both poles, with the result that is non-zero (and the contour is clockwise, which cancels the in the integrand). For , the contour is closed with a large semicircle in the upper half plane so that the contour excludes both poles, with the result that is zero.
We can choose 3 other ways of skirting the two poles: we could use semicircles that go under both poles, or over the pole at and under the pole at , or vice versa. The last choice (under and over ) gives the Feynman propagator. In this case, if we close the contour using a large semicircle in the lower half plane, which excludes the pole and includes the pole. If , we close the contour in the upper half plane, which includes the pole and excludes the pole. In either case, the integral includes only one pole, and the result is a propagator that we met when discussing causality.
Take first. Then the contour is in the lower half plane, and is clockwise. The residue at is
The integral is the same as 1, but with a different contour, so we’ll call it . Doing the integral around this contour gives
For , the contour is in the upper half plane and is now counterclockwise and the residue is at :
The integral to be done is now
Note that we now multiply the residue by because the contour is now counterclockwise. We can now write for the exponent
Since the spatial part of the exponent is integrated over all , we can replace by , or equivalently, by , to get
where is the time-ordering symbol which places the field with the later time first.
The Feynman propagator 15 is still a Green’s function for the Klein-Gordon operator, as we can show by following through the same steps we did earlier for . Applying the Klein-Gordon operator to the first term in 15 we get (remember that all derivatives are with respect to , not ):
Doing the same to the second term gives the same result with opposite signs on the delta functions because
Thus we get
Combining the two gives
which is the same as the result we got for .