I'm going through Mark Srednicki's Quantum Field Theory. Chapter 8 on The Path Integral for the Free Field Theory includes the following:
In the presence of a classical source, $J$, the ground state-to-ground state transition amplitude is given as:
$$ Z_{0}(J) = \exp \Bigg [\frac{i}{2} \int \frac{d^4 k}{(2\pi)^4} \frac{e^{ik(x - x')}}{k^2 + m^2 -i \epsilon} \Bigg ] , $$
which is equal to
$$ Z_{0}(J) = \exp \Bigg [\frac{i}{2} \int d^4 x \; d^4 x' J(x) \triangle (x - x') J(x') \Bigg ] , $$
where we have defined the Feynmann Propagator:
$$ \triangle (x - x') = \int \frac{d^4k}{(2\pi)^4} \frac{e^{ik(x - x')}}{k^2 + m^2 -i \epsilon}. $$
The Feynman propagator is a Green’s function for the Klein-Gordon equation,
See equations (8.10) to (8.12) on http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf.
To me, this step seems more of a mathematical manipulation; perhaps because I haven'thad a course on Mathematical Methods which has dealt with Green's Function. I'm not sure:
- What motivates the introduction of the Green's Function/Feynmann Propagator; and
- What is the mathematical basis for our introducing this function?
I'd appreciate if someone could give a comprehensive yet self-contained answer that enables me, and others alike, to digest such steps. If anyone could also introduce Green's function, that'd be great.
In addition:
- If you could recommend any good textbook that deals with Green's functions, that'd be great as well. I have had a horrid time taking a look at math methods textbook since I think they're more of "here's a formula/method, now attempt/solve questions" type textbooks.
- If you could also recommend a good QFT that discusses Green's functions (in the appropriate context of course), that'd be great as well. It'd provide context to the problem at hand. Srednicki's textbook simply introduces the function with nothing else whatsoever!
Thanks.
