Splitting up of functional integral (Peskin and Schroeder)

[center o bass]

I'm reading about path integrals in Peskin and Schroeder's Introduction to Quantum field theory and there is a few things in the text which I find puzzling. At page 283 in the section about correlation functions we are considering the object (equation 9.15)

$$\int D\phi(x) \phi(x_1) \phi(x_2) \exp(i\int_{-T}^T d^4x L(\phi)).$$

Then the author break up the integral into

$$\int D\phi_1(\vec x) \int D\phi_2(\vec x) \int_{\phi(x_1^0, \vec x) = \phi_1(\vec x), \phi(x_1^0, \vec x) = \phi_2(\vec x)} D\phi(x).$$

Now that I do not have that much experience with functional integration I was wondering about the justification and reasoning behind this step. Pesking and Schroeder can be a bit brief in places. Does anyone know of any other books with a bit more detail on path integrals which also derives the Feynman rules?

 

[vanhees71]

The correct formula reads
$$\int D\phi_1(\vec x) \int D\phi_2(\vec x) \int_{\phi(x_1^0, \vec x) = \phi_1(\vec x), \phi(x_2^0, \vec x) = \phi_2(\vec x)} D\phi(x).$$
The idea is to first integrate over all fields with the contraints given at times $x_1^0$ and$x_2^0$ and then to integrate over all the possible field configurations at these times. The only reason to do this is to prove the Feynman-Kac formula.

I think it's easier to comprehend working with the (naive) lattice version of the path integral. The calculation is shown in the first chapter of my QFT manuscript for the easier case of first-quantized non-relativistic mechanics. The proof for the quantum-field case is the very same:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

A great book using nearly only the path-integral formulation to relativistic QFT is

Bailin, Love, Gauge Theories

Other than the title suggests, it's a nice introduction to QFT not only to gauge theories (which of course are indeed the most important QFTs :-)).

 

[center o bass]

Thanks for your help! :) I will certainly check out the book and your manuscript.