Understanding the Breakdown of Path Integrals in Peskin

[malawi_glenn]

Hello

I am trying to follow how one can define a correlation function of two quantum fields using Path integrals.

I have stumbled on equation 9.16 in Peskin, where they states that the functional integral can be split into:

$$\int D \phi(x) = \int D\phi _1 (\vec{x}) \int D \phi _2 (\vec{x} ) \int _{\substack{\phi (x_1^0, \vec{x}) = \phi _1 (\vec{x} ) \\\phi (x_2^0, \vec{x}) = \phi _2 (\vec{x} )}} D \phi (x)$$

I was wondering how one can justify this break up? It is only an endpoint constraint imposed, but why do we have to integrate over the intermediate configurations?

Any additional insight or reading tips will be very welcomed! :-)

 

[CompuChip]

I'm not quite sure whether I understand your question correctly, but... isn't that the idea of a path integral?
We know the initial and final configurations, but not what happens in between. So we sum over all the possible configurations that lie between. For a point particle, for example, we fix the initial and final positions A and B but we must integrate over all possible paths that the particle could have taken to get from A to B. This includes paths very far from the classical path, which will on average cancel each other out because of the peculiar form of the integrand. So in effect, we're considering mainly configurations close to the classical one.

 

[jensa]

I'm not sure I understand the question either. I don't have Peskin with me at the moment but as I read the equation it simply states that

the integral over all possible field configurations (without constraints) = the integral over all possible field configurations with all possible boundary conditions.

As for reading tips: Negele-Orland Quantum Many-Particle Systems has some very nice technical discussions about functional integrals.

 

[malawi_glenn]

Well thank your for your explanations, I will try to digest this :-)