- #1
sizzleiah
- 15
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Hi there.
I've just finished reading chapter 2 of Peskin and Schroeder, and I managed to follow all of their calculations - with one exception:
I'm not sure how P&S arrive at the integral in equation (2.52) (page 27) from the previous step in the calculation of D(x-y).
We're trying to calculate [tex]D(x-y)=<0|\phi (x) \phi (y) |0>[/tex] for a real Klein-Gordon scalar field [tex]\phi[/tex], where [tex]x-y[/tex] is purely spatial.
Getting to the step right before eq. (2.52) is easy enough - it's just a standard integration in spherical coordinates. Then P&S make branch cuts to create a simply connected domain, so that they can apply path independence to the contour integration. I'm ok with all of that, but then they lose me when they write down the integral in eq. (2.52). It's confusing to me for a couple of reasons. One is that I'm not entirely sure how to deal with a contour that goes off to infinity in this way - where we can't restrict the variable of integration to be real (doesn't the complex plane only have one infinity?). Another is that it seems that for the lower limit of the integration to be valid, P&S are claiming that we have [tex]p=i m[/tex]. Are they implying that we should be integrating along the branch cut? This seems very strange to me. I'm obviously no complex analyst, but I knew enough to be able to understand fairly easily what they did on the next few pages with the Feynman propagator. So...what am I missing?
Thanks!
I've just finished reading chapter 2 of Peskin and Schroeder, and I managed to follow all of their calculations - with one exception:
Homework Statement
I'm not sure how P&S arrive at the integral in equation (2.52) (page 27) from the previous step in the calculation of D(x-y).
Homework Equations
We're trying to calculate [tex]D(x-y)=<0|\phi (x) \phi (y) |0>[/tex] for a real Klein-Gordon scalar field [tex]\phi[/tex], where [tex]x-y[/tex] is purely spatial.
The Attempt at a Solution
Getting to the step right before eq. (2.52) is easy enough - it's just a standard integration in spherical coordinates. Then P&S make branch cuts to create a simply connected domain, so that they can apply path independence to the contour integration. I'm ok with all of that, but then they lose me when they write down the integral in eq. (2.52). It's confusing to me for a couple of reasons. One is that I'm not entirely sure how to deal with a contour that goes off to infinity in this way - where we can't restrict the variable of integration to be real (doesn't the complex plane only have one infinity?). Another is that it seems that for the lower limit of the integration to be valid, P&S are claiming that we have [tex]p=i m[/tex]. Are they implying that we should be integrating along the branch cut? This seems very strange to me. I'm obviously no complex analyst, but I knew enough to be able to understand fairly easily what they did on the next few pages with the Feynman propagator. So...what am I missing?
Thanks!