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Change of variable for functional integration

+ 4 like - 0 dislike
107 views

First of all, could someone suggest a book on Functional Integration with emphasis on QFT? It's difficult to find a good one and had really only access to Peskin & Schroeder (in which it isn't discussed in much length) and Ramamurti Shankar (which is really good but only covers the basics).

Second, I'm actually having trouble with the change of variable that P&S does in p.285,

\[\cal{D}\phi\left(x\right)=\prod_{i} d\phi\left(x_{i}\right)\]

\[\cal{D}\phi\left(x\right)=\prod _{k^{0}_{n}\gt0}d\Re\phi\left(k_{n}\right)d\Im\phi\left(k_{n}\right)\]

The first line is simply the definition of the functional integration. But in the second one he did a change of variable \(\phi\left(k\right)\rightarrow\{\Re\phi\left(k\right),\Im\phi\left(k\right)\}\)and restricted the time part of the momentum to positive values. I understand why this is needed but I didn't understand formally how to go from one to the other. What I understood is that he doubled the number of differentials so he restricted the time component to obtain the same overall number number of differentials, and, in the exponential (which is the functional integration's integrand), he doubled the number of variables when \(|\phi\left(k\right)|^{2}\rightarrow |\Re\phi\left(k\right)|^{2}+|\Im\phi\left(k\right)|^{2}\).  Any way, I need more information about what is actually happening.

Third, I had an idea on how to proceed in another way to solve the same functional integral but I need to know how to change variables, so, how do I proceed to change the integration variable if I want to let \(\phi\left(x\right)\rightarrow f\left(x\right)+g\left(x\right)\)?

Thanks

asked Nov 19, 2015 in Theoretical Physics by pniau7 (25 points) [ no revision ]
recategorized Nov 19, 2015 by Dilaton

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