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  Doubt in Path integral equation

+ 1 like - 0 dislike

In pokorski's "Gauge Field Theories" book, page 108 we find equation (2.87)


where we are dealing with some (for our purposes unspecified) theory of a real scalar field $\phi$, where $S_0$ is the free part of the action, while $S_I$ is the interaction part; and $J$ is the current of that appears in the generating functional. Can anybody make explicit all the necessary steps to go from the left hand side to the right?

asked Dec 16, 2015 in Theoretical Physics by Dmitry hand me the Kalashnikov (735 points) [ no revision ]

There really aren't many intermediate steps, it's fairly straightforward to see $\int{}\mathcal{D}\phi{}F[\phi]e^{iS_0[\phi]+i\int{}d^4y\,\phi(y)J(y)}=F[-i\frac{\delta}{\delta{}J(x)}]\int{}\mathcal{D}\phi{}e^{iS_0[\phi]+i\int{}d^4y\,\phi(y)J(y)}$ for any functional F. Maybe it's helpful to first think of the case when F is a polynomial.

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