# Doubt in Path integral equation

+ 1 like - 0 dislike
162 views

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

$$\int{}\mathcal{D}\phi{}e^{iS_0[\phi]+iS_I[\phi]+i\int{}d^4y\,\phi(y)J(y)}=e^{iS_I[-i\frac{\delta}{\delta{}J(x)}]}\int{}\mathcal{D}\phi{}e^{iS_0[\phi]+i\int{}d^4y\,\phi(y)J(y)}$$

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?

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.