# Fermion propagator is not a Grassmann-odd object?

+ 0 like - 0 dislike
169 views

Is the following differentiation correct: $$\frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$

where $\eta$ is a Grassmann-valued field and $S_F$ is the Fermion propagator, or is the result actually with a minus sign?

This post imported from StackExchange Physics at 2014-04-13 14:30 (UCT), posted by SE-user Psycho_pr

+ 2 like - 0 dislike

The bounds of the integral have no dependence on any of the variables, and hence we may move the differential operator into the integrand,

$$\frac{\delta}{\delta \eta (z)} \int \mathrm{d}^4 y \, S_F (z-y) \eta(y) = \int \mathrm{d}^4 y \, S_F (z-y) \delta^{(4)}(z-y)$$

Evaluating the integral using the standard delta distribution identity, we obtain your result, namely $S_F(z-z)$. In this case, the final answer does not pick a minus sign, even though $\eta$ is Grassmann-valued. See Peskin and Schroeder's text on QFT for a summary of Berezin/Grassmann integration.

This post imported from StackExchange Physics at 2014-04-13 14:30 (UCT), posted by SE-user JamalS
answered Apr 11, 2014 by (895 points)

 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$\hbar$ysicsOve$\varnothing$flowThen 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.