Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 2 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.