0
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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