In many text books, in the mean-field treatment of weakly interacting Fermi gas, the standard operation is to first write down the interacting action in terms of fermion densities
$$ S_1 \propto \int D[\psi] \sum_q \rho_q \rho_{-q} $$
and then use Hubbard-Stratonovich transformation to decouple the density.
Here Hubbard-Stratonovich rely on Gaussian integral for complex variable, and most textbooks argued that
$$ \rho_q = \sum_p \bar{\psi}_{p+q} \psi_p $$
is a real variable because of the commutation relation. But strictly following the rule of Grassmann number, Grassmann number in the exponential is
$$ \exp{\bar{\psi} \psi} = 1 + \bar{\psi} \psi $$
How could we explain this discrepancy?
