For a Lagrangian written in fermionic (Grassmann) variables $\psi$ and $\bar{\psi}$, in a path integral formulation I am supposed to treat these two as independent variables with no relationship. However, if I consider transforming them under various symmetry operations (eg. parity, charge conjugation, time reversal), I am supposed to take the prescription $\bar{\psi} = \psi^\dagger \gamma^0$, where I interpret $\psi^\dagger$ to be the Hermitian adjoint of $\psi$.
How does one reconcile these seemingly paradoxical statements?
