As far as I understood, in the path integral formulation of QFT, a field configuration is modelled by a mapping
$$x \rightarrow \Psi(x)$$Where $\Psi(x)$ are 4 components, each represented by 4 grassmann numbers. We need 4 Grassmann-Numbers for every point in space, an all those numbers need to anticommute, so all in all we need 4 times infinity Grassmann-Generators. This means that $\Psi$ is a mapping from space time to an infinite dimensional exterior Algebra (I hope this is correct).
What I wonder now is: We require the Lagrangian
$$\bar{\Psi}(\partial_{\mu}\gamma^{\mu} - m)\Psi$$
To be some rather simple object, namely a Lorentz scalar, something that we can integrate over and that will give an action $S$ that can be used as a phase in the exponential function. We want to add for example products $ \bar{\Psi}A_{\mu} \gamma^{\mu} \Psi$ or terms that don't have anything to do with the exterior algebra, like $F^{\mu \nu}F_{\mu \nu}$.
How can we add products of the exterior algebra (which are somewhat two-forms) and terms that are zero forms, like the mentioned free electromagnetic field Lagrangian?If I understand correctly, $\bar{\Psi} \Psi$ is some kind of two-form, while $F^{\mu \nu}F_{\mu \nu}$ is some kind of 0-Form). How do we reconcile this?
I hope I understood the usage of Grassmann numbers correctly, If not, I'd appreciate if an answer would point out the flaw in the described usage.
