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Be careful that there are some integrals, which implies that $S_I^n\neq 0$ ! For instance $S_I^2=\int_{x_1,x_2,x_3,x_4} (\psi_1^\dagger\psi_1)V_{12}(\psi_2^\dagger\psi_2)(\psi_3^\dagger\psi_3)V_{34} …

The idea is correct (it's called the Hubbard-Stratonovich transformation), but I can't say more without the details of the action. It is discussed in any good textbook on quantum field theory for cond …

The problem was in assuming that because one has $$ (|b|^2-|a|^2)\theta^*=0,$$ then $|b|^2=|a|^2$. Indeed, the only thing it tells you is that $$|b|^2=|a|^2+\sigma \theta^*,$$ where $\sigma$ can be an …

When writing a path integral starting from a Hamiltonian, one must write down all operators in normal order, as the Grassmann variables correspond to coherent states of the fermionic creation/annihila …

I think it is a general fact about grassmannian field, and this has nothing to do with Lorentz invariance or other symmetries (you can invent a lot of QFTs without this kind of symmetry, but the VEV o …