Coherent-state time slicing

Insert the coherent-state resolution of identity between short-time evolution operators. For a normally ordered Hamiltonian, the kernel uses the next bra label and current ket label: $⟨{\psi }_{j+1}|e^{-ϵH}|{\psi }_j⟩=\exp ({\bar{\psi }}_{j+1}{\psi }_j-ϵH({\bar{\psi }}_{j+1},{\psi }_j)+O({ϵ}^2))$. Together with the Gaussian measure this yields the first-order derivative term $\bar{\psi }{\partial }_{\tau }\psi$. The adjacent-label prescription retains operator ordering information that a formal continuum expression alone can hide.

The thermal trace closes bosonic coherent-state paths periodically and fermionic coherent-state paths antiperiodically: $\psi (\beta )=\zeta \psi (0)$ and $\bar{\psi }(\beta )=\zeta \bar{\psi }(0)$, with $\zeta =1$ for bosons and $-1$ for fermions. The minus sign follows from the fermionic coherent-state trace, not from choosing a different Hamiltonian. These boundary conditions determine the allowed Matsubara frequencies.