= Grand-canonical coherent-state action
{title2=$S_E=\int_0^{\beta\hbar}d\tau[\hbar\bar\psi\partial_\tau\psi+H_N-\mu\bar\psi\psi]$}
Insert coherent-state completeness between short factors of $e^{-\beta(H-\mu N)}$. The adjacent bra and ket labels determine the normal-ordered symbol $H_N(\bar\psi_{j+1},\psi_j)$, while overlaps generate the derivative term. Bosonic labels are complex and periodic in imaginary time; fermionic labels are independent <Grassmann variables> and antiperiodic. The continuum shorthand is defined by that temporal ordering and measure, not an arbitrary substitution of commuting fields for operators.<Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§§3.1–3.2, coherent-state time slices and trace]{ref}
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