Grand-canonical coherent-state action ($S_E={\int }_0^{\beta \hslash }d\tau [\hslash \bar{\psi }{\partial }_{\tau }\psi +H_N-\mu \bar{\psi }\psi ]$)

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.^[ref]