= Solution
A <coherent state> is an eigenstate of an annihilation operator. Bosonic unnormalized states have exponential overlap and the Gaussian resolution of identity used above; their overcompleteness supplies a phase-space integration measure, not an orthonormal state count. For a normal-ordered many-body Hamiltonian, coherent-state insertion gives the <bosonic coherent-state path integral>
$$
Z=\int_{\rm periodic}\mathcal D(\bar\phi,\phi)\exp\!\left[-\int_0^\beta d\tau\left(\sum_i\bar\phi_i\partial_\tau\phi_i+H_N(\bar\phi,\phi)-\mu\sum_i\bar\phi_i\phi_i\right)\right].
$$
Constants in $H_N$ must be retained. The exact time-sliced prescription places creation and annihilation symbols on adjacent slices; discarding that prescription can change ordering corrections, as the oscillator example demonstrates. The derivative term is first order because the fields describe phase space.
Fermionic coherent states instead use <Grassmann variables>. Their Gaussian identity is $\int d\bar\psi\,d\psi\,e^{-\bar\psi A\psi}=\det A$, whereas complex bosons give $(\det A)^{-1}$. Anticommutation in the trace produces \b[antiperiodic] fields. Accordingly bosons have $\nu_n=2\pi n/\beta$, fermions $\nu_n=(2n+1)\pi/\beta$, in these inverse-energy time units.
For one mode of grand-canonical energy $\xi$, the products give $Z_B=(1-e^{-\beta\xi})^{-1}$ and $Z_F=1+e^{-\beta\xi}$; convergence requires $\xi>0$ for the bosonic Gaussian. Correlation functions follow by source differentiation. Quartic interactions can be treated perturbatively or decoupled with a <Hubbard–Stratonovich transformation>; a saddle then describes an order parameter and quadratic fluctuations its collective modes. Such saddles require stability and an appropriate control parameter; the functional notation alone is not an exact solution of an interacting problem.
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