 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 integralConstants 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 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.