= Time-slice ordering in a many-body coherent integral
In a <bosonic coherent-state path integral>, normal ordering determines which fields occur on the two neighboring time slices. For unnormalized coherent states, the short-step kernel obeys
$$
\langle z_{j+1}|e^{-\epsilon K}|z_j\rangle\simeq e^{z_{j+1}^*z_j}\exp[-\epsilon K(z_{j+1}^*,z_j)],
$$
where $K$ is the normal-ordered grand <Hamiltonian> and $\epsilon$ has inverse-energy units. Creation operators supply the later conjugate field and annihilation operators the earlier field.
The overlaps yield the temporal derivative term in the continuum <action>, but replacing the two times by one too early can lose equal-time ordering information. Retaining the discrete prescription fixes contact terms and the meaning of occupation insertions. A continuum expression is therefore used together with its time-slice convention, especially when evaluating products at coincident times.
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