 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 obeyswhere $K$ is the normal-ordered grand Hamiltonian and $ϵ$ 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.