We consider the partition function for a many-body model consisting of a weakly coupled gas of bosons at any temperature T > 0 and any chemical potential μ, but with both infrared and ultraviolet cutoffs imposed in both temporal and spatial directions. We take the limit as the ultraviolet cutoff in the temporal direction is removed and develop a representation for the limit that, hopefully, provides a suitable starting point for controlling the limit as the infrared cutoffs are removed.
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