“Hidden connections” and functional integral representation of the partition function of the canonical ensemble

A method is proposed for separating the “hidden connection” delta function in the functional integral by integration over a group in the cases when the action functional is transformed linearly with respect to the group parameters. As an example, the functional integral representing the partition function of the grand canonical ensemble of a quantum gas is considered. By means of integration with respect to the continuous gauge group it is possible to separate in this integral the connection that expresses the conservation in time of the total particle number, and then, by means of the discrete group, the connection that requires the particle number to be integral. As a result, the original integral is split into a countable sum of terms, each of which represents the partition function of the canonical ensemble with given particle number. In the Appendix, the problem of defining functional integrals in theories with derivatives of first order with respect to the time (integrals over phase space) is discussed.