Theory of noninteracting fermions and bosons in the canonical ensemble

We present a self-contained theory for the exact calculation of particle number counting statistics of noninteracting indistinguishable particles in the canonical ensemble. This general framework introduces the concept of auxiliary partition functions, and represents a unification of previous distinct approaches with many known results appearing as direct consequences of the developed mathematical structure. In addition, we introduce a general decomposition of the correlations between occupation numbers in terms of the occupation numbers of individual energy levels, that is valid for both nondegenerate and degenerate spectra. To demonstrate the applicability of the theory in the presence of degeneracy, we compute energy-level correlations up to fourth order in a bosonic ring in the presence of a magnetic field.