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Deviations from Wick's theorem in the canonical ensemble

Wick's theorem for the expectation values of products of field operators for a system of noninteracting fermions or bosons plays an important role in the perturbative approach to the quantum many body problem. A finite temperature version holds in the framework of the grand canonical ensemble but not for the canonical ensemble appropriate for systems with fixed particle number like ultracold quantum gases in optical lattices. Here we present new formulas for expectation values of products of field operators in the canonical ensemble using a method in the spirit of Gaudin's proof of Wick's theorem for the grand canonical case. The deviations from Wick's theorem are examined quantitatively for two simple models of noninteracting fermions.