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From d-dimensional Quantum to d+ 1-dimensional Classical Systems

I review the mapping from the partition function of a d-dimensional quantum system to that of a d+ 1-dimensional classical system and I apply it to several models, in particular the classical 1d Ising, 2d XY, and 3d Ising models. These examples respectively illustrate the general features of the classical/quantum correspondence, a Berry phase term, and the role of gauge constraints. The limitations of the mapping due to the Berry phase term from the quantum side are discussed in the context of the Heisenberg spin chain.