Density matrix of interacting boson-fermion systems as functional integrals in the holomorphic representation

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Abstract
We formulate the density matrix of many-body systems as functional integrals in the holomorphic representation, similar to the S-matrix formulation in field theory. Symmetry and antisymmetry requirements are built into the integration variables: c-numbers for bosons and Grassmann numbers for fermions. The integrals have well-defined normalization. They yield first-order differential equations for semiclassical extremal configurations and corrections to it, and do not require periodicity conditions on the individual integration paths, unlike the formulations found in the literature. We present some applications to simple boson and fermion systems.