We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. We identify the stochastic variable of the effective statistical theory that we derive as a boundary configur
ation and a zero mode relevant to the discussion of infrared physics. We illustrate our formulation by computing the partition function of an interacting one-dimensional quantum mechanical system at finite temperature from the path-integral representation for the density matrix. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident endpoints, and includes non-vanishing boundary terms. An appropriately modified expansion into Matsubara modes provides a natural separation of the zero-mode physics. This feature may be useful in the treatment of infrared divergences that plague the perturbative approach in thermal field theory.
We present a systematic semiclassical procedure to compute the partition function for scalar field theories at finite temperature. The central objects in our scheme are the solutions of the classical equations of motion in imaginary time, with spatia
lly independent boundary conditions. Field fluctuations -- both field deviations around these classical solutions, and fluctuations of the boundary value of the fields -- are resummed in a Gaussian approximation. In our final expression for the partition function, this resummation is reduced to solving certain ordinary differential equations. Moreover, we show that it is renormalizable with the usual 1-loop counterterms.
