Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. I. Algebraic Formulation

Abstract

This paper shows how the dynamical and thermodynamical properties of an interacting quantum mechanical system with many degrees of freedom may be expressed and calculated solely in terms of renormalized propagators and renormalized vertices or interactions. The formulation employed is sufficiently general to encompass systems which have several components, with Fermi or Bose statistics, whether or not they exhibit superfluidity or superconductivity. The process of renormalization is the functional generalization of the thermodynamic transformation from the chemical potential and temperature to the energy and matter densities. With each set of variables (here, functions) is associated a natural thermodynamic function (here a functional). The natural functional for the unrenormalized potentials which occur in the Hamiltonian is the logarithm of the grand partition function; the natural functional for the fully renormalized variables, the distribution functions, is the entropy. In particular, a stationarity principle for a functional F(2) of distribution functions subject to constraints is shown to provide a fully renormalized description of the system. The numerical value of this functional, at the stationarity point at which the distribution functions take their actual value, is the entropy of the system. The equations of stationarity are expressions for the unrenormalized ν-body potentials vν in terms of the ν′-body distribution functions Gν′. The functionals F(2) and vν (of the distribution functions Gν′) are expressed as the solutions of closed functional differential equations which may be used to generate their power-series expansions. For a superfluid Bose system, as for the electromagnetic field interacting with matter, it is necessary to consider expectation values of odd, as well as even, numbers of field operators. In particular it is necessary to employ the expectation values Gν for 2ν = 1, 2, 3, 4 field operators. For a fermion system, even if it is superconducting, only the functions Gν for 2ν = 2, 4 are required. In contrast to other thermodynamical functionals, the entropy functional F(2) makes no reference to equilibrium parameters such as temperature and chemical potential.