Journal of Mathematical Physics

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

C. De Dominicis, Paul C. Martin

Published Jan 1, 1964• 404 citations

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.

DOI: 10.1063/1.1704062

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Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. II. Diagrammatic Formulation

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Quantum statistical mechanics is renormalized, that is, the thermodynamical functions and the ``bare'' ν-body potentials vν occurring in the Hamiltonian are expressed as functionals of the ν-body distribution functions Gν (the expectation values of 2ν field operators). The field operators are considered as having two components (creation and annihilation), and Gν thus has (2)2ν components. The renormalization is carried out for superfluid Bose systems for which it is necessary to consider the expectation values of 2ν = 1, 2, 3, 4 operators. Superfluid Fermi systems and normal systems appear as particular cases, described by expectation values of 2ν = 2, 4 operators. This problem, dealt with by algebraic methods in part I, is attacked here by diagrammatic methods. These methods are more dependent on convergence properties but the resulting functionals are obtained explicitly as power series of the Gν′s, the general term being represented by a class of diagrams characterized by their topological structure. In the final result, the entropy is exhibited as an explicit functional of the distribution functions Gν, (2ν = 1, 2, 3, 4), or more precisely, of functions directly related to them. This functional no longer involves the potentials (nor the equilibrium parameters). It is stationary under independent variations of the Gν′s, subject to the constraints of constant energy and particle number. The four equations of stationarity exhibit each function vν as a functional of the Gν′s, self-consistently defining these distribution functions.

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