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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Field theory in classical phase-space Green’s functions and a general response theory

A. Matos Neto, J. D. M. Vianna

Published Apr 1, 1985• 5 citations
Publisher
DOI
10.1007/bf02721526

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A method of expressing statistical classical r esults in terms of mathematical entities ucually associated to quantum field theoreti - cal treatment of many particle 'systems (Fock space, commutators, field operators, state vector) is discussed. It is developed a linear response theory using the second quantized Liouville equation introduced by sch8nberg. The relationship of this method to that of Prigogine et aZ. is briefly analyzed. The chain of equations and the s pectral represen - tations for the new classical Green's functions a re presented. Gener- alized operators defined on Fock space are dlscussed. It is shown that the correlation functions can be obtained from Green's functions de- fined with generalized operators.
Computing response functions by following the time evolution of superoperators in Liouville space (whose vectors are ordinary Hilbert space operators) offers an attractive alternative to the diagrammatic perturbative expansion of many-body equilibrium and nonequilibrium Green's functions. The bookkeeping of time ordering is naturally maintained in real (physical) time, allowing the formulation of Wick's theorem for superoperators, giving a factorization of higher order response functions in terms of two fundamental Green's functions. Backward propagations and analytic continuations using artificial times (Keldysh loops and Matsubara contours) are avoided. A generating functional for nonlinear response functions unifies quantum field theory and the classical mode coupling formalism of nonlinear hydrodynamics and may be used for semiclassical expansions. Classical response functions are obtained without the explicit computation of stability matrices.
The present paper is concerned with the retarded many-time thermodynamic Green's function method in classical statistical mechanics. In the formulation of the nonlinear response of a dynamical variable, unaveraged Green's functions are specified and used to discuss the physical meaning of many-time Poisson-bracket nests. Equilibrium ensemble averages of these Green's functions are relevant to the classical Kubo response theory, the latter of which is interpreted from the viewpoints of both the classical ``Heisenberg'' and ``Schrödinger'' pictures. A method for evaluating the many-time thermodynamic Green's functions based on their definition in terms of the resolvent operator is suggested and discussed in relation to the linear dielectric function and the anharmonic oscillator. The many-time Green's function equations of motion are shown to be equivalent to identities based on the resolvent operator. It is shown how microscopic equations of motion can be used to derive the Bogoliubov-Sadovnikov (B-S) hierarchy from the lowest-order many-time Green's function equation of motion.