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Theoretical and Mathematical Physics

Dynamical structure factor and Green's functions for classical systems

N. N. Bogolyubov, V. V. Kamaeva, V. N. Plechko

Published Jul 1, 1977• 1 citation
Publisher
DOI
10.1007/bf01041433

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New projection operators are introduced in the theory of two-time Green functions. The self-energy part of the Green function is expressed in terms of the random force whose time development differes from usual one by a factor (1- P ), P being the projection operator. In the case of the Bose system, we obtained the time-relaxation expression for the width function Γ\varGammaΓ( k , ω) of the dynamical structure factor, which satisfies the sum rule by Puff.
The Thermal Matsubara Green Functions, and their zero temperature analogs, in many-fermion systems are shown to be built up from products, or sums of products, of factors each of which is determined by a Lie Algebra. These factors are: the structure constants of a mapping φ of the B2N+1 Fermion Algebra (consisting of singles and pairs of fermion operators) onto itself; and the structure constants of a mapping φ′ of the D2N+1 Dynamical Algebra of the Hamiltonian (consisting of pairs of fermion operators) onto itself; and eigenvalues of the generators of the Abelian Cartan subalgebra. Eigenstates of the many-fermion Hamiltonian typically relate to different types or numbers of elementary excitations (quasi-particles); these states are shwon to be bases for irreducible representations of the Dynamical Algebra. The contribution of each such irreducible set of states to the Green Functions is exhibited. These results are exact for mean-field Hamiltonians; generalization to include effects of fluctuations may be possible. The Green Functions for a model BCS singlet superconductor are determined (Dynamical Algebra SU (2)) to illustrate the various steps in the analysis.