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Journal of Mathematical Physics

Analyticity of Green's Functions of Dilute Quantum Gases

David Pierre Ruelle

Published Jun 1, 1971• 21 citations
Publisher

Abstract

Previous results on reduced density matrices of quantum gases are used to obtain theorems on the existence and analyticity of Green's functions.
DOI
10.1063/1.1665679

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In two previous papers, the reduced density matrices of quantum gases, in the grand canonical formalism and for suitably restricted interaction potentials, have been shown to be analytic vector-valued functions of the activity in a neighborhood of the origin, to tend in some sense to well-defined limits as the volume of the system becomes infinite, and to satisfy a cluster decomposition property. The same results are extended here by the same methods to a wider class of potentials, including hard-core, and allowing attractive interactions, which were excluded previously.
In previous works, the projected truncation approximation (PTA) was developed as a systematic and controlled method to truncate the equation of motion of Green's functions (GFs) for a given quantum or classical many-body Hamiltonian. The static averages are obtained self-consistently with the GF through the spectral theorem. In this work, PTA is reformulated as a self-consistent theory for the reduced density matrices (RDMs) without reference to GF. We separately discuss the issues of determining the dynamical matrix M{\bf M}M and solving the physical quantities from it. The properties of M{\bf M}M is clarified and the solution of PTA equations is cast into an over-constrained optimization problem. This makes connection of the present theory to the variational RDM theory. We discuss various issues of PTA under this formalism, including the scheme of alternative inner product, the generalized virial theorem, the generalized Wick's theorem, and the static component problem of PTA.