The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.

Journal of Mathematical PhysicsOct 1, 196610 citations

Many-Body Theory of Correlated Fermion Systems

José M. Arias, María Isabel Gallardo, Manuel Lozano

May 1, 19983 citations

Theory of Strongly Coupled Many-Fermion Systems. I. Convergence of Linked-Cluster Expansions

Leon N. Cooper

A strongly coupled system---the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface---has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{(H\ensuremath{-}z)}〉}_{0}$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.A strongly coupled system---the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface---has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{(H\ensuremath{-}z)}〉}_{0}$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.

Physical ReviewMay 15, 19615 citations

The Wick theorem for coupled fermion clusters

Jin-Quan Chen

Nuclear Physics ASep 1, 199365 citations

Linked-Clusters Expansion in Quantum Statistics: Petit Canonical Ensemble

R. Brout, F. Englert

A linked-cluster expansion for the free energy in the petit canonical ensemble is presented and proved. It has the advantage of avoiding the introduction of the unknown chemical potential in the perturbation series. As a consequence of correlations among the population numbers $n(\mathrm{K})$ , additional linkages representing these correlations appear. The result is used to find the ground-state energy of a many-body fermion system. This expression reduces to the Brueckner-Goldstone expansion only in the case of central forces in an isotropic system, a theorem due to Kohn, Luttinger, and Ward. It is also shown that in the random phase approximation, correlation bonds do not contribute. Finally the relation of our formalism to the Bloch-De Dominicis expansion for the grand partition function is discussed.A linked-cluster expansion for the free energy in the petit canonical ensemble is presented and proved. It has the advantage of avoiding the introduction of the unknown chemical potential in the perturbation series. As a consequence of correlations among the population numbers $n(\mathrm{K})$, additional linkages representing these correlations appear. The result is used to find the ground-state energy of a many-body fermion system. This expression reduces to the Brueckner-Goldstone expansion only in the case of central forces in an isotropic system, a theorem due to Kohn, Luttinger, and Ward. It is also shown that in the random phase approximation, correlation bonds do not contribute. Finally the relation of our formalism to the Bloch-De Dominicis expansion for the grand partition function is discussed.

Physical ReviewDec 1, 196027 citations

Variational Principle for Many-Fermion Systems

Élliott H. Lieb

Physical Review LettersJul 6, 198133 citations

n(\mathrm{K})

, additional linkages representing these correlations appear. The result is used to find the ground-state energy of a many-body fermion system. This expression reduces to the Brueckner-Goldstone expansion only in the case of central forces in an isotropic system, a theorem due to Kohn, Luttinger, and Ward. It is also shown that in the random phase approximation, correlation bonds do not contribute. Finally the relation of our formalism to the Bloch-De Dominicis expansion for the grand partition function is discussed.