The set of equations for the two-particle Green's functions in a nonequilibrium medium is derived in the diffusive approximation. Both the two- and single-particle Green's functions are calculated when matter is in the ground state. It is shown that the Landau Fermi-liquid consideration is correct when the interaction between particles in matter is rather weak or when matter is not too dense. In the opposite cases the energy spectrum of the particles is rearranged so that the step in the momentum distribution of the particles disappears. The applicability of the Landau Fermi-liquid theory is established in the general case.
The set of equations for the two-particle Green's functions in a nonequilibrium medium is derived in the diffusive approximation. Both the two- and single-particle Green's functions are calculated when matter is in the ground state. It is shown that the Landau Fermi-liquid consideration is correct when the interaction between particles in matter is rather weak or when matter is not too dense. In the opposite cases the energy spectrum of the particles is rearranged so that the step in the momentum distribution of the particles disappears. The applicability of the Landau Fermi-liquid theory is established in the general case.
A Landau-type Fermi-liquid theory for interacting electrons moving in a fixed periodic potential is derived. Two methods are used: the Green's-function method, which gives a more physical description of the system but does not give all the desired results, and the many-body perturbation-theory approach, which is complete but is more complicated. The Green's-function formalism of Kadanoff and Baym is adapted to inhomogeneous systems by rewriting it in operator form. A kinetic equation for an energy-dependent, single-particle density matrix is then derived. The quasiparticle concept is introduced and a formal solution of the kinetic equation is given and discussed. These developments are specialized to the periodic potential and a transport equation of the Landau-Boltzmann type is derived for quasiparticles moving within a single energy band. Expressions for the density and current density are then identified from a continuity equation. Although the present work does not treat real transitions between energy bands, the Green's-function method suggests a plausible form for a Landau-like theory which includes such transitions. The perturbation-theory treatment follows the work of Luttinger and Nozi\`eres, who derived Landau's theory for systems without a lattice potential. When the perturbation theory is expressed in terms of Bloch functions, the quantities involved become matrices between different energy bands. However, it is shown that many of the key results of Luttinger and Nozi\`eres can be suitably generalized. The perturbation-theory calculation shows that when the periodic potential is present the change in quasiparticle energy can still be expressed in terms of the variation of the quasiparticle distribution function, and that the results obtained from the Green's-function approach are exact in the case of slowly varying external fields.
On the basis of the Kadanoff-Baym (KB) version of the time-dependent Green's function method, an Ansatz for the approximation of a spectral function is offered. The Ansatz possesses all the advantages of quasiparticle and extended quasiparticle approximations and satisfies the KB equation for a spectral function in the case of slightly nonequilibrium system when disturbances in space and time are taken into consideration in the gradient approximation. This feature opens opportunities for the microscopic derivation of the Landau kinetic equation for the quasiparticle distribution function of the normal Fermi liquid and provides the widening of these equations' temperature range of validity.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of N particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of N particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.
Based on a class of exactly solvable models of interacting Bose and Fermi liquids, we compute the single-particle propagators of these systems exactly for all wavelengths and energies and in any number of spatial dimensions. The field operators are expressed in terms of Bose fields that correspond to displacements of the condensate in the Bose case and displacements of the Fermi sea in the Fermi case. Unlike some of the previous attempts, the present attempt reduces the answer for the spectral function in any dimension in both Fermi and Bose systems to quadratures. It is shown that when only the lowest-order sea-displacement terms are included, is the random-phase approximation in its many guises recovered in the Fermi case, and Bogoliubov's theory in the Bose case. The momentum distribution is evaluated using two different approaches, exact diagonalization and the equation of motion approach; the novelty being, of course, the exact computation of single-particle properties including short-wavelength behavior.
Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.