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.
The linked cluster expansion of the eigenvalues and eigenfunctions of a many-fermion system is extended to the case where the eigenvalues of the unperturbed system are degenerate. It is shown that the eigenvalues and eigenfunctions are to be obtained by solving a secular equation in the space of degenerate unperturbed wave functions; the matrix elements between these unperturbed functions take the form of linked clusters with energy denominators which are the differences of unperturbed energies. Some discussions are given on the relation of this paper to the formalism of Bloch and Horowitz, in which the energy denominators are the differences of an unperturbed energy and the true energy which is to be obtained as the solution to the secular equation.
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.
The methods employed in the previous papers of this series for the ground-state energy of a normal Fermion system are extended to finite temperatures. The basic assumption is that the density matrix of the actual system in thermodynamic equilibrium can be derived by adiabatic transformation from the density matrix of system of noninteracting particles at the same temperature, volume, and chemical potential. However, the spectrum of the noninteracting system must be dependent on the intensive parameters. The self-consistency of the scheme is exhibited by demonstrating that the effective potential characterizing this system can be related to the self-energy operator of the actual system. Two equivalent versions of the theory are derived. In one of these the basic equations are precisely those of the Landau theory of Fermi liquids with the difference, however, that all quantities are given explicitly from first principles.
Hartree-Fock theory is used in conjunction with the quantum Ursell expansion to obtain the grand potential $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and the momentum distribution
〈n(k)〉 for a Fermi fluid. It is shown that the perturbation expansion of the quasiparticle Ursell functions thus obtained yields a formal expansion of the grand potential which is easily related to the theory of de Dominicis. After the summation of certain ladder diagrams in the perturbation expansion, $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and
〈n(k)〉 take on forms identical to those obtained in the $\ensuremath{\Lambda}$-transformed theory derived by Mohling, RamaRao, and Shea. It is therefore possible to identify the Hartree-Fock single-particle energy with the quasiparticle energy of those authors' work.
New forms of the linked cluster expansion for the ground-state energy of a normal spherical system are given in terms of the exact single-particle propagator and of the associated self-energy operator. These forms are shown to be stationary with respect to variations of the latter quantities. A new concise proof of the Hugenholtz-Van Hove theorem on single-particle energies is then possible. The variational principle also explains the insensitivity of recent binding energy calculations to the choice of single-particle energies. It is shown how all results may be extended to systems with anisotropic features.