PAULI EXCLUSION PRINCIPLE

In terms of an exact equation for the thermodynamic potential due to interaction between two particles and based on Green's function method; we have derived the Landau expansion of the thermodynamic potentials in terms of the variation of the quasiparticle distribution function. We have also derived the expansion of the thermodynamic potential in terms of the variation of an exact single particle (not quasiparticles), this derivations lead to the relationship between the interaction function for two quasiparticles and the interaction energy between two particles as shown. Further, in terms of the four-point vertex part we are led to the Pauli exclusion principle.

Cooper Instability for a Given Field in a Quiescent Fermi Gas

The Cooper instability in a Fermi gas is examined using the perturbative diagram approach. A graphical functional derivative technique based on Ward's identity is developed to obtain two-particle interactions and then to calculate the vertex part. The pairing instability for a given interaction, such as a phonon (plasmon, etc.) field, occurs in a quiescent Fermi sea, i.e. without exciting or involving background particles (holes), only if the interaction is attractive, as first proposed by Cooper and adopted in the BCS (Bardeen–Cooper–Schrieffer) theory. The consequence from this technique provides a way to evaluate the effect of the vertex corrections and responses of the Fermi gas in both charge and spin channels incorporating the backward scattering process. The significance of the methodology presented in the present work lies in the fact that it can both reproduce the known results, and, more importantly, be extended to investigate the intermediate or strong coupling case, such as nuclear interactions, where a neglect of vertex corrections may not be a good approximation.

WARD IDENTITY FOR NONLOCAL QED

Evens et al.1 have given a gauge-invariant regularization scheme for QED which they have named nonlocal regularization. The present authors2 have worked out the QED vertex part in this scheme of regularization. In this paper we present a Ward identity for nonlocal QED to the order of two loops (order e4). In the limit of QED (Λ→∞), this identity reduces to the usual form of the Ward identity.

QED VERTEX PART IN NONLOCAL REGULARIZATION

Evens et al.1 have given a gauge invariant regularization scheme for QED which they have named nonlocal regularization. We have evaluated the QED vertex part in this scheme of regularization. This result agrees with the expression obtained using dimensional regularization apart from numerical constants.

On the linear response of Bardeen–Cooper–Schrieffer superconductors II: superconducting alloys

The linear response to longitudinal modulations in a superconducting alloy is calculated for the general case. Extension to transverse waves, point-charge Coulomb potentials, and magnetic dipoles is trivial. All the results are limited to Born approximation, s-wave scattering by the impurities embedded in the superconductor. Derivation of the equations for linear response is the same as for clean superconductors, with the appearing correlation functions containing impurity-averaged vertex parts. Only one vertex part is evaluated here; the remaining ones are easy extensions of our calculation.

Onset of Localized Correlation in a Narrow s Band

The correlation of electrons in a narrow s band is studied by examining the vertex part for the multiple scattering of two electrons of opposite spin. When bare electrons are used in the calculation, there exists a high-lying pole only for a band not approaching half filled. When the electrons are clothed by the simplest self-energy correction, a high energy pole exists even for the half-filled band, in which case, an insulating state is implied.

Application of many-fermion techniques to lattice statistical problems

The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.