Algebra of Noncommutative Riemann Surfaces

We examine several algebraic properties of the noncommutive $z$-plane and Riemann surfaces. The starting point of our investigation is a two-dimensional noncommutative field theory, and the framework of the theory will be converted into that of a complex coordinate system. The basis of noncommutative complex analysis is obtained thoroughly, and the considerations on functional analysis are also given before performing the examination of the conformal mapping and the Teichm\"{u}ller theory. (Keywords; Complex Analysis, Riemann Surfaces and Teichm\"{u}ller Space, Functional Analysis, Deformation Quantization, Non-Commutative Geometry, Quantum Groups)Comment: 25 page

BCS and Generalized BCS Superconductivity in Relativistic Quantum Field Theory. II. Numerical Calculations

We solve numerically various types of the gap equations developed in the relativistic BCS and generalized BCS framework, presented in part I of this paper. We apply the method for, not only the usual solid metal but also other physical systems, by using homogeneous fermion gas approximation. We examine the relativistic effects on the thermal properties and the Meissner effect of the BCS and generalized BCS superconductivity of various cases.Comment: 27 pages, 18 figures. Submitted to Phys. Rev. B at Oct. 19, 200

Deformation Quantization and Quaternions

The deformation quantization of Moyal-Weyl star product of functions of quaternions is investigated.Comment: 9 page

Supersymmetric Theory of (Color)superconductivity

The supersymmetric theory of (color)superconductivity is investigated.Comment: 1 page. A report for a YITP conference. To be published in Soryushiron Kenky

Moyal-Weyl Star-products as Quasiconformal Mappings

The relation between the Moyal-Weyl deformation quantization and quasiconformal mappings of Riemann surfaces of complex analysis are shown by several examples.Comment: 10 page

Generalized Seesaw Mechanism of Neutrino and Bose-Einstein Condensation in the Modified O'Raifeartaigh Model

The modified O'Raifeartaigh model from the context of the generalized seesaw mechanism of neutrino mass is investigated. In our evaluation of effective potentials of the theory, both the component field and the superspace formalisms to approach the problem are presented. In the component field formalism, we take into account the Bose-Einstein condensates in the scalar sector by the method of many-boson theory, i.e. we consider both the condensates and the Hartree-Fock-Bogoliubov-type self-energies of quantum fluctuations. The diagonalization of the mass matrix of the fermion sector gives the same functional forms of the mass eigenvalues in the generalized seesaw mechanism. The stability condition in the vicinity of the classical vacuum which shows the generalized seesaw situation is obtained by the examination of the mass eigenvalues of the scalar sector of the model. The superspace formalism will be devoted to a comparison between its result with that of the component field formalism.Comment: 15 pages, submitted for publicatio

Relativistic Model for two-band Superconductivity

To understand the superconductivity in MgB2, several two-band models of superconductivity were proposed. In this paper, by using the relativistic fermion model, we clearize the effect of the lower band in the superconductivity.Comment: 2 pages. To be published in RCNP (Osaka univ.) annual report 200

Theory of Quantum Electrodynamical Self-consistent Fields

To obtain the basis for combining various many-body techniques to QED in a consistent manner, we investigate the theory of quantum electrodynamical self-consistent fields. The reserch interest was born mainly of the electronic structure theory, thus we consider of atomic and moleculer systems as our main subjects. But the formalism is more fundamental. First, we derive the quantum electrodynamical Hartree-Fock theory by using the Green's function method. Then we construct a relativistic Hamiltonian written by creation-annihilation operators for electron and positron in a general form, and check that it reproduces the Hartree-Fock theory. We use this Hamiltonian to derive the time-dependent Hartree-Fock theory and random phase approximation, in the operator formalism. The relativistic Slater determinant of the Thouless parametrization is also used. Finally we discuss the applications and futher possible developments.Comment: 33 page

The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic Quantum Field Theory

The anomalous Nambu-Goldstone (NG) theorem which is found as a violation of counting law of the number of NG bosons of the normal NG theorem in nonrelativistic and Lorentz-symmetry-violated relativistic theories is studied in detail, with emphasis on its mathematical aspect from Lie algebras, geometry to number theory. The basis of counting law of NG bosons in the anomalous NG theorem is examined by Lie algebras (local) and Lie groups (global). A quasi-Heisenberg algebra is found generically in various symmetry breaking schema of the anomalous NG theorem, and it indicates that it causes a violation/modification of the Heisenberg uncertainty relation in an NG sector which can be experimentally confirmed. The formalism of effective potential is presented for understanding the mechanism of anomalous NG theorem with the aid of our result of Lie algebras. After an investigation on a bosonic kaon condensation model with a finite chemical potential as an explicit Lorentz-symmetry-breaking parameter, a model Lagrangian approach on the anomalous NG theorem is given for our general discussion. Not only the condition of the counting law of true NG bosons, but also the mechanism to generate a mass of massive NG boson is also found by our examination on the kaon condensation model. Furthermore, the generation of a massive mode in the NG sector is understood by the quantum uncertainty relation of the Heisenberg algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in the effective potential of the kaon condensation model. Hence the relation between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode coupling, and the mechanism of mass generation in an NG sector is established. Finally, some relations between the Riemann hypothesis and the anomalous NG theorem are presented.Comment: 71 page

Chiral Symmetry and Collective Excitations in p-wave, d-wave and f-wave Superconductors

We discuss the origin of charge density wave (CDW) and spin density wave (SDW) in p-wave, d-wave and f-wave superconductors. To describe the low-energy quasiparticle excitation of p-wave case, we introduce a two- (one for time and one for space) dimensional massless Dirac model. After the non-Abelian bosonization is performed, the charge and spin density waves emerge from the model. By using this scheme, we try to explain the characteristic aspect of phase diagrams of various compounds, oxides and organic superconductors. The purpose of this paper is to make an argument that the dimensionality of the nodal excitation in superconductors plays an important role in the determination of the structure of the phase diagram.Comment: 7 page