Seiya Nishiyama

Extended supersymmetric σ-model based on the SO(2N+1) Lie algebra of the fermion operators

Extended supersymmetric σ-model is given, standing on the SO(2N+1) Lie algebra of fermion operators composed of annihilation–creation operators and pair operators. Canonical transformation, the extension of the SO(2N) Bogoliubov transformation to the SO(2N+1) group, is introduced. Embedding the SO(2N+1) group into an SO(2N+2) group and using SO(2N+2)U(N+1) coset variables, we investigate a new aspect of the supersymmetric σ-model on the Kahler manifold of the symmetric space SO(2N+2)U(N+1). We construct a Killing potential which is just the extension of the Killing potential in the SO(2N)U(N) coset space given by van Holten et al. to that in the SO(2N+2)U(N+1) coset space. To our great surprise, the Killing potential is equivalent with the generalized density matrix. Its diagonal-block matrix is related to a reduced scalar potential with a Fayet–Ilipoulos term. The reduced scalar potential is optimized in order to see the behaviour of the vacuum expectation value of the σ-model fields and a proper solution for one of the SO(2N+1) group parameters is obtained. We give bosonization of the SO(2N+2) Lie operators and vacuum functions in terms of the SO(2N+2)U(N+1) coset variables, a U(1) phase and the corresponding Kahler potential.

Exact Canonically Conjugate Momenta to Quadrupole-Type Collective Coordinates and Derivation of Nuclear Quadrupole-Type Collective Hamiltonian ∗

Exact canonically conjugate momenta Π2μ in quadrupole nuclear collective motions are proposed. The basic idea lies in the introduction of a discrete integral equation for the strict definition of canonically conjugate momenta to collective variables ϕ2μ. A part of our collective Hamiltonian, the Π2μ-dependence of the kinetic part of the Hamiltonian, is given exactly. Further, ϕ2μ-dependence of the kinetic part of the Hamiltonian is also given.

Self-consistent field method from a τ -functional viewpoint ∗

A unified aspect of the self-consistent field (SCF) method and the τ-functional method is presented. SCF theory in the τ-functional space F∞ manifestly results in a gauge theory of fermions and then a collective motion appears as a motion of fermion gauges with a common factor. This provides a new algebraic tool for the microscopic understanding of fermion many-body systems.

The Buck–Sukumar model described in terms of su(2) ⊗ su(1, 1) coherent states

The Buck–Sukumar model, which describes an assembly of A identical two-level atoms in interaction with a monochromatic radiation field, is investigated using su(2) ⊗ su(1, 1) coherent states, in the framework of conventional mean-field many-body approaches. In particular, the super-radiant phase transition is studied. We find that results based on the mean-field method compare favorably with exact results. We also find that the results are much improved if the constant of motion of the model is implemented exactly, with the help of appropriate projection techniques, instead of being implemented only in the average. Since the Hamiltonian of the Buck–Sukumar model is unbounded from below, i.e., it lacks a ground state, a stabilized version of the model is also studied.

Schwinger representation approach to the Lipkin model

The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The groundstate, first excited state and RPA energies are obtained within several variant types of coherent states. It has been found that generalized coherent states defined in consonance with the parity symmetry of the model describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone.

Time Dependent Hartree–Bogoliubov Equation on the Coset Space SO(2N + 2)/U(N + 1) and Quasi Anti-Commutation Relation Approximation

An induced representation of an SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using the SO(2N + 2)/U(N + 1) coset variables, we develop an extended time dependent Hartree–Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner. The extended TDHB theory applicable to both even and odd fermion systems is a time dependent self-consistent field (TDSCF) theory with the same level of the mean field approximation as the usual TDHB theory for even fermion systems. We start from the Hamiltonian of the fermion system which includes, however, the Lagrange multiplier terms to select the physical spinor subspace. The extended TDHB equation is derived from the classical Euler–Lagrange equation of motion for the SO(2N + 2)/U(N + 1) coset variables in the TDSCF. The final form of the extended TDHB equation can be expressed through the variables as the representatives of the paired mode and the unpaired mode. We introduce the quasi anti-commutation relation approximation for the fermion. The parameters included in the Lagrangian multiplier terms are determined under the quasi anti-commutation relation approximation.

Description of collective motion in two-dimensional nuclei; Tomonaga's method revisited

Abstract Four decades ago, Tomonaga proposed the elementary theory of quantum mechanical collective motion of two-dimensional nuclei of N nucleons. The theory is based essentially on neglecting 1 N against unity. Very recently we have given exact canonically conjugate momenta to quadrupole-type collective coordinates under some subsidiary conditions and have derived nuclear quadrupole-type collective Hamiltonian. Even in the case of simple two-dimensional nuclei, we require a subsidiary condition to obtain exact canonical variables. Particularly the structure of the collective subspace satisfying the subsidiary condition is studied in detail. This subsidiary condition is important to investigate the structure of the collective subspace.

Exact canonically conjugate momenta approach to a one-dimensional neutron–proton system, I

Introducing collective variables, a collective description of nuclear surface oscillations has been developed with the first quantized language, contrary to the second quantized one in Sunakawas approach for a Bose system. It overcomes difficulties remaining in the traditional theories of nuclear collective motions: Collective momenta are not exact canonically conjugate to collective coordinates and are not independent. On the contrary to such a description, Tomonaga first gave the basic idea to approach elementary excitations in a one-dimensional Fermi system. The Sunakawas approach for a Fermi system is also expected to work well for such a problem. In this paper, on the

Resonating mean-field theoretical description of π and σ mesons by the Nambu–Jona-Lasinio model

isospin

Toward a unified algebraic understanding of concepts of particle and collective motions in fermion many-body systems*

space, we define a density operator and further following Tomonaga, introduce a collective momentum. We propose an