Interacting boson model of collective states I. The vibrational limit

The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems. The major application of these ideas is to the derivation, starting from the shell model, of the phenomenological models of nuclear collective motion, in particular, the geometric model of Bohr and Mottelson and the more recently developed interacting boson model of Arima and Iachello. A critical discussion of those applications is interwoven with the theoretical developments on which they are based; many other applications are included, some of practical interest, some simply to illustrate the concepts, and some to suggest new lines of inquiry.

Boson realizations of Lie algebras with applications to nuclear physics

We examine several phenomenological nonrelativistic\(q\bar q\) potentials, with different radial dependences, that fit the low-lying charmonium spectra. With the familar Coulomb-plus-ramp-type dependence the same potential can reproduce the spectra of the light mesons using the Schrodinger equation. With half-strength and the addition of a small three-body term the ground states of the baryons are also obtained.

A unified potential for mesons and baryons

The Bohr Hamiltonian, also called collective Hamiltonian, is one of the cornerstones of nuclear physics and a wealth of solutions (analytic or approximated) of the associated eigenvalue equation have been proposed over more than half a century (confining ourselves to the quadrupole degree of freedom). Each particular solution is associated with a peculiar form for the V(β,γ) potential. The large number and the different details of the mathematical derivation of these solutions, as well as their increased and renewed importance for nuclear structure and spectroscopy, demand a thorough discussion. It is the aim of the present monograph to present in detail all the known solutions in γ-unstable and γ-stable cases, in a taxonomic and didactical way. In pursuing this task we especially stressed the mathematical side leaving the discussion of the physics to already published comprehensive material. The paper contains also a new approximate solution for the linear potential, and a new solution for prolate and oblate soft axial rotors, as well as some new formulae and comments. The quasi-dynamical SO(2) symmetry is proposed in connection with the labeling of bands in triaxial nuclei.

Solutions of the Bohr Hamiltonian, a compendium

Publisher Summary This chapter presents a survey on the application of “geometric algebra” to the physics of electrons. Geometric algebra is the simplest and most coherent language available for mathematical physics and provides a single unified approach to a vast range of mathematical physics, and formulating and solving a problem in geometric algebra invariably leads to new physical insights. The chapter discusses aspects encompassing a wider range of topics relevant to electron physics. The idea that Clifford algebra provides the framework for a unified language for physics has been advocated most strongly by Hestenes, who is largely responsible for shaping the modern form of the subject. A list of some of the algebraic systems and techniques employed in modern theoretical physics (and especially particle physics) is presented in the chapter. The chapter focuses on the geometric algebra of spacetime—the spacetime algebra. The chapter explains that spacetime algebra, simiplifies the study of the Dirac theory, and discusses that the Dirac theory once formulated in the spacetime algebra is a powerful and flexible tool for the analysis of all aspects of electron physics—not just relativistic theory. The chapter begins with an introduction to the spacetime algebra (STA); concentrating on how the algebra of the STA is used to encode geometric ideas, such as lines, planes, and rotations.

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Spacetime Algebra and Electron Physics

Under the assumption that quarks are confined, in a first approximation, by a relativistic linear potentialV(r)=1/2 (1+β)(V0+λr), several properties of low-lying baryons have been calculated. Corrections to the mass spectrum have also been calculated, by taking the one-gluon exchange into consideration.

A linear-potential model for quark confinement

Gluon exchange corrections for baryons in a model with confining potentials

On a model for baryons based on a Dirac equation with confining potentials

The basic functions for a class of the irreducible representations of the rotation groups in n dimensions (Rn) are explicitly constructed by an algebraic method in which the basic functions are taken to be homogeneous polynomials in the variables of En. The solutions correspond to the hyperspherical harmonics of the mathematical literature and are of interest for problems exhibiting invariance under a certain Rn. The method is also applied to derive a basis for the infinite‐dimensional irreducible representations of the homogeneous Lorentz group if we then look for homogeneous functions in the variables of the corresponding pseudo‐Euclidean space.

P. Leal Ferreira

Papers

A linear-potential model for quark confinement

Gluon exchange corrections for baryons in a model with confining potentials

On a model for baryons based on a Dirac equation with confining potentials

Algebraic Construction of the Basis for the Irreducible Representations of Rotation Groups and for the Homogeneous Lorentz Group

The quantum pendulum