Boson realizations of Lie algebras with applications to nuclear physics

Abstract

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 freedorn 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 Arirna 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.

Nuclear structure, random interactions and mesoscopic physics

“…This is confirmed by many successes of the IBM [95], where the boson model is postulated including certain collective modes although their exact relation to the original fermion interaction is not rigorously derived. The regular methods of boson expansion of fermion operators were introduced in nuclear physics long ago, [96] for a single-j level and [97] for a general level scheme, see the detailed review article [98]. The application of the boson picture to our problem seems promising, especially because numerous studies of the IBM with random interactions [61,62,63,64,65,75] found a similar pattern of the predominance of J = 0 ground states.…”

A new matrix method for the Casimir operators of the Lie algebras and

“…Using the variable Z for the identity operator I, equations (6)- (7) and (11)- (16) are the brackets for the semidirect product wsp(N, R) of the symplectic algebra sp(2N, R) with the (2N+1)-dimensional Heisenberg-Weyl Lie algebra h N [7]. This type of construction for semidirect products is typical for the study of shift operator contractions and coherent state realisations of Lie algebras [7,11].…”

Some unusual algebraic structures and their applications in many-body problems

“…More specifically let us consider the generalized Marumori Boson Mapping [21]. IfÔ symbolizes a Fermionic Operator acting on the fermionic Hilbert space spanned by a finite basis of states, {|n },n= 0, 1, 2, ..., N , then the operatorÔ can be exactly represented in this basis as:…”

“…So, the Marumori mapping is an exact one [21]. Note that explicit expressions for the mapped operators are obtained, by using the expression for the vacuum Boson projector [22]:…”