The harmonic oscillator and nuclear physics

The three-dimensional harmonic oscillator plays a central role in nuclear physics. It provides the underlying structure of the independent-particle shell model and gives rise to the dynamical group structures on which models of nuclear collective motion are based. It is shown that the three-dimensional harmonic oscillator features a rich variety of coherent states, including vibrations of the monopole, dipole, and quadrupole types, and rotations of the rigid flow, vortex flow, and irrotational flow types. Nuclear collective states exhibit all of these flows. It is also shown that the coherent state representations, which have their origins in applications to the dynamical groups of the simple harmonic oscillator, can be extended to vector coherent state representations with a much wider range of applicability. As a result, coherent state theory and vector coherent state theory become powerful tools in the application of algebraic methods in physics

Coherent state triplets and their inner products

It is shown that if H is a Hilbert space for a representation of a group G, then there are triplets of spaces F_H, H, F^H, in which F^H is a space of coherent state or vector coherent state wave functions and F_H is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps F_H -> H -> F^H which facilitates the construction of the corresponding inner products. After completion if necessary, the F_H, H, and F^H, become isomorphic Hilbert spaces. It is shown that the inner product for H is often easier to evaluate in F_H than F^H. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.Comment: 33 pages, RevTex (Latex2.09) This paper is withdrawn because it contained errors that are being correcte

Letter from Jennifer Rowe to Assistant Dean Robert J. Reilly

Letter from Principal Private Secretary Jennifer Rowe of the Lord Chancellor\u27s Department (1990-1993) to Assistant Dean Robert J. Reilly of Fordham University School of Law regarding The Advocate: Should He Speak or Write? by Lord Chancellor James Mackay of Great Britain (1987-1997).https://ir.lawnet.fordham.edu/events_programs_sonnett_miscellaneous/1007/thumbnail.jp

Dual pairing of symmetry groups and dynamical groups in physics

This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired representations are powerful aids in understanding the dynamics associated with shell-model coupling schemes and in identifying the physical situations for which a given scheme is most appropriate. In particular, they suggest model Hamiltonians that are diagonal in the various coupling schemes. The dual pairing of group representations has been applied profitably in mathematics to the study of invariant theory. We show that parallel applications to the theory of symmetry and dynamical groups in physics are equally valuable. In particular, the pairing of the representations of a discrete group with those of a continuous Lie group or those of a compact Lie with those of a non-compact Lie group makes it possible to infer many properties of difficult groups from those of simpler groups. This review starts with the representations of the symmetric and unitary groups, which are used extensively in the many-particle quantum mechanics of bosonic and fermionic systems. It gives a summary of the many solutions and computational techniques for solving problems that arise in applications of symmetry methods in physics and which result from the famous Schur-Weyl duality theorem for the pairing of these representations. It continues to examine many chains of symmetry groups and dual chains of dynamical groups associated with the several coupling schemes in atomic and nuclear shell models and the valuable insights and applications that result from this examination.Comment: 51 pages, 5 figures and 5 table

Vector coherent state representations and their inner products

Several advances have extended the power and versatility of coherent state theory to the extent that it has become a vital tool in the representation theory of Lie groups and their Lie algebras. Representative applications are reviewed and some new developments are introduced. The examples given are chosen to illustrate special features of the scalar and vector coherent state constructions and how they work in practical situations. Comparisons are made with Mackey's theory of induced representations. For simplicity, we focus on square integrable (discrete series) unitary representations although many of the techniques apply more generally, with minor adjustment.Comment: 28 pages, 1 figur