Chapter 5 - The Boson Calculus Applied to the Theory of Turns

Summary

Introduction

The space of turns was shown in the preceding chapter to be the carrier space of the defining (fundamental) representation of the group SU(2) * SU(2) – the group of the symmetric top. We shall show in the present chapter how the complete representation theory of this structure can be obtained from a mapping (the generalized Jordan map) that maps the generic turn into an operator (the matrix boson operator) acting in a Hilbert space. By means of this construction we shall obtain a unified presentation of the angular momentum states, the rotation matrices, and the Wigner coefficients as well. An interesting aspect of this technique is that it leads automatically to a view of the Wigner coefficients as a form of discretized rotation matrix (Section 8 below), which in turn implies a relationship between the Wigner coefficients and Jacobi polynomials (Gel'fand et al. [1, Supplement III]).

The key idea behind this unified presentation – the Jordan mapping of the turn into a matrix boson – is of more general validity than the application made here. To put this structure in a larger context we first develop, in some detail, the techniques of the boson calculus. It is hoped that this digression does not break the thread of the development too badly.

Excursus on the Boson Calculus

The harmonic oscillator. The boson calculus originated in the treatment of a basic physical problem, the quantum mechanics of the linear harmonic oscillator.