The Hurwitz Map and Harmonic Wave Functions for Integer and Half-Integer Angular Momentum

Abstract

Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles $(\phi,\theta,\psi)$ that define a rotation in $SO(3)$, and the Euclidean norm $r$ in ${\Bbb R}^3$, keeping the usual meaning of the spherical coordinates $(r,\phi,\theta)$. They form a Hilbert (super)-space decomposed in the form \Cal H=\Cal H_0\oplus\Cal H_1. Following a classical work by Schwinger, $2$-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space \Cal H is approached from the double covering group homomorphism $SU(2)\to SO(3)$ and the topology involved is taken care of by using the Hurwitz map $H:{\Bbb R}^4\to{\Bbb R}^3$. It is shown how to reconsider $H$ as a 2-to-1 group map, $G_0={\Bbb R}^+\times SU(2)\to {\Bbb R}^+\times SO(3)$, translating into an assignment $(z_1,z_2)\mapsto (r,\phi,\theta,\psi)$ in terms of two complex variables $(z_1,z_2)$, under the appropriate identification of ${\Bbb R}^4$ with ${\Bbb C}^2$. It is shown how the Lie algebra of $G_0$ is coupled with the two Heisenberg Lie algebras of the $2$-dimensional (Schwigner's) harmonic oscillators generated by the operators $\{z_1,z_2,\bar{z}_1,\bar{z}_2\}$ and their adjoints. The whole set of operators close either into a $9$-dimensional Lie algebra or into an $8$-dimensional Lie superalgebra. The wave functions in \Cal H can also be written in terms polynomials in the complex coordinates $(z_1,z_2)$ and their complex conjugates $(\bar{z}_1,\bar{z}_2)$ and the representations are explicitly constructed via highest weight (or lowest weight) vector representations for $G_0$.Comment: 19 pages, 1 diagra