An Alternative Approach to Rotations

In the preceding chapters, we have discussed the standard view of the quantum theory of angular momentum. The present chapter is concerned with developing a new view of the subject starting from the beginning with a fresh look at rotations based on a viewpoint adapted primarily from Hamilton [1], but also partly from Klein and Sommerfeld [2] and from Wigner [3] (see Note 1). This new survey will be rewarded by the concept of a novel geometric entity – the turn – which in its quantal version (Chapter 5) will be realized by an elementary operator structure. In discussing this alternative viewpoint, we are primarily interested in motivation and in suggesting concepts. Accordingly, we shall proceed, at first, intuitively.

When reduced to fundamentals, the concept of an elementary (point) particle in physics (as discussed in Chapter 1) becomes synonymous with the fundamental symmetries assumed to characterize physical space. We have seen that, from the homogeneity and isotropy assumed for space, one induces the symmetries of spatial displacements (translations) and rotations; that is, one assumes that space is Euclidean and three-dimensional, having the isometry group, E(3), of rotations and translations.

It is quite easy to characterize translations, since three-dimensional translations necessarily (Artin [4]) form an abelian (commutative) group. For physics, the translation generator is the (Hermitian) momentum operator p, and the associated operator generating finite displacements is realized by U(a) = exp(–ia · p/ħ). This operator displaces the system by the (numerical) vector a.