CHAPTER VI
The groups SO(3) and SU(2) and their representations
Two continuous groups of transformations that play an important role in physics are the special orthogonal group of order 3, SO(3), and the special unitary group of order 2, SU(2), which are in fact related to each other, and to which the present chapter is devoted. We first recall in Secs. VI.1 and VI.2 their most useful properties, which the reader probably knows from previous lectures, and introduce their respective Lie algebras. We then discuss their representations (Sec. VI.3).
VI.1 The group SO(3) In this section, we begin with a reminder on the rotations in three-dimensional Euclidean space (Sec. VI.1.1) — which as the reader knows models the spatial arena in which the physical objects of non-relativistic physics evolve.
VI.1.1 Rotations in three-dimensional space Consider first the rotations in three-dimensional Euclidean space that leave a given point O invariant. For the usual composition, they form a continuous group, which is non-Abelian since rotations about different axes do not commute. By choosing a Cartesian coordinate system with O at its origin, one maps these rotations one-to-one to those on R3. To describe such a rotation, one mostly uses either a set of three angles, or the direction of the rotation axis and the rotation angle, as we now recall.
:::::::VI.1.1 a::::::::::::::Euler angles A given rotation R is fully characterized by its action on three pairwise orthogonal directions in three-dimensional Euclidean space — or equivalently, on an orthonormal basis of R3. Thus, R transforms axes Ox, Oy, Oz into new axes Ou, Ov, Ow, as illustrated in Fig. VI.1. Note that it is in fact sufficient to know the transformations from Ox to Ou and from Oz to Ow, since the remaining axes Oy and Ov are entirely determined by the first two directions of a given triplet. As was already mentioned before, in these lecture notes we consider “active” transformations, i.e. the coordinate system is fixed while the physical system is being transformed — here, rotated. Accordingly, one should not view the transformation from the axes Ox, Oy, Oz to the directions Ou, Ov, Ow as a change of coordinates, but as the rotation of “physical” axes — which possibly coincided with the coordinate axes in the initial state, in case the Cartesian coordinates are denoted x, y, z. The action of R can be decomposed into the composition of three rotations about axes that are simply related to Oy, Oz, Ou,andOw: First, one performs a rotation about Oz through an angle 0 ↵<2⇡, which is actually • determined by the second step, and which brings the axis Oy along a new direction On,the so-called line of nodes. This first rotation will be denoted Rz(↵). 70 The groups SO(3) and SU(2) and their representations
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