Chapter 7

Rotations and the Spin

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2

Particle in a Magnetic Field

The existence of a non-trivial double coverSpin(3) of the three dimensional rota-
tion group may seem to be a somewhat obscure mathematical fact. Remarkably
though, the existence of fundamental spin^12 particles shows that it isSpin(3)
rather thanSO(3) that is the symmetry group corresponding to rotations of fun-
damental quantum systems. Ignoring the degrees of freedom describing their
motion in space, which we will examine in later chapters, states of elementary
particles such as the electron are described by a state spaceH=C^2 , with rota-
tions acting on this space by the two dimensional irreducible representation of
SU(2) =Spin(3).
This is the same two-state system studied in chapter 3, with theSU(2) action
found there now acquiring an interpretation as corresponding to the double cover
of the group of rotations of physical space. In this chapter we will revisit that
example, emphasizing the relation to rotations.

7.1 The spinor representation

In chapter 6 we examined in great detail various ways of looking at a particular
three dimensional irreducible real representation of the groupsSO(3),SU(2)
andSp(1). This was the adjoint representation for those three groups, and
isomorphic to the vector representation forSO(3). In theSU(2) andSp(1)
cases, there is an even simpler non-trivial irreducible representation than the
adjoint: the representation of 2 by 2 complex matrices inSU(2) on column
vectorsC^2 by matrix multiplication or the representation of unit quaternions in
Sp(1) onHby scalar multiplication. Choosing an identificationC^2 =Hthese
are isomorphic representations onC^2 of isomorphic groups, and for calculational
convenience we will useSU(2) and its complex matrices rather than dealing with
quaternions. We thus have: