Quantum Mechanics for Mathematicians

and in general we have operators

−iL=−iQ×P

that provide the Lie algebra version of the representation (recall that at

the Lie algebra level,SO(3) andSpin(3) are isomorphic).

2. The action of the matrix Ω∈SU(2) on the two-component wavefunction
   by
   ψ ̃E,±(p)→Ωψ ̃E,±(p)
   ForRa rotation by angleφabout thex-axis one choice of Ω is

Ω =e−iφ

σ 1

2

and the operators that provide the Lie algebra version of the representation

are the

−iS=−i

1

2

σ

The Lie algebra representation corresponding to the action of these transfor-
mations on the two factors of the tensor product is given as usual (see chapter
9) by a sum of operators that act on each factor

−iJ=−i(L+S)

The standard terminology is to callLthe “orbital” angular momentum,Sthe
“spin” angular momentum, andJthe “total” angular momentum.
The second Casimir operator for this case is

J·P

and as in the one-component case (see section 19.3) a straightforward calculation

shows that theL·Ppart of this acts trivially on our solutionsψ ̃E,±(p). The
spin component acts non-trivially and we have

(J·P)ψ ̃E,±(p) = (

1

2

σ·p)ψ ̃E,±(p) =±

1

2

|p|ψ ̃E,±(p)

so we see that our solutions have helicity (eigenvalue ofJ·Pdivided by the
square root of the eigenvalue of|P|^2 ) values±^12 , as opposed to the integral
helicity values discussed in chapter 19, whereE(3) appeared and not its double

cover. These two representations on the spaces of solutionsψ ̃E,±(p) are thus the

E ̃(3) representations described in section 19.3, the ones labeled by the helicity
±^12 representations of the stabilizer groupSO(2).
Solutions for either sign of equation 34.6 are given by a one dimensional
subspace ofC^2 for eachp, and it is sometimes convenient to represent them as
follows. Note that for eachpone can decompose

C^2 =C⊕C