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).
- 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