whereLjacts onHEandSj=ρ′(lj) acts onC^2 s+1.
This tensor product representation will not be irreducible, but its irreducible
components can be found by taking the eigenspaces of the second Casimir op-
erator, which will now be
J·P
We will not work out the details of this here (although details can be found in
chapter 34 for the cases=^12 , where the half-integrality corresponds to replacing
SO(3) by its double coverSpin(3)). What happens is that the tensor product
breaks up into irreducibles as
HE⊗C^2 s+1=n⊕=sn=−sHE,nwherenis an integer taking values from−stosthat is called the “helicity”.
HE,nis the subspace of the tensor product on which the first Casimir|P|^2 takes
the value 2√ mE, and the second CasimirJ·Ptakes the valuenp, wherep=
2 mE. The physical interpretation of the helicity is that it is the component of
angular momentum along the axis given by the momentum vector. The helicity
can also be thought of as the weight of the action of theSO(2) subgroup of
SO(3) corresponding to rotations about the axis of the momentum vector.
ChoosingE >0 andn∈Z, the representations onHE,n(which we have
constructed using somessuch thats≥ |n|) give all possible irreducible repre-
sentations ofE(3). The representation spaces have a physical interpretation as
the state space for a free quantum particle of energyEwhich carries an “inter-
nal” quantized angular momentum about its direction of motion, given by the
helicity.
19.4 For further reading
The angular momentum operators are a standard topic in every quantum me-
chanics textbook, see for example chapter 12 of [81]. The characterization here of
free particle wavefunctions at fixed energy as giving irreducible representations
of the Euclidean group is not so conventional, but it is just a non-relativistic
version of the conventional description of relativistic quantum particles in terms
of representations of the Poincar ́e group (see chapter 42). In the Poincar ́e group
case the analog of theE(3) irreducible representations of non-zero energyEand
helicitynconsidered here will be irreducible representations labeled by a non-
zero mass and an irreducible representation ofSO(3) (the spin). In the Poincar ́e
group case, for massless particles one will again see representations labeled by
an integral helicity (an irreducible representation ofSO(2)), but there is no
analog of such massless particles in theE(3) case.
For more details about representations ofE(2) andE(3), see [94] or [97]
(which is based on [92]).