Quantum Mechanics for Mathematicians

and calculating the commutators

[b 1 ,b 2 ] = 0, [l 3 ,b 1 ] =b 2 , [l 3 ,b 2 ] =−b 1

we see that these three elements of the Lie algebra are a basis of a Lie subalgebra
isomorphic to the Lie algebra ofE(2).
Recall from section 19.1 that there are two kinds of irreducible unitary rep-
resentations ofE(2):

• Representations such that the two translations act trivially. These are
  irreducible representations ofSO(2), so one dimensional and characterized
  by an integern(half-integers when the Poincar ́e group double cover is
  used).


• Infinite dimensional irreducible representations on a space of functions on
  a circle of radiusr.

The first of these two cases corresponds to irreducible representations of the
Poincar ́e group labeled by an integern, which is called the “helicity” of the
representation. Given the representation,nwill be the eigenvalue ofJ 3 acting
on the energy-momentum eigenspace with energy-momentum (|p|, 0 , 0 ,|p|). We
will in later chapters consider the casesn= 0 (massless scalars, wave equa-
tion the Klein-Gordon equation),n=±^12 (Weyl spinors, wave equation the
Weyl equation), andn=±1 (photons, wave equation the Maxwell equations).
The second sort of representation ofE(2) corresponds to representations of the
Poincar ́e group known as “continuous spin” representations, but these seem not
to correspond to any known physical phenomena.
Calculating the components of the Pauli-Lubanski operator, one finds

W 0 =−|p|J 3 , W 1 =−(J 1 +K 2 ), W 2 =−(J 2 −K 1 ), W 3 =−|p|J 3

Defining

B 1 =

1

√

2

(J 1 +K 2 ), B 2 =

1

√

2

(J 2 −K 1 )

the second Casimir operator is given by

W^2 = 2(B 12 +B 22 )

which is the Casimir operator forE(2). It takes non-zero values on the contin-
uous spin representations, but is zero for the representations whereE(2) trans-
lations act trivially. It does thus not distinguish between massless Poincar ́e
representations of different helicities.

42.3.6 Negative energy null orbits

Looking instead at the orbit ofp= (−|p|, 0 , 0 ,|p|), one gets the negative energy
part of the null-cone. As with the time-like hyperboloids of non-zero mass
m, these will correspond to antiparticles instead of particles, with the same
classification as in the positive energy case.