Quantum Mechanics for Mathematicians

A straightforward calculation using the Poincar ́e Lie algebra commutation
relations shows thatP^2 is a Casimir operator since one has

[P 0 ,P^2 ] = [Pj,P^2 ] = [Jj,P^2 ] = [Kj,P^2 ] = 0

forj= 1, 2 ,3. HereJjis a Lie algebra representation operator corresponding
to thelj, andKjthe operator corresponding tokj.
The second Casimir operator is more difficult to identify in the Poincar ́e case
than in theE(3) case. To find it, first define:

Definition(Pauli-Lubanski operator).The Pauli-Lubanski operator is the four-
component operator

W 0 =−P·J, W=−P 0 J+P×K

By use of the commutation relations, one can show that the components of
Wμbehave like a four-vector, i.e.,

[Wμ,Pν] = 0

and the commutation relations with theJjandKjare the same forWμas for
Pμ. One can then define:

Definition(Second Casimir operator). The second Casimir operator for the
Poincar ́e Lie algebra is

W^2 =−W 02 +W 12 +W 22 +W 32

Use of the commutation relations shows that

[P 0 ,W^2 ] = [Pj,W^2 ] = [Jj,W^2 ] = [Kj,W^2 ] = 0

soW^2 is a Casimir operator.
To classify Poincar ́e group representations, we have two tools available. We
can use the two Casimir operatorsP^2 andW^2 and characterize irreducible
representations by their eigenvalues. In addition, recall from chapter 20 that
irreducible representations of semi-direct productsNoKare associated with
pairs of aK-orbitOαforα ∈Nˆ, and an irreducible representation of the
corresponding little groupKα.
For the Poincar ́e group,Nˆ=R^4 is the space of characters (one dimensional
representations) of the translation group of Minkowski space. Elementsαare
labeled by
p= (p 0 ,p 1 ,p 2 ,p 3 )

where thepμare the eigenvalues of the energy-momentum operatorsPμ. For
representations on wavefunctions, these eigenvalues will correspond to elements
in the representation space with space-time dependence.

ei(−p^0 x^0 +p^1 x^1 +p^2 x^2 +p^3 x^3 )