Quantum Mechanics for Mathematicians

To construct representations ofP, we would like to generalize this construc-
tion fromR^3 to Minkowski spaceM^4. To do this, one begins by defining an
action ofPonn-component wavefunctions by

ψ→u(a,Λ)ψ(x) =S(Λ)ψ(Λ−^1 (x−a))

This is the action one gets by identifyingn-component wavefunctions with

(functions onM^4 )⊗Cn

and using the induced action on functions for the first factor in the tensor
product, on the second factor takingS(Λ) to be anndimensional representation
of the Lorentz group.
One then chooses a differential operatorDonn-component wavefunctions,
one that commutes with the group action, so

u(a,Λ)Du(a,Λ)−^1 =D

Theu(a,Λ) then give a representation ofPon the space of solutions to the wave
equation
Dψ=cψ

forca constant, and in some cases this will give an irreducible representation.
If the space of solutions is not irreducible an additional set of “subsidiary con-
ditions” can be used to pick out a subspace of solutions on which the represen-
tation is irreducible. In later chapters we will consider several examples of this
construction, but now will turn to the general classification of representations
ofP.
Recall that in theE(3) case we had two Casimir operators:

P^2 =P 12 +P 22 +P 32

and
J·P

HerePj is the representation operator for Lie algebra representation, corre-
sponding to an infinitesimal translation in thej-direction.Jjis the operator for
an infinitesimal rotation about thej-axis. The Lie algebra commutation rela-
tions ofE(3) ensure that these two operators commute with the action ofE(3)
and thus, by Schur’s lemma, act as a scalar on an irreducible representation.
Note that the fact that the first Casimir operator is a differential operator in
position space and commutes with theE(3) action means that the eigenvalue
equation
P^2 ψ=cψ

has a space of solutions that is aE(3) representation, and potentially irreducible.
In the Poincar ́e group case, we can easily identify:

Definition(Casimir operator).The Casimir (or first Casimir) operator for the
Poincar ́e group is the operator

P^2 =−P 02 +P 12 +P 22 +P 32