Quantum Mechanics for Mathematicians

Given an irreducible representation, the operatorP^2 will act by the scalar

−p^20 +p^21 +p^22 +p^23

which can be positive, negative, or zero, so given bym^2 ,−m^2 ,0 for various
m. The value of the scalar will be the same everywhere on the orbit, so in
energy-momentum space, orbits will satisfy one of the three equations

−p^20 +p^21 +p^22 +p^23 =







−m^2

m^2

0

The representation can be further characterized in one of two ways:

• By the value of the second Casimir operatorW^2.


• By the representation of the stabilizer groupKpon the eigenspace of the
  momentum operators with eigenvaluep.

At the pointpon an orbit, the Pauli-Lubanski operator has components

W 0 =−p·J, W=−p 0 +p×K

In the next chapter we will find the possible orbits, then pick a pointpon each
orbit, and see what the stabilizer groupKpand Pauli-Lubanski operator are at
that point.

42.3 Classification of representations by orbits

The Lorentz group acts on the energy-momentum spaceR^4 by

p→Λp

and, restricting attention to thep 0 p 3 plane, the picture of the orbits looks like
this