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