Quantum Mechanics for Mathematicians

p 0

m p 3

m

−m

−m

O 0

O(m, 0 , 0 ,0)

O(1, 0 , 0 ,1)

O(− 1 , 0 , 0 ,1)

O(0, 0 , 0 ,m)

O(−m, 0 , 0 ,0)

Figure 42.1: Orbits of vectors under the Lorentz group.

Unlike the Euclidean group case, here there are several different kinds of
orbitsOp. We’ll examine them and the corresponding stabilizer groupsKpeach
in turn, and see what can be said about the associated representations.

42.3.1 Positive energy time-like orbits

One way to get negative values−m^2 of the CasimirP^2 is to take the vector
p= (m, 0 , 0 ,0),m >0 and generate an orbitO(m, 0 , 0 ,0)by acting on it with the
Lorentz group. This will be the upper, positive energy, sheet of the hyperboloid
of two sheets
−p^20 +p^21 +p^22 +p^23 =−m^2

so
p 0 =

√

p^21 +p^22 +p^23 +m^2

The stabilizer group ofK(m, 0 , 0 ,0)is the subgroup ofSO(3,1) of elements of
the form (
1 0
0 R

)