Quantum Mechanics for Mathematicians

whereR∈SO(3), soK(m, 0 , 0 ,0)=SO(3). Irreducible representations of this
group are classified by the spin. For spin 0, points on the hyperboloid can
be identified with positive energy solutions to a wave equation called the Klein-
Gordon equation and functions on the hyperboloid both correspond to the space
of all solutions of this equation and carry an irreducible representation of the
Poincar ́e group. This case will be studied in detail in chapters 43 and 44. We
will study the case of spin^12 in chapter 47, where one must use the double cover
SU(2) ofSO(3). The Poincar ́e group representation will be on functions on
the orbit that take values in two copies of the spinor representation ofSU(2).
These will correspond to solutions of a wave equation called the massive Dirac
equation. For choices of higher spin representations of the stabilizer group, one
can again find appropriate wave equations and construct Poincar ́e group repre-
sentations on their space of solutions (although additional subsidiary conditions
are often needed) but we will not enter into this topic.
Forp= (m, 0 , 0 ,0) the Pauli-Lubanski operator will be

W 0 = 0, W=−mJ

and the second Casimir operator will be

W^2 =m^2 J^2

The eigenvalues ofW^2 are thus proportional to the eigenvalues ofJ^2 , the
Casimir operator for the subgroup of spatial rotations. These are again given
by the spins, and will take the valuess(s+ 1). These eigenvalues classify
representations consistently with the stabilizer group classification.

42.3.2 Negative energy time-like orbits

Starting instead with the energy-momentum vectorp= (−m, 0 , 0 ,0),m >0,
the orbitO(−m, 0 , 0 ,0)one gets is the lower, negative energy component of the
hyperboloid
−p^20 +p^21 +p^22 +p^23 =−m^2

satisfying

p 0 =−

√

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

Again, one has the same stabilizer groupK(−m, 0 , 0 ,0)=SO(3) and the same
constructions of wave equations of various spins and Poincar ́e group represen-
tations on their solution spaces as in the positive energy case. Since negative
energies lead to unstable, unphysical theories, we will see that these represen-
tations are treated differently under quantization, corresponding physically not
to particles, but to antiparticles.

42.3.3 Space-like orbits

One can get positive valuesm^2 of the CasimirP^2 by considering the orbit
O(0, 0 , 0 ,m)of the vectorp= (0, 0 , 0 ,m). This is a hyperboloid of one sheet,