Definition(Stabilizer group or little group).The subgroupKα⊂Kof elements
k∈Ksuch that
̂Φk(α) =α
for a givenα∈N̂is called the stabilizer subgroup (by mathematicians) or little
subgroup (by physicists).
The groupKαwill act on the subspaceVα, and this representation ofKαis a
second piece of information that can be used to characterize a representation.
In the case of the Euclidean groupE(2) we found that the non-zero orbits
Oαwere circles and the groupsKαwere trivial. ForE(3), the non-zero orbits
were spheres, withKαanSO(2) subgroup ofSO(3) (one that varies withα). In
these cases we found that our construction of representations ofE(2) orE(3) on
spaces of solutions of the single-component Schr ̈odinger equation corresponded
under Fourier transform to a representation on functions on the orbitsOα.
We also found in theE(3) case that using multiple-component wavefunctions
gave new representations corresponding to a choice of orbitOαand a choice
of irreducible representation ofKα=SO(2). We did not show this, but this
construction gives an irreducible representation when a single orbitOαoccurs
(with a transitiveKaction), with an irreducible representation ofKαonVα.
We will not further pursue the general theory here, but one can show that
distinct irreducible representations ofNoKwill occur for each choice of an
orbitOαand an irreducible representation ofKα. One way to construct these
representations is as the solution space of an appropriate wave equation, with the
wave equation corresponding to the eigenvalue equation for a Casimir operator.
In general, other “subsidiary conditions” then must be imposed to pick out a
subspace of solutions that gives an irreducible representation ofNoK; this
corresponds to the existence of other Casimir operators. Another part of the
general theory has to do with the question of the unitarity of representations
produced in this way, which will require that one starts with an irreducible
representation ofKαthat is unitary.
20.5 For further reading
For more on representations of semi-direct products, see section 3.8 of [84],
chapter 5 of [94], [9], and [39]. The general theory was developed by Mackey
during the late 1940s and 1950s, and his lecture notes on representation theory
[58] are a good source for the details of this. The point of view taken here,
that emphasizes constructing representations as solution spaces of differential
equations, where the differential operators are Casimir operators, is explained
in more detail in [47].
The conventional derivation found in most physics textbooks of the opera-
torsUL′coming from an infinitesimal group action uses Lagrangian methods and
Noether’s theorem. The purely Hamiltonian method used here treats configu-
ration and momentum variables on the same footing, and is useful especially in
the case of group actions that mix them (such as the example of section 20.3.2)