The second-order differential operatorL^2 in theρrepresentation on functions
can explicitly be computed, it is
L^2 =L^21 +L^22 +L^23=(
i(
sinφ∂
∂θ+ cotθcosφ∂
∂φ)) 2
+
(
i(
−cosφ∂
∂θ+ cotθsinφ∂
∂φ)) 2
+
(
−i∂
∂φ) 2
=−
(
1
sinθ∂
∂θ(
sinθ∂
∂θ)
+
1
sin^2 θ∂^2
∂φ^2)
(8.4)
We will re-encounter this operator in chapter 21 as the angular part of the
Laplace operator onR^3.
For the groupSU(2) we can also find irreducible representations as solution
spaces of differential equations on functions onC^2. In that case, the differential
equation point of view is much less useful, since the solutions we are looking for
are just the homogeneous polynomials, which are more easily studied by purely
algebraic methods.
8.5 For further reading
The classification ofSU(2) representations is a standard topic in all textbooks
that deal with Lie group representations. A good example is [40], which covers
this material well, and from which the discussion here of the construction of
representations as homogeneous polynomials is drawn (see pages 77-79). The
calculation of theLjand the derivation of expressions for spherical harmonics
as Lie algebra representations ofso(3) appears in most quantum mechanics
textbooks in one form or another (for example, see chapter 12 of [81]). Another
source used here for the explicit constructions of representations is [20], chapters
27-30.
A conventional topic in books on representation theory in physics is that of
the representation theory of the groupSU(3), or even ofSU(n) for arbitrary
n. The casen= 3 is of great historical importance, because of its use in the
classification and study of strongly interacting particles, The success of these
methods is now understood as due to an approximateSU(3) symmetry of the
strong interaction theory corresponding to the existence of three different light
quarks. The highest weight theory ofSU(2) representations can be generalized
to the case ofSU(n), as well as to finite dimensional representations of other
Lie groups. We will not try and cover this topic here since it is a bit intricate,
and is already very well-described in many textbooks aimed at mathematicians
(e.g., [42]) and at physicists (e.g., [32]).