Quantum Mechanics for Mathematicians

We now have an explicit highest weight vector, and an explicit construction
of the corresponding irreducible representation. If one chooses a basis ofVn,
then the linear operatorπ′(X) will be given by an+ 1 byn+ 1 matrix. Clearly
though, the expression as a simple first-order differential operator is much easier
to work with. In the examples we will be studying in later chapters, the repre-
sentations under consideration will often be on function spaces, with Lie algebra
representations appearing as differential operators. Instead of using linear al-
gebra techniques to find eigenvalues and eigenvectors, the eigenvector equation
will be a partial differential equation, with our focus on using Lie groups and
their representation theory to solve such equations.
One issue we haven’t addressed yet is that of unitarity of the representation.
We need Hermitian inner products on the spacesVn, inner products that will
be preserved by the action ofSU(2) that we have defined on these spaces. A
standard way to define a Hermitian inner product on functions on a spaceMis
to define them using an integral: forf,gcomplex-valued functions onM, take
their inner product to be

〈f,g〉=

∫

M

fg

While forM=C^2 this gives anSU(2) invariant inner product on functions (one
that is not invariant for the full groupGL(2,C)), it is useless forf,gpolyno-
mial, since such integrals diverge. In this case an inner product on polynomial
functions onC^2 can be defined by

〈f,g〉=

1

π^2

∫

C^2

f(z 1 ,z 2 )g(z 1 ,z 2 )e−(|z^1 |

(^2) +|z 2 | (^2) )
dx 1 dy 1 dx 2 dy 2 (8.2)
Herez 1 =x 1 +iy 1 ,z 2 =x 2 +iy 2. Integrals of this kind can be done fairly easily
since they factorize into separate integrals overz 1 andz 2 , each of which can be
treated using polar coordinates and standard calculus methods. One can check
by explicit computation that the polynomials
zj 1 zk 2
√
j!k!
will be an orthonormal basis of the space of polynomial functions with respect
to this inner product, and the operatorsπ′(X),X∈su(2) will be skew-adjoint.
Working out what happens for the first few examples of irreducibleSU(2)
representations, one finds orthonormal bases for the representation spacesVn
of homogeneous polynomials as follows

• Forn=s= 0
  1


• Forn= 1,s=^12
  z 1 , z 2


• Forn= 2,s= 1
  1
  √
  2

z 12 , z 1 z 2 ,

1

√

2

z 22