Quantum Mechanics for Mathematicians

• A Lie subalgebra with basis elementszjzk, which has dimensiond^2. Com-
  puting Poisson brackets one finds

{zjzk,zlzm}=zj{zk,zlzm}+zk{zj,zlzm}

=−izjzmδkl+izlzkδjm (25.2)

In this chapter we’ll focus on the third subalgebra and the operators that arise
by quantization of its elements.
Taking all complex linear combinations, this subalgebra can be identified
with the Lie algebragl(d,C) of alldbydcomplex matrices. One can see this
by noting that ifEjkis the matrix with 1 at thej-th row andk-th column,
zeros elsewhere, one has

[Ejk,Elm] =Ejmδkl−Elkδjm

and these provide a basis ofgl(d,C). Identifying bases by

izjzk↔Ejk

gives the isomorphism of Lie algebras. Thisgl(d,C) is the complexification of
u(d), the Lie algebra of the unitary groupU(d). Elements ofu(d) will corre-
spond to, equivalently, skew-adjoint matrices, or real linear combinations of the
quadratic functions
zjzk+zjzk, i(zjzk−zjzk)

onM.
In section 16.1.2 we saw that the moment map for the action of the symplectic
group on phase space is just the identity map when we identify the Lie algebra
sp(2d,R) with order two homogeneous polynomials in the phase space coor-
dinatesqj,pj. We can complexify and identifysp(2d,C) with complex-valued
order two homogeneous polynomials which we write in terms of the complexi-
fied coordinateszj,zj. The moment map is again the identity map, and on the
sub-Lie algebra we are concerned with, is explicitly given by

A∈gl(d,C)→μA=i

∑

j,k

zjAjkzk (25.3)

We can at the same time consider the complexification of the Heisenberg Lie
algebra, using linear functions ofzjandzj, with Poisson brackets between these
and the order two homogeneous functions giving a complexified version of the
derivation action ofsp(2d,R) onh 2 d+1.
We have (complexifying and restricting togl(d,C)⊂sp(2d,C)) the following
version of theorems 16.2 and 16.3

Theorem 25.1.The map of equation 25.3 is a Lie algebra homomorphism, i.e.

{μA,μA′}=μ[A,A′]

TheμAsatisfy (for column vectorszwith componentsz 1 ,...,zd)

{μA,z}=−Az, {μA,z}=ATz (25.4)