Proof.Using 25.2 one has
{μA,μA′}=−
∑
j,k,l,m
{zjAjkzk,zlA′lmzm}
=−
∑
j,k,l,m
AjkA′lm{zjzk,zlzm}
=i
∑
j,k,l,m
AjkA′lm(zjzmδkl−zlzkδjm)
=i
∑
j,k
zj[A,A′]jkzk=μ[A,A′]
To show 25.4, compute
{μA,zl}={i
∑
j,k
zjAjkzk,zl}=i
∑
j,k
Ajk{zj,zl}zk
=−
∑
j
Aljzj
and
{μA,zl}={i
∑
j,k
zjAjkzk,zl}=i
∑
j,k
zjAjk{zk,zl}
=
∑
k
zkAkl
Note that here we have written formulas forA∈gl(d,C), an arbitrary com-
plexdbydmatrix. It is only forA ∈u(d), the skew-adjoint (AT =−A)
matrices, thatμAwill be a real-valued moment map, lying in the real Lie al-
gebrasp(2d,R), and giving a unitary representation on the state space after
quantization. For suchAwe can write the relations 25.4 as a (complexified)
example of 16.22 {
μA,
(
z
z
)}
=
(
AT 0
0 AT
)(
z
z
)
The standard harmonic oscillator Hamiltonian
h=
∑d
j=1
zjzj (25.5)
lies in thisu(d) sub-algebra (it is the caseA=−i 1 ), and its Poisson brackets
with the rest of the sub-algebra are zero. It gives a basis element of the one
dimensionalu(1) subalgebra that commutes with the rest of theu(d) subalgebra.
While we are not entering here into the details of what happens for polyno-
mials that are linear combinations of thezjzkandzjzk, it may be worth noting