Quantum Mechanics for Mathematicians

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