Quantum Mechanics for Mathematicians

15.5 The dual of a Lie algebra and symplectic geometry

We have been careful to keep track of the difference between phase space
M=R^2 dand its dualM=M∗, even though the symplectic form provides
an isomorphism between them (see equation 14.6). One reason for this is that
it isM=M∗that is related to the Heisenberg Lie algebra by

h 2 d+1=M⊕R

withMthe linear functions on phase space,Rthe constant functions, and the
Poisson bracket the Lie bracket. It is this Lie algebra that we want to use in
chapter 17 when we define the quantization of a classical system.
Another reason to carefully keep track of the difference betweenMandMis
that they carry two different actions of the Heisenberg group, coming from the
fact that the group acts quite differently on its Lie algebra (the adjoint action)
and on the dual of its Lie algebra (the “co-adjoint” action). OnMandMthese
actions become:

• The Heisenberg groupH 2 d+1acts on its Lie algebrah 2 d+1=M⊕Rby
  the adjoint action, with the differential of this action given as usual by
  the Lie bracket (see equation 5.4). Here this means

ad

(((

cq

cp

)

,c

))(((

c′q

c′p

)

,c′

))

=

[((

cq

cp

)

,c

)

,

((

c′q

c′p

)

,c′

)]

=

((

0

0

)

,Ω

((

cq

cp

)

,

(

c′q

c′p

)))

This action is trivial on the subspaceM, taking

(

c′q

c′p

)

→

(

0

0

)

• The simplest way to define the “co-adjoint” action in this case is to define
  it as the Hamiltonian action ofH 2 d+1onM=R^2 dsuch that its moment
  mapμLis just the identification ofh 2 d+1with functions onM. For the
  cased= 1, one has

μL=L=cqq+cpp+c∈h 3

and

XμL=−cq

∂

∂p

+cp

∂

∂q

This is the action described in equations 15.3 and 15.4, satisfying

q

(((

x

y

)

,z

)

·m

)

=q(m) +y, p

(((

x

y

)

,z

)

·m

)

=p(m)−x(15.13)

Here the subgroup of elements ofH 3 withx=z= 0 acts as the usual

translations in positionq.