This isomorphism preserves the Lie bracket relations since
[X,Y] =Z↔{q,p}= 1
It is convenient to choose its own notation for the dual phase space, so we
will often writeM∗=M. The three dimensional space we have identified with
the Heisenberg Lie algebra is then
M⊕R
We will denote elements of this space in two different ways
- As functionscqq+cpp+c, with Lie bracket the Poisson bracket
{cqq+cpp+c,c′qq+c′pp+c′}=cqc′p−cpc′q
- As pairs of an element ofMand a real number
((
cq
cp
)
,c
)
In this second notation, the Lie bracket is
[((
cq
cp
)
,c
)
,
((
c′q
c′p
)
,c′
)]
=
((
0
0
)
,Ω
((
cq
cp
)
,
(
c′q
c′p
)))
which is identical to the Lie bracket forh 3 of equation 13.1. Notice that
the Lie bracket structure is determined purely by Ω.
In higher dimensions, coordinate functionsq 1 ,···,qd,p 1 ,···,pdonMpro-
vide a basis for the dual spaceM. Taking as an additional basis element the
constant function 1, we have a 2d+ 1 dimensional space with basis
q 1 ,···,qd,p 1 ,···,pd, 1
The Poisson bracket relations
{qj,qk}={pj,pk}= 0, {qj,pk}=δjk
turn this space into a Lie algebra, isomorphic to the Heisenberg Lie algebra
h 2 d+1. On general functions, the Poisson bracket will be given by the obvious
generalization of thed= 1 case
{f 1 ,f 2 }=
∑d
j=1
(
∂f 1
∂qj
∂f 2
∂pj
−
∂f 1
∂pj
∂f 2
∂qj
)
(14.2)
Elements ofh 2 d+1are functions onM=R^2 dof the form
cq 1 q 1 +···+cqdqd+cp 1 p 1 +···+cpdpd+c=cq·q+cp·p+c