Quantum Mechanics for Mathematicians

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