group elements expressible as exponentials) by knowing the Lie bracket. For
the full formula and a detailed proof, see chapter 5 of [42]. One can easily
check the first few terms in this formula by expanding the exponentials, but the
difficulty of the proof is that it is not at all obvious why all the terms can be
organized in terms of commutators.
For the case of the Heisenberg Lie algebra, since all multiple commutators
vanish, the Baker-Campbell-Hausdorff formula implies for exponentials of ele-
ments ofh 3
eAeB=eA+B+
(^12) [A,B]
(a proof of this special case of Baker-Campbell-Hausdorff is in section 5.2 of [42]).
We can use this to explicitly write the group law in exponential coordinates:
Definition(Heisenberg group,d= 1). The Heisenberg groupH 3 is the space
R^3 =R^2 ⊕Rwith the group law
((
x
y
)
,z)((
x′
y′)
z′)
=
((
x+x′
y+y′)
,z+z′+1
2
(xy′−yx′))
(13.2)
The isomorphism betweenR^2 ⊕Rwith this group law and the matrix form of
the group is given by
((
x
y)
,z)
↔
1 x z+^12 xy
0 1 y
0 0 1
Note that the Lie algebra basis elementsX,Y,Z each generate subgroups
ofH 3 isomorphic toR. Elements of the first two of these subgroups generate
the full group, and elements of the third subgroup are “central”, meaning they
commute with all group elements. Also notice that the non-commutative nature
of the Lie algebra (equation 13.1) or group (equation 13.2) depends purely on
the factorxy′−yx′.
The generalization of this to higher dimensions is:
Definition(Heisenberg group).The Heisenberg groupH 2 d+1is the spaceR^2 d+1
with the group law
((
x
y
)
,z)((
x′
y′)
,z′)
=
((
x+x′
y+y′)
,z+z′+1
2
(x·y′−y·x′))
wherex,x′y,y′∈Rd.
13.3 The Schr ̈odinger representation
Since it can be defined in terms of 3 by 3 matrices, the Heisenberg groupH 3
has an obvious representation onC^3 , but this representation is not unitary and
not of physical interest. What is of great interest is the infinite dimensional
representation on functions ofqfor which the Lie algebra version is given by
theQ,P, and unit operators: