13.1 The Heisenberg Lie algebra
In either the position or momentum space representation the operatorsPjand
Qjsatisfy the relation
[Qj,Pk] =iδjk 1
Soon after this commutation relation appeared in early work on quantum me-
chanics, Weyl realized that it can be interpreted as the relation between oper-
ators one would get from a representation of a 2d+ 1 dimensional Lie algebra,
now called the Heisenberg Lie algebra. Treating first thed= 1 case, we define:
Definition(Heisenberg Lie algebra,d= 1).The Heisenberg Lie algebrah 3 is
the vector spaceR^3 with the Lie bracket defined by its values on a basis(X,Y,Z)
by
[X,Y] =Z, [X,Z] = [Y,Z] = 0
Writing a general element ofh 3 in terms of this basis asxX+yY +zZ,
and grouping thex,ycoordinates together (we will see that it is useful to think
of the vector spaceh 3 asR^2 ⊕R), the Lie bracket is given in terms of the
coordinates by
[((
x
y
)
,z)
,
((
x′
y′)
,z′)]
=
((
0
0
)
,xy′−yx′)
Note that this is a non-trivial Lie algebra, but only minimally so. All Lie
brackets ofZwith anything else are zero. All Lie brackets of Lie brackets are
also zero (as a result, this is an example of what is known as a “nilpotent” Lie
algebra).
The Heisenberg Lie algebra is isomorphic to the Lie algebra of 3 by 3 strictly
upper triangular real matrices, with Lie bracket the matrix commutator, by the
following isomorphism:
X↔
0 1 0
0 0 0
0 0 0
, Y↔
0 0 0
0 0 1
0 0 0
, Z↔
0 0 1
0 0 0
0 0 0
((
x
y)
,z)
↔
0 x z
0 0 y
0 0 0
and one has
0 x z
0 0 y
0 0 0
,
0 x′ z′
0 0 y′
0 0 0
=
0 0 xy′−x′y
0 0 0
0 0 0
The generalization of this to higher dimensions is: