Definition(Schr ̈odinger representation, Lie algebra version).The Schr ̈odinger
representation of the Heisenberg Lie algebrah 3 is the representation(Γ′S,L^2 (R))
satisfying
Γ′S(X)ψ(q) =−iQψ(q) =−iqψ(q), Γ′S(Y)ψ(q) =−iPψ(q) =−
d
dq
ψ(q)
Γ′S(Z)ψ(q) =−iψ(q)
Factors ofihave been chosen to make these operators skew-adjoint and the
representation thus unitary. They can be exponentiated, giving in the exponen-
tial coordinates onH 3 of equation 13.2
ΓS
(((
x
0
)
, 0
))
ψ(q) =e−xiQψ(q) =e−ixqψ(q)
ΓS
(((
0
y
)
, 0
))
ψ(q) =e−yiPψ(q) =e−y
dqd
ψ(q) =ψ(q−y)
ΓS
(((
0
0
)
,z
))
ψ(q) =e−izψ(q)
For general group elements ofH 3 one has:
Definition(Schr ̈odinger representation, Lie group version). The Schr ̈odinger
representation of the Heisenberg Lie groupH 3 is the representation(ΓS,L^2 (R))
satisfying
ΓS
(((
x
y
)
,z
))
ψ(q) =e−izei
xy
(^2) e−ixqψ(q−y) (13.3)
To check that this defines a representation, one computes
ΓS
(((
x
y
)
,z
))
ΓS
(((
x′
y′
)
,z′
))
ψ(q)
=ΓS
(((
x
y
)
,z
))
e−iz
′
ei
x′ 2 y′
e−ix
′q
ψ(q−y′)
=e−i(z+z
′)
ei
xy+x′y′
(^2) e−ixqe−ix′(q−y)ψ(q−y−y′)
=e−i(z+z
′+ (^12) (xy′−yx′))
ei
(x+x′)(y+y′)
(^2) e−i(x+x′)qψ(q−(y+y′))
=ΓS
(((
x+x′
y+y′
)
,z+z′+
1
2
(xy′−yx′)
))
ψ(q)
The group analog of the Heisenberg commutation relations (often called the
“Weyl form” of the commutation relations) is the relation
e−ixQe−iyP=e−ixye−iyPe−ixQ
This can be derived by using the explicit representation operators in equation
13.3 (or the Baker-Campbell-Hausdorff formula and the Heisenberg commuta-
tion relations) to compute
e−ixQe−iyP=e−i(xQ+yP)+
(^12) [−ixQ,−iyP]
=e−i
xy
(^2) e−i(xQ+yP)