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
dqψ(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−izeixy(^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′
eix′ 2 y′
e−ix′q
ψ(q−y′)=e−i(z+z′)
eixy+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−ixQThis 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)