For group elements,gθ=eθZ∈SO(2)⊂SL(2,R) and the representation is
given by unitary operators
Ugθ= ΓBF(eθZ) =e−iθ 2 (aa†+a†a)which satisfy
Ugθ(
a
a†)
Ug−θ^1 =(
eiθ 0
0 e−iθ)(
a
a†)
(24.5)
Note that, using equation 5.1
d
dθ(
Ugθ(
a
a†)
Ug−θ^1)
|θ=0=
[
−
i
2(aa†+a†a),(
a
a†)]
so equation 24.4 is the derivative atθ= 0 of equation 24.5. We see that, on
operators, conjugation by the action of thisSO(2) subgroup ofSL(2,R) does
not mix creation and annihilation operators. On the distinguished state| 0 〉,Ugθ
acts as the phase transformation
Ugθ| 0 〉=e−
2 iθ
| 0 〉Besides 24.3, there are also the following other Poisson bracket relations
between order two and order one polynomials inz,z
{z^2 ,z}= 2iz, {z^2 ,z}= 0, {z^2 ,z}=− 2 iz, {z^2 ,z}= 0 (24.6)The function
μ=i
2(z^2 −z^2 )will provide a moment map for theR⊂SL(2,R) subgroup studied in section
20.3.4. This is the subgroup of elementsgrthat forr∈Ract on basis elements
q,pby (
q
p
)
→
(
er 0
0 e−r)(
q
p)
=
(
erq
e−rp)
(24.7)
or on basis elementsz,zby
(
z
z
)
→
(
coshr sinhr
sinhr coshr)(
z
z)
This moment map satisfies the relations
{
μ,
(
z
z)}
=
(
0 1
1 0
)(
z
z)
(24.8)
Quantization gives
Γ′BF(μ) =1
2
(a^2 −(a†)^2 )which satisfies [
1
2
(a^2 −(a†)^2 ),(
a
a†)]
=
(
0 1
1 0
)(
a
a†