118
F. Barbaresco
Appendix
Bargman Parameterization of SU(1,1)
SU (1, 1) is isomorphic to SL(2, R) = Sp(2, R) through the complex unitary matrix
W:
SL(2, R) =
g =
a b
c d
/ det g = ad − bc = 1
(5.A1)
Sp(2, R) =
g =
a b
c d
/g Jg
T
= J, J =
0 +1
−1 0
(5.A2)
W =
1
√
2
ω
−1
ω
−1
−ω ω
=
W
+
−1 with ω = e
iπ/4
=
1
√
2
(1 + i)
(5.A3)
If we observe that W
−1 J W = −i M, the isomorphism is given explicitely by:
a b
c d
= g(u) = W uW
−1
=
Re(α + β) −Im(α − β)
Im(α + β) Re(α − β)
(5.A30)
α β
β
∗
α
∗
= u(g) = W
−1 gW =
1
2
(a + d) − i(b − c) (a − d) + i(b + c)
(a − d) − i(b + c) (a + d) + i(b − c)
(5.A4)
We can also make also a link with SO(2, 1) of “1 + 2” pseudo-orthogonal
matrices:
SO(2, 1) =
⎧
⎨
⎩
∈ G L(3, 3)/ det() = 1, ,K
T
= , K =
⎛
⎝
+1 0 0
0 −1 0
0 0 −1
⎞
⎠
⎫
⎬
⎭
(5.A5)
(g) =
⎛
⎝
1
2
a
2
+ b
2
+ c
2
+ d
2
1
2
a
2
− b
2
+ c
2
− d
2
−cd − ab
1
2
a
2
+ b
2
− c
2
− d
2
1
2
a
2
− b
2
− c
2
+ d
2
cd − ab
−bd − ac
bd − ac
ad + bc
⎞
⎠
(5.A6)
with (g 1 )(g 2 ) = (g 1 g 2 ), ,(I ) = I, ,
g
−1
= (g)
−1
The SO(2, 1) matrix corresponds to any SU (1, 1):
(u) =
⎛
⎝
|α|
2
+ |β|
2
2Reαβ
∗
2Imαβ
∗
2Reαβ
Re
α
2
+ β
2
Im
α
2
− β
2
−2Imαβ −Im
α
2
+ β
2
Re
α
2
− β
2
⎞
⎠
(5.A7)
and α = ±
1
2 ( 11 + 12 ) + i( 12 − 21 ), β =
1
2α ( 10 − i 20 )