D.3 Unitary Matrices
445
D.3.3 Symplectic Unitary Matrices
Symplectic unitary matrices, ¯
U Q , are obtained via Eq. (D.51) from quaternion real
matrices, ¯
H Q . They are diagonalized by a symplectic unitary matrix ¯
V = ¯
S, whose
matrix elements are 2×2 quaternions. If we use Eq. (D.61), we find that the invariant
metric for the symplectic unitary matrix ¯
U Q takes the form
(ds)
2
U Q
= 2
N
j =1
(dθ j )
2
+ 4
1≤i |e
iθ j − e
iθ i |
2
3
α=0
(δs
α
i,j )
2
(D.75)
(see Eqs. (D.43) and (D.45)). Only off-diagonal elements of δ ¯
S contribute to the
invariant metric. The determinant of the metric tensor becomes
Det[ ¯
g] = 2
4N 2 −3N
1≤i |e
iθ j − e
iθ i |
8 .
(D.76)
The invariant measure then takes the form
dd U Q = e
N/2
1≤i |e
iθ j − e
iθ i |
4 dθ 1 ×. . .×dθ N δδ S ,
(D.77)
where
δδ S = 2
2N(N−1)
1≤i 3
α=0
δs
α
i,j .
(D.78)
If we use Eqs. (D.56), the invariant measure can also be written in terms of the
eigenvalues of ¯
H Q . We find
dd U Q = 2
(4N 2 −N)/2
N
j =1
1
(1 + e 2
j ) 2N −1
1≤i |e j − e i |
4
×de 1 ×. . .×de N δδ S = 2
N(2N −1)
dd H Q
Det[ ¯
1 + ¯
H 2
Q ] 2N −1
,
(D.79)
where the invariant measure, dd H Q , is defined in Eqs. (D.40) and (D.47). The
volume, U Q , of this measure is computed in Sect. D.4.