D.2 Hermitian Matrices
439
(ds)
2
H Q
= 2
N
j =1
(dh
(0)
j,j )
2
+ 4
1≤i 3
α=0
(dh
(α)
i,j )
2 .
(D.39)
The determinant of the metric tensor is Det[ ¯
g] = 2 N 4 4[N(N−1)/2] = 2 4N 2 −3N . The
invariant measure can be written
dd H Q = 2
N/2 2
2N(N−1)
N
j =1
dh
(0)
j,j
1≤i 3
α=0
dh
(α)
i,j
.
(D.40)
D.2.3.1 Polar Form of Invariant Measure for N×N Quaternion
Real Matrix
The N×N quaternion real matrix, H Q , is diagonalized by a unitary symplectic
matrix, ¯
S, which is also an N×N quaternion real matrix. We can write
¯
H Q = ¯
S· ¯
E Q · ¯
S
† ,
(D.41)
where ¯
E Q is the diagonal N×N quaternion real matrix containing the 2N twofold
degenerate real eigenvalues of ¯
H Q so that
¯
E Q =
⎛
⎜
⎜
⎜
⎝
e 1 ¯
τ 0 ¯
0 . . . ¯
0
¯
0 e 2 ¯
τ 0 . . . ¯
0
. . .
. . .
. . .
. . .
¯
0
¯
0 . . . e N ¯
τ 0
⎞
⎟
⎟
⎟
⎠
, where ¯
τ 0 =
1, 0
0, 1
and ¯
0 =
0, 0
0, 0
.
(D.42)
Let us now take the differential of Eq. (D.41). We find that
δ ¯
F Q ≡ ¯
S
†
·d ¯
H Q · ¯
S = δ ¯
S· ¯
E Q − ¯
E Q ·δ ¯
S + d ¯
E Q ,
(D.43)
where δ ¯
S = ¯
S † d ¯
S = −d ¯
S † ¯
S = −δ ¯
S. If we look at the (i, j )th element of the
matrix δ ¯
F Q , we find
(d ¯
F Q ) i,j = (e j − e i )δ ¯
S i,j + de j δ i,j ¯
τ 0 ,
(D.44)
where δ i,j is the Kronecker delta function, δ ¯
S i,j is a 2×2 quaternion real matrix,
δ ¯
S i,j =
δs
(0)
i,j + iδs
(1)
i,j δs
(2)
i,j + iδs
(3)
i,j
−δs
(2)
i,j + iδs
(3)
i,j δs
(0)
i,j − iδs
(1)
i,j
,
(D.45)