F.3 Gaussian Symplectic Ensemble (GSE)
487
The determinant of the metric tensor is Det[ ¯
g] = 2 4N 2 −3N
1≤i Notice that the diagonal elements of δ ¯
S do not contribute. The invariant measure
can be written
dd H Q = 2
N/2
1≤i (e j − e i )
4
de 1 de 2 ×. . .×de N δδ S ,
(F.113)
where
δδ S = 2
2N(N−1)
1≤i 3
α=0
ds
(α)
i,j
(F.114)
is a measure associated with the off-diagonal elements of the matrix, δ ¯
S.
F.3.3 Probability Density for GSE
The normalized probability density for the Gaussian Symplectic Ensemble (GSE)
can be written
P ( ¯
H Q ) =
1
2πδ 2
N(2N −1)/2
exp
−
1
2δ 2 Tr[ ¯
H Q · ¯
H
†
Q ]
,
(F.115)
and it extremizes information.
The probability density P ( ¯
H Q ) can also be written in polar form. Because
Tr[ ¯
H Q · ¯
H
†
Q ] is invariant under a unitary transformation of ¯
H Q , we can also write
it as Tr[ ¯
H Q · ¯
H
†
Q ] = 2
N
j =1 e 2
j . The probability density, in polar form, takes the
form
P ( ¯
H Q ) dd H Q = 2
N/2
1
2πδ 2
N(2N −1)/2
1≤i |e j − e i |
4
×exp
−
1
δ 2
N
j =1
e
2
j
de 1 ×. . .×de N δδ S ,
(F.116)
where δ determines the variance of the distribution.
If we let x 2
j = e 2
j /(4δ) and integrate over δδ S , we obtain
P Q (x 1 , . . ., x N ) =
2 2N 2 2 −N/2
π N/2
N
k=1 [(2k)!]
1≤i |x j − x i |
4
×exp
−2
N
j =1
x
2
j
(F.117)