E.2 Quaternions in the GOE
463
where ¯
Q n (N) is an n×n quaternion composed of matrix elements ¯
σ N (x i , x j ) and
∞
−∞
dx 1 . . .
∞
−∞
dx N Qdet[ ¯
Q N ] = N !
1 0
0 1
.
(E.44)
It is important to remember that the n×n quaternion ¯
Q n (N) has matrix elements
¯
σ N 1 (x i , x j ) that depend on N. The extra factor ¯
A N (x i , x j ), which appeared when
we integrated over products of matrix elements, ¯
σ N 1 (x i , x k ) and ¯
σ N 1 (x k , x j ), in
Eq. (E.37) cancels out of the final expression for the quaternion determinant after
all integrations are done.
Let us now relate the quaternion matrix ¯
Q N to the matrix ¯
K N in Eq. (E.24).
The first step is to write S N
1,1 (x i , x j ), S N
1,2 (x i , x j ), and N (x i , x j ) as the product of
matrices
S
N
1,1 (x i , x j ) = ¯
φ
T
i · ¯
φ j − ¯
p
T
i · ¯
I j ,
(E.45)
S
N
1,2 (x i , x j ) = ¯
p
T
i · ¯
φ j − ¯
φ
T
i · ¯
p j ,
(E.46)
and
N (x i , x j ) = ¯
I
T
i · ¯
φ j − ¯
φ
T
i · ¯
I j ,
(E.47)
where
¯
φ j =
⎛
⎜
⎜
⎜
⎝
φ 0 (x j )
φ 2 (x j )
. . .
φ N −2 (x j )
⎞
⎟
⎟
⎟
⎠
¯
p j =
⎛
⎜
⎜
⎜
⎝
φ
0 (x j )
φ
2 (x j )
. . .
φ
N −2 (x j )
⎞
⎟
⎟
⎟
⎠
, ¯
I j =
⎛
⎜
⎜
⎜
⎝
x j
0 dtφ 0 (t)
x j
0 dtφ 2 (t)
. . .
x j
0 dtφ N −2 (t)
⎞
⎟
⎟
⎟
⎠
.
(E.48)
The quaternion matrix ¯
Q N can be written
¯
Q
(1)
N =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
¯
φ T
1 · ¯
α + ¯
p T
1 · ¯
β
¯
I T
1 · ¯
α + ¯
φ T
1 · ¯
β
¯
φ T
2 · ¯
α + ¯
p T
2 · ¯
β
¯
I T
2 · ¯
α + φ T
2 · ¯
β
. . .
¯
φ T
N · ¯
α + ¯
p T
N · ¯
β
¯
I T
N · ¯
α + ¯
φ T
N · ¯
β
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
+ ¯
,
(E.49)
where ¯
α and ¯
β are 2N ×N/2 matrices defined as
¯
α =
¯
φ 1 , − ¯
p 1 , ¯
φ 2 , − ¯
p 2 , . . ., ¯
φ N , − ¯
p N
(E.50)