D.3 Unitary Matrices
443
Note that only off-diagonal elements of δ ¯
O contribute to the invariant metric. The
determinant of the metric tensor becomes
Det[ ¯
g] = 2
N(N−1)/2
1≤i |e
iθ j − e
iθ i |
2 .
(D.63)
The invariant measure then takes the form
dd U R =
1≤i |e
iθ j − e
iθ i |
dθ 1 ×. . .×dθ N δδ δO ,
(D.64)
where
δδ δO = 2
N(N−1)/4 δO 1,2 ×. . .×δO N −1,N
(D.65)
(see Eq. (D.18)).
We can also write the invariant measure in terms of eigenvalues of ¯
H R . Using
Eq. (D.56), it is straightforward to compute the Jacobian of the transformation
between phase angles, θ j , and eigenvalues, e j . We find
dθ 1 ×. . .×dθ N = 2
N cos
2 (θ 1 /2)×. . .×cos
2 (θ N /2)de 1 ×. . .×de N .
(D.66)
If we use Eqs. (D.56) and (D.66), we obtain (after some algebra) the expression for
the invariant measure
dd U R = 2
N(N+1)/2
N
j =1
de j
(1 + e 2
j ) (N +1)/2
1≤i |e j − e i |
δδ O
= 2
N(N+1)/2
dd H R
Det[ ¯
1 + ¯
H 2
R ] (N +1)/2
,
(D.67)
where the invariant measure dd H R is defined in Eqs. (D.8) and (D.17). The total
volume of this measure, U R =
dd U R , is computed in Sect. D.4 (see also Hua
1963, page 33) and is
U R = 2
N(N+1)/2 2
N(N−1)/4 π
N(N+1)/4
N
j =1
[j/2]
[j ]
.
(D.68)