420
E. J. Brändas
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 λ S
0 0
· · ·
0 λ L
0 0
. . .
. . .
. . .
0 0
0 0
· · ·
0 λ S
0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
= λ L J
n−1
+ λ S J
(6.19)
The matrix B, see Eq. (6.9), can simply be represented as
B =
1
√
n
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1 ω
ω
2
·
ω
n−1
1 ω
3
ω
6
· ω
3(n−1)
· ·
·
·
·
· ·
·
·
·
1 ω
2n−1
ω
2(2n−1)
· ω
(n−1)(2n−1)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(6.20)
with ω = e
iπ/n . The transition density matrix, Eq. (6.19), signifies a steady state
with d S = 0. Since erasing information, see the discussion in the previous section
and Eq. (6.18), according to Eq. (4.11) generates entropy, one obtains an over-all
agreement with Fig. 5 and the conditions for an U D(n). The steady state carries
information in terms of the matrix B, in analogy with the knowledge given by
the eigenvectors of the Schrödinger equation, its symmetries and other properties
obtained in quantum chemical applications for instance accurately predicting the
properties of a hypothetical polyatomic molecule.
The matrix in (6.20) has a special property that we will return to in the next section.
The matrix B, see Eqs. (6.9) and (6.10), transforms the second order reduced density
matrix to diagonal form, which under specific circumstances such as a sufficiently low
temperature leads to a dominating high eigenvalue that signifies Yang’s ODLRO and
superconductivity. At the same time, B
−1
= B
† transforms the correlated dissipative
structure, CDS, at temperature T, Eq. (6.18), to classical canonical form in terms of a
Jordan block of Segré characteristic n. Defining the dual preferred basis of n nuclear
oscillators, |h = |h 1 , h 2 , h 3 . . . h n , mirroring the N /2 electron pairs, the canonical
form (6.19) is represented by the transformation
58
| f =| hB
†
(6.21)
In terms of | f the irreducible part of the CDS, see (6.18), (6.19), obtains as
| h Qh | =| f J f | =
n−1
k=1
| f k f k+1 |
(6.22)
58 The present mapping appears incompatible since in general N < 2n. This is automatically taken
care of by the mirror theorem, since the extra eigenvalues in the larger space dimension are filled
out by zeros.
E. J. Brändas
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 λ S
0 0
· · ·
0 λ L
0 0
. . .
. . .
. . .
0 0
0 0
· · ·
0 λ S
0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
= λ L J
n−1
+ λ S J
(6.19)
The matrix B, see Eq. (6.9), can simply be represented as
B =
1
√
n
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1 ω
ω
2
·
ω
n−1
1 ω
3
ω
6
· ω
3(n−1)
· ·
·
·
·
· ·
·
·
·
1 ω
2n−1
ω
2(2n−1)
· ω
(n−1)(2n−1)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(6.20)
with ω = e
iπ/n . The transition density matrix, Eq. (6.19), signifies a steady state
with d S = 0. Since erasing information, see the discussion in the previous section
and Eq. (6.18), according to Eq. (4.11) generates entropy, one obtains an over-all
agreement with Fig. 5 and the conditions for an U D(n). The steady state carries
information in terms of the matrix B, in analogy with the knowledge given by
the eigenvectors of the Schrödinger equation, its symmetries and other properties
obtained in quantum chemical applications for instance accurately predicting the
properties of a hypothetical polyatomic molecule.
The matrix in (6.20) has a special property that we will return to in the next section.
The matrix B, see Eqs. (6.9) and (6.10), transforms the second order reduced density
matrix to diagonal form, which under specific circumstances such as a sufficiently low
temperature leads to a dominating high eigenvalue that signifies Yang’s ODLRO and
superconductivity. At the same time, B
−1
= B
† transforms the correlated dissipative
structure, CDS, at temperature T, Eq. (6.18), to classical canonical form in terms of a
Jordan block of Segré characteristic n. Defining the dual preferred basis of n nuclear
oscillators, |h = |h 1 , h 2 , h 3 . . . h n , mirroring the N /2 electron pairs, the canonical
form (6.19) is represented by the transformation
58
| f =| hB
†
(6.21)
In terms of | f the irreducible part of the CDS, see (6.18), (6.19), obtains as
| h Qh | =| f J f | =
n−1
k=1
| f k f k+1 |
(6.22)
58 The present mapping appears incompatible since in general N < 2n. This is automatically taken
care of by the mirror theorem, since the extra eigenvalues in the larger space dimension are filled
out by zeros.