s rel ¼ l À 1
ð
Þs l ¼ s 2 ¼ ns lim ¼ 4pns corr ; l ¼ 2; 3. . .n
ð8:6Þ
From Eqs. (7.6), (8.5–8.7) one gets
b l ¼
2p l À 1
ð
Þ
n
ð8:7Þ
which inserted into Eq. (7.5) yields, C T ¼ e
ÀbL B . = . T
C T ¼ . T ¼ k L
X n
k;l¼1
h k
j ie
i
p
n ðkþlÀ2Þ h l
h j þ k S
X n
k;l¼1
h k
j ie
i
p
n ðkþlÀ2Þ
ðd kl À
1
n
Þ h l
h j
ð8:8Þ
The result (8.8) may not appear very exciting unless one carries out the transformation B
À1
; i.e. introduce the new basis h
j iB
À1
¼ f
j i the outcome becoming
C T ¼ . T ¼ k L J
ðnÀ1Þ
þ k S J
ð 8:9Þ
with J being the nilpotent operator defined by J
ðnÞ
¼ 0; J
ðnÀ1Þ
6 ¼ 0 signifying the
quantum transitions below
J ¼
X nÀ1
k¼1
f k
j i f kþ1
h
j
ð8:10Þ
Incidentally the matrix representation of J is an n-dimensional matrix with
one’s above the diagonal and the remaining elements equal to zero. The largest
dimension of the so defined Jordan block is called the Segrè characteristic of the
degenerate eigenvalue. This thermalized configuration has previously been designated a coherent-dissipative structure [31, 36]. However, in order to signify its
thermal characteristics and relevance for free energy principles, see below, we will
denote Eqs. (8.8–8.10) as the Correlated Dissipative Ensemble, CDE. At the same
time the energy given by Eq. (6.2) over the CDE defines a correlated free energy
configuration being a authentic proxy of the (Helmholtz) free energy, here signified
as zero energy gauge.
The result above is of crucial importance implying that C
ðNÞ
¼ jW g
ð ÞihWðgÞj; after
reduction, analytic continuation and thermalization yields C
ðNÞ
T ¼ jWðf ÞihWðf
Ã
Þj,
f ¼ f 1 and with W f
ð Þ and W f
Ã
ð Þ orthogonal to each other as imparted by Eqs. (8.8–
8.10). We will use this result in two ways, (a) to develop relevant building blocks for
biological systems and (b) to use the properties of the transformation B in (6.4) as a
means for communication between entities on the molecular level. Once it is realized
that communication is ascertained between molecules and cells, this will extend to
higher order levels of semiotic interactions and communications.
A Zero Energy Universe Scenario: From Unstable Chemical …
269
ð
Þs l ¼ s 2 ¼ ns lim ¼ 4pns corr ; l ¼ 2; 3. . .n
ð8:6Þ
From Eqs. (7.6), (8.5–8.7) one gets
b l ¼
2p l À 1
ð
Þ
n
ð8:7Þ
which inserted into Eq. (7.5) yields, C T ¼ e
ÀbL B . = . T
C T ¼ . T ¼ k L
X n
k;l¼1
h k
j ie
i
p
n ðkþlÀ2Þ h l
h j þ k S
X n
k;l¼1
h k
j ie
i
p
n ðkþlÀ2Þ
ðd kl À
1
n
Þ h l
h j
ð8:8Þ
The result (8.8) may not appear very exciting unless one carries out the transformation B
À1
; i.e. introduce the new basis h
j iB
À1
¼ f
j i the outcome becoming
C T ¼ . T ¼ k L J
ðnÀ1Þ
þ k S J
ð 8:9Þ
with J being the nilpotent operator defined by J
ðnÞ
¼ 0; J
ðnÀ1Þ
6 ¼ 0 signifying the
quantum transitions below
J ¼
X nÀ1
k¼1
f k
j i f kþ1
h
j
ð8:10Þ
Incidentally the matrix representation of J is an n-dimensional matrix with
one’s above the diagonal and the remaining elements equal to zero. The largest
dimension of the so defined Jordan block is called the Segrè characteristic of the
degenerate eigenvalue. This thermalized configuration has previously been designated a coherent-dissipative structure [31, 36]. However, in order to signify its
thermal characteristics and relevance for free energy principles, see below, we will
denote Eqs. (8.8–8.10) as the Correlated Dissipative Ensemble, CDE. At the same
time the energy given by Eq. (6.2) over the CDE defines a correlated free energy
configuration being a authentic proxy of the (Helmholtz) free energy, here signified
as zero energy gauge.
The result above is of crucial importance implying that C
ðNÞ
¼ jW g
ð ÞihWðgÞj; after
reduction, analytic continuation and thermalization yields C
ðNÞ
T ¼ jWðf ÞihWðf
Ã
Þj,
f ¼ f 1 and with W f
ð Þ and W f
Ã
ð Þ orthogonal to each other as imparted by Eqs. (8.8–
8.10). We will use this result in two ways, (a) to develop relevant building blocks for
biological systems and (b) to use the properties of the transformation B in (6.4) as a
means for communication between entities on the molecular level. Once it is realized
that communication is ascertained between molecules and cells, this will extend to
higher order levels of semiotic interactions and communications.
A Zero Energy Universe Scenario: From Unstable Chemical …
269