GENERALIZED QUANTUM ENTANGLEMENT FAMILY IN CONNECTION TO BLACK HOLES AND NANOTECHNOLOGY

Abstract

We present a new entanglement relativity theory by dividing Hardy’s entanglement P(H)= into two parts, a global part given by and a local part . For different n we obtain a generalized quantum entanglement family , , , .
We introduce the Fibonacci-like dimension sequence as an infinite geometric sequence and we extend the Fibonacci-like dimension sequence into the negative side.
The present work makes a leap from E-Infinity dissection of Einstein’s equation into two parts, the ordinary energy E(O) plus the dark energy E(D) , to the connection by the E-Infinity scenario of the Kerr black hole.
The connection between the E-Infinity theory with the spinning Kerr black hole leads to a paradox. The ordinary and dark energy of the universe could be used as a guiding principle in the design of a nano-Casimir dark energy reactor.

Full text

In: Quantum Gravity
Editor: Brandon Mitchell
Chapter 2
GENERALIZED QUANTUM ENTANGLEMENT
FAMILY IN CONNECTION TO BLACK HOLES
AND NANOTECHNOLOGY
Leila Marek Crnjac
Technical School Center, Maribor, Slovenia
1. ABSTRACT
We present a new entanglement relativity theory by dividing Hardy’s entanglement
P(H)=
3
φ
n
φ
into two parts, a global part given by
3
φ
and a local part
n
φ
. For different n
we obtain a generalized quantum entanglement family
3
φ
,
4
φ
,
5
φ
,
6
φ
.
We introduce the Fibonacci-like dimension sequence as an infinite geometric
sequence and we extend the Fibonacci-like dimension sequence into the negative side.
The present work makes a leap from E-Infinity dissection of Einstein’s equation into
two parts, the ordinary energy E(O)
≈
22/
2
mc
plus the dark energy E(D)
≈
)22/21(
2
mc
, to the connection by the E-Infinity scenario of the Kerr black hole.
The connection between the E-Infinity theory with the spinning Kerr black hole
leads to a paradox. The ordinary and dark energy of the universe could be used as a
guiding principle in the design of a nano-Casimir dark energy reactor.
2. 1. INTRODUCTION
Quantum entanglement is a physical phenomenon that occurs when pairs (or
groups) of particles are generated or interact in a way that the quantum state of
each member must subsequently be described relative to each other.
There are two different and equally important facets to Hardy’s classical work
on entanglement [1, 2, 3]. He demonstrates in an almost perfect way that quantum

2
Leila Marek Crnjac
mechanics is non-local [1-8]. This is what most researchers concentrated upon [4-
8]. However, Hardy’s probability of 9.0169945% for quantum entanglement must
be looked upon as an incredible result as soon as one realizes that 9.0169945% is
exactly equal to the inverse of the golden mean
2
15
−
=
φ
to the power of five
5
φ
[5, 6, 7]. This value is not profound because it is the most irrational number
φ
which is ubiquitous in art, science and natural forms [9, 10] but because it stands
for the Hausdorff-Besicovitch dimension of a zero measure random Cantor set [9,
10].
In the Cantorian space-time theory the quantum particle is represented by a
Cantor zero set while the quantum wave is represented by an empty Cantor set.
3. 2. MISSING DARK ENERGY OF THE UNIVERSE
Dark energy or the missing energy in the universe constitutes the most
challenging problem in physics and cosmology [14-18]. Accurate measurement
has shown that only 4.5% of the total energy thought to be contained in the
universe is detectable. The simple conclusion for these results, which were
awarded the 2011 Nobel Prize in Physics, is that either Einstein’s equation
contains some errors, or 95.5% of the energy in the universe is due to the
mysterious dark matter and dark energy which cannot be detected with any known
methods. Einstein’s famous equation
2
mcE
=
consists of two parts and is the sum
of the ordinary energy E(O)
≈
22/
2
mc
and the missing dark energy E(D)
≈
)22/21(
2
mc
[9, 14, 15, 16, 17, 18].
Adding both expressions we find that
.)()()(
2
mcEinsteinEDEOEE
==+=
(1)
By dividing Hardy’s entanglement into two parts P(H) =
3
φ
n
φ
, a global
counterfactual part given by
3
φ
(where
2
15
−
=
φ
) and a local part
n
φ
where n is
the number of quantum particles, Hardy’s quantum topological entanglement
5
φ
is
found for n = 2. It is therefore closely related to the Unruh temperature
4
φ
where n

3
Generalized Quantum Entanglement Family in Connection to Black Holes …
= 1 and the Immirzi parameter
6
φ
for n = 3 [19, 20]. We obtain a generalized
quantum entanglement family
3
φ
,
4
φ
,
5
φ
,
6
φ
, for n = 0, 1, 2, 3.
The global part
3
φ
of Hardy’s entanglement P(H) =
3
φ
n
φ
leads to the ordinary
part of the space-time topological energy
2/2/))(()(
523
φφφ
==
OE
T
and this
leads further to the ordinary energy density
.22/)2/()(
225
mcmcOE
≈=
φ
Similarly, dark energy is clearly the part of the topological energy of the space-
time and is equal to
2/5)2/(1)(
25
φφ
=−=
DE
T
which leads to
)22/21()2/5()(
222
mcmcDE
≈=
φ
[9, 14, 15].
We obtain
.)2/5()2/()()(
22225
mcmcmcDEOEE
=+=+=
φφ
(2)
4. 3. THE RELATION BETWEEN NEUMANN-CONNES’
NON-COMMUTATIVE GEOMETRY DIMENSION FUNCTION AND
E-INFINITY BIJECTION FORMULA
Consider the dimension function of the non-commutative quotient space
representing the well-known Penrose tiling [6],
φ
babaD +=),(
; where a, b
∈
Z
−
and
2
15
−
=
φ
.
This is necessarily a fractal universe resembling a compactified holographic
boundary. Our aim is to show that under certain conditions this dimension
function will yield the bijection formula of E-infinity [9, 10, 21-24],
.)/1( 1)(
−
=
nn
c
d
φ
Let us set Dn (an, bn) to be first D(0) ≡ D0 (0, 1) and D(1) ≡ D1 (1,
0).
Subsequently we add ai and bi following the Fibonacci scheme: an = an-1 + an-2
and bn = bn-1 + bn-2
D(0) = D0 (0, 1) = 0 +
φ
=
φ
D(1) = D1 (1, 0) = 1 + (0)
φ
= 1

4
Leila Marek Crnjac
D(2) = D2 (0 + 1, 1 + 0) = 1 +
φ
= 1/
φ
D(3) = D3 (1 + 1, 0 + 1) = 2 +
φ
= (1/
φ
)2(3)
D(4) = D4 (1 + 2, 1 + 1) = 3 + 2
φ
= (1/
φ
)3
D(5) = D5 (2 + 3, 1 + 2) = 5 + 3
φ
= (1/
φ
)4
.
.
D(n) = Dn (an, bn) = (an-1 + an-2 ) + ( bn-1 + bn-2 )
φ
= (1/
φ
)n-1
By induction we conclude that
D(n) = (1/
φ
)n-1. (4)
We obtain a Fibonacci- like dimension sequence Ff (n)
Ff (n) = {
φ
, 1, 1+
φ
, 2+
φ
, 3+2
φ
, 5+3
φ
, …}. (5)
The classical Fibonacci sequence Fn is defined by the recurrence relation
Fn+1= Fn + Fn-1, n≥1 (6)
where F0 = 0, F1 = 1, F2 = 1. The first few Fibonacci numbers of the classical
Fibonacci sequence are given {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … }.
The n-th Fibonacci number is given by the formula which is called the Binet
form, named after Jaques Binet
( )
( )
φφ φφ
+−−
=
−
−
1
1n
n
n
F
(7)
where
φ
-1 and -
φ
are the solutions of the quadratic equation x2 = x + 1. The
solutions we can write as
φ
1
2
15
1
=
+
=x
and
φ
−=
−
=
2
51
2
x
.
The Binet form of the n-th Fibonacci-like number of the Ff (n) sequence can
be expressed similarly to the classical Fibonacci sequence [21, 22]

5
Generalized Quantum Entanglement Family in Connection to Black Holes …
( )
( )
( )
( )
φ
φφ φφ
φφ φφ
φ
+−−
+
+−−
=
−
−
−
−
−
−
1
1
1
1
1
1
)(
n
n
n
n
nF
,n≥0
( )
( )
( )
( )










−−+−−
+
=−
−
−−
−
φφφφφφ
φφ
φ
1
1
11
1
1
)(
n
n
n
n
nF
( ) ( )
( ) ( )





−−−−+
+
=−
−
−−
−
φφφφφφ
φφ
φ
1
1
11
1
1
)( nn
nn
nF
(8)
( ) ( )
( ) ( )
( )





−+−−+
+
=
−
−
φφφφφ
φ
φ
φ
1
21
2
11
1
)(
n
n
nF
( )
1
1
)( −
−
=n
nF
φ
φ
.
The Fibonacci-like dimension sequence Ff (n) can be presented as an infinite
geometric sequence
{
φ
, 1, 1+
φ
, 2+
φ
, 3+2
φ
, 5+3
φ
,…}=





...,
1
,
1
,
1
,
1
,
320
φφ
φ
φ
φ
(9)
The Golden Section Principle that connects the adjacent powers of the golden
mean is seen from the infinite geometric sequence. The formula for the n-th
Fibonacci number and the bijection formula are the same. This is the bijection
formula of E-infinity theory [9, 10], as shown in
1)(
)/1(
−
=
nn
c
d
φ
, where
φ
=
)0(
c
d
.
However, we see that the bijection notation is more compact and economical and
we recognize two dimensions at once; the n is the Menger-Urysohn dimension
while
)( n
c
d
is the Hausdorff-Besicovitch dimension. Our Fibonacci-like dimension
series could be extended into the negative side using the same logic as before [25]
D(1) = D1 (1, 0) = 1 + (0)
φ
= 1
D(0) = D0 (0, 1) = 0 +
φ
=
φ

6
Leila Marek Crnjac
D(-1) = D-1(1 – 0, 0 – 1) = 1 –
φ
=
2
φ
D(-2) = D-2(0 – 1, 1 – (– 1)) = –1 + 2
φ
=
3
φ
D(-3) = D-3(1 – (– 1), – 1 – 2) = 2 – 3
φ
=
4
φ
(10)
.
.
D(-n) = D-n(an, bn) = (an-1 – an-2 ) + ( bn-1 – bn-2 )
φ
=
1
+
n
φ
By induction we conclude that
D(-n) =
.
1
+
n
φ
(11)
The Binet form of the n-th Fibonacci-like number of the F-f (n) sequence can
also be expressed similarly to the classical Fibonacci sequence [21, 22]
( )
( )
( )
( )
φ
φφ φφ
φφ φφ
φ
+−−
−+
+−−
−=
−
+
+
−
−
−
−
−
1
1
1
1
1
1
1
)1()1()(
n
n
n
n
n
n
nF
, n≥0
( )
( )
( )
( )










−−−+





−−−
+
=+
+
−−−
−
−
φφφφφφ
φφ
φ
1
1
111
1)1()1(
1
)( n
n
n
n
n
n
nF
( )
( )
( )
( )
( ) ( ) ( )( )
( )





−−+−−−





−+−
+
=+−
+
−−
−
−
−
φφφφφφ
φφ
φ
11
1
11
1
11111
1
)( nnn
nn
n
nF
(12)

7
Generalized Quantum Entanglement Family in Connection to Black Holes …
( )
( )
( ) ( ) ( )
( )





+−−+−−
+
=
−
−
−21
1
2
11111
1
)(
φφφ
φ
φ
φ
nn
n
n
nF
1
)(
+
−
=
n
nF
φ
φ
.
We obtain a Fibonacci-like dimension sequence F-f (n)
{1,
,
φ
1–
,
φ
–1 + 2
,
φ
2 – 3
,
φ
–3 + 5
,
φ
…} = {1,
,
φ
,
2
φ
,
3
φ
,
4
φ
,
5
φ
…}
(13)
Consequently, it is easy to extend the bijection formula
1)( )/1(
−
=
nn
c
d
φ
to
negative dimensions so that we would have for instance [26-30]
)1(
c
d
= (1/
φ
)-1-1 = (1/
φ
)-2 =
φ
2
(14)
which is the empty set dimension binary and it can be written with the
Connes-El Naschie bi-dimension formula in the following way D(-n) = D(-n,
1
+
n
φ
). The empty set models the quantum wave and is given as [24]
D(–1) = (–1,
2
φ
)
(15)
where –1 is the topologically invariant Menger-Urysohn dimension while
φ
2
is the Hausdorff-Besicovitch dimension which is not topologically invariant but
extremely useful.
The zero set on the other hand models the quantum particle [31-36]
dc(0) = (1/
φ
)0-1 = (1/
φ
)-1 =
φ
(16)
and can be written as
D(0) = (0,
φ
)

8
Leila Marek Crnjac
(17)
which is well known and in full agreement with the dimensional function of
non-commutative geometry [23].
The zero set dc(0) separates the sets dc(n) from the empty sets dc(-n) and we can
determine the degree of emptiness of an empty set as we move from n = -1, n = -2
… to n = -∞ which leads to zero. We see clearly that the totally empty set, by a
short verification, must be [24, 26-30]
dc(-∞) = (1/f)-∞ -1 = 0
(18)
5. 4. THE POSSIBILITY OF A CORRESPONDENCE BETWEEN A
ROTATING KERR BLACK HOLE AND E-INFINITY CONCEPTION TO
ORDINARY AND DARK ENERGY
The E-infinity model of dark energy relies on the dissection of E = mc2 into
E(O) =
22/
2
mc
plus dark energy E(D) =
)22/21(
2
mc
where E(O) is the ordinary
measurable cosmic energy of the quantum particle modelled by the zero set, while
E(D) is the dark cosmic energy density of the quantum wave modelled by the
empty set. Further, dark energy E(D) can be divided into two parts: dark matter
E(DM) and pure dark energy E(PD). It has been shown [9, 14, 15, 19] that E(D) =
)22/21(
2
mc
which constitutes 95.5% of total dark energy, consists of E(DM)
%7.2222/5
≈≈
dark matter and E(PD)
%7.7222/16
≈≈
of pure dark energy.
Einstein’s equation E = mc2 can be divided into three parts
E =
22/
2
mc
+
)22/21(
2
mc
=
22/
2
mc
+
+
)22/5(
2
mc
)22/16(
2
mc
= mc 2
(19)
or with the expression of the golden mean [15]
.)2/10()2/5()2/(
2242525
mcmcmcmcE
=++=
φφφ
(20)

9
Generalized Quantum Entanglement Family in Connection to Black Holes …
The connection from E-infinity scenario to the spinning Kerr black hole is
presented in the following way. The spinning Kerr black hole has three regions; it
has two event horizons and not only one as the static black hole. There is an inner
event horizon surrounding the circular black hole pipe at the core and a second
outer event horizon separating the ergosphere from the rest of the Kerr black hole
[19, 37, 38]. The horizon is the region from which no signal can escape.
In the present work we rely heavily on the Kerr space-time geometry of
rotating black holes. Kerr’s geometry and its ergosphere tie almost perfectly with
our dark energy theory.
Figure 1. Black hole regions.

10
Leila Marek Crnjac
Figure 2. Black hole regions.
As a direct consequence of this new insight E = mc2 can be written as E = E(O)
+ E(D), where the rational approximation E(O) =
22/
2
mc
is the ordinary energy
density of the cosmos and E(D) =
)22/21(
2
mc
is the corresponding dark energy of
the ergosphere of the Kerr energy [9, 14, 15, 19]. In this sense we have a Kerr
black hole nucleus having all the ordinary energy in it and that could be seen as a
mini black hole model for elementary particles.The paradox of the black holes is
leading to the satisfactory resolution confirming that at the minimum of 95.5% of
energy and information of the ergosphere will never be lost while 4.5% in the
Kerr black hole nucleus will not be directly accessible for us. We can conclude
that 95.5% of the information of a black hole is the ordinary information and the
remaining 4.5% is the dark information [39].

11
Generalized Quantum Entanglement Family in Connection to Black Holes …
6. 5. TOPOLOGICAL INTERPRETATION OF THE CASIMIR EFFECT
AS A PROPERTY OF THE GEOMETRICAL TOPOLOGICAL STRUCTURE
OF THE QUANTUM- CANTORIAN MICRO SPACE-TIME
The Casimir effect is a small attractive force that acts between two close
parallel uncharged conducting plates. It is due to quantum vacuum fluctuations of
the electromagnetic field.
The effect was predicted by the Dutch physicist Hendrick Casimir in 1948.
According to the quantum theory, the vacuum contains virtual particles which are
in a continuous state of fluctuation. Casimir realised that between two plates, only
those virtual photons whose wavelengths fit a whole number of times into the gap
should be counted when calculating the vacuum energy. The energy density
decreases as the plates are moved closer, which implies that there is a small force
drawing them together. Although, the Casimir effect can be expressed in terms of
virtual particles interacting with the objects, it is best described and more easily
calculated in terms of the zero-point energy of a quantized field in the intervening
space between the objects.
The Casimir effect is a natural consequence of the quantum field theory. There
are at least two fundamental interpretations of this effect. The first is connected to
boundary conditions and the zero-point quantum vacuum fluctuation which may
be the common way of looking at the Casimir effect. The second is to see the
Casimir effect as a source in the mould of Schwinger’s way of thinking [20, 37-
41].
In the present paper we opted for a rather different point of viewing the
Casimir effect as a natural necessity of a Cantorian space-time fabric that was
woven from an infinite number of zero Cantor sets and empty Cantor sets. The
zero set is taken following von Neumann- Connes’ dimensional function to model
the quantum particle while the empty set models the quantum wave.
The quintessence of the present theory is easily explained as the
3
φ
intrinsic
topological energy, where
2
15
−
=
φ
is produced from the zero set
φ
of the
quantum particle when we extract the empty set quantum wave
2
φ
from it.
The Casimir energy, the universal fluctuation
3
φ
, is the difference between the
Hausdorff dimension of the particle zero set
φ
and the empty set
.
2
φ
The result is
almost equal to double the value found by Zee [42]. He used an imaginative

12
Leila Marek Crnjac
modification of the classical Casimir experiment and found the dimensionless
Casimir energy equal to
.1308.024/
≈
π
Surrounding the zero set quantum particle we have the quantum wave empty
set with the Connes-El Naschie bi-dimension D(-1) = (-1,
2
φ
) acting as a surface
of the quantum particle, i.e. zero set D(0) =(0,
φ
). The infinite number of zero and
empty sets have an average bi- dimension D(-2) = (-2,
3
φ
). This triadic picture of
a quantum particle zero set wrapped in a propagating quantum wave empty set
and floating in a quantum space-time, which has
3
φ
average topological Casimir
pressure, is more satisfactory than any previous picture which was presented in
the past [9, 20, 40, 41].
7. 6. NANOTECHNOLOGY
In recent years nanotechnology invaded all scientific fields and played a
significant role in Casimir effect experiments.
Figure 3. Casimir effect.

13
Generalized Quantum Entanglement Family in Connection to Black Holes …
We know, thanks to E-infinity theory, that there exists a physical-
mathematical connection between dark energy and ordinary measurable energy on
the one side and the Casimir effect on the other side. A natural consequence of
this discovered reality of the quantum wave is rendering it a relatively simple task
to find a way to harness dark energy or Casimir energy. The difference between
Casimir energy and dark energy is a difference of boundary condition where the
boundary of the holographic boundary of the universe is a one sided Möbius-like
manifold [20]. This seems simple but it is extremely difficult and in the moment
impossible. There are many ideas about how to start, irrespective of the
connection to Kerr black holes.
We can start for instance with a highly complex sub-structuring of space using
nano-tubes and nano-particles and in that way create fractal-like nano-spheres
packing. We stress in this connection that we have a clear model for our nano-
reactor based on two important facts. The first is the equivalence between
branching polymer clusters and Cantorian-fractal space-time [25, 40, 41]. The
second is that we replace the Casimir plates of our model with Casimir spheres
and model these spheres with real nano-particles and in principle this is our
reactor. The main idea is a construction of a nano-universe and extracting dark
energy from its nano-boundary of its holographic boundary. That means
extracting energy from such a nano-reactor. It is at the edge of the universe that
95.5% of the energy resides as dark energy. This follows from the incredible
measure theoretical theorem of Dvoretzky [18] which explains why energy is
concentrated at the edge of the universe. The Dvoretzky theorem states that the
volume of a sphere is concentrated at the surface, more accurately, 95.5% of the
volume would be at the surface while in the so called bulk we have only 4.5%.
However, we could create many nano-universes from which its 95.5% energy
concentration could be extracted without actually reaching to the boundary of our
universe which is of course factually impossible [16, 18]. On the other hand if we
could produce nano-Kerr black holes, then a Penrose process could be feasible
after all following broadly the preceding lines of speculation.
8. CONCLUSION
We introduced the generalized entanglement family and the Fibonacci-like
dimension sequence which was extended into the negative side.
Our model of the universe is very simple. Applying the Dvoretzky theorem we
can reason that E = mc2 can be split into a quantum wave energy density E(D) =
)22/21(
2
mc
concentrated at the holographic boundary. This is the surface of the

14
Leila Marek Crnjac
universe which we call dark energy. E(D) cannot be measured in any direct way
with the present-time technology. The ordinary energy density E(O) =
22/
2
mc
,
the core of the quantum particle universe can be measured directly. The
connection from E-infinity scenario to the spinning Kerr black hole leads to a
paradox. The dark energy and information in the ergosphere of the black holes is
accessible because the ordinary energy in the horizon, where no information can
escape, is lost. We can conclude that 95.5% of the information of a black hole is
the ordinary information and the remaining 4.5% is dark information. The
situation is analogous to that of the ordinary and dark energy of the universe and
could be used as a guiding principle in the design of a nano-Casimir dark-energy
reactor.
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15
Generalized Quantum Entanglement Family in Connection to Black Holes …
particle physics, Chaos, Solitons and Fractals, 41(5), 2635- 46, 2009.
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dark energy using a Transfinite version of Nottale’s scale relativity,
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Casimir-Dark Energy Nanotech Reactor Proposal. Open Journal of Applied
Sciences, 5, 313-324, 2015.
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famous equation in physics and its hidden quantum entanglement origin.
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cosmic holographic dark energy. International Journal of High Energy
Physics, 2(1), 13-21, 2015.
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16
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the possibility of a Casimir nano reactor – A preliminary design. American
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[38] M. S. El Naschie, The Casimir topological effect and a proposal for a
Casimir-dark energy nano reactor. World Journal of Nano Science and
Engineering, 5(1), 26-33, 2015.
[39] M. S. El Naschie, If Quantum “Wave” of the Universe then Quantum
“Particle” of the Universe: A Resolution of the Dark Energy Question and
the Black Hole Information Paradox. International Journal of Astronomy
and Astrophysics, 5, 243-247, 2015.
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of Nano Research and Applications, 3(2), 12-16, 2015.
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Gauss-Bonnet term. The European Physical Journal C, 75, 6, 2015.
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