Journal of Quantum Information Science, 2013, 3, 23-26
http://dx.doi.org/10.4236/jqis.2013.31006 Published Online March 2013 (http://www.scirp.org/journal/jqis)
A Resolution of Cosmic Dark Energy via a Quantum
Entanglement Relativity Theory
Mohamed S. El Naschie
Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt
Received January 22, 2013; revised Februa ry 26, 2013; accepted March 9, 2013
ABSTRACT
A new quantum gravity formula accurately predicting the actually measured cosmic energy content of the universe is
presented. Thus, by fusing Hardy’s quantum entanglement and Einstein’s energy formula, we have de facto unified
relativity and quantum mechanics in a single equatio n applicab le to predictin g the en erg y of the en tire univ erse. In addi-
tion, the equation could be seen as a simple scaling of Einstein’s celebrated equation when multiplied by a
scaling parameter
2
Emc
2
5


, where 5

is Hardy’s quantum entanglement and 51
2


. Further-
more, 51
2 22.18033989


 could be approximated to 1
22
  and thus may be interpreted as the inverse of the
compactified bosonic strings dimension .
26 422
Keywords: Golden Mean; Quantum Entanglement; Probabilistic Quantum Entanglement; Quantum Relativity Energy
Formula
1. Introduction
By way of indirectly equating the fundamental equation
of the probability of quantum entanglement with that of
Einstein’s maximal energy of special relativity, an exact
intersection between relativity and quantum mechanics is
obtained. The quintessential result of this quantitative
intersection is an effective quantum gravity formula re-
lating energy (E) to mass (m) and speed of light (c):
 


2
2
1
1.
21
QR QR
Emc m




 



2
c (1)
The formula generalizes Einstein’s famous equation
via simple multiplication by
2
EmcQR

.
Setting 0

 or 3 in

2

1
1
21
QR







Newton’s
kinetic energy,
 
2
12Emvc is obtained while
3
4

  or 3

 , where

125 1   leads to
Einstein’s non quantum but relativistic formula .
Finally and most importantly, setting
2
Emc

  or
1

  in
 


mc
2
22
1
1
21
QR
Emc





QR
 , we
obtain the effective quantum gravity formula
25
2
22.18033989 2
QR mc
Em


c
5
where

is the well-
known Hardy probability of quantum entanglement. Fig-
ures 1 and 2 summarize the basic assumption and main
results of the present analysis.
2. Preliminary Remarks
It is well known that at the esoterically small scale of the
Planck length


33
~10 cml

, the feeble gravity which
can normally be ignored as far as quantum mechanics
and the standard model of particle physics is concerned
[1-4], becomes important again and comparable in
strength to that of the rest of the fundamental forces, i.e.
electromagnetic force, weak force and strong force [3-9 ].
On the other hand and as could be reasoned using Wit-
ten’s T-duality [3,4], at the other extreme of unimagina-
bly large distance comparable to the Hubble radius [6-11]
the effect of quantum corrections accumulate (see Fig-
ures 1 and 2) and relativity could not ignore the quantum
[7-9]. In other words the beauty of Einstein’s relativity
must be a candidate for major overhaul when
the quantum mechanical effect of entanglement is taken
into account [12,13]. In the present work we develop the
2
Emc
C
opyright © 2013 SciRes. JQIS

M. S. EL NASCHIE
24
Newton Classical
Mechanic
2
N1
Emv
2

Einstein Relativity
2
R
Ecm
52
QR
E(/2)(mc)
Quantum
Relativity
Quantum Entanglement
for m =c =1
5
Q
PE
Figure 1. Quantum Relativity theory as an intersection of
the three major Fundamental theori es of Physics. Note that

521 220.45
 . Consequently EQR predicts 4.5% only
of the energy which the classical equation of Einstein
2
E
mc predicts. In other words EQR does not contradict
the cosmological measurement but rather confirms the data
of References [17-19]. This is a clear cut resolution of the
mystery of Dark Energy.
Newtonian mechanics
(Kinetic Energy)
2
N
1
Emv
2

Einstein Relativity
2
R
Emc
Quantum Rel a tivity
52
(/2)()
QR
Emc  
Probability of Quantum Entanglement
for two particles (when c=m=1)
5
Q
PE


Figure 2. The Effective Quantum Gravity Energy Formula




5 2
QR 2
E
mc 
as a synthesis of Newton, Einstein and
Quantum Theories. Note that EQR/E (Einstein) ~ 0.045. This
agrees with the correct energy content of the cosmos meas-
ured by WMAP and the supernova analysis of References
[17-19].
preceding basic ideas into an effective theory of quantum
gravity leading to a revision of Einstein’s to
2
Emc
2
22.18033989
QR mc
E, where the factor 122.18033989
is simply half that of Hardy’s probability of quantum en-
tanglement

5
5
1
11
P




 where

2
15
 . In
Figures 1 and 2 our methodology is represented gra-
phically. In other words E according to Einstein overes-
timates the energy by almost 95.5% in situations where
quantum effects play a major role like when considering
the mass and energy content of the entire universe [14-
20]. Thus the agreement of the energy prediction of the
new equation with the sophisticated cosmological meas-
urements of dark matter and dark energy [14] could be
regarded as a clear, simple and rational explanation of the
missing dark energy of the universe [14-20].
3. Theory
The analysis generalizing of special relativity
to quantum relativity [21,22] i.e. effec tive quan tum grav -
ity formula
2
Emc




222.1803989Emc
QR consists of four
main steps. The first is to transform space, time and mass
to a probabilistic space, time and mass using quantum
mechanics leading to

2
2
P
EPmc
2
mc
where P is a
quantum entanglement probability. Second we devise a
special form of R where
E  

is a function of a
unit interval boost

. Third we equate
P
E to
R
E and
find the exact value of

for which E becomes a ma-
xi mum. (see Figures 1 and 2).
3.1. Probabilistic Quantum Entanglement E
In [12] Mermin gives unrivalled lucid derivations and
interpretations of quantum non-locality and entanglement
of two quantum particles relevant to the movement from
a point 1 to a point 2. The probability P of the generic
Hardy entanglement [12,13] is given by Equation (10) of
[12] as

 
1122
12
11
1
pppp
Ppp

 (2)
For dpp


21 one finds
21.
1
d
Pd d







(3)
Now we introduce the following probabilistic trans-
formation [2 1, 22]




Space
Time
Mass
X
xp
Ttp
M
mp



(4).
Inserting into Newton’s kinetic energy one finds the
following probabilistic energy for v
c

22
1.
22
Pxp mp
Empvc
tp



 (5)

M. S. EL NASCHIE 25
That means [12,13]
2
11 .
21
Pd
Ed mc
d





2
(6)
3.2. Relativistic E
From relativity theory we are familiar with three phe-
nomenological effects namely: 1) time delineation; 2) rod
shortening; and 3) mass increase when the velocity v
tends to the speed of light c [21,22]. Theoretically all the
three effects are beyond doubt while exper imentally th ere
is reasonable evidence for the reality of all these effects
[21,22].
Now we introduce an unspecified boost 1


and
anti boost 1

 in conjunction with the following
sp ace, time and mass transformation akin to Lorentz trans-
formation [2 1- 23].



1
1
1
xx
tt
mm






(7).
Consequently Newton’s “Relativistic” kinetic energy
becomes
 
2 2
2
1
111
1
2121
R
Emc mc




 




2
.
(8)
3.3. Determining the Magnitude Probabilistic
Quantum Entanglement d and the
Relativistic

The next step in our strategy to arrive at an effective
quantum gravity E is to require that both
P
E and
R
E
be equal (for further elucidation, see Figures 1 and 2).
That means
P
R
EE (9)
Therefore we have

2
22
21
1.
21 21
mcd mc
dd





(10)
Clearly this is only possible for d

 and inserting
back in (9) one finds that

2
21
1
11








(11)
This leads to a simple quadratic equation
210,
  (12)
with the well known and rather expected solution
12
1
,



(13)
where 51
2


 is the golden mean as in the work of
Mermin [12] and Styer [13].
3.4. The Quantum Relativity Energy Formula
Now we have reached the fourth and final step to obtain
the generalization of to an effective quantum
gravity formula by setting
2
Emcd

  in the correspond-
ing expression and find that

 

22
52
23 22
2
11
21
1
22 22.18033989
quantumentanglement2
QR
Emc
mc
mc mc
Pm







c
(14).
4. Discussion, Conclusion and Future Work
Looking closely at our generalization of to
2
Emc
2
22.18033989
QR mc
E, we notice that the integer approxi-
mation
222
QR
Emc (15)
is amenable to different simple interpretations of which
we give two obvious ones. Firstly, the factor 22 can be
intuitively viewed as what remains of the 26 dimensions
of string spacetime of the original bosonic strong interac-
tion theory after subtracting Einstein’s 4 dimensions
[1,3,4]. Then, the 26 422

 dimensions “dilute” the
energy content of the cosmos and reduce it from 100% to


10022% i.e. to ~4.5%, in full agreement with the
well known cosmological measurement of the three 2011
Nobel Laureates [18,19]. Secondly, we cou ld interp ret the
factor 22 as the 11 elementary gauge bosons of the stan-
dard model not included in Einstein’s one photon degree
of freedom theory [21-23]. It is well known that the
standard model is based on
 


3211SUSU U2

,
while in 1905 Einstein knew only one of the 12 namely
the photon leaving the rest out in one way or another
[21,22]. Adding super-symmetric partners, this leads to
(2) (11) = 22. See Refe rences [1, 3].
There are numerous other intuitive as well as strictly
mathematical ways to show that 52
2
QR
Em

c is in-
deed the correct energy formula that includes both the
relativistic as well as the quantum effects in one equation
which was analyzed by the present author.
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M. S. EL NASCHIE
26
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