A Resolution of Cosmic Dark Energy via a Quantum Entanglement Relativity Theory

doi:10.4236/jqis.2013.31006

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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

Email: [email protected]

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

Emc





, where 5



is Hardy’s quantum entanglement and 51





. Further-

more, 51

2 22.18033989





 could be approximated to 1



 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):



QR QR

Emc m



















c (1)

The formula generalizes Einstein’s famous equation

via simple multiplication by

EmcQR



Setting 0



 or 3 in



















Newton’s

kinetic energy,

 

12Emvc is obtained while





 or 3





, where



125 1



 leads to

Einstein’s non quantum but relativistic formula .

Finally and most importantly, setting

Emc





 or





 in



Emc











, we

obtain the effective quantum gravity formula

22.18033989 2

QR mc





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





~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

Emc

M. S. EL NASCHIE

Newton Classical

Mechanic

Emv



Einstein Relativity

Ecm

E(/2)(mc)

Quantum

Relativity

Quantum Entanglement

for m =c =1

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

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)

Emv



Einstein Relativity

Emc

Quantum Rel a tivity

(/2)()

Emc





Probability of Quantum Entanglement

for two particles (when c=m=1)





Figure 2. The Effective Quantum Gravity Energy Formula









5

QR 2



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

Emc

22.18033989

QR mc

E, where the factor 122.18033989

is simply half that of Hardy’s probability of quantum en-

tanglement







 where





. 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

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



EPmc

where P is a

quantum entanglement probability. Second we devise a

special form of R where







is a function of a

unit interval boost



. Third we equate

E to

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

pppp

Ppp



 (2)

For dpp



21 one finds

21.

Pd d















(3)

Now we introduce the following probabilistic trans-

formation [2 1, 22]







Space

Time

Mass

Ttp



(4).

Inserting into Newton’s kinetic energy one finds the

following probabilistic energy for v

c



Pxp mp

Empvc







 (5)

M. S. EL NASCHIE 25

That means [12,13]

11 .

Ed mc











(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].









(7).

Consequently Newton’s “Relativistic” kinetic energy

becomes

 

111

2121

Emc mc



















(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

E and

be equal (for further elucidation, see Figures 1 and 2).

That means

EE (9)

Therefore we have



21 21

mcd mc









(10)

Clearly this is only possible for d



 and inserting

back in (9) one finds that











(11)

This leads to a simple quadratic equation

210,



 (12)

with the well known and rather expected solution







(13)

where 51





 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

Emcd





 in the correspond-

ing expression and find that





23 22

22 22.18033989

quantumentanglement2

Emc

mc mc















c

(14).

4. Discussion, Conclusion and Future Work

Looking closely at our generalization of to

Emc

22.18033989

QR mc

E, we notice that the integer approxi-

mation

222

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



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.

REFERENCES

[1] P. Davies, “The New Physics,” Cambridge University

M. S. EL NASCHIE

Press, Cambridge, 1989.

[2] P. Halpern, “The Great Beyond: Higher Dimensions, Pa-

rallel Universes and the Extraordinary Search for a The-

ory of Everything,” John Wiley & Son Inc., Hoboken,

2004.

[3] R. Penrose, “The Road to Reality,” Jonathan Cape, Lon-

don, 2004.

[4] M. Kaku, “Introduction to Superstrings and M-Theory,”

Springer, New York, 1999.

doi:10.1007/978-1-4612-0543-2

[5] B. Booss-Bavnbek, et al., “New Paths towards Quantum

Gravity,” Lecture Notes in Physics, Springer, 2010.

[6] B. G. Sidharth, “Frontiers of Fundamental Physics,” Uni-

versity Press, Hyderabad, 2007, pp. 1-32.

[7] J. Mageuijo, “Faster than the Speed of Light,” William

Heinemann, London, 2003.

[8] L. Nottale, “Fractal Space-Time and Microphysics: To-

wards a Theory of Scale Relativity,” World Scientific,

Singapore, 1998.

[9] L. Smolin, “Three Roads to Quantum Gravity,” Weiden-

feld and Nicolson, London, 2000.

[10] P. Steinhardt and N. Turok, “Endless Universe,” Weiden-

feld and Nicolson, London, 2007.

[11] Y. Baryshev and P. Teerikorpi, “Discovery of Cosmic

fractals,” World Scientific, New Jersey, 2002.

[12] D. Mermin, “Quantum Mysteries Refined,” American Jour-

nal of Physics, Vol. 62, No. 10, 1994, pp. 880-887.

doi:10.1119/1.17733

[13] D. F. Styer, “The Strange World of Quantum Mechanics,”

Cambridge University Press, Cambridge, 2000, pp. 54-55.

[14] L. Amendola and S. Tsujikawa, “Dark Energy: Theory and

Observations,” Cambridge University Press, Cambridge,

2010.

[15] R. Panek, “Dark Energy: The Biggest Mystery in the Uni-

verse,” 2010.

http:/www.smithsonianmagazine.com/science.Nature/Dark-

Energy

[16] Planck-Spacecraft, “Wikipedia,” 2012.

http://en.wikipedia.org/wiki/planck

[17] E. Komatsu, et al. (WMAP Collaboration), “Seven Years

Wilkinson Microwave Anisotropy probe (WMAP) Ob-

servations: Cosmological interpretation,” The Astrophy-

sical Journal Supplement Serie s, Vol. 192, No. 2, 2011, p .

18. doi:10.1088/0067-0049/192/2/18

[18] A. G. Reiss, et al. (Supernova Search Team Collabora-

tions), “Observation Evidence from Supernova for an

Accelerating Universe and a Cosmological Constant,”

Th e Astronomical Journal, Vol. 116, No. 3, 19 98, pp. 1009-

1038. doi:10.1086/300499

[19] S. Perlmutter, et al. (Supernova Cosmology Project Col-

laboration), “Measurements of Omega and Lambda from

42 High-Redshift Supernova,” The Astronomical Journal,

Vol. 517, No. 2, 1999, pp. 565-585.

doi:10.1086/307221

[20] C. Toni, “Dark Matter, Dark Energy and the Fate of Ein-

stein’s Theory of Gravity,” AMS Graduate Student Blog.

Mathgrablog.williams.edu/dark-matter-darkenergy= fate-eins -

teins-theory=gravity/#print

[21] W. Rindler, “Relativity,” Oxford University Press, Ox-

ford, 2001, pp. 111-112.

[22] W. Rindler, “General Relativity before Special Relativity:

An Unconventional Overview of Relativity Theory,” Ame-

rican Journal of Physics, Vol. 62, No. 10, 1994, p. 887.

[23] J. Mageuijo and L. Smolin, “Lorentz Invariance with an

Invariant Energy Scale,” 18 December 2001, arXiv: hep-

th/0112090V2.