Journal of Quantum Informatio n Science, 2011, 1, 50-53
doi:10.4236/jqis.2011.12007 Published Online September 2011 (http://www.SciRP.org/journal/jqis)
Quantum Entanglement as a Consequence of a Cantorian
Micro Spacetime Geometry
Mohamed S. El Naschie*
Departme nt of Physi c s, University of Alexandria, Alexandria, Egypt
Received June 20, 2011; revised July 18, 2011; accepted August 2, 2011
Abstract
Building upon the pioneering work of J. Bell [1] and an incredible result due to L. Hardy [2] it was shown
that the probability of quantum entanglement of two particles is a maximum of 9.0169945 percent [2]. This
happens to be exactly the golden mean

to the power of five (ϕ5) [3-7]. Although it has gone largely un-
noticed for a long time, this result was essentially established independently in a much wider context by the
present author almost two decades ago [3-6]. The present work gives two fundamentally different derivations
of Hardy’s beautiful result leading to precisely the same general conclusion, namely that by virtue of the zero
measure of the underlying Cantorian-fractal spacetime geometry the notion of spatial separability in quantum
physics is devoid of any meaning [7]. The first derivation is purely logical and uses a probability theory
which combines the discrete with the continuum. The second derivation is purely geometrical and topologi-
cal using the fundamental equations of a theory developed by the author and his collaborators frequently re-
ferred to as E-infinity or Cantorian spacetime theory [3-7].
Keywords: Hardy’s Quantum Entanglement, Golden mean, Cantor sets, Fractal Spacetime, E-Infinity Theory,
Quantum Mechanics, J. S. Bell
1. Introduction
In his equally ingenious as it is beautiful Gedanken ex-
periment, Hardy showed using orthodox quantum me-
chanics and Dirac’s notation that a two particles entan-
gled state will have a maximum probability of

55 112 [2]. In particular for a basis state i

for
two particles 1 and 2 the entanglement is given by the
Schmidt decomposition [2].
12 1
  
2
(1)
Here

and

are two real constants satisfying [2]
22
1   (2)
Proceeding in this way Hardy’s subtle analysis arrives
at a general expression for nonlo cality which need not be
associated with spin but rather any other measurable
quantity as in our two-slit experiment for example [3,4].
Using Hardy’s notation the expression is [2]

2
1









(*) Distinguished Fellow of the Frankfurt Association
for the Advancement of Fundamental Research in Phys-
ics, Faculty of Physics, University of Frankfurt, Ger-
many.
Subsequently Hardy shows that

, i.e. the probability
of entanglement is maximum when [2]
09070.   (4a)
and 0 4211.
 . (4b)
Being substantially larger than

= 0, this is a clear
proof for the refutation of naïve classical realism. Apart
of being rigorous, this result was experimentally verified
many times. Despite its almost perfection and lucidity
something went unnoticed in Hardy’s paper due to his
rounding of the numerics involved, for instance looking
at the untruncated exact expressions [7] for

and

namely 09069996487
  (5a)
and


(3) 0 4211313776.
  (5b)

51 M. S. E. NASCHIE
we notice these are all golden mean functions with defi-
nite connections to the fundamental equations of E-in-
finity Cantorian spacetime theo ry. For instance [3-7]
2,

  (6a)
22 ,



 
(6b)

5
55 112,

 (6c)
and [3,4]

23,
 
 (7)
where

512   is the golden mean [3-7 ].
It is now an almost trivial matter for those familiar
with A. Connes noncommutative Penrose fractal tilin g [3]
or E-infinity theory [3-7] to surmise that Hardy’s quan-
tum entanglement probability is a consequence of the
Cantorian-fractal structure of micro spacetime topology
[3-7]. In particular the fact that the Hausdorff dimension
of the zero set is the golden mean

and that of the
empty set is 2

is the reason for the appearance of the
golden mean in Hardy’s Gedanken experiment [2]. It is
the purpose of the present paper to make this fundamen-
tal connection with far reaching consequences for quan-
tum physics as crystal clear as possible.
2. Derivation of Hardy’s Quantum
Entanglement Using Pure Logic
and a Transfinite Probability
Theory
In some of his deepest papers Hardy recently called for
the essential need for a probability theory which is both
discrete and continuous at the same time but of course in
different senses. This is essentially and indirectly echo-
ing the same sentiment expressed by the present author
long ago using the language of transfinite set theory [3-4].
Indeed in E-infinity theory we use Cantor sets which are
totally disjointed and discrete and yet they have the car-
dinality of the continuum [3,4]. Combinato ric probability
can only be finite and rational. Irrational probability ex-
ists only for geometrical probability wh ich we cannot use
or for topological (Hausdorff) probability which we do
use [3-4]. This complies with what Hardy called for
[3-7].
Let us start our logical analysis by systematically de-
nying the existence of any meaning for spatial separation.
We consider two particles or two different points in some
space to be defined later on by the output of our re-
quirements and analysis. The probability to be at point 1
will be denoted d1 while the probability of being at po int
2 is d2. Consequently the probability of not being at 1 is
obviously 1 – d1 and similarly not being at 2 is 1 – d2.
Let us create the maximum local muddle possible result-
ing from denying classical common sense and calculate
the total probability o f being all of the above at the same
time. In other words, the probability of being in 1 and no t
in 1 as well as in 2 and not in 2 in addition to be ing in all
of that at the same time. Following the multiplication
theorem or the intersection rule, the total entanglement
probability would be

 
11 122
P1 1dddd  (8)
To appreciate the value of P1 it has to be compared
with the simplest local realism. To obtain in an analo-
gous way the probability for simple local realism, we
reason that this must negate being at 1 and at 2 at the
same time. This non-entangled state is clearly 1 – d1d2
and must be substantially larger than P1. Thus we have
established P2 of non-entanglement, namely
21
P1dd
2

 (9)
and consequently we can work with a “relative” prob-
ability defined for two particles in the most general way
possibl e, na mely
12
PPP

(10)
or

 

112 2
12
11
P1
dddd
dd

 (11)
The next step is crucial because we are searching for
an extremum for P which will turn out to be a maximum.
This maximum is defined by
1
P0
d


 (12a)
and
2
P0.
d

 (12b)
It is easily shown that the two equations result in one
cubic algabraic equation with three solution s, namely for
d1 = d2 = 1, –1/

and

.
The third solution d =

= 0.61833989 is clearly a
confirmation of the E-infinity theory result where
=

0
c
d

is both the Hausdorff dimension of a random tri-
adic Cantor set as well as being the topological probabil-
ity of finding a Cantorian point in this set [3,4]. To ob-
tain Hardy’s result explicitly we insert d1 = d2 =

in P
and find [7]
5
P


(13)
exactly as found by Hardy [2]. In a sense E-in finity is the
limit set of the quantum geometry corresponding to
quantum mechanics [3-7].

M. S. E. NASCHIE
52
3. Derivation of Hardy’s Result Using
E-Infinity Theory.
Those familiar with E-infinity theory for which two
convenient summaries and reviews may be found in [3,4]
know that the probability of finding a point in this space
is 3

. This is the inverse of its average Hausdorff dimen-
sion 4 + 3

. The general formula for the dimension is
[3,4]


0
0
1
1
c
c
d
nd

 (14)
Consequently the probability is [3,7]



0
0
1
11
c
c
d
n
d

. (15)
This probability may be regarded as the result of
“counterfactual” influence [1,2,7]. To find the total
probability of two entangled points in this space we re-
call first that for each point on its own, the probability
not counting “counterfactual” is [3,4]. Conse-
quently the entanglement not counting counterfactual is
( )2. The total probability is con sequently the product
of

0
c
d

0
c
d1n with ()2 which means [7]

0
c
d




0
2
0
0
1
P1
c
c
c
d
d
d






.


(16)
Now either we trust the E-infinity result that =

0
c
d

or we maximize P with respect to and find a quad-
ratic equation

0
c
d



2
00
10
cc
dd (17)
from which we obtain two solutions
 
00
1
and
cc
dd 

 (18)
Inserting =

0
c
d

in P we find Hardy’s result again
[2,7]
5
P


To obtain the result P = 0 befitting the classical ex-
pectation of classical mechanics we just need to set
= = 1 of a classical one dimensional continu-
ous line rather than a Cantor transfinite set of points in
our symmetric expression for P.

0
c
d

1
c
d
From the preceding derivation and unlike the first
derivation, it is absolutely clear that Hardy’s result is
geometrically and topologically rooted in the Cantorian
nature of micro quantum spacetime [3-7]. Consequently
quantum entanglement is not counter intuitive but rather
intuitive when we adopt the appropriate intuition of the
zero measure transfinite point set geometry of Cantorian
geometry [3-7].
4. Connection between the Logical and the
Topological Derivations of Hardy’s
Quantum Entanglement
Let us return to the general P obtained from nonlocality
logic





 
112
12
11
P1
dddd
dd


2
(19)
Let us make the solution totally symmetric by setting
12
ddd

 (20)
then we find

22
2
1
P1
dd
d

 (21)
This means
 



2
22
11
P11 1
dd
dd.
(22)
dd d


 
Not surprisingly, setting d = we find our formula
for P obtained using E-infinity namely [7]
 0
c
d




0
2
0
0
1
P1
c
c
c
d
d.
(23)
d


In fact we can generalize P for n particles or locations
easily by writing [7]




0
0
0
1
P1
nc
c
c
d
dd

 (24)
where 2,3,n

.
5. Discussion and Conclusion
A random Cantor set with its golden mean Hausdorff
dimension interpreted as a topological Hausdorff prob-
ability is a synthetic a priori for a topology and a corre-
sponding probability theory which unites the ununiteable,
namely the discrete and the continuum [3,4].
A Cantor set has a zero Lebesgue measure. That
means zero length. In a sense it is physically not there
and yet it has a substantial Hausdorff dimension, namely

= 0.618033 [3-7]. Therefore it is ‘there’ to consider
and work with. A Cantor set, in a way surprising to the
naïve intuition, is there and not there at the same time.
Measure zero and the emptiness of an empty set are as-
pects which quantum mechanics does not address di-
rectly. E-infinity is based on these subtle concepts [3-7].
Hardy’s magnificent work [2] reached in a formal way
using orthodox quantum mechanics the same result

M. S. E. NASCHIE
53
. References
] J. S. Bell, “Speakable and Unspeakable in Quantum Me-
Two Particles without Ine-
which we demonstrated to be natural and non spooky
once suitable mathematics and the associated Cantorian
based geometry are utilized to interpret and understand
the results.
6
[1 chanics,” Cambridge, 1987.
[2] L. Hardy, “Nonlocality for
qualities,” Physical Review Letters, Vol. 71 No. 11, 1993,
pp. 1665-1668. doi:10.1103/PhysRevLett.71.1665
[3] M. S. El Naschie, “A Review of E-Infinity Theor
Naschie, “The Theory of Cantorian Spacetime
y and
the Mass Spectrum of High Energy Particle Physics,”
Chaos, Solitons & Fractals, Vol. 19, No. 1, 2004, pp.
209-236.
[4] M. S. El
and High Energy Particle Physics (an Informal Review),”
Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp.
2635-2646. doi:10.1016/j.chaos.2008.09.059
[5] J.-H. He, et al., “The Importance of the Empty Set and
ollapse of Wave Interfer-
“Quantum Golden Mean Entanglement
Noncommutative Geometry in Underpinning the Founda-
tions of Quantum Physics,” Nonlinear Science Letters B,
Vol. 1, No. 1, 2001, pp. 15-24.
[6] M. S. El Naschie, “Quantum C
ence Pattern in the Two-Slit Experiment: A Set Theoreti-
cal Resolution,” Nonlinear Science Letters A, Vol. 2, No.
1, 2011, pp. 1-9.
[7] J.-H. He, et al.,
Test as the Signature of the Fractality of Micro Space-
time,” Nonlinear Science Letters B, Vol. 1, No. 2, 2011,
pp. 45-50.