Journal of Modern Physics, 2011, 2, 289-300
doi:10.4236/jmp.2011.25038 Published Online May 2011 (
Copyright © 2011 SciRes. JMP
Granular Space and the Problem of Large Numbers
V. I. Konushko
Protvino, Moscow, Russia
Received February 21, 2010; revised February 28, 2011; accepted March 2, 2011
Two and a half thousand years ago the ancient atomists made a suggestion that space has a cellular structure,
is material and consists of elementary cells. In 1900 Plank found the elementary length L* = 10–33 cm. This
notion has been widely used in modern physics ever since. The properties of granular space are studied in
this article on the assumption that a three-dimensional material cell with the size of Planck’s elementary
length is the only material for the construction of the whole Universe. This approach allows one to account
for such mysterious phenomena as inertia, ultimate velocity of transfer of material body interactions and
huge difference between gravitational and Coulomb forces - the so called “Large Numbers Problem”, as well
essence of electric charge and Pauli exclusions principle.
Keywords: Problem of Large Numbers
1. Introduction
The concept of space is one of the most important con-
ceptions forming the system of our knowledge.
Is the space infinite or not? Is it just a relation between
material bodies or does it exist independently? Is the
space a container for matter observable even in the ab-
sence of material bodies? Is the space uniform from one
point to another or are there some selected directions? Is
it neutral or does it direct bodies inside it? Do we know
its properties intuitively without any external influence
on our brain or do we acquire these properties from ex-
perience? These are the questions made in different times
with respect to a phenomenon named space.
The conception of discreteness is as old as that of con-
tinuity. It goes back to ancient atomists and can be re-
garded as one of the first solutions of Zeno’s of Elea
aporias. However, it should be noted that in spite of suc-
cessful application of the idea of discreteness to describe
the structure of matter, the operating conception for the
space and time structure was nevertheless that of conti-
A big step in solving the problem of space structure
was made in 1900 by Plank [1]. In that year Plank’s con-
stant h was born. Plank researched the irradiation of
black bodies. He was attracted by the universality of this
irradiation, which turned out to be independent of the
size as well as of the shape of the irradiating body or of
the properties of the vessel walls. While the reasons of
this universality were searched for, the problem of the
standards of length, mass and time appeared. These
standards were to be established from the principles not
appealing to any substance including elementary parti-
cles. They were only to be expressed through the funda-
mental constants, i.e. the speed of light c, Newton gravi-
tational constant G and the quantum of action h found
from the irradiation law. Only these three constants Plank
could take as fundamental ones. From those three con-
stants only one value with the dimension of the length
, fundamental mass and time quantum
be constructed:
*3 33
** 43
1.610 cm,
10 g,
10 sec.
An elementary clot of matter with the mass got
the name of a plankeon. Three values , and
play a major role in the theory of elementary particles as
well as in the Big bang model. However, physical mean-
ing of the elementary length has still not become
There is a wide-spread opinion that the Plank’s length
could bring light to numerous mysteries of micro- and
macro-worlds. “Only one value has in the existing theory
a clear and available interpretation - that is the Plank’s
length. Whence still, if not from here, is it possible to
start researches of particles? It is quite possible that only
the physics of the 1033 cm region will help us to under-
stand the physics of elementary particles”, - wrote
Wheeler [2]. Okun’ also saw the significance of this
value [3]. “It seems more and more probable that physics
on a Plank’s scale determines not only all the physics of
low energies but the whole picture of the Universe as
well”, - he remarked.
The problems of the micro world make us consider the
picture of the universe where the idea of discreteness
must play the role not less than that of continuity. It is
necessary that the discreteness should participate in the
description of quantum objects quite naturally without
being artificially brought in the context of continuum
A serious step in comprehension of the space structure
was made by Beckenstein [4]. Considering the thermo-
dynamics of black holes he supposed that the entropy of
a black hole was proportional to its square: S~A. But the
square A has the dimension of the length squared and
there appeared a problem how to make the expression for
the entropy dimensionless. A hypothesis was put forward
that the entropy of a black hole must have the following
where the coefficient α must be calculated basing on
some other ideas. This coefficient was later on calculated
by Hawking [4]. It turned out to be 1/4. The value
is the minimal square of an elementary object. Whereas
Plank found the minimum value of the elemen- tary
length, Beckenstein and Hawking indirectly intro- duced
the minimum size of the elementary square. Using the
above-mentioned, we only have to take the next step
towards the generalization up to a minimum
three-dimensional object in order to establish the space
We suppose that the meaning of the Plank length
lies in the fact that physical space has a cellular struc-
ture and consists of material three-dimensional cells with
the size of cm. Thus, we suppose that eternal,
invariable, primary matter- protomatter - exists in the
form of an elementary cell of the size of fundamental
length .
According to Wheeler [2], “it is cells of this size that
make up space on its deepest level”. All our observations
and experiments have been and are performed in a ma-
terial Universe. It is quite unreasonable to expect that a
theory developed under such conditions will be applica-
ble in an empty Universe.
As we have already mentioned, no matter what the na-
ture of original mater is, if cannot produce either a point,
or an infinite thin line or an ideal plane. The only and the
simplest object which can be created by the material
Universe is a bubble. When coming in contact with one
another these bubbles turn into polyhedrons, i.e. three-
dimensional geometries or cells.
There are five types of polyhedrons which, when ar-
ranged in parallel, can cover a three-dimensional space
so that they would be franked to one another by their
whole faces (Figure 1):
The most economical geometry is a fourteen-sided
polyhedron: the volume being the same, it takes the least
material to make its face.
The entropy of the black hole acquires quite a trans-
parent physical meaning – it is equal to the number of
elementary cells forming the surface of this object:
. (2)
Therefore, a mysterious and amazing quantity - en-
tropy - appears to be connected with the structure of
space. Further on, this fact will help us to see the reason
of irreversibility of physical processes though all con-
servation laws are convertible in time. “The arrow of
time” will be considered in more detail in our subsequent
2. The Size of Elementary Particles
Before considering the above-mentioned problem let us
raise one of the “native” questions which are most diffi-
cult to answer. Doesn’t the assumption, that the Universe
consists of only one element, inappropriately simplify
the reality? To answer it we should keep in mind that the
Figure 1. Three-dimensional polyhedrons. (1. parallelepi-
peds; 2. prisms with centr ally-symmetric six-sided bases; 3.
twelve-sides polyhedrons; 4. twelve-sides polyhedrons; 5.
fourteen-sides polyhedr ons. )
Copyright © 2011 SciRes. JMP
world constants give us a notion of the size of an ele-
mentary particle - Comption’s wavelength. For protons,
it is
2.110 cm
. (3)
Experiments give a somewhat larger value of proton
size: cm.
Simple calculations give us the number of elementary
cells of which this particle consists, ~1060. The number
of nucleons in the whole Universe is ~1080. It is rather
amazing, but in the number of structural elements any
elementary particle is hardly inferior to the Universe.
The creation of an elementary particle requires a sup-
plementary quantity of matter of mass. Since the whole
space is filled with cells, the rest mass when it forms a
particle shell, must deform both internal and external
cells. The internal cells form the body of the particle
which literally “occupies” or confines these space cells.
The deformation of external cells makes the essence of
physical fields. Radial deformation creates electrostatic
and gravitational fields. Tangential (torsion) deformation
results in magnetic, gravitational - magnetic and vortex
electrostatic fields, as well as a particle spin.
Matter means the substance which is used to form
elementary cells. Particle mass is the amount of matter
used to form this corpuscle.
In creating a particle the outer cells have to be slightly
pushed apart thus forming an excess of matter in the
surrounding space. This excess is exactly equal to the
mass of the particle itself and, according to the Einstein
formula for excessive energy, . It is just this
excess of matter that provides the basis for introducing
the concept of potential energy: it becomes quite clear
from where the Space gets excessive energy as a particle
moves in different physical fields.
Kinetic energy means the amount of matter carried by
a particle; this matter moves ahead of the particle car-
rying it forward and makes the motion wavy by nature.
Moreover, it is this matter that forms new particles as
two corpuscles collide. In our article “Weak Interaction
and the nature of virtual of particles” we have discussed
more comprehensively the motion of photons and parti-
Nowadays we haven’t the slightest idea of what the
electric charge and the spin mean. The reason lies in our
erroneous view of leptons as point particles. Erroneous is
the interpretation of experiments on lepton - lepton scat-
tering. In this case the matrix element does not contain
any form-factors of these particles which would take into
account their complex structure. The absence of such a
structure is closely connected with the lepton size. Now
let us consider, as a counterexample, the process of bil-
liard balls elastic scattering which is considered as a col-
lision of material points though their structure is much
more complex than that of particles.
According to the present-day concepts, the size of an
electron 17
cm, and this comes into conflict when
the density of matter inside an electron and a proton is
under consideration. The electron mass is 2000 times less
than the mass of a proton, and the density of electron
matter in this case is 109 times as much as that proton.
All this leads to an absurd chain: the less the mass of an
elementary particle, the smaller its size and the more the
density of matter inside it.
The rule - the more the mass of a particle, the smaller
its size - is supported by an experiment performed on a
relativistic heavy ion collider (RHIC). The mass of a
is more than three times larger than that
of a proton and, as it follows from the experiment, its
size is three times smaller.
The size of a particle is its most important characteris-
tic which must be determined only by the world con-
stants ħ, c, G and by the mass of a particle m. Three
quantities pretend to be the radius of electron: the
Compton wavelength , the classical radius and
gravitational radius rg:
3.86 10cm,
2.8 10cm,
10 cm.
rmc cmc
 
Any theory having a claim on a correct description of
the microcosm must be able to calculate the fine struc-
ture constant α, which acts as the electromagnetic inter-
action constant, and in the radius 0 it serves just as a
scale factor. Since the electric charge is the same for all
elementary particles, the value of e² cannot determine the
size of numerous elementary corpuscles which differ
greatly in mass. The gravitational radius of an electron is
much smaller than that of an elementary cell, and there-
fore is not discussed here. It inevitably follows that the
size of any structure - free particle is only dictated by its
Compton wavelength. The structure of a particle, how-
ever, increases its size just slightly, like in case of a pro-
ton. Over fifty years ago both Yukawa and Landau pro-
posed that the size of an electron is equal to its Compton
On arriving at this decision we finally can understand
such notions as the mass, the electric charge and the spin
of elementary particles. These problems will be studied
in detail in our article “Weak Interaction and the Nature
of Virtual Particles” where we calculated both the mass
of one elementary cell mcell and its density pcell and found
Copyright © 2011 SciRes. JMP
a unique relation between an elementary cell and a W-
boson. An elementary cell is a generalized image of an
object which nature gives us though its world constants.
In imaginary experiments used widely by Galileo,
Newton and Einstein we can see with our own eyes all
these mysterious natural phenomena thus doing away
with the tragedy of blindness. Being prisoners of “point-
ness” we would never solve these fundamental problems.
Besides, there are infinities which have been poisoning
the life of theorists for about a hundred years provided
that particles are considered to be points.
Let us go back to the proton. Its large mass creates
surface tension strong enough to produce inside a parti-
cle clusters of deformed internal cells. Such formation
has already acquired the name of quarks. Quark “con-
finement” becomes now quite transparent, and the simi-
larity of quarks and leptons can be accounted for by the
fact that inside either of them there are no clusters of
deformed cells.
α proton consists of Np deformed cells:
Nmc L
The deformation of such a huge number of cells is so
queer that it gives grounds to introduce into theory such
objects as gluons and a sea of virtual quart-antiquark
pairs despite the fact that all these objects have a material
basis, i.e. they consist of material cells.
Even in collisions of an electron with another particle
its internal cells are just elastically deformed without
creating any new internal formations, and this point is
considered nowadays as electron “pointness”.
3. Particle Motion. Ultimate Velocity.
Enigma of Inertia
To observe the motion of a particle we must make an
imaginary experiment by increasing the elementary cell
up to the size of a small cube. The particle begins to
move only when there is a difference in cell deformation
behind and in front of the particle, i.e. a deformation
gradient. The amount of matter required for it and sur-
rounding the particle in asymmetric way is called kinetic
energy and the cell deformation gradient is referred to as
Since cells process elastic properties, the motion of
this additional matter has a wave nature creating some
kind of “a centaur”: a wave-particle predicted by the
Broglie. An electron is only carried by the wave never
becoming part of it. After colliding with another particle
the electron loses the prefix “wave”. The process of
transmitting either a part of matter (kinetic energy) or the
whole matter has an exchange character. In collision the
matter carried by the shell-particle having reached the
target-particle having reached the target-particle finds
itself between a hammer and an anvil.
The enormous quantity of deformed cells participating
in the collision leads us to introducing into our theory the
notion of a “virtual” exchange particle. In this immense
“pot” a strong deformation of space cells makes up all
kinds of cluster providing the right to introduce such
notions as a sea of quark-antiquark pairs, current quarks
and gluons. That is why there is such a strong difference
between the masses of current and constituent quarks.
This real collision process enables us to understand this
“spin disaster” as well.
The introduction of structural functions into the matrix
element is the first raw attempt of describing a complex
collision act where up to 1080 deformed space cells take
part. Such a huge amount of deformed cells participating
in a collision act is responsible for the fact that all the
events in the Universe are not local and our mathematical
description of an event will always be just approximate.
The nonlocal character of elementary particles and inter-
actions gives rise to a false concept about the violation of
the principle of casuality and the principle of equation
invariance under Lorentz’s transformation.
But both a line, a point and plane are started mathe-
matical concept, and in real space such objects do not
exist at all. Any event in the Universe occupies a certain
space-time area, and the fact that we attribute the coor-
dinates X, Y, Z and t to this event just says that this event
has really happened, that is, something has occurred in
the Universe, something has changed in a certain region
of space in a definite time. In all our equations we men-
tally reduce this region to a point and the use of form-
factors is just a weak attempt to account for a colossal
complexity of the collision process. We have so far pre-
pared the ground for discussing the following fact: no
one has ever observed the motion of a physical object
with respect to other objects with its velocity exceeding
at a definite moment of time the speed of light in vacuum
310 cmsc excluding the giant “scissors” effect.
In three-dimensional cell space all motion is con-
ducted by cells themselves and the speed of light is the
parameter of their elastic properties. In other words, for
the cells there exists a maximum deformation that de-
fines the maximum possible velocity. The deformed cells
cannot achieve the velocity of deformation transfer above
the maximum. Consequently no particle can exceed this
velocity. The history of tachyons repeat that of flogiston
and caloric. The planned experiment on the discovery of
neutrino velocity exceeding the speed of light is in our
opinion doomed to failure.
The phenomenon of inertia has been known for many
Copyright © 2011 SciRes. JMP
centuries and Galilei, Huygens, Descartes and Newton
polished the wording of this mysterious phenomenon.
Any macroscopic body moves through liquid or gaseous
media as one whole pushing the molecules of these ma-
terials. However, in order to break the bonds between the
water molecules a body, e.g. a submarine, must spend a
part of its kinetic energy (in other words, matter) that it is
carrying to compensate for the binding energy between
these molecules. Along with this, the momentum of the
submarine is diminished, which leads to the decrease of
the deformation gradient of the cells surrounding the
submarine. Finally the body stops.
An elementary particle, which itself consists of cells,
is moving in the cell space also pushing and deforming
the other cells in front of it without spending any kinetic
energy, since in free space the notion of the binding en-
ergy between the adjacent cells is absent.
If a liquid or a gas were not viscous at all, a body
moving in these media would not meet any resistance
(“Eiler’s paradox”). It is in this way that an elementary
cell moves through the cells of field-free space. The
space viscosity is equal to “0”!
For the last five years the experiments with the use of
a relativistic heavy ion collider (RHIC) have allowed us
to reproduce at a microscopic scale quark-gluon plasma
formed by collisions of gold nuclei flying almost at the
speed of light.
Some physicists are surprised to see that a quark-gluon
media is practically free of viscosity and so presents the
most ideal liquid among all the known liquids [8]. It is
rather difficult to get rid of the idea that this ideal liquid
accounts for the absence of viscosity in space.
But there is a fundamental difference between the
wave motion of elementary particles and photons and a
sound wave or a wave in a liquid.
The propagation of sound in the air is the motion of
elastic deformations caused in the air rather than the mo-
tion of air masses, for example, the wind as a photon or a
particle moves, space carries a mass which is the mass of
a virtual object, i.e. the corner-stone of quantum theory.
Even this peculiarity alone of the space body motion
established a crushing psychological barrier under the
necessity to allot the invisible and imperceptible object,
i.e. the space, a real material structure. Mysterious char-
acter of space makes some physicists go back to the no-
tion of ether, others - to create a new generalized image
of the space-vacuum, endowing both notions with fantas-
tic properties.
The basis of the inertia principle appears to be the ab-
sence of the absorption of matter connected with a mov-
ing object, which changes the deformations of cells situ-
ated at each time moment near the given body. However,
after the body leaves this region, the space cells obtain
their previous form if there are no other bodies or fields
there. Even sweeping all the stars out of the Universe,
nevertheless the space and the inertia will still exist.
Thus, the cell space contradicts the Mach’s principle.
4. Enigma of Large Numbers
Any theory is only worthy of notice when it contains
For further investigation of the properties of the ele-
mentary cell we revert to two fundamental laws, i.e.
Newton’s gravitation law and the Coulomb’s law:
12 12
, .
mm qq
 (5)
Here physical mechanisms veiled by false simplicity
of their mathematical expression are of interest. The laws
are very similar in form. Noticeable is a similar depend-
ence both on distance and charges. But the most inter-
esting is the relative value of these forces. From the pre-
vious experience it is known what key role the dimen-
sionless values like the Reynolds’s number, Knudsen’s
number, Mach’s number, etc. play in understanding phy-
sical phenomena.
Let us find the ratio of these forces for two electrons:
24.17 10
. (6)
The value is amazing and there hardly exists a physic-
cist who has never thought what it means. Many promi-
nent scientists attempted to get this number [4]. Most
known is the Dirac’s attempt. He divided the age of the
Universe by the time during which a light beam passes
the distance equal to Compton’s length of a proton:
10 10
. (7)
Feynman [7] joked that this number could be obtained
by dividing the Earth volume by the volume of an aphid,
but what have they to do with this number? In fact the
situation becomes much more complicated if to recollect
that even nowadays the number of elementary particles is
about one thousand and for each particle there is its infi-
nitely large number and, besides, the number of particles
is growing with disastrous speed.
All these infinite numbers must be obtained from one
and the same assumption. Perfect will be a solution when
theory will operate only with fundamental constants G, c,
To make the solution more clear we shall do an opera-
tion, the meaning of which will further on become evi-
dent. Let us consider an interaction where the constant is
1 137
times bigger that the electromagnetic con-
Copyright © 2011 SciRes. JMP
stant (α is the fine structure constant):
. (8)
Now let us find the ratio of this interaction force to
Newton’s force. Then for e, μ and p, respectively, we
10 ,
10 ,
10 .
Fe Gm Gm
F pGmGm
We do not give numbers for other particles and only
note that this ratio is rapidly decreasing with the increase
of the particle mass. Intriguing is the result for the
heaviest clot (if it exists), i.e. plankeon:
**2 1.
GcFm Gm
 
Now let us recollect that matter (particle mass) forms
only a steady stable surface shell, but a “particle body”
consists of space cells. Or else, the mass of a particle
deforms both internal and external cells as if “cutting”
from space a mini-volume, which later on we call an ele-
mentary particle. Therefore, only the cells of the surface
layer participate in the interaction.
Let us find the square of one elementary cell using the
ideas of Beckenstein and Hawking:
310 сm
SL c
. (11)
Now let us recollect that the size of the particle, as it
was previously found, is equal:
const ,rmc
and the coefficient const can only be of an order of a unit
(for a proton const is equal to 3).
Let us estimate the number of cells for one layer
cm thick, which makes the surface of a particle:
22 2
 
. (13)
The number of cells on the particle surface is equal to:
44 40
38 *
10 ,10 ,
10, 1.
Se cSc
Ne N
Sc ScGm Gm
Sp cSm
Np Nm
Sc Sc
 
 
for e, μ, p and m*, respectively.
Comparing the ratio of the interaction forces (9) with
the number of cells making the particle surface (
obtain an overwhelming result: these numbers are equal
eaning of this amazing
14), we
What is then the physical m
equality? It appears to be rather transparent. Let us con-
sider the simplest case when particles are pressed to each
other and their surfaces touch. Then in our contrived
strong interaction the matter of all cells, making the
nearest to the surface layer, participate; and in the gravi-
tational interaction there is only one cell which partici-
Turning back to the Coulomb’s force, we obtain the
 (15)
which is a key to the solution of the mystery of large
Now the physical meaning of the fine structure con-
stant α becomes clear: in the Coulom
charged particles at a distance equa
b interaction of two
l to the size of the
particle only a part of cells of the surface layer equal to
of their total number takes part. Besides, the
cells form layers around the particle and in each layer the
deformation of cells is identical, since the cells of one
layer are at the same distance from the particle. As the
difference in force in these interactions lies only in the
number of participating cells, then the dependence of
these two fundamental forces on the distance becomes
similar because the forces are saturated with matter from
one and the same layer. Moreover, an elementary cell
will “allow” us later on to prove that the potential of both
gravitational and Coulomb’s interactions is inversely
proportional to the distance:
and, consequently, the force:
r (17)
Physical meaning of other i
clear: the more cells from the layer participate in the in-
interaction. However, the
constant of any interaction canno
nteractions also becomes
teraction, the stronger is the
t be more than a unit:
( 18 )
since at 2
all cells of the layer take part in the
interaction, which, in its turn, imposes a limit on the
binding energy of two particles:
Copyright © 2011 SciRes. JMP
12 Ec mc, (19)
where m is the mass of a lighter particle of the pair.
rous Co
the interaction. We
shall call this phenomenon the eff
. A positron has the same amount of
electricity as an electron. A more striking example is that
ctly equal in
value, for instance, the positive charge of electron.
fferent reference
A vigoulomb interaction brings us to the as-
sumption that there must be a profound reason for such a
large number of cells to participate in
ect of elementary cells
The Essence of Electric Charge
Milliken and other scientists have shown in their expe-
riments that electric charges consist naturally of discrete
constant portions
all other charged particles have charges exa
According to the present-day views, there is a differ-
ence between an electron and proton, probably, the big-
gest for elementary particles. But their charges are equal
to a high degree of accuracy. It is not clear yet what
doesn’t permit an electron to decay as well as what de-
termines the exact value of its charge.
The existence of an electric charge in two forms is, of
course, its fundamental property. The conservation law
and relativistic invariance are also its mysterious features.
In an isolated system the full electric charge, i.e. the al-
gebraic sum of positive and negative charges, remains
constant. If we measuring a charge in di
stem, we get the same number which drastically differs
from measuring the total mass carried by a moving par-
ticle: the higher velocity, the larger the total mass,
01mm vc .
It is difficult to give up the idea that if we had an ac-
celerator with 19
10E GeV and discovered experi-
mentally all the elementary particles which space can
create we would not answer any of the above questions.
Thus, we have to study the real structure of space already
nly be e
that the second part has a concave surface,
t mistaken with the signs, these clusters are
A colossal difference between the Coulomb force and
gravitation can oxplained by geometries. Let us
consider the simplest analogy. Cut a convex lens of a
whole piece of glass. Make a convex line on the glass
with one move of cutter. Then, after breaking the glass
one can see
d the radii of curvature of these surfaces are absolutely
the same.
When creating electric charges Nature acts in a similar
wave, in an imaginary experiment, if we increase an
electron to the size of a football we can see that the sur-
face of this particle is made up of clusters of deformed
cells resembling segments of the football design. And, if
we are no
ncave for the electron (negative curvature) and convex
for the positron (positive curvature). The question arises:
how many clusters are sited on the electron surface? We
can answer this question by considering the constant fine
structure a:
This unattractive number has been agitating the minds
of physicists for nearly a hundred years. Feynman said:
“… this is one of the greatest accursed mysteries of
physics - a magic number we have got and don’t under-
stand at all. It might be said that thiumber has been
written by ‘God’s hand’ but we don’t know what moved
t the role of the Load was played by Nature.
s n
s pencil”.
If the only construction for the Universe is a three-
dimensional elementary cell, the Universe “knows” only
natural numbers. E. Cronecker, a mathematician, was
rather sagacious when he said that the Lord had created
natural numbers, and all the rest was man’s handiwork
meaning tha
this is the case, the quantity α should be an echo of a
whole number. Since the shape of an elementary particle
differs a little from being spherical, one can suggest that
the number π is involved here. Then,
The number 1722 is accurate enough but it should be
taken into account that α has been varying rather widely
for the last hundred years, beginning with 136. The
number 1722 has a simple physical meaning: it means
that the surface of any charged par 1722
clusters, whereas only the cells of one cluster take part in
ticle consists of
ulomb interaction. Hence it follows that all charged
particles have equal electric charges even though their
masses differ greatly. This fact verifies again an absolute
dependence of the particle size on its mass. This de-
pendence is vividly reflected by the Compton length :
In our articles we paid attention time and again to the
false concept of an electron as a point particle with re <
10–17 cm. There are not any point objects in the wld or
and they cannot exist. Pointness means the absence of a
structure, i.e. indivisibility into coponents. The number
10–17 cm only means that if there are structural forma-
ns inside an electron, they are less than 10–17 cm, and
this figure has nothing to do with the size of the electron
Although a more massive particle has a smaller sur-
face, its cluster contains fewer cells, and a big mass of
one cell compensates for the shortage of cells materially.
The electron surface consists of N cells:
Copyright © 2011 SciRes. JMP
4π7.3 10
where 3
- is the Planck length or the parti-
cle size, and every cluster contains 4.17 × 1042 cells. The
qulomb interac-
tion of an electron and positron if they ould be brought
into contact. In this case the convex suace of one clus-
ter (positron) fully coincides with the concave surface of
tron cluster. I
antity of cells would take part in the Cou
an elecn gravitational interaction, however,
only one cell would participate in this case! So we are
able to unravel the secrets of extreme weakness of gravi-
tational interaction between elementary particles as well
as the mystery of big numbers the solution of which
Dirac was looking for the Universe.
Our studies into the granular space structure enabled
us to find all these infinitely big numbers in the structure
of elementary particles without turning to the Universe.
It should be noted that a positron in a real experiment
cannot be at the distance re = 3.86 × 10–11 cm when it is
in contact with electron because the number of cluster
cells is 4.17 × 1042 and the positron surface consists of
~1045 cells. The positron in this case simply cannot be
placed in “the bed of Procrusten” of electron cluster. The
deformed space cells with inevitably push aside the posi-
tron at the distance a determined again by the constant α:
20.529 10cm
mc e
 
 ,
where 2α is the positron size, α is the hydrogen atom
6. Nature of Energy Levels
It is of high interest to observe in an imaginary experi-
planceon is the simplest and the most prominent
ment the structure of deformed cells as they gradually
move away from the particle. The structure of the cells
near a
since it only consist of one cell.
The mass of this particle is:
210 gmcG
and its size
*3 33
1.610 cmLcG
It should be stressed again that L* is nothing but the
Compton length of the planceon whichmonstrates a
rigid relation between the particle mass and size men-
tedly in [6].
Moving away from this particle we can observe a won-
derful phenomenon - a gradual decrease space cell
r can be densely covered with
at the distance rn from the planceon:
tioned repea
formation gives rise to cluster of collectivized space
This may occur in a layer lying at the distance r2 = 4L*
because this nearest laye
uster consisting of four cells. The next cluster has nine
cells in a layer spaced at the distance r3 = 9L* from the
planceon. In the most natural way we can detect energy
For the first time energy levels were referred to by N.
Bohr in 1913. It is easily seen that the number of clusters
on the first energy level is two, on the second - eight, on
the third - eighteen, and so on. As we have already noted,
this is due to the fact that clusters must closely fill their
relevant layer of cells. Every cluster is characterized by
ite being spectacularity confirmed by number
that this is a deficiency. The impres-
ers n, l and m reflect a com-
eir specific features in the structure embodied in the
quantum numbers: n, l and m, thus revealing the mystery
of Pauli’s exclusion principle. On the first energy level
space forms only two clusters, and that is why only two
electrons can be sited on it, irrespective of the nuclear
charge. On the second level - eight electrons; this is the
exact number of clusters formed by space on this level,
and so on.
The Pauli exclusion principle plays a pivotal role in
our understanding of countless physical and chemical
phenomena, ranging from the periodic table of elements
to the dynamics if white dwarfs and neutron stars. It has
defied all attempts to produce a simple and intuitive
proof, desp
d accuracy of it.
Wolfgang Pauli remarked in his Nobel Prize lecture
(13 December 1946): “Already in my original paper I
stressed the circumstance that I was unable to give a
logical reason for the exclusion principle or to deduce it
from more general assumption. I had the feeling, and I
still have it today,
n that the shadow of some incompleteness fell here on
the bright light of success of the new quantum mechanics
seems to me unavoidable”.
The intrigue connected with Pauli’s principle is that
the second level can have no more than eight electrons
not because these particles “avoid” each other if they have
the same quantum numbers but because on the second
energy level space allots only eight “seats” to the elec-
trons and the quantum numb
ex structure of clusters which consist of a huge number
of deformed elementary cells.
A characteristic feature of science is that we must be
able to describe phenomena so that we could say some-
thing intelligible without exhaustive data and hand. It is
worth nothing that every new theory asks more new
questions than answers the old ones.
Copyright © 2011 SciRes. JMP
o planceons αpl is: The interaction constant of tw
Consequently, with *2 19
10 GeV
Emc , all the
interactions become united and the energy levels can be
call the
forgoing enables us to draw the following conclusions.
1) All energy levels are characterized by the forma-
tion of clusters (collectivization) of deformed space
stance from a particle.
e cluster
should be introduced for a smaller
lled both gravitational, Colombian and nuclear. A
cells at a definite di
2) Any body generates gravitational energy levels
around itself.
The formation of energy levels near a charged particl
is similar to the process which occurs in the case of the
planceon, with the only difference that the role of one
elementary cell (planceon) is played by the whol
containing N = 4.17 × 1042 cells in the case of electron.
Also, a correction
lue of the electromagnetic interaction constant α. Hence,
electron energy levels are placed at the distance rn from
the proton in a hydrogen atom:
 .
Of interest is that the ordinary proportionality to
squared distance can be found in both Colombian and
gravitational interactions, and this regularity has a deep
meaning. Feynman writes [7]:”... Nobody has so far
managed to represent gravitation an
different manifestations of one and the same essence”.
d electricity as two
As it has repeatedly been mentioned, in forming a par-
ticle the additional matter, i.e. the rest mass, which tries
to be sited in a space closely filled with cells, has to de-
form both internal and external cells of space. The ele-
mentary cells pushed outside make up the reserved mat-
ter, which can be transferred by space to other particle
d are referred to as potential energy. The deformed
outer cells from the substance called a physical field. It is
quite obvious that this mass cannot be larger than the rest
mass of a particle being formed - a peculiar law of Ar-
chimedes in the microcosm. In its turn, it means that
the constant of any interaction cannot exceed unity,
Many physicists have already realized that the value
215gc is just a phenomenological parameter of a
strong interaction at low energies rather than the constant
of this interaction. In the article concerned with nuclear
forces we can find the constant of nuclear interaction on
othe basis f the deuton binding energy, 2
0.09736 .
e dependence of interactions on distance,
Feynman spoke about, has a deep meaning which con-
sists in forming clusters of deformed outer cells. A gra-
vitational cluster begins with one elementary cell and
then gradually grows as it moves away from th.
The sam
e particle
In its turn,e electric cluster of, say, the electron in hy-
ogen atom begins with an object containing 4.17 × 1042
elementary cells and then it grows in the same manner as
the gravitational cluster. And only at the distance a =
0.529 × 10–8 cm the electric clusters area increases by 1/a
times and the electron can be placed at last in this bed of
Procrustes. The gravitational clusters will increase simi-
larly. Then, as the electron moves far away, the cluster
areas of both the interactions grow up forming a stringent
sequence, like for planceon, 123
:: 1:4:9SSS 
According to the granular space theory, the electron is
not only as a definite distance a from the proton but,
besides, it is at rest though it has a huge velocity, a mo-
mentum and kinetic energy. At first glance, this statement
is paradoxical. The electron velocity in heavy atoms is as
high as the velocity of length
To unravel this paradox we revealed the true physical
meaning of velocity in about ten articles considering
numerous examples: 22
is the relative value of
deformation of elementary space cells. So, the following
three quantities: the velocity, momentum and kinetic
energy of an electron on the ground energy level of any
nomena c
, too. In our work “Gravitational levels and
. In mathematics the properties
om are characteristic not of the motion but the defor-
mation of elementary space cells.
All the physical pheonsidered by us from the
standpoint of material discrete space fully confirm Ein-
stein’s position “Got doesn’t play dice” and save us for
ever Bohr’s attempt to ascribe indeterminism and uncer-
tainty to space. In more detail it will be discussed in an-
other article.
To our surprise we can observe almost visually the
quantization of not only Coulomb fields but also gravita-
tional ones. In experiments we cannot feel gravitational
levels of particles because of their small sizes but, as the
mass of the object increases, the gravitational clusters
become larger
e Problem of Microwave Background” we found out
that the gravitational levels of the Earth are responsible
for the formation of quasi-black-body radiation near the
Earth with T 2.7 K which is competitive with the “rel-
ict” radiation hypothesis.
One of the fundamental properties of electric charge,
its existence in two forms, is related, as we have estab-
lished, to the fact the deformed cells on the surfaces of
two particles form clusters consisting of a huge number
of cells and having the same concave and convex sur-
faces for both the particles
curve surfaces were studied by Lobachevsky, Gauss,
Rhyman, et al. It is rather amusing that curve surface
mathematics enters physics now through the front door
Copyright © 2011 SciRes. JMP
ant just to note
on is characterized by large
does not possess this property of invariance. The
as the transferred mass of matter is increased
and not through the back one because we are able to see
the curvature of surfaces with our own eyes removing the
nonobservability of theory. As we cannot cut only one
convex surface of glass without the other part of glass
getting a concave surface, so it is impossible to generate
a positive charge without a negative one. No doubt,
charges can disappear only in pairs, too.
So, we can observe practically visually the fundamen-
tal property - the law of charge conservation and its ex-
istence in two forms.
In our next articles we are going to consider more
comprehensively the mechanisms of both gravitational
and Coulomb interactions, and here we w
at gravitation can be realized when the deformation of
elementary cells beyond two bodies is larger than be-
tween them whereas repulsi
formation of cells between the bodies. The difference
in cell deformation results in a deformation gradient called
force. The curvature of the layers of the deformed cells
around a particle or a macroscopic body is a secondary
There is exhaustive experimental evidence that the to-
tal charge of a system remains constant as the charge
charge carriers move. We have got used to it so that we
don’t often think about such a wonderful and fundamen-
tal fact.
atter carried by kinetic energy forms an object referred
to in modern theory as a virtual photon, its structure will
be considered in a section concerned with the motion of
elementary particles. The kinetic energy of a particle
11vc times. The space and a particle carry
fast additional substance used to form new particles
when a shell particle collides with a target particle. The
mass of the electron does not change in this case, its size
remains constant, too. As for the moving virtual object
(photon), the following rigid ratio is valid: rmc
where 22
means the total mass of the
electron and the virtual photon.
Let a moving electron has Ee = 938 MeV which is
equal to the rest energy of proton. In this case the wa-
velength of the virtual photon (not the electron)
210 cm
, i.e. it is equal to the size o.
It shou surface of the virtual photon
f rest proton
noted that the
is made up of segments,
face, and the d
our eyesight were keener, these secrets would have been
laughed at our difficulties.
sult again honeycomb. Every
riming surface.
en it reaches the end of a notch, it falls
space cells form clusters of
ld be
duplicates that of the electron: it
clusters of negative curvature, too. Since the mass
transferred by it is equal to that of the proton, the cluster
will be exactly the same as the cluster on the proton sur-
ecreased area of the cluster is compen-
sated for by a larger mass. To our surprise we have to
state that this virtual photon has a charge of proton and,
hence, the charge of a moving electron is equal to that of
a rest electron thus making the electric charge invariant.
7. The Birth of an Elementary Particle
The process of formation of an elementary particle re-
mains as mysterious as the number 1722. But space cre-
ates a particle in miserable portions of a second, and if
revealed long ago. Once someone said well that
Now we can set forth just some preliminary considera-
tions having a claim on rigorous proof but, nevertheless,
containing a number of important ideas.
It is the shape of the cell, a polyhedron, which evi-
dently plays a leading role in forming a particle. To sim-
plify the problem, let us con
ll is hexahedral in shape, and all the subsequent layers
that “dress” are the same hexahedrons. Thus, almost all
elementary particles have the same p
wing to this needle-shaped broken structure, the addi-
tional matter - the particle mass - can cover part of space
cells deforming them and, with certain conditions met,
can form a surface film at least for a short instant, like in
the case of resonances. We are, without fail, to reveal
these mysterious conditions already today not waiting for
the construction of a 1019 GeV accelerator, because even
this energy will not tell us anything new about the elec-
tron structure.
The formation of a particle is a real disaster; the ma-
thematical theory of catastrophes is under rapid devel-
opment now. Let’s consider a simple analogy. The birth
of a particle only vaguely resembles the work of clock
ratchet-and-pawl. The pawl slides quietly over the ratchet
for a while but, wh
d a catastrophe happens.
Similarly, a gradual increase in substance quantity, the
mass in this region, causes both the radial and the torsion
deformations of cells to increase. The latter defines the
spin structure. Charged particles are born in pairs, and
this process ands with a catastrophe all of a sudden when
the previously independent
sitive and negative curvatures on a closed surface. The
process of clusterization and collectivization of elemen-
tary cells is the main mystery of elementary particle birth.
As the particle mass increases, the deformation of the
inner cells reaches such a point that it is useful for space
to create clusters inside the particle, clusters called quarks.
The cluster confinement in this case is quite a transparent
phenomenon: without external deformation of space cells
the quarks simply disappear.
This unusual deformation makes a prerequisite for in-
troducing the conception of color and odor to theory.
Though these processes are still far from being fully
Copyright © 2011 SciRes. JMP
to approach the secret of
f a
understood, the birth of a particle does not need either
additional fields or mediator-particles. An analogy with a
soliton will probably help us
rmation of a stable surface of elementary particles. The
soliton is assumed to be kept stationary at the cost o
lance between the action of a nonlinear medium and
We should add another point to understand the gravi-
tational potential φ. Let us estimate φ on the surface of
an electron, a muon and a proton:
44 2
10 ,
 
40 2
10 ,
 
38 2
10 .
 
We have already come across the large numbers 1044,
1040 and 1038 which denote the numbers of ces making
up the surfaces of an electron, a muon and a proton.
Since the squared velocity is a dimension of poten-
tial, the value
vc shows wh
layer is inv
ich part of cells of their
total number in a givenolved in the interact-
ly on
and fantasy off quite a number of fundament
physical phenomena. The Universe is found to be infi-
is only made up of one element, and
infinitely complex, as any of its matter clusters consists
irac and
ity. The func-
Here we want to illustrate with a proton one specific
feature of cell behavior: though one cell of proton sur-
face requires substance 108 lrger than one cell of elec-
tron surface, one cell participates again in the gra-
vitational interaction of two protons at an ultimately close
8. Conclusion
In conclusion it should be noted that even the first
“steps” of an elementary cell have removed a touch of
nitely simple, as it
of an infinite number of elementary cells. So, is space
continuous or discrete? It would be safe to say, neither of
the two. But this naked negation does not feed our
thoughts and thoughts and negatively affects theory.
Therefore, the more exact answer is: both.
Space is discrete because it consists of discrete ele-
mentary cells, it is also continuous since any deformation
of cells in a continuous manner (and not by steps) spreads
from one cell to another establishing absolute hundred-
percent causality and definiteness our great predecessors
Planck, Lorentz, de Broglie, Schrödinger, DEin-
ein believed in till the end of their lives.
To solve the problem of large numbers filly explains
the smallness of gravitation as compared to other forces
acting in the microcosm and does not need a fantastic
hypothesis for the existence of extra spatial dimensions.
Only a three-dimensional elementary cell enables the
Universe to create objects of any complex
nal dependence of gravity on distance 2
r must
remain constant up to distances 33
10 cmr
Atom stability is one of the most burning problems of
theoretical physics, and any attempt of solving this prob-
lem with the use of the Geisenberg inequality is invalid-
dated by numerous experiments. In our next articles we
shall subsequently consider atom stabilienberg
inequalities, the foundation of pon
ty, Geis
robability csideration
various scales form a cellular-netted large-scale
hared the opinion that the real space is to
ler, “Einsteins Vision,” SPRINGER - VERLAG,
roduction to Elementary Particle Physics,”
A. Dirac,
oncepts of Granular Space Theory,”
n Lectures on Physics,” Addi-
son-Wesley Publishing Company, London, 1963.
the ψ-function and the curvature of four-dimensional
space-time in Einstein’s theory of gravity, the nature of
the microwave background of the Universe, the time
“arrow” and other subjects making use of the material
structure of granular space and only three world con-
More and more physicists are aware that space is
granular in structure and sets an absolute system of ref-
erence [8].
Comparatively a short time ago astronomers discov-
ered a wonderful star picture: groups and clusters of gal-
axies of
ucture of the Universe. In our opinion, this large astral
cell is born by the cellular structure of the space itself.
Newton s
me degree an empty box where material bodies move
about without affecting the space at all. Einstein’s theory
of gravity has invalidated this assumption by supporting
the view that matter and space are directly interrelated.
The theory of granular space is making another step
e unification of matter and space: any cluster of matter
is a complex of fantastically deformed elementary cells
of the space itself thus symbolizing the Great Unity of
Nature [5].
[1] M. Planck, “The Unity of the Physical Patter of the
World,” NAUKA, Moscow, 1996, p. 108.
[2] J. Whee
New York, 1968
[3] L. Okun’, “Int
NAUKA, Moscow, 1985.
[4] A. Eddington, “The Mathematicl Theory of Relativity,”
Cambridge University, Cambridge, 1923. P.
“Directions in Physics,” Wiley, New York, 1978.
[5] V. Konushko, “C
SPUTNIK, Moscow, 1999.
[6] R. Feynman, “The Feynma
Copyright © 2011 SciRes. JMP
Copyright © 2011 SciRes. JMP
[7] S. W. Hawking, “The Formation of Particles on Black
Holes,” Communications in Mathematical Physics, Vol.
43, No. 3, 1975, pp. 199-220. doi:10.1007/BF0234
[8] T. Jacobson and R. Parentani, “The Echo of the Black
Holes,” Scientific American, No. 3, 2006, p. 17. A.
Smolin, “Atom’s Space and Time,” Scientific American,
No. 4, 2004, p. 20.