Rafael Javier Martínez Olmo
2011-07-05 14:08:36 UTC
Model of Unification of General Relativity and Quantum Mechanics
Author: Rafael Javier Martínez Olmo
All rights © Rafael Javier Martínez Olmo
Certainty: “Not even human stupidity is infinite, although it seems
so, is not physically possible.”
Updated: in Vigo, on Friday, the 27th February, 2011 to 06:42 A.M
Physics and the Infinite
For centuries the hypothesis that the universe is continued that has
been considered a clear and unquestionable truth. Not only matter and
energy have been considered so but also the proper spacetime which
With the birth of quantum mechanics, we understood that matter is not
continuum (represented by the set of real numbers (R). For example, an
iron bar cannot be divided into smaller pieces and these in turn into
smaller ones and so on. If we kept dividing it only an atom of iron
would remain and this one cannot be divided in “1/n atoms of iron ".
However, since an atom is not elementary, we can even split an atom of
iron. (In the context of this work ‘elementary’ means that it has no
parts and consequently is indivisible).
The atom can be divided in protons, neutrons and electrons.
There is a strong assumption that protons and neutrons are not
elementary. They have an internal structure composed of three quarks.
At present, there is no one who has raised the hypothesis that a quark
is infinitely divisible.
The electron does not have an internal structure (it has no parts) and
is indivisible. We can say that tit is elementary in the sense
expressed in this work.
Quantum Mechanics also ensures that energy is emitted and absorbed in
the form of all discreet Quantum (represented only by the set of
Natural numbers (N)).
We can affirm that the hypothesis of a continuum universe is neither
fulfilled with matter nor with energy. When I use the term ‘continous’
or 'continuity', I mean that the physical reality of the phenomenon to
describe can only be represented by the set of the Real numbers (R).
Here a question arises: What would happen if spacetime was not
continuum? What will happen if spacetime cannot be represented by (R)?
The purpose of this work is to unify Quantum Mechanics and the
General Theory of Relativity, eliminating the randomness in Quantum
Mechanics through a model of geometric variables of curvature O_Pi ,
replacing the set of the Real numbers R [1 by the set of the "Real -
natural" numbers R (N)  which would help the physical description
of nature. Under the "atomic" hypothesis, that there is a natural
physical unit of Spacetime (VTN), tetra dimensional, elementary, the
smallest and indivisible.
Max Planck devised a dimensional procedure to determine the absolute
units of the Nature, since they are obtained from universal constants.
As the Universe is four-dimensional we seek the unique combination of
constants which gives us a Volume in Time (VT) = G • h • c^-2 = cm^+3
• sc^-1. The hypothesis is that this volume-time is the basic atom of
VTN = G • h • c ^-2 = 4,9205 • 10E-55 cm. ^ + 3 • sc ^-1. 
 The Real number (R) always has an infinite number of decimal
 The Real - natural number R (N) always has a finite number of
 Fundamental physical constants, (G) of gravity, (h) Planck's
constant (c) speed of light.
Geometry of the Spacetime
We know that the two fundamental theories on our understanding of the
universe, General Relativity and Quantum Mechanics, are not right,
since they describe two antagonistic worlds, therefore one of them or
perhaps both must be wrong.
The cause for the infinites to appear when we join General
Relativity and Quantum Mechanics without the possibility of being
eliminated (General Relativity is not renormalizable), has its origin
in the use of the set (R) to describe physical nature.
This is in my opinion a fundamental and previous mathematical reason,
which must be solved to achieve the unification of both models,
replacing the set of the Real numbers (R) by the set of the Real -
natural numbers R (N) to represent physical reality.
We use the set of Real numbers, for both theories, in fact for all
Physics. This is completely wrong, absurd and unnecessary, since it
introduces mathematical concepts into Physics. These are not
measurable (Physical) such as moment, point, infinitely small,
infinite. We should remember that R is an “actual infinite” set
(Cantor), that is to say that it fulfils the following:
1. Its elements lack “good order”, that is, an ordinal number (N)
associated with each element of R, which would indicate your order
number according to size (number value). This property implies that
there is no number R previous or posterior to any other R number.
2. The use of R implies that any physical measurement, such as
distance, is infinitely divisible. It means that a distance can take
3. In R the part is equal to the whole
All these properties, in my opinion, do not exist in nature. If we use
the Real - natural numbers R(N) these three ants natural properties
disappear. For this purpose, it is enough if we give up the idea that
real numbers have infinite decimal places (that physically we do not
need) and we replace it by R (N) which has a sufficiently large number
of decimal places.
This choice (R) allows a physical variable to take any value, but it
also implies that the value of the variable may be worth infinite
(singularity) and be confined to a point that lacks dimensions.
Gravitational and electromagnetic forces are inversely proportional to
the square of the distance between the masses and the electrical
particles. These forces increase when distance decreases. If distance
is zero the value of the Physical variable (gravitational force or
electromagnetic force) is infinite (division by zero). These are the
infinites that prevent the unification, this is the error that we
continue to make, an error in the essence of Logic.
This error appears through the mathematical procedure of abstraction,
which causes geometric objects of smaller dimensions than those of
Physical Nature to appear, that is to say four, three spatial
dimensions and a temporary dimension. I mean the mathematical concepts
of point, zero dimension, line (dimension one), plane (dimension
two)... I also mean the mathematical concepts and physical meaning of
zero and infinite.
This Error is in the ambiguity of the mathematical point, when applied
to physical concepts as distance, speed, acceleration...
The concept of point is physically absurd, since it has two exclusive
properties; it exists or does not exist.
It is absurd to think that physical objects exist and do not exist.
However, the mathematical concept of point of dimension = 0, when it
is applied to the physics presents this paradox:
The point exists (physically). Because it has a real and exact
position in any system of spacetime reference which is determined by
its four coordinates in space time. In fact the point is used to
indicate a position in spacetime (R4).
The point does not exist (physically). Because it does not have
dimensions (dimension = 0). It has no height, no length, no width, and
does not exist in time. This incoherence is subtle, but clear after a
Physical Hypotheses and Prediction Model
The universe and all his contents has four dimensions.
Spacetime has four dimensions.
Matter has always has three spatial dimensions (volume) and exists in
time; energy always occupies a volume of space and exists in time,
then both have four dimensions.
Absolute vacuum lacking both matter and energy does not exist, since
there is no way to isolate the gravity of a “spacetime volume”.
Relative vacuum, which at least contains gravity, has therefore, four
dimensions. There is no physical example, (that exists and could be
measured), of objects in our universe which has not four dimensions.
Following Euclides, it is not possible to construct, simply by adding
(addition sum), geometric objects of a different dimension from its
construction elements, for example, to build a line adding a large
enough number of points.
If in Nature there are only four dimension elements, the smallest and
indivisible element of our universe must have four dimensions, that
is, a volume that changes with time.
There are no objects in Nature of less than four dimensions; they only
exist in the Platonic world of ideas as mathematical abstractions of
To calculate the value of the lowest spacetime we use the fundamental
constants of nature, c, h, y G, in the same way as Max Karl Ernst
Ludwig Planck did a century ago. We combined them looking in this case
for a time volume, i.e. the volume of a box during an hour, for
example, an unit of volume during an unit of time. This can be done
with the following combination of fundamental constants, which is also
G • h • c^-2 = 4,920 551 532 644 910 • 10-55 cm ^+ 3 • sc ^-1
Lowest volume-time (without parts), indivisible, elementary.
As we know, the universe is isotropic, does not have favourite spatial
directions, it presents the same appearance and properties in any
directions. The distinctions we make between length, width and height
are merely semantic, since we can share their names and they still
represent the same physical reality.
If we associate the idea of isotropy to the elementary and lowest
order of the universe, this can only be a sphere.
As a sphere it is easy to calculate the radio: Lmo = 4,897 506 921 037
260E-19 centimeters minimum distance and any measurement of distance
is equal to the Lmo product by a Natural number (N)
This distance is covered at the speed of light in Tmo = 1,633 632 464
842 480 E-29 seconds. This is the lowest time interval, and any
measurement of time is equal to the product of Tmo by a Natural number
As the distance is very small, there cannot exist a wavelength shorter
than Lmo, 4,897 506 921 037 260E-19 centimeters. As it is a the
smallest wavelength it is the highest value of Emo energy = 253,177
660 585 902 TeV. Any wavelength is equal to the product of Lmo by a
Natural number (N).
The value of any measurement of a quantity of energy is achieved
dividing Emo by a Natural number (N).
According the equivalence between mass and energy of Einstein E = m •
c2, we have to divide by c2 the highest mass of a smallest event
(elementary, without parts) Mmo = 4,512 946 783 762 060E-19 grams.
Table of Energy Levels Predicted by the Model
First hundred levels of energy, predicted by the model, in color
energy levels detected in the LHC (CERN).
Maximum 7 Tev (1st phase) and 14 Tev (2nd phase).
Levels of Energy Energy in TeV Level Energy in TeV Level
Energy in TeV
1 253,1776605859 26 9,73760233 51 4,964267855 76 3,331285008
2 126,5888302929 27 9,376950392 52 4,868801165 77 3,288021566
3 84,39255352863 28 9,042059307 53 4,776936992 78 3,245867443
4 63,29441514647 29 8,730264158 54 4,688475196 79 3,204780514
5 50,63553211718 30 8,439255353 55 4,603230192 80 3,164720757
6 42,19627676431 31 8,167021309 56 4,521029653 81 3,125650131
7 36,16823722655 32 7,911801893 57 4,441713344 82 3,087532446
8 31,64720757323 33 7,672050321 58 4,365132079 83 3,05033326
9 28,13085117621 34 7,446401782 59 4,29114679 84 3,014019769
10 25,31776605859 35 7,233647445 60 4,219627676 85 2,978560713
11 23,01615096235 36 7,032712794 61 4,150453452 86 2,943926286
12 21,09813838215 37 6,842639475 62 4,083510655 87 2,910088053
13 19,47520466045 38 6,662570015 63 4,018693025 88 2,87701887
14 18,08411861327 39 6,491734887 64 3,955900947 89 2,844692816
15 16,87851070572 40 6,329441515 65 3,895040932 90 2,813085118
16 15,82360378661 41 6,175064892 66 3,83602516 91 2,782172094
17 14,89280356387 42 6,028039538 67 3,778771054 92 2,751931093
18 14,06542558810 43 5,887852572 68 3,723200891 93 2,722340436
19 13,32514003083 44 5,754037741 69 3,669241458 94 2,693379368
20 12,65888302929 45 5,626170235 70 3,616823723 95 2,665028006
21 12,05607907551 46 5,503862187 71 3,565882543 96 2,637267298
22 11,50807548117 47 5,386758736 72 3,516356397 97 2,610078975
23 11,00772437330 48 5,274534596 73 3,468187131 98 2,583445516
24 10,54906919107 49 5,166891032 74 3,421319738 99 2,557350107
25 10,12710642343 50 5,063553212 75 3,375702141 100 2,531776606
Notice how the energy levels are closer to each other as the number of
energy level increases. We can see that there are 21 levels of energy
in the range of 3 TeV (from level 64 to 84) and only a level of 13
TeV (19 in red), according to the above, these discreet levels of
energy (if we look for them) will become more evident in the second
phase (yellow levels). I suggest the search of the level 19 as it is
the only level that exists in the status of 13 TeV. These levels are
so close to the energy levels in our daily life that they are not
detectable and they seem to have the continuity that real numbers
idealize. Remember that these energies are of quantum elementary
A century ago Max Planck took into account length, mass, time, the
electrical load and the temperature as fundamental elements to
describe Nature, using Coulomb's constant and Boltzmann's constant in
addition to c, h, G.
These last two constants are not used in this work, since they are
considered unnecessary for the description of Quantum Geometry
(elementary = not divisible) of the Spacetime.
The difference between Max Planck's dimensional procedure and that
established in this work, is in what we consider elementary. For
Planck these were length, mass and time.
This can be argued about, against the idea accepted at present, as it
is reflected in the following paragraph:
“The system measures several of the fundamental magnitudes of the
universe: time, length, mass, electrical charge and temperature.
Planck's units are often called (in joke) “God’s units” by physicists.
This eliminates any anthropocentric arbitrariness of the system of
Extracted from Wikipedia, Planck's units.
This work considers a four dimension “spacetime volume” to be an
elementary magnitude (without parts, indivisible.
In the following table we see the differences between both
Dimensional formulae Value (System c, g, s) Dimensional formulae
Value (System c, g, s)
Max Planck =5.39121 • 10E -44 seconds This Work>> t = 1,63363 •
Max Planck =1.61624 • 10E-33centimeters This Work>> l = 4,89750 •
Max Planck =2.17645 • 10E-5 grams This Work>> m = 4,51294 • 10E-19
The magnitudes calculated by Max Planck for length and time are
extremely small and they involve quantities of inaccessible energy to
On the contrary the magnitudes calculated in this work are
experimentally contrasted at the levels of energy of the current
particle accelerators (14 TeV LHC; CERN).
The highest quantity of energy 253,177 TeV is only to two orders of
magnitude from the previous particle accelerators, quantum geometry
has probably been photographed, but it has remained unnoticed between
the trillions of events and data obtained in the experiments of these
For the first time, Quantum Gravitation will be within reach of the
experimental physicists, particularly those currently working at the
CERN (LHC), which are those that will be able to determine if the
prophecies of this work are in accordance with Nature or are wrong
On the words and concepts that I use I would like to avoid any
ambiguity, firstly because these terms are easily mistaken due to
their colloquial use, secondly the smallest because this ambiguity is
also implicit in the physical - mathematical concepts of continuum,
infinite and vacuum (zero), which we handle in General Relativity and
Quantum Mechanics, when we refer to Space or to Time (Quantum), or to
1st This text is essentially mathematical. When I say 'point' I refer
to its physical-mathematical concept, geometric object of zero
dimension [0,0,0,0]. In this geometrical object it has neither height,
width, nor depth and does not exist in time.
2nd When I use the term continuous, continuum or continuity, I mean
that the physical reality of the phenomenon to describe, can only be
represented by the set of Real numbers (R).
3rd For all matters concerning the concept of infinite, types of
infinite sets, sets of Real numbers (R) and Natural numbers (N), I
base on the work "The Theory of Transfinite Sets" by Georg Cantor.
Extract of the model.
If we understand the Differential calculus as a Physical Theory
(reality), not as a mathematical model, which brings us near to that
one infinitely, but without reaching it using the Mythological and
physically absurd set of Real numbers (R), we will verify that that is
the logical way that the elementary and quantum nature of Spacetime
Geometry shows us. Its elementary components (without parts), are
volumes in time (VT). They have four dimensions (D4) and are smallest
(e> 0), of curved topology (p). They can only be represented by the
set of natural numbers R (N).
This route based on the Differential calculus is what I call the
Quantum Geometry of Spacetime.
The work tries to lay the foundations and theoretical essentials for
this route, which unifies the experimental results of Quantum
Mechanics with the theory of the General Relativity, by including the
invisible, hidden variable O_Pi (Model of hidden variables).
O_Pi = spacetime curvature of an elementary quantum event.
The model is relational and independently from the beginning (Quantum
Relativity), where both the General Theory of Relativity and Quantum
Mechanics are modified. Neither the structure, nor the variables or
the results of both models can be expressed as belonging to the set of
the Real numbers (R), all these belong to the set of the numbers R
General Relativity can be expressed as an elliptical geometry with
spherical topology(p) and radial dynamics. Quantum Mechanics is
completed introducing causality, replacing the statistical treatment
(chance, randomness) by a geometric treatment (causal, determinist)
since we include the variable O_Pi. This variable O_Pi gives a causal
explanation of Quantum Mechanics, since it establishes one to one
(Bijection) connection, between the statistical results and the
elementary geometries (without parts) of Spacetime that contain the
above mentioned quantum events.
Likewise, it also determines the geometry (its form) and the metrics
of Nature in its elementary, smallest or indivisible scale. It
specifies a model of Quantum Relativity (Quantum Mechanics Geometry)
The attempts of unification of both theories, have always failed due
to the appearance of the infinites. The origin of these
indeterminations is in a mistaken reading of the "Calculus" as a
consequence of the error of expressing the continuity from points of
zero dimension (D = 0; R).
The set of real numbers (R) has allowed us to represent the physical
reality to our scale, but it is not right in the description of the
elementary or indivisible nature of "Spacetime"; this can only be
described in elementary terms, by four dimensional geometric
objects, Temporary Volumes (VT) and only represented by the set of the
natural numbers (N).
The term continuum is only opposed to discreet if we use the concept
of mathematical point to create the continuum (D=0; R) (infinite
There are no physical singularities in Nature; they are the result of
using the set of the real numbers out of context (quantum scale).
There are therefore mathematical singularities (division by zero) and
1.- Quantum Mechanics describes the universe when the radio of volume-
times containing the event, stretches to 4,897506921037260 E-19 cm.
lowest Lmo distance. At this distance of an elementary mass Mmo, the
universe has the highest curvature, or equivalently, is the highest
energy of a quantum event. This wavelength corresponds at an energy
level of 253,177660585902000 TeV. This level of energy has an
asymptotic nature as unattainable or supreme.
The highest energy of a quantum elementary event = 253,177660585902
2.-The hidden variables of Quantum Mechanics are in the geometry, a
very small scale, of the spacetime that contains the quantum event. Up
to date we have thought that the spacetime curvature on this scale was
almost flat, therefore with very little influence. On the contrary,
the Quantum Geometry places the origin of the curvature and therefore
of geometry, in any particle with mass. Consequently, the highest
curvature or maximum energy is at a Lmo smallest distance, of any
"volume-time" occupied by an elementary mass (Mmo). The highest
1/Lmo = 4,897 506 921 037 260 E+19 cm-1
3.-As we know from Feynman, the quantum electrodynamics (Q.E.D.) was
characterised in its beginnings because all its results were infinite.
The reason for this is that the sum of stories must bear in mind all
the possible ways, and these ways depend on the distance between
particles. This distance becomes zero; the zero introduces the
infinite in the results.
Feynman determined not to take the calculations up to zero to avoid
the indetermination, replacing it by a very small number 10E-100 cm.
and stopping the sums on stories on this value. This supposed a
solution to indeterminations, but other problems arised. The unity of
the probability is lost and infinitesimal terms with negative energy
These problems disappear, if instead of using a very small and
arbitrary distance, we use the nature lowest distance.
Lmo=4,897506921037470 E-19 cm. to interrupt the sums on stories. Both
problems appear when adding probabilities of nonexistent interactions,
particularly all those calculated for distances lower than 4,8975
4.-The tetra dimensional geometry at a quantum level behaves as if it
were two-dimensional, (only two grades of freedom), since the radio
determines three spatial dimensions and time the dynamic evolution.
Moreover, the radio has the same ordinal that as time, (the tag of the
spherical ring and the time since when it was radiated coincide).
Historically this incoherence has appeared before us three times. It
is always concealed under the concept of continuity.
The first time that the incoherence showed itself was in the four
sophisms by Zeno of Elea, 2.600 years ago. His logic is perfect,
closed and conclusive. These sophisms were presented to the local
philosophers in Athens, perhaps before the proper Socrates. We know
that Aristotle and Plato knew the speech given by Zeno perfectly well.
The second time that the contradiction appeared again, was with the
invention of the infinitesimal calculation. At that time many
intellectuals were against the logic of such a theory, since there is
no way of explaining the movement or its associated variables, speed
and acceleration, in term of zero dimensional points
The mathematical point does not have another point coming after it,
neither a previous one. There is no second point neither a fifth one,
no previous one to any given point. Because the set of the Real number
(R) lacks an ordinal number associated to its elements. In spite of
being a completely tidy set, it lacks good order. How can we explain
the movement from the point A, to the point B, if we cannot go out of
A because the following point does not exist?
This contradiction is implicit in the incoherence of the mathematical
point (dimensional = zero) with which the mathematical "continuum" is
But since the calculation worked and it still works, and the subtle
contradiction was not discovered, those who did not agree had to keep
Due to this, the movement and its associated magnitudes, speed and
acceleration entered the club of the incoherence, since the set of the
numbers Real (R) is continuous (by definition) but the infinitesimal
Calculation cannot be demonstrated mathematically in terms of points,
dimension = zero, it can only be demonstrated mathematically in terms
of intervals (e > 0, dimension = 1).
The third time the incoherence appeared was with the birth of Quantum
Mechanics (Beginning of Heisenberg uncertainty). The incoherent
reality of the point concept went unnoticed again and divided Physics
in two, introducing randomness in the quantum world.
Since then the universe possesses simultaneously two mutually
exclusive properties: it is random and it is causal, depending on the
size of the object of the Nature that we are studying. It is random if
this one belongs to the microcosm and determinist if the object
belongs to the macrocosm.
This model is finite, quantitative and predictive. It is
experimentally contrastable at levels of energy between 1 and 14 TeV.
Level 19=13,325 140 TeV. The first level of energy is 253,177 660 585
902 TeV. (Supreme value of energy of an Elementary Quantum event)
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