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Quantum Geometry of the Spacetime.pdf

Geometría Cuántica del Espaciotiempo.htm Quanten Geometrie der Raumzeit.htm La Géométrie Quantique de l'Espace-temps.htm Квантовая геометрия пространства Время.htm Quantum Geometry of the Spacetime.htm


Quantum Geometry of the Space-time


Model of Unification of General Relativity and Quantum Mechanics

Authors: Rafael Javier Martínez Olmo & Pilar Ávila Barredo (Piska)




Vigo, Monday, 2 July 2012

Certainty: “Physically nothing is infinite.

Not even human stupidity is infinite, although it seems so, is not physically possible.”


Physics and the Infinite


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 "hidden variables", geometric of curvature variables "Op", replacing the set of the Real numbers R[1] by the set of the "Real-Natural" numbers R(N) [2] which would help the physical description of nature; under the hypothesis that there is a natural physical unit of Spacetime (VTN), tetra dimensional, elementary, indivisible and the smallest one.

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 spacetime.

VTN= G • h • c-2 = 4,9205 ·10E-55 cm+3· sg-1.[3]

[1] The Real number (R) always has an infinite number of decimal numbers.

[2] The Real-Natural number R(N) always has a finite number of decimal numbers.

[3] Fundamental physical constants, (G) of gravity, (h) Planck's constant (c) speed of light.


For centuries the hypothesis that the universe is continuous that has been considered a clear and unquestionable truth. Not only matter and energy have been considered so but also the proper spacetime which contains them.

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 it 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 “continuous”, “continuum” 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)?

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 (Georg Cantor), that is to say that it fulfils the following:

1. Its elements lack ordinal number, that is, an number (N) associated with each element of (R), which would indicate your order number according to structure of construction. 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 any value.

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 unnatural 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 brief reflection.

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, therefore both of them 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, it is not possible to build a line adding a large enough number of points.

If in Nature there are only four dimensions 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 reality.

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, for example, a unit of volume during a unit of time, i.e. the volume of a box during an hour. This can be done with the following combination of fundamental constants, which is also unique:

G • h • c-2 = 4,920 551 532 644 910 • 10 -55 cm + 3 • sc -1

Smallest 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 = 4,8975E-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 = 1,6336E-29 seconds. This is the lowest time interval, and any measurement of time is equal to the product of Tmo by a Natural number (N).

As the distance is very small, there cannot exist a wavelength shorter than Lmo, 4,8975E-19 centimeters. As it is a the smallest wavelength it is the highest value of Emo energy = 253,1776TeV.

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).

First hundred levels of energy, predicted by the model, in color energy levels detected at the LHC (CERN). Maximum 7 TeV (1 st phase) and 14 TeV (2nd phase).

Table: Quantum´s Elementary Levels of Energy

Energy Level Energy in TeV E. Level Energy in TeV E. Level Energy in TeV E. Level Energy in TeV
1 253,177660585902 26 9,73760233 51 4,964267855 76 3,331285008
2 126,588830292951 27 9,376950392 52 4,868801165 77 3,288021566
3 84,392553528634 28 9,042059307 53 4,776936992 78 3,245867443
4 63,294415146476 29 8,730264158 54 4,688475196 79 3,204780514
5 50,635532117180 30 8,439255353 55 4,603230192 80 3,164720757
6 42,196276764317 31 8,167021309 56 4,521029653 81 3,125650131
7 36,168237226557 32 7,911801893 57 4,441713344 82 3,087532446
8 31,647207573238 33 7,672050321 58 4,365132079 83 3,05033326
9 28,130851176211 34 7,446401782 59 4,29114679 84 3,014019769
10 25,317766058590 35 7,233647445 60 4,219627676 85 2,978560713
11 23,016150962355 36 7,032712794 61 4,150453452 86 2,943926286
12 21,098138382159 37 6,842639475 62 4,083510655 87 2,910088053
13 19,475204660454 38 6,662570015 63 4,018693025 88 2,87701887
14 18,084118613279 39 6,491734887 64 3,955900947 89 2,844692816
15 16,878510705727 40 6,329441515 65 3,895040932 90 2,813085118
16 15,823603786619 41 6,175064892 66 3,83602516 91 2,782172094
17 14,892803563877 42 6,028039538 67 3,778771054 92 2,751931093
18 14,065425588106 43 5,887852572 68 3,723200891 93 2,722340436
19 13,325140030837 44 5,754037741 69 3,669241458 94 2,693379368
20 12,658883029295 45 5,626170235 70 3,616823723 95 2,665028006
21 12,056079075519 46 5,503862187 71 3,565882543 96 2,637267298
22 11,508075481177 47 5,386758736 72 3,516356397 97 2,610078975
23 11,007724373300 48 5,274534596 73 3,468187131 98 2,583445516
24 10,549069191079 49 5,166891032 74 3,421319738 99 2,557350107
25 10,127106423436 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 in blue) 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 events.

A century ago Max Planck took into account length, mass, time, electric charge and the temperature as  fundamental elements to describe The Nature, using Coulomb's and Boltzmann's constants in addition to G, h, c.

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 units”.

Extracted from Wikipedia, Planck's units.

The Hypothesis of work Tetra Dimensional, considers a volume of “spacetime” to be the elementary magnitude (without parts, indivisible).

VT = [G • h • c-2]

 In the following table we see the differences between both calculations.

Max Planck Hypothesis Tetra-Dimensional
Dimensional formulae Values(System c,g,s) Dimensional formulae Values(System c,g,s)


tt = 1.63·10-29seconds


lt = 4.89·10-19centimeters


mt = 4.51·10-19 grams

The magnitudes calculated by Max Planck for length and time are extremely small and they involve quantities of inaccessible energy to our technology.

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 big accelerators.

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 Spacetime (Relativity).

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) but never zero , 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 hidden variable Op (Model of hidden variables).

Op = spacetime curvature of an elementary quantum event.

The model is relational and independent from the system of reference (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(N).

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 Op.

This variable Op 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 (to give geometrical form to Quantum Mechanics).

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, Volume-time (VT) and only represented by the set of the natural numbers (N).

The term continuous is only opposed to discreet if we use the concept of mathematical point to create the continuum (D = 0; R) (infinite divisibility).

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 not physical.

Other Predictions

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 = 253,1776 TeV

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 curvature is: 1/Lmo = 4,8975E+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 indeterminacy, replacing it by a very small number 10-100 centimeters. and stopping the sums on stories on this value. This supposed a solution to indeterminations, but other problems arise. The uniqueness of the probability is lost and infinitesimal terms with negative energy appear.

These problems disappear, if instead of using a very small and arbitrary distance, we use the nature lowest distance. Lmo = 4,8975E-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 4,8975E-19 centimeters.

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 created (R).

But since the calculation worked and it still works, and the subtle contradiction was not discovered, those who did not agree had to keep silent unwillingly.

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 possible to contrast experimentally at levels of energy between 1 and 14 TeV. The level 19=13,325 140 TeV is the only level of energy that exists between 13 and 14 TeV . The highest level of energy is 253,177 660 585 902 TeV. (Supreme value of energy of an Elementary Quantum event)


Index and Bibliographical Base of the Model Tetra Dimensional. Gentleness of the Bachelor Maria del Pilar Ávila Barredo



Hypothesis Tetra-Dimensional of the Geometry of the Space-time (1 tetrahedron and 4 spheres) of a neutron, 1 up quark (blue) and 2 down quarks, (3 spheres with the center in three vertex of the tetrahedron), and his associate lepton, 1 sphere (clearer color) with his center in the quarter vertex.



Autor: Rafael Javier Martínez Olmo

Quantum Geometry of the Space-time

Updated on Monday, 13 de Abril de 2015

All rights © Rafael Javier Martínez Olmo