Rafael Javier Martínez Olmo

2011-07-05 14:08:36 UTC

Quantum Geometry of Spacetime

Model of Unification of General Relativity and Quantum Mechanics

Author: Rafael Javier Martínez Olmo

All rights © Rafael Javier Martínez Olmo

http://www.gravityquantum.com

Madrid

Spain

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

Introduction.

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

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 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)?

Hypothesis

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) [2] 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

spacetime.

VTN = G • h • c ^-2 = 4,9205 • 10E-55 cm. ^ + 3 • sc ^-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.

Geometry of the Spacetime

Foreword

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

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 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

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,

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

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, 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

unique:

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

(N).

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

events.

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

units”.

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

calculations.

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 •

10E-29 seconds

Max Planck =1.61624 • 10E-33centimeters This Work>> l = 4,89750 •

10E-19centimeters

Max Planck =2.17645 • 10E-5 grams This Work>> m = 4,51294 • 10E-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

Details

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), 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

(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 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

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 = 253,177660585902

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 = 4,897 506 921 037 260 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

appear.

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

E-19 cm.

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).

History

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

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)

All rights © Rafael Javier Martínez Olmo

++++++++++++++++++++++

LIBRARY OF THIS WORK IN FORMAT PDF

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Wednesday, 12 October 2005

All rights © Rafael Javier Martínez Olmo

E-mail: ***@gravityquantum.com

Model of Unification of General Relativity and Quantum Mechanics

Author: Rafael Javier Martínez Olmo

All rights © Rafael Javier Martínez Olmo

http://www.gravityquantum.com

Madrid

Spain

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

Introduction.

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

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 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)?

Hypothesis

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) [2] 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

spacetime.

VTN = G • h • c ^-2 = 4,9205 • 10E-55 cm. ^ + 3 • sc ^-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.

Geometry of the Spacetime

Foreword

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

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 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

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,

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

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, 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

unique:

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

(N).

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

events.

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

units”.

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

calculations.

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 •

10E-29 seconds

Max Planck =1.61624 • 10E-33centimeters This Work>> l = 4,89750 •

10E-19centimeters

Max Planck =2.17645 • 10E-5 grams This Work>> m = 4,51294 • 10E-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

Details

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), 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

(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 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

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 = 253,177660585902

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 = 4,897 506 921 037 260 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

appear.

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

E-19 cm.

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).

History

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

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)

All rights © Rafael Javier Martínez Olmo

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Wednesday, 12 October 2005

All rights © Rafael Javier Martínez Olmo

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