IV. The development of quantum geometry

In the grounds of the theory have made these statements: no tiene partes

1) The existence of a natural principle infinitesimal VT (OLmo) (not reducible, or divisible, it has no parts).

2) That the infinite or uncertainties that are obtained when working with RG and MC, have their origin in the use, in both, total R to model physical reality.

3) physical absence (no really) of mathematical concepts, zero and infinity.

These three issues are closely interrelated, and as we shall explain it lies in understanding and set some ambiguous concepts unambiguously. To do this we go back as a basis in logic, number theory and calculus and as such the sophistry of Zeno and Parmenides.

a. The Real Numbers (R)

We know from the theory of numbers that division by zero produces an absolute uncertainty in any calculation.

We also know from the number theory properties R act as an infinite set or second order (Cantor). All R has three properties associated with its infinity, which are impossible at physical events, unparalleled in nature.

           1. His points are not ordinal (good order). Despite being a completely ordered.

           2. Infinite divisibility of all sizes. R

           3. The part is equal to everything. (You can put one correspondence with all the elements R)

Being R the origin of the infinite, like the mythological taking hold properties of the " Continuum " absent in nature, its use implies that we might think that the uncertainties that appear in the calculations (physical singularities), are the result of a property of Nature (Continuity) when they are only logical and inevitable consequence of using the set (R)

The above reasons seem to suggest the immediate abandonment of all R to describe (quantify) the quantum reality. However, such a claim is viable without incontrovertible evidence, or more importantly, provide the solution.

This solution exists naturally, and then reflect on their logic.

b. A Walk from Runtil N. The Art of Measure

What rationale can then be generalized to any measurement process, but we will limit the study of a magnitude useful for our purpose, the distance or length between two physical objects.

           1. To measure is a necessary condition the existence of a unit of measure.

           2. The unit of measure on our scale, need not be basic (no sides), and we can use multiples or fractions with the same generality as the unit chosen to compare (measure).

           3. This indicates that any choice of the size (length) of the unit of measurement is completely arbitrary in the sense that it established the ratio of two elections unit length (cm and inch) and faced with a measure of reality, distance from A to B, both measurements represent the same physical reality.

           4. But there is a logical limit, necessary and sufficient condition in physics to measure the length of any unit must always be greater than zero. In addition, this unit of measure must be unique in the sense that before any measurement must be chosen from the infinite set of possible units of measurement, one and only one, the result is the number of times that this unit is chosen contained in the length measured.

           5. Nature imposes on the calculation or measurement of any physical distance, the previous assumption of the existence in nature of a minimal length unit - tiny ε > 0. Failure to meet this condition is impossible to measure (comparison).

c. Continuity

First note that there are two concepts of continuity, a colloquial, physical (D = 4) and rather ambiguous. You can specify an example, a physical line, drawn between the ends A, B points in a single stroke with a pencil on paper. This line is continuous, meaning that, when plotted We have gone through all the intermediate points of AB. The second is the mathematical concept of continuity, unambiguous, based on the geometric concept of zero-dimensional point (D = 0). Exemplifies dimensional abstraction of the previous example. The motion of a point from A to B.

In the first case (slang) all items listed in the description are infinitesimal VT (D = 4), and the precise concept of continuity is expressed also in the form D = April. If you make a careful observation, we see that the line drawn on paper with the pen is in physical terms a volume -time, because this line has wide (graphite mine), along AB, height (the layer of graphite deposited on the paper) and exists in time.

      d. The Sophie, Parmenides and Zeno of Elea

Retain some eighty examples of sophistry due to Parmenides and Zeno, all with the same central idea, continuity, whether of space or time, the infinite divisibility that entails, and the resulting paradoxes, which violate the laws of logically.

Taking the common factor of all may be summarized and updated with the following statement :

Premise 1S: The " Spacetime " is continuous.

2S Premise : If continuous is infinitely divisible in intermediate steps of length zero.

Conclusion 1 : There is no movement, because to move from A to B, there is always an infinite number of intermediate steps, not zero.

Each step requires a non-zero time (since the speed c, is finite), going from A to B, it becomes a task of infinite duration, impossible.

The paradox is that two true premises (in R) Indicates an absolutely wrong conclusion (in everyday reality), which involves the breaking of the laws of reason and logic.

Personally I believe more in the logic, that the possible truth or reality, contained in the two assumptions about continuity. On whether there is movement, not doubt because it is verifiable by experimental contrast. (Everything is in motion with respect to all.)

Moreover, if we change the two premises for their denials :

1N Premise : The " Spacetime " is not continuous.

2N Premise : If there is continuous, is not infinitely divisible in intermediate steps of length zero.

Conclusion 2: Yes there is a movement, because to move from a position A to another B, there exists a finite number of intermediate steps, as every step requires a non-zero time, pass from any point A to B is any other task duration finite and therefore doable.

The conclusion is true, but it means the negation of the continuity of the " Spacetime ".

If we consider the longitudinal movement from A to B as a solid line, is that such continuity, necessarily, pass through all points in between.

If, however, is not continuous, it means that we move from A to B going only for a finite or infinite number of them. In any case, there and everywhere in between. All of which, it is an idea contrary to our physical experience of motion.

Also paradoxically, after a little reflection, it implies that any phone that goes from A to B, just passing by the points on the path, not the countless others.

When premises certain false conclusions are drawn, and vice versa is that something is very wrong.

e What 's wrong?

Note both the reasoning on which we affirm the continuity of the " Spacetime ", as when we reason, his non- continuity, are deeply unsatisfactory, in the first case because it denies the obvious (the movement). In the second, because the non-continuity implies that we pass from a position A to another B, without passing through all points in between (physically absurd).

Between 1S and 1N premises, taking into account the length displayed in R. (Continuum), the H1 is the correct hypothesis. The " Spacetime " is continuous.

The Premise 2S refers to any length can be divided into infinite sets, each consisting of infinitely many points, which is true. In (R.)

Premise 2S supports two types of uncertainty or infinite :

Type I. What intermediate steps are of a length greater than zero ε > 0, bringing the number of intermediate steps will :

a. Finite if all steps are of equal or similar length and the distance AB is finite.

b. Infinite first order : if the steps are a decreasing series and convergent at the point B and the distance AB is finite.

Both cases are solvable by the calculus

Type II Although all  Examples of sophistry imply infinite first-order can easily raise endless sophistry which is of second order, since R is. In this case the intermediate steps are of length equal to zero, (puntos.), bringing the number of intermediate steps will be a second order infinite (R" Continuum "), whether the distance AB is finite, as if it is not.

This case is unsolvable for the calculus.

The calculus does not work if it is true that, ε > 0.

f. The implicit assumption

Because except for all the examples of sophistry have exact analytical solution using calculus, we only deal with the latter or type II. = 0ε

Note 2That the premise contains two assumptions not specified or justified:

                        1 The definition of continuity implies that any " thing " so defined, such as the " Spacetime " R must exist at all points. therefore is based on a concept of "continuous time "

                        2 The continuity on the basis of zero- dimensional points (D = 0) only can be modeled (represented) on R.

It is the concept of continuity, the origin of the absurd conclusion that the movement does not exist, but these implicit assumptions, which impose the choice of geometric object point zero -dimensional (D = 0), assuming, based only points can be explained or modeled geometric -mathematical concept of continuity. (About R.)

By stating, "If it is continuous is infinitely divisible in intermediate steps off ", we say that any distance contains infinitely many points. Steps of length equal to zero, the only way that this result can give, is to divide by zero length AB, and we always get, infinite zero-length steps (points D = 0). Regardless of the magnitude of the length.

We know from the theory of numbers that divide by zero means absolute indeterminacy. That is why it is not working and that the calculation can not measure the lack of the necessary and sufficient condition to measure the existence of a unit ε > 0.

Therefore, the statement " if continuous is infinitely divisible in intermediate steps of zero length " is a premise, of course, false.

Following this, we should modify the second premise, we know false, incorporating information obtained from the above reasoning. For which there is only one logical possibility : "If it continued, can not be infinitely divided into intermediate steps of zero length " with a degree of certainty 1.

As a result the concept of continuity (of a length AB) can not be based on physical terms, the geometric concept of zero- dimensional point (D = 0) because it involves so obvious, division by zero.

                        g. The ambiguity of the term "Point"

The concept of point involves a paradoxical nature, ubiquitous and ambiguous one hand, we have a very precise definition of what is in mathematical terms : a geometric object without dimensions (D = 0). That is, an object that has no high or long or wide, and they do not exist in time, therefore, no reality or physical representation.

On the other hand, has a colloquial sense that we use every day to represent a physical place more or less concrete, and thus has physical reality, for example, the end of a beam. Here we can not say that the point is zero -dimensional. Then there is a " volume-time "of points that satisfy, referring to a measurement of length, be the end of a beam. The size of the volume-time depends on the accuracy of measurement, but since there is a latent error in any measurement, the set of points admissible as ends of a beam, always form a " volume- time. " It has four dimensions (D = 4), however we say that the length of a beam is determined by measuring between the ends. And its volumes- time ends.

We can see this ambiguity in the following paragraphs:

(D = 0) " (If it is continuous is infinitely divisible into intermediate steps of length zero. It refers to any length can be divided into infinite ensembles of infinite points, which is true (in R, D = 4) "Considering the distance (length) from A to B as a solid line, is that this continuity implies necessarily pass through all points in between. "

If you have to go through all points in between, they must exist, and physically there are only objects of Dimension = 4 (Volumes in time). Furthermore, it is impossible to express or describe the continuity of movement from A to B with points (zero-dimensional) because the Royals do not have ordinal numbers, ie located at any point, we can not go to the next, because there is no point next or the next of the next, to any R given. This means, paradoxically, that R not go through these intermediate points, therefore, is not continuous, has no physical parallel also, by definition, having no dimensions.

However, based on volume -time infinitesimal (colloquial points D = 4). These are discrete finite, contiguous and continuous (incremental, without parts), have an associated ordinal, which is a Natural number (N) That indicates the number of good order, therefore, there is an infinitesimal first, fifth, previous, next, penultimate, and so on. ε > 0

By establishing a partition of the universe in VT discrete and infinitesimal adjacent, restored the "physical" dimension to the points (changed = 0 for D D = 4), no continuity is lost and regain space-time ordinal so this whole no part R.

The physical meaning have the above expression "Going through all the intermediate points, if we wrap the solid line by a very thin cylinder (radio infinitesimal) and divide it into compartments by perpendicular planes, a short distance each (infinitesimal).

Changing the definition of continuity "Considering the length AB as a solid line, is in the sense, that this continuity involves going through all the intermediate -temporal volumes (D = 4), (VT) in infinitesimal cylinder radius and height "

The mathematical abstraction has been considered a physical infinitesimal volume- time " four dimensional ", which we do not care about their size, only its physical situation, for zero-dimensional object of interest or zero-dimensional. The procedure is no different from the abstractions when we are interested only one dimension of the physical object, we speak of length (D1) and give zero value to the dimensions that do not interest measurement, but all physical reality always has four dimensions. There is no zero in nature.

Note that any point (D = 0) becomes (D = 4), if it can be observed or measured by some physical means.

Euclid know

      i. One point is that no party or you have dimension (D = 0)

      ii. Infinite points do not add (form) a line.

      iii. Endless lines do not add up (are) a plane.

      iv. Infinite planes do not add (form) a volume.

We admit this, say:

           i. A point is zero.

           ii. That a line has infinitely many points.

           iii. in a plane that there are infinite lines.

           iv. That volume contains infinite planes.

If all are true, there is only one way to explain it. If no infinite points form a line, and yet a line contains infinitely many points, you will agree with me that in addition to points (D = 0) must contain anything else so that there is a line. In particular, a value for the dimension infinitesimal length (D1), associated with each point.

From this we can infer, according to Euclid, which can only be built in terms of geometrical objects of the same dimension as its structural component. The same reasoning can be applied to other dimensions.

As we see, there are two conceptions of the point: a mathematician, and zero-dimensional (D = 0) without any physical reality, the other " four dimensional "(D = 4), but use zero- dimensional, physical reality is equipped with (D = 4), has a volume and time-space situation, if only we are interested in their position and size.

h. The ambiguity of the Void

The vacuum also has an ambiguous treatment. In mathematical terms the void, nothingness, absence, is represented by zero. Physically there is also empty, we define the vacuum as the volume -time that are not occupied by any subject. An example can be found in outer space, imagine a cubic meter of " Spacetime " empty of matter and energy.

I affirm that the first absolute gaps in the sense of non-existence, is zero (nothing) can not be occupied because there is a heuristic example would be the backyard of my house, if we consider that my home not have a yard, therefore, can not be occupied, or measured, no nothing.

However, the second is a volume-time, situated in a particular place and given for all other parts of the world (VT). Giving it a unique identity, as their position compared to everything else (VT) is different from that of any other cubic meter of " Spacetime " (VT), has physical reality, can be occupied, measured, located relation to other existing objects, then this gap exists physically, the energy associated with zero -dimensional mathematical vacuum is zero, the energy associated with a physical vacuum must be provided. δ > 0

i. The Ambiguity of "Calculus "

The paradox between the calculus and the continuity of motion in specific terms R, which is none other, than those contained in the previous section of the paradoxes of Zeno and Parmenides, more specifically, the case, type II, ε = 0 (R)

Not being solved, is obscured by ambiguity type, instantaneous speed, instantaneous acceleration, time... that seem to indicate that these values are at a point of " Spacetime " R, where it is obvious that there is no speed or acceleration, if time is zero.

This ambiguity is transferred to the concepts, as infinitesimal and limit, to describe an infinitesimal as " infinitely "small, and the limit as " infinitely "close, and therefore indeterminate R.

Although the conditions to be very small and greater than zero, do not specify a value, this does not mean that this value does not exist. But to say that it is infinitely small RWe are arguing that there is, because as we know R there is a negligible number.

V. The basic hypothesis of the quantum geometry of "Space-Time "

a. Infinitesimal Calculus

The birth of the calculus is parallel to the need to explain the physical movement of material objects and their associated variables, velocity and acceleration.

The idea that the object motion is continuous and therefore so is the space and time containing it R are assumed a priori. Therefore, all applied mathematics framework is R.

If we study the fundamentals of calculus, we see that based on the existence of two concepts, that of continuity R and existence of an infinitesimal ε > 0 to split and use as a measurement unit, and that choice is essential, unique, and prior to the subsequent process of calculation, being the real value of the result, depending on the chosen ε value in the sense, the smaller ε is the result more accurately.

In this scenario R Newton and Leibniz conceived the infinitesimal calculus realized, that they could better explain in mathematical terms the physical reality of Natural curved objects (4D), if assumed that variations in the magnitudes of the "space-time "involved were very small, infinitesimal value whenever ε > 0, this assumption can be made new calculations, and they agree with a physical measurement (experimental) Nature.

The calculus is not only a mathematical tool, is primarily a physical theory, that part of the intuition that the curved volumes of nature can be better explained, assuming that their dimensional components are infinitesimal. fully matched because as physical theory, its predictions may be compared with physical measurements, it is indeed the case. The calculation works fine.

The calculation as a physical theory is clearly a very successful hypotheses about quantum geometry " Spacetime "in two senses :

                  1. Its discreet and quantifier to describe the physical reality of the curved volumes of nature.

                   2. May -dimensional geometric components are negligible, but never zero.

It came therefore a fundamental and insurmountable contradiction in reconciling the calculus with the continuity of movement (in R.) Because there is no way to prove mathematically (in R) calculus based on the continuity point, steps in length. In spite of being studied by the best mathematicians and physicists, the only way to show is sure that the infinitesimal is used to divide and calculate compliance (e > 0) that is, other than a point (0). It is a clear demonstration of continuous movement in the " Spacetime " can not be described by means of Zero-Dimensional points or even in (R.)

Note these infinitesimal bits of path or road, and have ordinal dimension (D1), e ∉ R

this implies that we make a transformation R4 in R4, Then substitute the infinite points (D = 0), which contains the infinitesimal distance ε by a single valueThe length of each of the pieces ε (Dimension = 1) that is divided as finite form (i-times) the distance AB.

From Leibniz to Cauchy, many tried to reconcile the calculation with continuous stage movement RBecause it is a necessary condition that the functions are continuous, to apply the calculation, which paradoxically is a tool for quantifying, ordinal, ordinal have no points, the infinitesimal, yes, clearly finite and discrete, where the concept can not be point, because its lower part is always e > 0.

Given all this, the logical thing would have been putting the proven results that point to a constructed nature with very small pieces (D = 4) (infinitesimal in mathematical terms) and rules out any suggestion that involves precise description R. Saying something like the model of continuity based on points R and used until today, has become obsolete, with the results of the Infinitesimal Calculus, which clearly point to dimensional constituents of nature are very small, but in no case equal to zero (point). One must add, that continuity is not lost, if we assume the universe divided into "volumes " time " incremental, contiguous and infinitesimal.

Happened historically, on the contrary, it prefaced the obvious reality that the movement is continuous, but continuity in interpreting the specific terms RAs only (?) mathematically describing the continuity of the movement, led them to a very different reading of the facts:

Leibniz concludes that the calculus is a mathematical approach that allows us to calculate an infinitesimal error, always greater than zero, the reality of " Spacetime " (movement) which is continuous R.

For Leibniz the only way possible was logical continuity point of " Spacetime " (movement), not override the tool of calculus, which is based precisely on this concept, to be shown in mathematical terms (in R.) Consequently the answer to two questions about continuity and divisibility given by Leibniz and updated to our theme:

                        1. The " Spacetime " (movement), if it continues. R

                        2. Any distance and time (motion) Yes is infinitely divisible. R

This is only a consensus view or hypothesis about the nature impossible to prove but apparently more consistent, because it allows to continue to build mathematics. While its opposite, similarly based on the concept of zero dimensional, and therefore equally absurd and unprovable, does not. It was therefore the lesser evil. "

b. Inventory

Before starting to construct the model of quantum geometry, we must take stock of our knowledge on the " Spacetime ", including those contributed by the previous entries and discard assumptions made by mathematical abstraction and contrary to reality physical nature, which always shows " four dimensional "and finite.

Likewise, it is also necessary to ignore the outcome of which involve mathematical abstractions that a physical space-time variable takes the value zero or infinite value, since both terms without parallel (sense) physical in nature.

If Physics is the art of measurement and through these measurements to establish models that quantitatively describe the nature, zero and infinity, are outside the scope of this art, because they are by definition immeasurable concepts, the first for his absence (D = 0) and the second by its infinity (Infinity = beyond any measure, immeasurable). If you could measure the infinite would be, and if the zero measured something, it would not be zero, all of which is absurd.

Geometric information provided to us R.G. and M.C.

1. Both theories contain the constant geometric π, R.G. M.C. tensor in equation by normalized Planck 's constant or reduced.

2. Since the end of this writing, is the unification of the two theories RG and M.C. and both are contradictory, it is clear that undoes the road this contradiction.

The essential difference is in the causal or probabilistic nature attributed to the quantum level, each of the theories. I consider the R.G. as fundamental to the description of quantum geometry, although need to be modified and specified in some respects, these do not change the essence of geometric and relational model, but the concrete, eliminating ambiguities. Regarding quantum mechanics, fully admit its very broad experimental results, and interpret these results in terms of probability and not causal, not taking into account the variable " quantum geometry, O (π)"That surrounds and includes the event (hidden variables), and that can transform or express the random outcome in geometric terms, in the following sense (heuristic)

Suppose with Feynman, that the behavior of nature at small scales is like watching a chess game, for someone who is not aware of this game. It's about learning to play by observation and systematic tabulation of all the events : movements of parts, which has allowed us to assign probabilities to every possible movements very accurately, but we know at a given time, which of these events occur. The result is uncertain because it was not taken into account the variable geometry "checkerboard "in the context of which happen all the items, which affects in a fundamental way in moving parts.

                 3. A key information that gives us quantum mechanics is the inverse relationship between distance explored and the energy required to achieve it. This amount increases with decreasing distance to explore.

                 4. Other important information is the discrete and probabilistic nature presents infinitesimal distances.

                 5. General Relativity gives us that the geometry is curved, the origin of this curvature is in the matter (mass), and that this geometry varies with the inverse square of the distance to that body.

Information from previous entries.

For this we say that it is continuous only elementary or without parts, and that the universe (D = 4) in which we live, basics, as elemental is not always " four dimensional ", being the least, an infinitesimal tetra also dimensional. Therefore, the concept of continuity must be established in terms of volume -time D = 4 points infinitesimal and not totally devoid of physical reality, where nothing can be, for nothing can be or can contain or stop by a place, not wide or long, or high and also there in time.

Information calculus.

                        1. The calculus is a physical hypothesis about the nature of quantum geometry " Spacetime " wholly and happily contrasted to reality, allowing us to send to hell the whole of physics R. How? Simply applying the calculus, not a specific problem, but caused by the metric structure of R4.

            2. To avoid death at the stake of heretics, who have lost faith in the real numbers or "True " proceed responsibly to its justification.

                        3. Infinitesimal Calculus As a physical theory of quantum geometry, validated by experimental contrast, and therefore, a valid picture of the infinitesimal character of the variables of "space- time " we can apply the " Calculus " to the line of real numbers, So that the elements of R (Points, D = 0), are replaced by infinitesimal dimension 1 (D1), so that elements of the new set, ordinal and have lost the infinite divisibility, since the lower part of the axis is an infinitesimal very small value, negligible (Lmo = 4.897506921037260 E -19) but never zero (point).

Note the profound changes we make in the metric of the reference system R4As an example and using a Cartesian coordinate system, the axes are no longer made of dots (D = 0), unable to take any zero dimensional components are replaced on the axis (line of real numbers) by infinitesimal ε > 0 (D1), which is the smallest part of this transformation RApril to R4 Is an infinitesimal (D = 4) (regular cube edge and infinitesimal unit in the Cartesian case).

In this system can only be represented in geometric terms, volumes and time. Disappearing points, lines and planes as geometric entities represented, with our own reality, as mere mathematical abstractions being a volume-time, without parallel reality or physical in nature

However, we know that those dimensional abstractions in some cases are relevant, if we interpret the size ignored as unnecessary for a measurement or calculation. Note our use of physical concepts of points, lines, planes and volumes, considering these as " Dimensional Four ", giving value 1 (infinitesimal unit), the dimensions do not think, being the lowest unit value in NAnd never zero (0∉N), by removing the dimension.

c. Objection

We whole life working with real numbers and have never had problems, the real numbers are used continuously and function.

A number is any real number just by definition, but must comply with the associated properties by Cantor to R as an infinite set of second order.

As the property that the part is equal to all meaningless for a single item. Analyze the property that any real number is ordinal, ie, that there is a number immediately before or after any given real number, and ownership of the infinite divisibility of any real number.

These properties may be posed equivalently, with a single statement that every real number has infinitely many decimal places, which already satisfies both properties.

Having infinite decimal implies that we can only use (as input into a computer, or to write a real number and can operate with him), a subset of R whose elements satisfy all your decimals are zero after some finite decimal. (Evidently, we are not eternal).

That said, we return to the issue of whether the numbers we use every day are certainly real or not. We pose the following question: you distinguish a difference in our use of numbers, if we change the scenario?

                        a. In a scenario with the length represented by the set R

                        b. In a scenario in which we choose as an infinitesimal unit small enough

                        ε = 10E -100 cm.

                        And with it we divide the real number line into segments, which, numbers (before and now points segments) ordinal recover and lose the infinite divisibility.  Being a minority 10E -100 cm and length is a multiple all natural (N) Of ε.n

There is no difference in everyday use, enabling us to discern which of the two scenarios are. For indeed, we use the numbers "Royal" just as you would natural numbers, which is an infinitesimal unit of length, ie, a negligible number ε > 0, but not zero (point).

It is easy to understand, that when "I really used the real numbers, as such ", ie to satisfy the property of their lack of ordinal items (Cantor's theory of numbers.) It is impossible to calculate anything with them, given the infinite has its numerical expression. Any approach, as only use the first trillion of decimals, is in fact the same as dividing the line of the real numbers with the infinitesimal ε = 10- 10E +12 cm recovering the proper order (ordinal). They are not zero- dimensional points "R"(Real Math), but R (Royal Physical or Natural).

We call this set, the result of applying the " Calculus " to the line of R, and in good order, with the provisional name of numbers Raphaelite R.

VI. Quantum Geometry. Part.

The idea of representing the " Spacetime " in mathematical terms, with real numbers, is untenable, if we make a description of quantum geometry, which he saw as the study of elementary geometry of " space - time. "

The starting point of quantum geometry is the "Calculus "as a physical theory (reality) and not as a mathematical tool (approach to the reality of R), tested on experimental measurements of Nature.

The theory "Calculus "confirms the physical hypothesis that can be described with much greater precision and mathematical generality, the geometric nature of the " Spacetime " curved (π), assuming that the dimensional value of the components is very small, infinitesimal ε>0.

The mode of operation with calculus, without respecting the necessary condition ε>0Because it directly substitute variables tend to zero for zero, independent of the measurement scale, confirms that the dimensions have to be really very small and that in cases where this zero is adding, subtracting or multiplying no obviously No replacement of the infinitesimal uncertainty by zero. When this substitution causes problems is when the zero is splitting, but this indeterminacy disappears as we turn to calculate, now, respecting the necessary condition ε>0.

It is true that we can, due to calculus, to bring as much as we estimate the exact result based on reducing the value of the infinitesimal, as this is only true in the scenario R, Where, in the absence of a tiny number, any number less is closer to R, and this scenario goes, when we replace the mathematical points (D = 0) with contiguous infinitesimal length (ε >0 (D1) in the axes of R4 (Cartesian coordinates) XX, YY, ZZ, TTi. (Cartesian coordinates)

The absence of zero, this means geometrically partition R⁴» R⁴ Being consistent with the above, we understand that there must be an infinitesimal, which is an ordinal and insignificant value, which serves as the unit Assuming that all the infinitesimal intervals of the partition are infinitesimal equivalent R⁴and N are almost indistinguishable.

The only difference is the cardinal value of the unit, but construction of the coordinate axes, with subsequent translation of the basic unit of measurement from 1 to N, is identical, and the difference in the cardinal value is zero, with just a convenient change the unit of measure, saying the cardinal ε is 1, in units ε. In this case Natural (N) and (R) Rafaelianos are indistinguishable.

a. Elementary Geometry of Nature.

The number π, Which as noted, appears in an essential way on both theories RG and M.C. It is the relationship between linear and long lengths, areas and volumes of curved geometric objects, which generates this length to rotate around its center point or its end points, circle, circle and sphere. Since our geometry (D = 4) π points to a spherical geometry and radial dynamics.

We also know that the law of the inverse of the square, relates, two masses gravitationally depending on the distance between them, two electric charges is also associated with this law. Objects that are close to or away from us physically, increase or decrease apparent size before our eyes, following the same law. This property is clearly geometric in the sense that it is based on a dimensional scale of " Spacetime " distance. We see that 1 / r2 also points to a spherical geometry and radial dynamics.

The Isotropy of the " Spacetime" is the physical observation of the homogeneity of the directions of"space- time " within the meaning of non- preferred directions exist in nature. If you move to any space-time dimension has a negligible value, implies that this value must be equal for all directions (dimensions), indicating a spherical geometry and radial dynamics. We see that any expansion in nature, in the sense of volume increase in time is radial, then all directions are equally likely, and consequently evolving and dynamic radial spherical geometry.

Astronomical observation provides many examples of the preference which has the nature of the spheres and the axes of rotation and whether internal or external to the object itself. The physical observation of the structure of an atom, the expansion of the universe, the Big Bang theory, the physical existence of the center of gravity, also points to a spherical geometry and radial dynamics.

The construction of this model reverses the process, usually followed, to infer the structure of the very small, our scale model, ie in the direction of the direction in this model is from, 1→ ∞

The origin of the geometry of the " Spacetime " is in every elementary particle with mass (Mmo), in both senses of the center of the sphere and the energy source used in the geometry generation process, which involves the creation of " Space- time. "

b. The " Spacetime " negligible, the dynamic (4D).

To describe the dynamics of this generation working on the following geometric assumptions.

All VT elemental mass, creates around gravitational radiation emission and on account of its mass, a spherical field, and concentric strength, length tiny radio (Lmo) 4.897506921037260 E-19 cm. in time also negligible Tmo 1.633632464842480 E -29 Sg. This establishes, as moments go by Tmo unit, a foliation volumetric spherical concentric rings of " Spacetime " and section Lmo tiny, around it, which obliges every other elemental VT (with mass) in the neighborhood, to occupy one of the elementary spherical rings, being therefore at a distance VT both natural multiple (N Lmo | n ∈ N.)

The full foliation " Spacetime "around the masses (Mmo) VT provides a partition on infinitesimal, contiguous, basic and unobtrusive. These VT can be in two states, as containing an elemental mass (origin of the curve) or not containing it. Each state has an associated geometry being spherical, VTS that of the first, ring and spherical geometry, VTA the second i = {1, 2, 3,..., n- 1 } n..foliation Each mass and volume, is a separate inertial system of " Spacetime " within the meaning of " Spacetime " of the particle itself, and through which relates to other inertial systems.

Each ring is labeled clearly, with a single number, the ordinal i ring counting from the origin (center mass). r = (1, 2, 3,..., n- 1, n)

quantum geometry is a spherical polar coordinate system.

At the quantum level, four - dimensional geometry has two degrees of freedom as if it were two dimensional, because the radio determines the three spatial dimensions and the dynamic evolution over time, but also the radio, has the same ordinal time (matches the number that describes the ring and the time since it was radiated).

Note that knowing a single value, the curve number or label, which is always a natural number, we have defined all the parameters of the " space-time " of this " volume-time " scale spherical quantum

VII. Quantum Geometry Part Two

a. An outline of Source Geometry

Let's see where it takes us the geometric assumptions of " Spacetime " infinitesimal N= 1, Next to the tiny information hypothesis N= 1 of every particle of " Spacetime " elemental (not parties).

A. All distance must be a multiple Lmo natural. (N)

B. The geometric shape of "volume -time " very small, therefore it must be a sphere, as it is generated naturally, without assuming any new assumption. If the sphericity of the geometry we add a dynamic radial, we see that this geometry is very good about the results of nature.

    i. geometry is a " four-dimensional - invariant. "

    ii. The radial spherical geometry and dynamics contains the Principle of Action and Reaction

    iii. The radial spherical geometry and dynamics describes physically the center of gravity.

    iv. Includes the formulation by replacing Newton 's gravitational mass and distance for a sum of products of elementary VT curvatures.

    v. spherically symmetric geometry and dynamics coupled with radio allows you to narrow the differential equations of Einstein, which together with discrete and elementary intrinsic geometry (spherical) removes the singularities.

    C. The elemental mass radiates spherical waves of " Spacetime " very energetic, which establishes a management (foliation -concentric spherical) of the " Spacetime " in a structure of spherical rings originating from the particle mass that issued them.

    D. As a discrete process, and therefore countable, we label each of the rings with a natural number (N) For the ordinal of the ring counting from the origin, which coincides with the proper time elapsed since their issuance.

E. If we include more particles with mass in the model, keeping strictly to all distances between elemental masses are there is a geometric limit maximum particle number four. If your collation corresponds to the vertices of a tetrahedron, such that each of the ring particles labeling occupies two of the three other particles. Then the " Spacetime " is closed, meaning that there is in the partition volume set of the four particles, any " volume-time "that meets the minimum requirement (LMO) for four. And so, that may be occupied by new particles with mass.

F. All Mmo elementary particle, contains a tiny information, ie, a single instruction, receives the " Spacetime " contiguous information curvature and falls to the lowest tag number, the interaction is radiating energy in proportion to curvature variation between the " volume-time "busy before and after the fall. For its part each VT also verifies the hypothesis of negligible information, has an instruction, receives the information Mmo gravitational radiation and curves in proportion to it (change tag)

    G. The model with three particles (1, 2, 3), irrespective of their initial space-time, reflected in the amount of labels for the "volume - time "(ring) that occupies about the other two geometries. The labels will be 1 for geometry (anillo_1) and another number for each of the other two generating particles, (ring N with respect to 2 and M about 3). natural in M and N (N), whatever M and N. Information on the hypothesis tiny, elementary bodies fall occupying the corners of an equilateral triangle (regular) so as to be separated by the minimum distance at position 1 with respect to geometry and 2 for the other two particles. Following the same argument there are two special volumes- time complete the tetrahedron, on both sides of the plane of the equilateral triangle, are located in the line perpendicular to the plane above and at the center of an equilateral triangle such that verify the conditions of existence to be occupied and close the tetrahedron, these two volumes have a label time zero (agreement) to signify that its geometry is empty.

    H. Any of the three particles tend to fall into this considering that VT 's own tag value, at that point is lower, will be 0 for the geometry (not to be idle). To -temporal volume occupied is 1. Whenever a particle radiation and fall, freeing up your site and form an equilateral triangle on another face of the tetrahedron. The site is now no longer the lowest and will be occupied by another particle. If you follow the idea of the interaction will produce all combinations of three particles to occupy the five possible sites, always forming an equilateral triangle, ie, taking the three points of one side of the two possible tetrahedra. The number of possible states is 8, one for each side of the two tetrahedra

    I. If we compare this picture with a proton, the color charge can be paired with the vertices of equilateral. Just as no two quarks of the same color, (Pauli exclusion principle), it can be two particles in the same vertex. (We explain the Pauli exclusion principle in terms of geometry), and eight gluons that also can be paired to form eight equilateral faces of the two tetrahedra formable.

    J. The constant changes of plane to radiate (2 tetrahedra = 8 levels) is what we value as a spin of the proton.

    K. For geometrical considerations the three particles form an equilateral, by having an identical volume and spherical, may be in one another Mmo ring was found far enough so that the curvature of the ring coincide with the triangle being tangent to the surface interior and exterior of the ring to the three areas. And this mass, far enough, the sidebar with the electron, this will call tetrahedral geometry extended. The basis of the three quarks are bound in the shape of an equilateral and the electron is the apex, which is away until a sufficient distance to meet in geometric terms tangency conditions with respect to areas of the base, described above.

    L. Following the previous point, the electron is bound to a position of " Spacetime " is given in geometric terms the focus perpendicular to the plane of radiation of the three Mmo (face one of the two tetrahedra) and determines the direction and distance to the electron must be located (with respect to the reference of the three masses Mmo). This we interpret as the curl of the electron.

    M. For the foregoing explains the electronic structure of the atom as a spherical expansion with a slight lag (spatial and temporal) of the geometrical structure of the nucleus.

    N. The next step is to describe all the isotopes in terms of geometric configurations established between neutrons and protons in the different layers of the core and should inform their stability and their decay processes with the matching data already obtained by experiment, plus its causality given in geometric terms.

b. Compatibility with Newton's gravity

We can see that this geometry is compatible with Newtonian gravitation in the sense of geometric principles can be explained by the gravitational force term.

As we know the gravitational force between two masses, is quantified by the formula:

If we assume two masses M and N, consisting of elementary masses myn Mmo, we can express Gravitational force between two masses in terms of product G by the sum of products of curvatures between m M particles the n particles N.

c. Modifications in the " Calculus "

The fundamental change is the scenario that goes from R4  to be R 4 and coordinate systems of nature are spherical polar.

This coordinate system has two degrees of freedom, as is specified by only two variables volume and longer dimensional spherical geometry. The variable volume is fully specified with constant curvature π and a single value, the radio. As the dynamics determined by the time variable. This system is not possible to represent any object with dimensions other than 4.

calculus as a theory to physical or " natural computing "we have to add the information we have obtained the real value of the tiny natural (constant) as LmoO" volume-time " tiny, elementary, Mmo mass. Given its sphericity and radial dynamics, also get Lmo and Tmo.

d. The General Theory of relativity

The theory of relativity is a model of physical intuition about the existence of a dynamic connection between " space - time "and the substance contained. This relationship is geometric in the sense that matter bends the " Spacetime " around him. Finally, it establishes a relationship by ten differential equations, between the distribution of matter in the " Space- time "and the geometry of spacetime, this distribution generates.

These equations have only been possible to solve analytically, where simplifying assumptions have been made based on spherical symmetry. A spherical geometry to describe a homogeneous distribution of matter leads to the solution of Schwarzschild.

General Relativity needs to be redefined in the new physical frame numbers Raphaelite R and introduce the intrinsic geometry (π) Produced by the subject (Mmo) around it, which is spherical, elementary, contiguous, discreet and volume, with only two dimensional variables : volume and time.

This means, in fact, a full quantization " Spacetime "using spherical geometry, the dynamic radial and the " Calculus Natural -time elementary volumes, contiguous, discreet, around the entire elemental mass. (Origin of geometry.)

e Quantum Mechanics

to Quantum mechanics provides us with a valuable experimental fact, the linear correlation between energies and distances explored. If we add the principle of isotropy of the " Spacetime ", we get used to collide particles spherical geometry, where the outside has the physical sense of being accessible to the level of energy used and the inside remains hidden. The inside will be accessible in the rate at which we increase the energy of the collision.

This allows us to establish a relationship of identity between the physical variable energy and variable geometric curvature (1 / r), in the sense that we can express also from the point of view of energy and curvature.

VIII. Cosmological Geometry

All the above have consequences for our ideas of the universe. The energy associated with each ring and that is unitary, its density decreases rapidly with increasing radius, the density decreases as a cubic function of the radius, so that relatively small distance elemental mass Mmo, the geometry of the " Spacetime " energy is very flat, which is why, we see the world at our level as almost Euclidean or plane. Eighteen orders of magnitude above the tiny radio, or an inch away from an elementary mass Mmo, the energy density has decreased by a cubic exponential rate, ie 54 orders of magnitude.

Dark energy gravitational radiation is emitted every particle with mass and is responsible for the universe's expansion. The value of the particle expansion Mmo and second is the " volume-time " infinitesimal

VTmo =O = 4.920 213 551 532 644 910 E -55 cm⁺³· Sg^-1

Dark matter the mass of gravitational radiation (geometric extension in the " Spacetime " of matter), by physical foliation above and completely fill the universe.

On our scale and for all observers, The curvature of the universe is always convex, it is located within the area you are observing. Conversely, the observation of a minimum area is from the outside, its geometry is always concave to the observer.

Rafael Javier Martínez Olmo

IX. Elementary Geometry and Quantum

 

Spacetime lowest of three quarks (proton or neutron), in dark and its associated lepton, in clear, occupying the centers of the spheres of a tetrahedron edge.N·Lmo.

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Revised copy on Thursday, October 29, 2009 at 21:57

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[1 ] inertial reference syste