PREFACEWhy has space three dimensions? What role plays the 3-dimensional nature of space in

the fundamental laws of physics? What does exactly indicate the 3-dimensional nature of space? What are the possibilities for describing the concept of the dimensionality of space in mathematics? How did contemporary concepts about dimensionality of space arise in physics and mathematics? Is it possible to assume that the dimensionality of space is not simply some well-defined number, but a physical quantity, whose value can differ from three under extreme conditions?

This book is dedicated to all these questions. First two of those questions served as the names of the remarkable works a. of Poincare [1] and p. of Ehrenfest [2], from where the contemporary discussion of dimensionality begins.

In order to understand the contemporary state of the problem of dimensionality more precisely and more deeply, it is best to examine this problem in its historical development. The dimensionality of space is one of the most general common properties of space-time, and the evolution of ideas about the space and the time has a centuries-old history.

However we will call the period from the beginning 20th century the „history of contemporary concepts about the dimensionality of space“. The entire previous period we will call „prehistory“. This separation is to a certain extent deliberate, but nevertheless precise. In 20th century the most essential elements of contemporary concepts about dimensionality were elaborated.

In physics these ideas passed their way from a mere number, which characterizes the entire material world, to the physical concept, connected with the properties of physical phenomena and allowing experimental, empirical substantiation. The concepts, have been generalized and formalized.

The book takes into account the contemporary state of the problem of dimensionality and discusses a possible value of this problem for the future development of physics.

The fact is that the state of physics is at present characterized not only by the massive use of a concept of dimensionality, but also by the expectation of a more important role, which dimensionality can play during the construction of the unified theory of fundamental interactions. However, history always gives the possibility to understand the present more deeply and more precisely.

In the contemporary science the concept of the dimensionality of space has important significance in two substantially different areas.

First, in physics the 3-dimensional nature of space, or (taking into account the theory of relativity 3+1) dimensionality of space-time, as the fundamental property determines the most general common physical laws.

In the second area, the dimensionality is the central concept of the topological theory and has been one of the areas of importance of topology and is constructed for arbitrary values of dimensionality.

The importance of ideas about the dimensionality in physics and topology is determined by the following circumstances:

Contemporary concepts about the dimensionality of physical space and the topological concept of dimensionality arose approximately simultaneously (beginning 20th century).

Poincare formulated the physical concepts (chapter 1). But the first, really physical analysis of the fact of the 3-dimensional nature of space was carried out by Paul Ehrenfest, who was stimulated

by the development of mathematics. The contemporary concepts about dimensionality are the most interesting examples of interaction between physics and mathematics (chapter 1,4).

The concept of dimensionality in the mathematical models of space is utilized by physics. It benefits from generalization within the framework of topology (although in mathematics there are geometric models, in which the dimensionality can not be topological, for example in the so-called final geometry).

The analysis of physical ideas about the dimensionality must, it goes without saying, consider the available mathematical descriptions - But topological dimensional theory did not contribute to a more fundamental understanding of the physical problem of the dimensionality. From the point of view of the deepest today physical theory of the space-time of - the general theory of relativity - physical space has a of the so-called four-dimensional pseudo-Riemann structure. In the small environment of each point, it coincides with the structure of usual Euclidean space. Therefore the topological dimensional theory is rather trivial. Usually it proves to be a completely sufficient idea about the space as a set, where each element, is described by numbers (coordinates), and to consider the dimensionality as the minimum number of parameters, necessary for “the numeration” of the points of space. Such ideas correspond to the mathematical level of past century (19th century that is).

Furthermore, topological approach to the dimensionality of physical space-time is incompatible with the interesting hypotheses of discrete space and fundamental length, which characterizes the limit of the applicability of usual ideas about the continuous space-time.

With these ideas, as is known, were connected large hopes for the solution of the fundamental problems of elementary-particles-physics. However, within the framework of topological model of physical space the discrete model (on the microscale) cannot be reconciled with a continuous three-dimensional model (on the macroscales). One-dimensional strings and two-dimensional bags in the theory of strong interactions are another different manifold of dimensionalities.

In quantum field theory the dimensionality is not equal to three (even nonintegral values of dimensionality are considered). In the cosmology of the early universe completely different considerations lead also to the examination of spaces with a dimensionality, different from three. All these circumstances nourish our interest in the fundamental concept of the dimensionality of space in contemporary physics.

The author is deeply grateful to Kuznetsov and the university of Frankfurt for help with the work on this book.

INTRODUCTION

Briefly about the „prehistory“

The property, which is called the dimensionality in the modern language, or the number of dimensions, was invented very early. The concept of geometric space began to be formed by Plato in 4th century B.C. The establishment of geometric objects of different dimensionality (point, line, surface, body) and of connections between these objects goes back to Plato and Pythagoreans.

Such interrelations of the geometric objects of the different dimensionality were the basis for one of the first attempts to prove the 3-dimensional nature of space. Actually, the motion of a point forms line (i.e. one-dimensional figure). The motion of a line forms the surface of the totality “of the tracks” of all points of the moving line (i.e. two-dimensional figure). The motion of a surface in the same sense forms a body (i.e. three-dimensional figure), but the motion of a body can lead only to the formation of another body. Thus, it would seem, 3-dimensional nature proves to be chosen already for the purely mathematical reasons.

However: Can this observation of mathematical nature be the substantiation of the 3-dimensional nature of space or be the answer to the question: “Why is space three-dimensional?” Certainly, no. It is first of all easy to note an inaccuracy in the given reasoning. Not any motion of a line forms a surface, motion of a (straight) line along itself gives the same line.

If we limit ourself to motions in a certain plane, then let us see, that any motion of a point as before forms a line, the motion of line can form flat (two-dimensional) figure, but the motion of the plane figure cannot already form nothing different from the plane figure (let us recall that we allow only motions in the fixed plane).

In this way inhabitants, which dwell on a certain plane could “prove” the two-dimensional character of their environment, being incapable to form such phrases as, for example, perpendicular to this plane, the intersection of two planes, etc.

We see that “the proof” of this type only indicates the property of 3-dimensional nature and the way it is separated from the set of other properties of the material world. It is worthwhile to, however, note that this is fruitful, establishing connection with other concepts: with line, with surface (i.e. unidimensionality) and with the most important concept of motion.

To man of 20th century, who becomes acquainted with many abstract notions even in the childhood, large efforts would be required to estimate the salient achievements of the ancient Greece thinkers, and in particular Plato, who for the first time clearly realized the power of theoretical thinking and special features of such strange “objects” as “concept” or “idea”.

Having only recreated for ourself the sensation of that level of culture, which was starting with Plato and his associates, we can appreciate the jump in the development of civilization, which occurred in ancient Greece. By the results of that epoch, the concept of point, line, surface in our days seem trivial for every schoolboy (however, only they seem).

Aristotle, who was the most outstanding student of Plato elaborated the fact of 3-dimensional nature. Criticizing correctly the confidence of Plato in the complete independence, and the eternal existence of ideas (in particular of geometric ideas) and emphasizing the second-rateness of ideas with respect to the sensually received objects, Aristotle, as this frequently was the case in the history of developing knowledge, went to the other extreme, denying independence of ideas almost completely.

As far as 3-dimensional nature is concerned, Aristotle considered that the number we take from nature as one of its laws, and “assuming nature itself as our leader”, there is no sense whatever to think, why the material world has precisely three dimensions (dimensions).

But the contribution of Aristotle to the history of the concept of dimensionality was not reduced to this, in a sense tautological position. First, it explicitly connected dimensionality with the continuity, understanding continuity as the infinite divisible: [4]. After mentioning that already the Pythagoreans attributed special importance to the number 3, Aristotle gives even the certain philological proof of this assertion: “… about two things or people we speak of “both”, but not of “all”. This term is for the first time utilized, when the discussion deals with three…”.

Furthermore, Aristotle first identified lines, surfaces and body with the numbers 1, 2 and 3 in spite of the Pythagoreans, who assigned the number 1 to the point (which is, from their point of view, the simplest geometric object), and 2 to lines, etc.

Simplifying the situation, it is possible to say that the Pythagoreans simply wrote down “the inventory” of qualitatively different geometric objects and numbered them (in the correct order). However, Aristotle advanced to a more complete definition: “… body… is determined by extent in three directions.

Other objects are divided in one or two directions depending on the number of directions, by which their divisibility and continuity is characterized. One is continuous in one direction, another in two and third in all”.

Contemporary mathematics follows Aristotle, when they call the line one-dimensional, the plane two-dimensional, and the point zero-dimensional. However Aristotle did not speak about the point. He considered only three-dimensional objects valueable, while for mathematics all values are equally valuable.

It is here necessary to emphasize that to speak about the ideas of Pythagoras, Plato, Aristotle and even Euclid, using words like “the number of dimensions” and “dimensionality”, is not entirely correct. Geometric idea of the measurement of length, the metric point of view is of much later origin. Aristotle speaks about the values, about the direction, about the continuity and the divisibility, but not about measurement. Since the topology (in more detail see below), is the field of mathematics, dedicated to study the different sides of the concept of continuity, it is possible to say that the Aristotelian definition, based on the concept of continuity, actually has topological nature. However, the history of science preferred not to consider metric ideas before 20th century. In the beginning of 20th century the first sprouts of topology were grown and the present topological definition of dimensionality appeared. Specifically topology as the connection of the concepts of the dimensionality and of continuity obtained the most general expression.

Let us move now from Aristotle two thousand years forward, into the middle of 18th century., to one of greatest philosophers of all times, to Immanuel Kant. This does not mean that between them there was no one, whose name would be worthwhile to mention in connection with the history of ideas about dimensionality.

Euclid, who summed up the geometric achievements of antiquity and the framework of Euclidean geometry overwhelmed the achievements of the future centuries, up to 19th century they were Galileo, Newton, Leibnitz, whose work marked the beginning physics of new time. There were other remarkable physics, mathematics, philosophers.

But nevertheless precisely to Kant belongs a really new idea. In the work of Kant the concept of dimensionality was for the first time connected with the concrete physical law and proved to be participating to one of the most great ideological opposition in the history of physics - the rivalry of the concepts of absoluteness and relativity of space.

Briefly speaking, the first of them assumes that the space exists somehow absolute, as a finished scene, on which physical phenomena happen, but which does not depend on these

phenomena.

The idea of the relativity of space means that the three-dimensional relations of space are only relations between physical bodies; that if the space can be likened to scene, then this scene is created in the course of play itself. It is created by physical phenomena, interactions between the bodies. And it cannot be of even thought of, as existing independent of interactions.

The concept of absolute space conquered the mechanic of Newton. It reigned in physics up to the beginning of 20th century, when the general theory of Einstein replaced it with (although furthermore not completely) the idea of the relativity of space, the first convinced supporter of which was Leibnitz.

Kant was under the influence of Leibnitz's views, when he turned himself to the problem of the dimensionality of space. Accurately one should say that Kant was then not yet the great German philosopher, but only a student at the university in Koenigsberg. His first published work was called “thoughts about the true estimation of kinetic energies and the selection of the proofs, which used Mr. Leibnitz and other experts of mechanics in this debatable question, and also some preliminary considerations, which are concerned the force of bodies generally”. It was dedicated in essence to a question, what value is the true measure for the motion: mv or mv²/2 (i.e. pulse or kinetic energy), and is antiquated already at the moment of its publishing, since the question, which caused heated arguments, was resolved several years earlier.

This book, written in clear and energetic language, would preserve the value only as an interesting biographical document, which vividly demonstrates young fervor and courage of early Kant's spirit, if not some of its 180 pages were dedicated to the fact of the 3-dimensional natures of space. These three pages relate to “preliminary considerations, which are concerning the force of the bodies generally”, and the following words are the main topic on these pages: “3-dimensional nature occurs, apparently, because substances in the existing world act on each other in such a way that the effective force is inversely proportional to the square of distance” . But the following reasonings led Kant to the crucial assumption. (It begins with the idea about relative nature of space according to Leibnitz): “It is easy to prove that there would be no space and any elongation, if substances possessed no force to act outside. Since without this force it would lack any connection, without the connection of bodies there is no order and finally without the order there is no space… However it is with more difficulty to understand, how from the law, according to which this force of substances acts outside, follows the plurality of the dimensions of space”.

Then under the title “the base of the 3-dimensional nature of space not yet known" Kant tells about his one unsuccessful attempt to find this base: “Since in the proof, based on Leibnitz in one place of his “theodicy” on a quantity of lines, which can be carried out from one point perpendicular to each other, ... I decided to derive the 3-dimensional nature of elongation from the fact what we observe about the „powers of the numbers“. However, the first three powers of the numbers are completely simple, ... but the fourth, being by the square of square, there is nothing else but the repetition of the second power”. Then Kant acknowledges that, “however useful this property seemed”, it was impossible with its aid to explain the 3-dimensional nature of space.

And only then follow the reasonings, which lead to the assumption about the connection of the 3-dimensional nature of space and the fact that “the substances in the existing world act on each other” with the force, inversely proportional to the square of distance.

The following reasonings are based on the connection of the dimensionality of space on the law of force: Space is ordering, ordering the totality of bodies. Space is the relation of bodies. However, same of these relations are manifested in the forces, which act between the bodies. But forces (gravitation being the only fundamental force known at that time) change inversely

proportional to the square of the distance between the bodies. Therefore “2” in the law of force and “3” the dimensionality of the space must be connected somehow (although Kant does not speak about the concrete form of this connection).

In this reasoning of Kant the idea of the relative nature of space proved to be fruitful (space as the relation of bodies originating from Leibnitz). A similar reasoning was impossible for “rigid” supporters of Newtonian absolute space (not depending on interaction of bodies).

Kant speaks in conclusion: “These thoughts can serve as a sketch for a certain study that I intend to perform. I cannot, however, deny, that the equality of dimensionality to the maximum number of the mutually perpendicular lines, drawn from one point, can not explain 3-dimensional nature to a larger degree, than the earlier “proof”, based on the interrelation of geometric figures of different dimensionality. This is simply one additional form of the statement of the 3-dimensional nature. These ideas came to my mind, without the required authenticity of a more detailed study. I will come back to them again, as soon as I can remedy their weakness ".

So it occurred. In the later, so-called critical period, which includes the basic philosophical works of Kant, he arrived at the idea about the fact that the space is a priori, i.e. it does not depend on experience, it precedes any experience and, it goes without saying, it cannot depend on the concrete law of forces.

So he came to a “dogmatic sleep”. The examination of validity of this ideological evolution of Kant would require too much place and would take us away from the theme of this book. Therefore it is necessary to limit us to the observation, that the brilliant thought does not become less brilliant, if it will subsequently rejected (by ist author).

Let us return now to the very hypothesis of Kant, which actually posed the problem of explaining the 3-dimensional nature of the space on the basis of the concrete law of forces between bodies.

(Living in last quarter of 20th century and knowing that among fundamental interactions there are such, which are not subordinated to the simple law of inverse square (i.e. strong and weak interactions), it is easy to doubt that this task has a solution. In 1747., which marked Kant's book, only the gravitational law of interaction was known, and there was no basis for such doubts. Kant's hypothesis was backed however, in the year 1785, when Coulomb established that the electric charges interact according to the same law. And up to the 30s of our century, when the short-range fundamental forces were discovered, it remained that way. One should, however emphasize: If the construction of the unified theory of all fundamental interactions will be successfully completed, then it is possible that in 21st century Kant’s problem will be considered as the correctly formulated problem.)

Let us assume that this task would be solved, would this solution of the problem be recognized as a satisfactory answer to the question: “Why is space three-dimensional?” Hardly many people would agree. Immediately the new question arises: “Why vary forces inversely proportional to the square of distance?” Therefore let us try generally to understand how science answers questions starting with “why”.

Questions starting with “why” in the science

Due to fundamentality and unusualness of the topic of our book, let us examine any seemingly entirely simple question, for example: “Why is there winter and summer?”

This question answers contemporary science approximately thus. „The Earth has ball-shaped form, it revolves around its axis, inclined toward the plane of its motion around the sun.

At the different points of orbit solar rays fall on this place for the earth's surface at different angles, and the thermal effect of light is the greater, the less the angle, i.e. the nearer the beam's direction to the perpendicular. Therefore that section of the orbit is most cold (winter), where the solar rays fall on the earth's surface at the greatest angle (relative to the perpendiculat), and most warm (summer) in that section of the orbit, where the angle of incidence is smallest.“

This answer to be comprehensive, it is necessary to include in it: the laws of mechanics and the law of universal gravitation, controlling motion of Earth around the sun and the motion of the spin axis of the Earth; the laws of electromagnetic field, which control the propagation of light and its interaction with the substance. For these laws it would be required to introduce many concepts of mechanics, theory of gravity and electromagnetism, including the extensive mathematical apparatus (derivatives, integrals, vector analysis, tensor analysis, etc, etc).

It would be necessary to know the so-called initial conditions (angle of the slope of axis to the plane of motion, the luminosity of the sun). Some of these initial conditions could depend on “fundamental laws”. For example, the theory of the evolution of stars could explain today's luminosity of the sun, cosmogony the angle of the slope of axis, etc.

Thus, to answer the question: “why there is winter and summer?” the entire contemporary physical science and even something from the future are required. But we obtained an answer not only to the initial question, but at the same time to many others; and not only “why from winter to the winter passes approximately identical time”, “why on some planets there can not be the change of winter and summer (if their axes of rotation are not inclined)". If we would want to understand the change of the seasons on a planet, which revolves around a neutron star at small distance, it would be necessary to refine many of the already used concepts: the mass of planet would prove to be depending on the speed, its orbit would prove to be not elliptical, and even the period of the time from the winter to the winter would prove to be decreasing in the course of time.

The most complete answer, which science can give to a question about some physical phenomenon, consists of embedding this phenomenon in the physical picture of the world, which describes the studied phenomenon (together with a set of other phenomena) on the basis of the most fundamental elements of the physical picture of the universe.

Matters concerning any other question “why?” would always go along the same line.

Let us note now that for entire answer to the question: “Why there is winter and summer?” the fact was implied (tacitly) that space has a property of dimensionality, be it the theory of Newton or that of Einstein, which replaced it. However during this replacement the concept of dimensionality was somehow changed, but as before, it was equal to three.

But this only indicates the special fundamentality of the concept of dimensionality and of its 3-dimensional nature, i.e. the dimensionality of space is located in the foundation of the building of physical science.

This building, as is known, constantly undergoes changes: “completions”, “repairs”, “re-plannings” and even radical “reconstructions”. Foundation, of course, is rarely affected by any reconstruction of building. But those cases, when changes in the foundation prove to be necessary, especially large action is involved, both on science itself and on humanity as a whole.

And, it goes without saying, there are no foundations for the concept of dimensionality of real physical space, not taking into account physical fact.

In connection with all this the question: “Why space is three-dimensional?” poses the following problems:

1. It is necessary to build a mathematical theory of dimensionality, suitable for the use in the general physical theory. This task although ambiguous, is indispensable, because only with the aid of the mathematical language physics can build its theories and describe nature by sufficiently specific means.

2. In order to consider the 3-dimensional nature of space as actually physical, it is necessary to develop the methods of its testing and the methods “of the measurement” of dimensionality (testability).

This is not a simple task either. In the course of investigations we come across questions like: “With what accuracy the dimensionality of space is equal to three?”. This (modified) question has completely definite meaning, for example, with respect to the equality of gravitational and inert masses. This fact (equality of of gravitational and inert of masses), which lies at the basis of general theory of relativity (in the form the principle of equivalence), is tested at present with an accuracy to 10-12.

When it comes to testing, dimensionality d (at least if we proceed from the usual ideas about the dimensionality), satisfies an inequality of the type |d – 3| < ε, where ε is a certain small number. The problem of the empirical validity of this assertion against the pure 3-dimensional nature of space, in spite of its unusualness, is well-defined.

3. And finally: a complete answer to the question: “why is space three-dimensional” must be in one oft two categories: Either the concept of dimensionality must be included in a physical theory, where the fact of 3-dimensional nature (at least with respect to the known phenomena, already studied by physics) together with other fundamental physical facts will prove to be the consequence of even more general considerations or it must be proven that the 3-dimensional nature of space belongs to set of certain initial conditions and cannot be reduced to deeper roots; In this case the nature of these initial conditions must be researched.

In the first two directions (mathematical theory and testability) science moved sufficiently far. The third direction (which, naturally, must rest on the results the first two) is yet to be explored.

Into these three directions our study will go. However, before examining the formation of contemporary concepts about dimensionality, two expressions need to be explained, which will be frequently used. It is necessary, in the first place, to refine, how the words “usual” or “classical” are to be understood within the model of space, and second, to give a certain explanation of «topology“.

Classical model of the space

This most well-known physical model of space is taught in the secondary school, and this model exhausts all needs of macroscopic physics. Macroscopic are called those physical phenomena on “human” scales (according to the sizes, the energy, the temperature and in terms of the values of other characteristic parameters). However, in actuality the classical model of space is sufficient for describing the much larger scale of phenomena, which includes atomic spectra, and the motion of planets. And this is perhaps one of the most surprising facts in the history of science.

Actually the geometric description of this model of space was already contained in Euclid's “elements”, which appeared in the 3rd century B.C. In twenty-three centuries, that passed prior to the beginning 20th century., occurred immense changes in the physical ideas and in the whole nature of physics as sciences. The ideas of Aristotelian physics about the motion, which divided into celestial and terrestrial mechanics, were replaced by Newtonian laws, controlling the motions of celestial and terrestrial bodies. The Aristotelian ideas, according to which the

motion of a body ceases, when forces cease to act on it, they were substituted with law of inertia: the body, on which no forces, act preserves the state of uniform and rectilinear motion. The idea about the visual rays, which emanate from the eye and “feeling” objects, was replaced by physical optics. Interaction of the rubbing pieces of amber and magnetic bodies, that was being explained by the actions of the corresponding soul, was later understood as the manifestation of Maxwell’s law.

And in spite of all this, the geometric model of physical space remained virtually unchanged. A very outstanding event (especially for the future development) was creation of analytical geometry in 17th of century, usage of coordinate space. But actually this was only a new form of the same Euclidean model of space.

Here it is worthwhile to emphasize that the word combination itself “the model of space”, would seem absurd for the mathematicians up to the middle of the 19th century (when the of value of the non-Euclidean of geometry was understood, opened by Lobachevsky, Boljai and Riemann), and it would also seem absurd for the physicists up to the beginning 20th century, when in the special theory of relativity the Euclidean geometry of space was replaced by the geometry of the Minkowski-space-time and then in the general theory of relativity by the general Riemann-geometry of space-time.

And is it not amazing. to accept a description as model in spite of the ambiguity of this description? To this moment the scientists were certain, that the Euclidean geometry was identical with physical space.

The physical model of space is not reduced to one geometry. Indeed in the geometry nothing is said how to carry out straight lines etc. Therefore the physical interpretations of geometric concepts must be included into the physical model of space besides underlying geometry. For example, „straight line“ is compared to the tightly drawn rope or to light beam. The mass-point, the dimensions of which can be disregarded in comparison with other significant dimensions does correspond to the theoretical description of reality.

What is now the classical model of space? Let us define the so-called axes of rectangular coordinates, i.e. three mutually perpendicular lines, passing through one point, called the origin of coordinates. Let us establish a specific unit of length, let us put off on the axes the numbers, equal to the distances from the origin of coordinates, to one side from the origin of coordinates with the plus sign, and to the other with the minus sign.

After this, it is possible to attribute to each point in space three numbers. Let us lower from the selected point three perpendiculars to the coordinate axes, each of them will correspond to a point on the coordinate-axis, i.e. a certain number of positive or negative value. As a result each point of space corresponds now to three numbers, which are called the coordinates of this point. And vice versa, any troika of numbers corresponds to one and only one point of space, coordinates of which are exactly these three numbers.

The possibility of all triples (troikas) can be checked experimentally. (This checking for the regions of space on the order of 1 m with an accuracy to 1% (i.e. to 1 cm) is easily performed with usual domestic physical instruments. But the issue of verification in the cases, let us say, of atomic or astronomical phenomena is not so simple.)

Thus, the introduction of the coordinate system converts space into the totality of all possible troikas of numbers. In this case the Euclidean nature of geometry, as it occurs, it is completely determined by the fact that the square of the distance between the point P with the coordinates (x1, x2, x3) and the point P’ with the coordinates (x1’. x2’. x3’) is equal to:

(1) r² = (x1-x1’)² + (x2-x2’)² + (x3-x3’)²

This equality follows from the Pythagorean theorem, and, in turn, the consequence of this

equality delivers the entire Euclidean geometry, each assertion of which can be reformulated, using only the concept of the distance between two points. Formula (1), as they say, assigns the metric (in this case Euclidean metric) to the structure of space.

In mathematics Euclidean metric space En is called the totality of all possible collections of n numbers (x1,...,xn). Each such collection is called a point of space, and the distance between two points is determined by the equality similar to (1):

(2) r ² = (x 1 - x1') ² + …..+ (xn - xn') ²

Thus, from a mathematical point of view the classical model of space coincides with the three-dimensional Euclidean space E3. It is clear that the coordinates, assigned to a certain point of space, do not have absolute sense, but they depend on how the origin of coordinates and reference directions are chosen. It is not compulsory to also use an exactly rectangular coordinate system; For other coordinate systems equality (1), expressing the value of the distance between two points through their coordinates, will change its form. It should preserve the Pythagorean theorem. Otherwise we will obtain a non-Euclidean geometry.

However, in any event it is necessary to assign precisely three coordinates in order to cover all positions of points in the space completely. This circumstance is received as the manifestation of the fact of 3-dimensional nature. The following discussion will deal with the insufficiency of this understanding of dimensionality, and thus far let us recall that the mathematical model of the space E3 proved to be completely sufficient for the enormous region of physical phenomena and the value of this simple model will remain for physics, no matter how ideas about the space changed.

What is topology?

It was earlier said that topology is the field of mathematics, which studies the concept of continuity from different sides. However, this definition is too fuzzy. It remains obscure, how mathematics simulate the general properties of continuity. It is intuitively sufficiently clear, but we cannot rely on psychological arguments.

Epithet “topological” is applicable to such mathematical facts, and such properties of the mathematical objects, which are based only on the contiguity of the points of this object. In order to speak about the topological properties of a certain point set, it is necessary for each point of this set to know, how they adjoin to other points this set. With this knowledge the topology of this set is assigned. Any set with an assigned topological structure is called topological space.

Let us examine, for example, a number scale axis and certain point set on it. Let us use the topological property „connectedness“ as an example. A set is called “connected“, if it cannot be broken into two parts so that any point of each part would not adjoin to any point of the other part. Another example of topological concept is the „boundary“ of a set, which is called the totality of all points, adjoining simultaneously this set and its complement, i.e. the remaining part of topological space.

On a number scale axis each of the sets [0, 1], (0,1], [0, 1), (0,1) has a boundary, which consists of two points: 0 and 1. It is easy to see that each of the sets is connected, because they cannot be broken into two parts, whose boundaries do not have common points.

Other topological concepts are closedness and openness. A set is called closed, if it contain alls points of its boundary, and open if it does not contain any point of its boundary. From four the sets of number-intervals indicated are closed only [0, 1], and open is only (0, 1), the remaining two are neither open nor closed.

In the previous paragraph the relation of contiguity, which is introduced on the numerical representation of space, was assumed to be intuitively clear. But in the general case the

question arises: how can topological structure be introduced? In topology itself this question boils down to the question, „how it is possible to assign distances for every two points, in other words how to reach a metric geometry?“.

Metric geometry may be reduced to the definition a certain infinite table, in one column of which all possible pairs of points are listed, and in one column non-negative numbers representing the distance between the corresponding two points. In exactly the same manner topology can proceed from the table, in one column of which all possible pairs are found, belonging to a certain set of points, but in another column to each pair corresponding words “it adjoins” or “it does not adjoin”.

In reality other methods are used instead of tables. One of the important and historically the first method assigns topology with the aid of the formula. Assume that in a certain set the of metric of structure is assigned, i.e. for any two points the distance between them is known. Then we say that the point P adjoins the set M, if in M there are for each numerical value ε there are points, which have a smaller distance from P than ε. This topological structure, as they say, is induced by a mapping. The topological structure on number scale axis, which was being implied above, is induced by the usual distance, which maps the number scale axis into the one-dimensional Euclidean space E1.

Another method of the task of topological structure consists of the indication of all environments for each point. Point P adjoins the set M, if any environment of point P contains at least one point, which belongs to M. The natural topology on number scale axis can be defined, if we introduce the environment of a given point P as the set of intervals (a, e) with a < P < e. As we see, here it is no longer required to know the distances between the points.

Let us recall now Aristotle, who attempted to give (without knowing about these concepts) the topological definition of dimensionality on the basis of the concept of continuity. It would be difficult to explain what topology understands under the „continuity of space“.

It is amazing, but there is no concept “continuous space” in topology. The intuitive idea about the continuous space proved to be too capacious, that it it would be possible to personify in one mathematical concept. In topology there are several concepts, which reflect different aspects of continuous space. The already mentioned connectedness speaks about sets, which are not decomposed into several individual parts. The topological concept of dimensionality, at which Poincare arrived and which will be discussed in chapter 1. There are other topological concepts, which refine the continuities of space.

However, in the topological language there is the concept of continuous mapping of one set onto another. Mapping is called continuous, if it transfers the adjacent points into adjoining points again. If two figures can be connected with bicontinuous mapping (or conversion), then they are indistinguishable from the point of view of topology. The topological structure is reduced to the relation of contiguity. If we imagine, that the geometric figure is made from some elastic material, then any deformation of it without breaks will prove to be topological conversion.

Let us give two examples of topological theorems.

Assume that a certain closed surface is given. Let us isolate a certain quantity of points, let us connect these points to sections of another set, so that the entire surface would be broken into triangles (curvilinear in the general case). Let us denote the total quantity of points by the letter B, the lines of points by letter P, the number of triangles by Γ . Then for any topological conversion for a certain surface the value Χ=B-P+Γ has one and the same value and is called the Euler characteristic of this surface. Euler's name is here used nonaccidentally. This is one of the first topological theorems and was proven by Euler in 18th century.

The second example is already directly connected with the theme of this book. In 1911 it was proved that there does not exists any topological mapping, which connects two Euclidean spaces of different dimension.

Now it is already possible to answer the question what topology does. Topology studies different types of topological structures and their properties, which are not changed during the topological mappings.

Chapter I

THE TOPOLOGICAL CONCEPT OF DIMENSIONALITY

Beginning of the contemporary history of dimensionality

The history of dimensionality of geometric objects and 3-dimensional nature of physical space includes, as has already been discussed, spans at least 2000 years. However, it makes only sense to examine approximately the last one hundred years of this history, after the beginning of Henri Poincare’s work (1854-1912), and history preceding this century's events can be named “the prehistory” of dimensionality.

Base for this can be seen in the following.

The greatest generalization of physical ideas today about dimensionality gives topology. But the appearance of topology as an independent field of mathematics, happened in the time interval in question and it is, in the second place, connected to a high degree with Poincare's work. He is one of the creators of topology, to him belongs the first, (formally not completely strict) topological definition of dimensionality.

Poincare was not only the greatest mathematician of his time, but also an important physicist. His active role in the creation of the special theory of relativity is well known, which lead to radical changes in the idea about space and time. But independent of this Poincare, “repeatedly turned himself to the explanation of the initial beginnings of geometry and the concept of space” [1].

The manifold of Poincare's interests is amazing. His deep and bright ideas, the abundance of which sometimes could barely be managed, ventured practically into all the fields of mathematics, into celestial mechanics and physics. This physico-mathematical universality of Poincare was right on target with the examination of the problem of dimensionality which is a physico-mathematical problem.

Poincare - one of the greatest representatives of classical mathematics and classical physics lived in the epoch, when in these sciences revolutionary changes occurred, just to mention the appearance of the theory of sets and the theory of relativity.

He contributed much for the victory of both theories. With his aid according to Aleksandrov, the traditions of classical science “were exploded from within”. He cooperated with Cantor (1845—1918), fruitfully using set theory. By the creation of Poincare's topology opened for mathematics “an entire world of new problems… inaccessible to classical mathematics…”.

When the paradoxes of set theory were revealed and began its reconstruction on the axiomatic basis, Poincare rejected this axiomatic approach, because the theory of sets and actual infinity are indispensable. In the case of the theory of relativity Poincare created the mathematical apparatus for the theory and earlier than Einstein he analyzed the concept of simultaneity. He concludes his last article on this theme by the words: “Now some physics want to accept a new agreement (that is the association of the concepts of „space” and „time” into the concept of „space-time”) . This does not mean that they were forced to make this step; they consider this new agreement more convenient, and those, who do not adhere to this kind of thoughts, can completely lawfully preserve their old habits” [4].

Why did Poincare turn himself to the problem of dimensionality? Why do we have to call him the creator of topology? These questions almost coincide, since for Poincare the most basic property of space is the topological “property of continuity of three dimensions” .

But can Poincare really be called the creator of topology? Indeed, for example, well-known

French mathematician Nicolas Bourbaki (under this name an association of the most outstanding French mathematicians wrote several important mathematical books) calls Riemann the founder of topology and convincingly substantiates this. Yes even Poincare's himself spoke about Riemann as the founder of topology.

If we use the comparison of knowledge with a torch (unfortunately, worn out), then the initial development period of a science can be compared with a relay race, in which this torch is transferred from hand to hand by participants in the relay race (for the topological dimensional theory these are Riemann, Betty, Cantor, Poincare).

And finally, from this one torch ignites immediately several torches and there is a moment of transition from “linear”, “one-dimensional” development of science into “multidimensional” development.

This moment is determined by intra-scientific, and external, socially caused reasons. It is possible to consider each of the scientists as the creator of a new region, since the participation of each is vitally important for the fate of this new region. Important is of course, the detection of a new field of research (in the case of topology his was Riemann), and its formation into an independent science. The merit of Poincare was to reveal the fruitfulness of topological approach in different fields of mathematics.

Especially great was the advance of Poincare in the explanation of concept of measured space. Bernhard Riemann (1826—1866) also contributed to the concept of dimensionality, in his famous lecture “about the hypotheses, which lie at the base of geometry” he was confronted with the task to design the concept of repeatedly extensive value”.

We will see soon, how far Poincare got in the definition of the concept of the dimensionality of space. But what led Poincare to the question: “Why space has three dimensions”. An answer can be the following. The second half of 19th century. from the point of view of the history of mathematics was the epoch of non-Euclidean geometry. The non-Euclidean geometry obtained acknowledgment in mathematics. Failure of the a priori uniqueness of Euclidean geometry, the possibility of several (after the work of Riemann even of an infinite number) non-Euclidean-geometries produced huge impression on the mathematicians of that time, and in particular on Poincare. He repeatedly chose the ideas of non-Euclidean geometry for problems of space. Creating his non-Euclidean geometry took some time, and he already could glance “from the height of bird flight”.

Poincare progressed to the analysis of other, even more fundamental properties of space, which had “survived” the transfer from the Euclidean geometry to the non-Euclidean, in particular its

3-dimensional nature.

Poincare's psychologism

In the survey of his own work in 1901 Poincare writes: “I dealt… with the analysis of the psychological foundations of the concept of space”. The psychological element was the essential element of his methodological approach and it was extended beyond geometric concepts. Thus, for instance, connecting his failure with the concept of actual infinity with psychological considerations, he writes: “Russel will undoubtedly say to me that he is not occupied with psychology, but with logic and theory of knowledge; I will answer, that there is no logic and theory of knowledge, independent from psychology”.

The meaningful result of this analysis is his conclusion about the influence of external world on the geometric ideas. In his chapter “space and geometry” in the book “science and hypothesis” (1902) Poincare writes, “There is a small paradox, that imaginary creatures - living in a four-dimensional world - with mind and sensory organs similar to ours, could obtain such impressions from their external world, that to them it would make sense to build a geometry with

four dimensions”.

From a materialist point of view in this example there is nothing paradoxical. This is the starting point of the materialist theory of knowledge. However, apparently, Poincare considered it paradoxical to speak about the non-Euclidean or four-dimensional geometry as real, at least for the imaginary creatures. Poincare based his conclusion about the influence of the external world on the detailed psychological examination of “visual and tactile spaces”. (represented spaces). Too detailed, because his attempt to base the analysis of the influence of external world on molding of scientific ideas seems somewhat naive.

For this purpose the concept of practice is more useful, which rises from the theses of Feuerbach and Marx, which includes not only the practice of the individual researcher, but also social-historical practice. Certainly, the problem of formation of three-dimensional idea by an individual man is important, but nevertheless this problem the essence of psychology.

In “the paradox”, Poincare's examined, the assumption about similarity of mind and sensory organs for imaginary creatures is doubtful. The physical analysis of Ehrenfest showed that even if it is possible to assume the existence of creatures in any four-dimensional world, then their “devices”, most likely, must radically differ from ours.

Furthermore the analysis of “represented space”, is faulty, the way it was carried out by Poincare. He begins with “visual space” and writes that this space “differs from the tactile space”, in particular because “when it is desired to give three dimensions” (two dimensions correspond to the two-dimensional character of retina, and third - the depth - is given by the sensation of the angle between the lines of sight from both eyes), while “tactile space”, in the opinion of Poincare, is two-dimensional (since it feels always surfaces). He raises the question, “why the geometry of blind is the same as ours” and why combination of visual and tactile spaces does not give to us 5 (3+2) of dimensions. It is difficult to call his analysis completely convincing. Poincare does not examine, for example, “auditory space” as very important for the blind man.

All this shows again that the attitude of man toward space is very complex. If psychological analysis could solve the task of molding of ideas about the space, then only for individual persons. Science as a whole is a very complex form of culture, corresponding to high levels of the development of society.

And only the concept of practice is capable to solve the problem of molding scientific ideas about space and the problem of the truth of these ideas. _

Poincare's analysis leads to the conclusion about the guiding role of experience, and the second-rateness of geometric ideas, in particular, about the dimensionality of space with respect to the external world. And although the substantiation of dimensionality, like: “Space is three-dimensional, since the elements of our visual sensations are completely determined, if three dimensions are known - the indication of dimensionality is very fruitful.“

It seems surprising that Poincare considered it possible to combine the assertion about the determining influence of external world with the assertion about the great significance of a priori elements: “.. since we have a ability to build continuity both physical and mathematical; … this ability exists in us before any experience, since without it experience in the true sense of word would be impossible…” [10].

For the history of dimensionality Poincare's psychologism is very important. They usually rate the article of Poincare 1912. “why space has three dimensions?” to be the first appearance of the inductive topological definition of dimensionality (on the basis of this work Brouwer in 1913 gave the precise inductive definition of dimensionality).

In reality this idea appeared ten years earlier. And it appeared in “psychological” direction of

the reflections of Poincare, when he attempted to design “physical continuity of many dimensions” only on the basis of “direct data of our sensations”. However, the psychological origin of this idea prevented Poincare's from immediately converting it into the precise mathematical formalism. It would be otherwise incomprehensible, why Poincare for ten years did not tackle the precise mathematical formulation of this idea. This question would not arise, if this idea appeared for the first time in the work of 1912, the year of Poincare's death.

Poincare would not be physicist, if, after being interested in the problem of dimensionality, he limited himself to the psychological approach and would not ask questions about the physics of the dimensionality of space.

Dimensionality of space and topology

In Poincare’s work we find two substantially different approaches to the concept of dimensionality - parametric and topological. In the first of his books on general questions of science “science and hypothesis”, that was published in 1902, he presented both approaches.

Parametric dimensionality: Poincare writes that the complete visual space “has exactly three dimensions; i.e. the elements of our visual sensations… will be completely determined, when three of them are known; being expressed by mathematical language, they will be the functions of three independent variables”.

A similar approach is based on a parametric “definition” of dimensionality as the minimal number of parameters, which are necessary to distinguish the points of space from each other mathematically. This became clear after Georg Kantor;(1845—1918) created a famous example of correspondence between sets of points and a sqare.

This example is as simple, as important for the problem of the mathematical description of dimensionality. The method, by which Cantor arrived at this result, was approximately the following: The creation by Cantor utlizing set theory began with the concept of the cardinal number as infinite set (generalizing the concept „quantity of elements“ of final sets). It was necessary to find infinite sets, between which correspondence could be established, when each element of one set is corresponding to exactly one element of another set.

The geometric figures of different dimensionality seemed to be suitable candidates for the role of the infinite sets of different power. In the plane figure, for example a square of points, it would seem the infinite number is greater than that in a section of a line. Cantor for three years attempted to prove this intuitively obvious fact. This difference in two infinity, if it actually existed, could become a basis for determining the dimensionality. However, as a result of his attempts Cantor surprisingly revealed that these two sets are equally powerful: he succeeded in establishing a one-to-one correspondence between the elements of these sets. Reporting in 1877 his discovery to Dedikind, Cantor in the letter wrote “I see this, but I do not believe it”.

Correspondence itself is arranged simply. Without approaching special strictness, it is possible to describe it thus. From the point of view of mathematics single section can be represented as the set of all real numbers between 0 and 1 while the sqare may be represented as all pairs of real numbers between 0 and 1. Using the decimal representation the pairs of real numbers can be combined into a singe real number, thus every point of a square maps to a certain point of a section, moreover different points of section correspond to different points of square (just reverse the process). Thus, it turned out that the sets of different dimensionality have identical power. This meant that with the aid of the concept of power alone it cannot be defined what dimensionality is.

However, parametric definition makes sense, together with the concept of the number of degrees of freedom, which is known in physics. Poincare in 1902 could not give this intuitively clear definition. He uses it only for “complete visual space” and does not attempt to give the correct mathematical definition, based on this idea.

Parametric representations of dimensionality could satisfy physics, which used sufficiently simple geometric figures, but not the general geometry, which studies all possible figures. At the end of 19th century the insufficiency of simple parametric representations was revealed. It seemed clear, for example, that any line must be one-dimensional. But does that apply to “any line”? For a long time mathematicians were satisfied with this definition: a line of points is any figure, which can be obtained by the continuous transformation of the unit line segment. This definition refines only the ancient idea: “The line is the track of a moving point”.

It turned out that mathematical intuition can make mistakes like any another. It was necessary

to search for the replacement of parametric representations of dimensionality.

Inductive topological definition of dimensionality: The new approach (being radically differed from the parametric) of Poincare to the dimensionality of space arose with the analysis of the concept of continuity, in particular “physical continuity”. Specifically, with the analysis “physical continuity of many dimensions” appeared the idea, which became the first topological definition of dimensionality. The definition of “mathematical continuity of many dimensions” leaves apparently some uneasinesses in Poincare, because it in 1902 he considers the parametric definition of this concept to be sufficient: “The point of a similar continuity seemed to be defined with the aid of a system of the separate values, called its coordinates” [11].

Even in 1912 Poincare gives the definition: “The continuity of n dimensions is described by a totality of n coordinates, i.e., the totality of n quantities, which change independently from each other” [12]. In this work the tracks of the parametric definition of dimensionality remained, but it is emphasized that “the question of the number of dimensions is tightly connected with the concept of continuity”. And really as we will see, Poincare considers multidimensionality as the generalization of the simplest case of continuity, which corresponds to a real axis. The definition given above, according to Poincare, is flawless from the point of view of mathematics, but not for the philosopher ", little is left of the intuitive origin of the concept of continuity. And therefore Poincare searches for another definition of dimensionality.

In 1902 Poincare, contemplating the concept of physical continuity as primary, he connects it with the so-called law of Fechner. This law can be illustrated as follows. Let a certain “experimenter” estimate the relative length of different sections. Let us assume that the minimum difference, which this man can note (when, let us say, the discussion deals with the sections with a length of from one to two meters), is equal to 1,5 cm. Now to this man there are given three sections: A=100 cm, B=101 cm and C=102 cm. Then our experimenter, examining sections in pairs, can say that A=B , C=D but A < C. Poincare writes : “The naked results of experience can be, therefore, expressed by the following relationships: A=B, B=C but A < C, which can be considered as the formulas of physical continuity” [13]. He deals with the fact that physically only those values can be distinguished, which are differed not less than to a certain finite quantity (threshold of discernability). It is clear that the discussion deals with the physico psychological or even (if we are not afraid of long words) with the physico-psycho-physiological continuity.

Poincare analyzes the concept “physical continuity of many dimensions”, and his idea of the future topological definition of dimensionality appears here. It characterizes one-dimensional physical continuity by the fact that it is possible “to subdivide… by moving away from it a finite number of distinguishable elements”. If the subdivision of continuity C is reached by the cuts, which is the limit of one (physical) measurement, then we will say that it has two dimensions”, etc [14].

In 1902 Poincare does not yet substitute word “distinguishable elements“ by “points“ and does not transfer this definition to the case of the mathematical continuity of dimensions. Although he writes that the mathematical continuity differs from the concept of physical continuity, he stated that “the point of a similar continuity is presented to us by the aid of the system of n separate values, called its coordinates”.

The 3-dimensional nature of Poincare's space he asserts as follows: “When we say that the space has three dimensions, we want to say simply that the totality of these elements (sensations as prototypes of points) have for us the characteristic features of the physical continuity of three dimensions”.

In mathematics those properties are called recurrent or inductive (and also proofs, relationships, etc), where the property, characterized by the number n, is defined through the property,

characterized by the number n - 1. In this case the initial point of the inductive chain, of course, must be assigned.

Poincare’s initial point was the unidimensionality of the simplest form of continuity. However, as it was explained more lately, it is more convenient as the initial point of induction to take 0, and so that the definition would preserve its form, to the empty set (i.e. “the set”, which does contain no element) it is necessary to formally assign (minus one).

Further Poincare writes that the faith in this definition gave him, (ascending even to Euclid's idea about the fact that surfaces are the boundary of bodies, line are the boundary of surfaces and points are the boundary of lines. Then he emphasizes the importance of the concept of the section in topology (“on the section everything based”) and in connection with this concept he recalls that, according to Riemann, a difference from the sphere to the torus is expressed with the aid of sections: on the torus (in contrast to the sphere) not any closed curve divides it into two parts.

Comparing the definition of Poincare with the contemporary topological definitions of the dimensionality (see addition), it is possible to see that the construction of Poincare corresponds to the so-called large inductive dimensionality: the dimensionality of the space X is equal to n, if between any two closed disjoint sets in the X a partition of the dimensionality of n-1 is located, in this case the dimensionality of empty set is counted as n-1 (the concept of partition corresponds to what Poincare calls section).

Thus, we see that the dimensionality of the space as a fundamental concept of topology arose (even if not directly) from physics.

Dimensionality and the property of Tiling/Covering.

It is usually believed, that the idea of the inductive topological definition of dimensionality first appeared in the work of Poincare in 1912. In reality, as we saw, the idea of this definition was already conceived in the work of Poincare 1902, and, apparently, this idea appears here for the first time. In any case, in the survey of its own works, which Poincare wrote in 1901 [16], there are no tracks of inductive approach what so ever.

If we turn to one of a great previous work of Poincare about the bases of the geometry in his article from 1898 [17], we find there no hints for the inductive definition, but in the article of 1898 there is something not less surprising. Poincare here raises the question about the internal (topological) characteristic of n-dimensional space, which is not dealing with the idea of independently changing values of n coordinates:

“Let us visualize the totality of plane figures, which partially cover one another in such a way that the plane proves to be completely covered by them; like a cabbage soup let us visualize something analogous in the space with three dimensions. If such figures, formed a kind of one-dimensional tape, we could recognize this, because the connection between these figures obeys the following law: if A it is connected simultaneously with B and C and D, then C is connected with B. This law would not be valid, if figures, being assigned, covered plane or spaces with dimension larger than two. Poincare attempts to connect the concept of dimensionality with the properties of tilings. Certainly, there are no final results, and if one-dimensional “totality” is described by some means, then about the larger number of dimensions it is said that “analogous laws” must be carried out.

These laws were expressed in 1911 by the remarkable French mathematician Lebesgue (1875—1941), who revealed the connection between dimensionality and „multiplicity of tilings/coverings“. It speaks about the greatest number of elements of a tiling, which have at least one common point. Lebesgue's theorem (german: „Pflastersatz von Lebesgue”) asserts that

the minimum multiplicity of tiling in n-dimensional Euclidean space is equal to n+1.

Subsequently this theorem became one more topological definition of the dimensionality: the dimensionality of a figure is equal to n, if the minimum multiplicity of its tilings is equal to n+1.

Although, as we saw, Poincare's attempt to connect dimensionality with the properties of tilings was not successful, but the direction of his reflections could inspire Lebesgue's idea.

Dimensionality of space and physicsAlthough Poincare calls experimental facts the results of his psycho-physiological analysis,

he clearly sees the possibility of another point of view: “The laws of physics are expressed by differential equations, these equations use three coordinates for material points. Perhaps it is possible to express these laws by other equations, with material points, which have four coordinates (in an imaginary four-dimensional world)?”

Immediately after the formulation of this problem he gave one possible answers: “But perhaps if this is possible, those equations will be more complex? Or finally if they prove to be as simple, then we do not reject them simply because they contradict our mental habits?”

Poincare begins with the analysis of the phrase “expression of the same laws with other equations” and writes: “We can say that both worlds are subordinated to the same laws” if there is “parallelism”, the correspondence between the totalities of all phenomena, possible in these worlds. Poincare assumes this correspondence to be possible, and even always possible; But we know today that parallelism of two “worlds” of different dimensionality is impossible. The radical difference of the physical laws in the spaces of different dimensionality composes the basic result of Ehrenfest.

Poincare writes that “it suffices to examine the simple case of astronomical phenomena and Newton's law”. According to him, into the laws of celestial mechanics only their relative distances must enter. These distances, which are in 3-dimensional space derived from 6 ( = 2 X 3 ) coordinates, are substituted under the assumption of a 4-dimensional space with 8 ( = 2 X 4 ) coordinates. Then he asserts: “It is clear that these equations will be more complex than our ordinary equations. Certainly, the same will occur also with the other laws of physics”. Confident words like “it is clear” and “certainly” do not compensate the absence of real arguments.

The general methodological approach of Poincare is reduced to the following: the geometric model of reality is ambiguous, since in the experiments we do not deal with the model itself, but with the combination of a specific geometric model and the specific physical theory; therefore the model is selected utilizing criteria of simplicity and convenience.

As Poincare attempted to illustrate the possibility to assign numbers of dimensions not equal to three. This illustration, strictly speaking, does not refer to physics, although it is necessary to discuss such questions as inertial system, principle of Mach and law of relativity under these conditions. Actually, Poincare writes: “We observe not the coordinates of stars, but only their relative distances; the natural image of the laws of their motion must be the differential equations, which connect these distances to the course of time”.

In Newton's dynamics there is besides relative distances both the concept of inertial reference system and the concept of absolute space. That these elements of Newtonian mechanics caused doubts, beginning with Leibnitz, in many physicists (including Einstein, for whom these doubts played the constructive role for the creation of SR= special relativity), does not give the justification for ignoring them.

Here is the simplest counterexample to the assertion of Poincare. Let us examine two situations:

1) two “stars” are located at a constant distance from each other because of rotation (relative to inertial reference system);

2) let us place the same two stars up to the same distance, let these stars be at rest, and we will control their motion from the revolving (noninertial) frame of reference.

Despite the fact that “relative distances” and speeds in both cases at first coincide, subsequently the distance between the stars changes differently: in the first case it remains the same, in the second case it decreases. So that the laws of motion include not only relative distances.

But Poincare does not consider an even more important circumstance. He actually implies that upon transfer from the three-dimensional to the four-dimensional space the law of gravity remains the same (force is proportional to r-2, where r is the distance between two points), since he speaks only about substituting r into the formula with (4 X 2) coordinates, i.e. in 4-dimensional space.

But in 4-dimensional space law F~ r-2 is incompatible with the principle of the superposition (i.e. this law no longer obeys the linear equation of Laplace), it is incompatible, in particular with the fact that gravitational field of a sphere is equivalent to the field of a point located in the center with a mass, equal to the mass of sphere. Thus, the parallelism of the totalities of phenomena is not achieved, which indicates the failure of the physical case study of dimensionality, conducted by Poincare. In reality in the n-dimensional space the dependence of force F from r must be: F∼1/rn-1

The calculation of this law allowed Ehrenfest five years to later achieve the final success in the case study of the dimensionality of space.

Quantization in quantum physics and the concept of the dimensionality of the space-time

The last 12 years of Poincare's life were also the first 12 years of quantum physics. Poincare was interested in quantum physics only the last two years of his life, nevertheless this interest deeply penetrated the essence of his ideas. In spite of the primitive state of the quantum theory at that time (there was not even the atom model of Bohr), he realized the impact of quantization the set of possible magnitudes of a physical quantity and fruitfully participated in the consideration of quantum ideas.

The work of Poincare 1912 dealt with the quantum concept. His ideas are interesting, in the first place, because here for the first quantization was connected with the question of the number of dimensions of space, which according to Poincare, “is tightly connected with the concept of continuity and loses any meaning if you abandon continuity” [20]. But in the second place, these works give the possibility to explain the important question in the history of the special theory of relativity concerning Poincare's role in its creation.

The association of the concepts of three-dimensional space and one-dimensional time into the concept of the four-dimensional space-time, which will be discussed in more detail in the following chapter, became one of the most important changes in the physical picture of the world.

In his article “hypothesis of quanta”, analyzing Planck's way to calculate the spectrum of thermal radiation, he comes to the conclusion that Planck's assumption about the possible states of elementary oscillators could be generalized to any physical system: “Physical systems

possess final (discrete) numbers of different states: it jumps from one state into another, without penetrating the continuous row of intermediate states” [21].

Let us see what example is discussed by Poincare: “Let us assume for simplicity, that the state of a system depends only on three parameters, so that we can present it geometrically by a point in the space. The ensemble of the points, which depict different possible states, does not fill space completely or any region of space, as it is usually assumed, but there is only a large number of points, isolated in the space. True, these points are distributed so very densely, that they create the illusion of continuity”.

Is this not parametric representation of the dimensionality of space? Despite the fact that mathematician Poincare knows about the insufficience of parametric definition, physicist Poincare uses parametric language for describing the 3-dimensional nature of state space. He examines the possibility not only for a “totally disconnected”, zero-dimensional set of possible states, but also one-dimensional or two-dimensional, as before assuming that the state of the physical system depends only on three parameters.

Can it be, that Poincare did have in mind only state-space, but not physical space? This is a substantial question, since the rule of physical instruments for measuring the length of a physical of system, which consists atoms, the set of states of which must be discrete. But more important is the fact that Poincare approached very closely to the idea about the possible discretion/quantization of space.

The complexity of this problem is confirmed by the extensiveness of the literature, dedicated to it [2]. In the work of Poincare from 1906, completely independent from the famous work of Einstein 1905, practically the entire mathematical apparatus is contained. This work overlaps, as it seems from a mathematical point of view with the work of Einstein, and the work of Minkowski 1907—1908, with which the introduction of the four-dimensional space into physics took place - also called space of Minkowski. The supporters of Poincare's priority in the discovery of four-dimensional space-time, leaving to Minkowski only large arbitrariness and enthusiasm in comparison with Poincare, substantiate their point of view by, for example, such words from the article of Poincare in 1906: the conversions of Lorenz “are the linear substitutions, which do not change quadratic forms x²+y²+z²-t². We will examine x,y,z,t√-1 as coordinates… in the space of four dimensions. It is easy to see that the conversion of Lorenz presents nothing else but turning in this space around the origin of coordinates, …„[23].

It seems that commentaries are superfluous. However, let us return to the work of Poincare 1912 “hypothesis of quanta”. Assertion about the discretion of the set of the possible states of any isolated physical system, as Poincare indicates, is applicable also to the universe: “Consequently, the universe must abruptly pass from one state to another, but in the spaces between the jumps it remains constant, and different moments, during which it preserves its state, could not be already differed from each other; we come, thus, to the discontinuous course of time, to the atoms of time”. A certain paragraph concludes with these words, and Poincare passes to other questions.

Let us focus attention on conclusion about the discontinuity of time, about the atoms of time. But why only time? Relativists are confident about the fact: “space and time are not two completely different essences, which can be represented separately, but the are two parts of the same whole” [24]. Poincare does not raise the question about the quantization of space and, thus he deprives us of the possibility to learn if he considered it impossible to speak about the 3-dimensional nature of space, being distracted from its continuity. But we can be convinced of the fact that Poincare's position is not completely relativistic.

According to Poincare, “physics cannot manage without mathematics, which presents to it the only language, in which it can speak”. A difference lies in the fact that in the concept of

physical theory these concepts have empirical status and thus have the capability to change with the refinement of experiments; all this, of course, does not apply to the mathematical model. The view on the history of physics from the positions of the developed theory, when the component of this theory becomes to a certain degree trivial or at least customary, can lead to the identification of physical theory and its mathematical apparatus.

Chapter II

THEORY OF RELATIVITY AND DIMENSIONALITY OF SPACE-TIME

Concept of the dimensionality of space and general theory of relativity.

In the contemporary science the deepest physical theory of space and time is the general theory of relativity, created by Albert Einstein (1879 — 1955). Since the dimensionality is one of the most fundamental properties of space-time, it is not possible to in examine the problem of dimensionality out of its context within the general theory of relativity.

The term “space-time”, which appeared in the previous paragraph, and expression “dimensionality of space-time”, that occurs in the name of this chapter, require special explanation, since the essence of the theory of relativity is connected with them.

Somewhat simplifying the situation, it is possible to say that the space and time in prerelativistic physics was identified (sometimes unconsciously) with the three- and one-dimensional Euclidean spaces of E3 and E1, and only toward the endof the prerelativistic epoch inside of physics arose (in connection with the discovery of non-Euclidean geometry) the need for the substantiation of such ideas. Space and time, as it was considered, make absolute sense separately, i.e. it was considered that for any observer - experimenter - the distance between two points has one and the same value and time interval between two events it has also one and the same value. In the space, since it identified with E3, the distance between the points p and p' is expressed by the formula:

(1) δρs² = (δx1)² + (δx2)² + (δx3)²

where δxi ist the difference of the Cartesian coordinates of the points and the time, understood as member of E1; It would be possible to characterize by certain “formula”:

(2) δρt² = (δt)²

where δt is the time, which passed from the event p to the event p', formula (2) appears, of course, very artificial, but it will be justified by the following presentation.

In classical physics the description of any physical phenomenon or physical process had to include a three-dimensional and a time characteristics. Therefore the concept “event” as the idealization of the physical phenomenon, proceeding in the very small region of space and being lasted very small time interval (flash of light), would be completely natural to the theory of relativity. But in prerelativistic physics it was possible to get along without the concept “event”, since it was split by single-valued and absolute means into concepts “position” and “moment of time”.

Space and time in nonrelativistic physics could be combined formally into 4-dimensional space E4. This it repeatedly emphasized by Einstein, stating that 4-dimensionality was not connected solely with the theory of relativity: “How the relationship of the special theory of relativity to the problem of space? First of all we must warn against the opinion that the for numbers for the discription of reality were introduced for the first time by this theory. Even in the classical mechanics “the position” of an event is determined by four numbers: by three space coordinates even one time coordinates; Thus, physical “events” were always thought of as being embedded in the four-dimensional continuous manifold…” [1]. Actually, the space E3 with formula (1) and E1 with formula (2) could be combined into the time-spatial manifold, which has the structure E4 with the formula

(3) (δρ)² = (δx1)² + (δx2)² + (δx3)² + a²(δt)²

where a is a certain parameter, as can easily be seen, with the dimensionality of speed. This association, although it did not have application in physics, was formally possible. The question about the sense of that formula arose, but in any case the topologies, generated by formula (3) with different a were equivalent.

However, in history of science the concept “space-time” appeared entirely differently. At the beginning 20th century as the result of a number of experimental and theoretical works it was explained, that the known classical mechanics law of addition of velocities is only approximate. The deviation from this law is the greater, the nearer the velocity to the speed of light c.

In physics the idea about existence of a fundamental constant with the dimensionality of speed for the metric association of space and time seems unfamiliar. However, in this case it was simultaneously revealed that it is not possible to attach absolute values to quantities (1) and (2); for example, the results of measuring the distance between the ends of a certain rod or measurement during the time interval between the flashes of a certain lamp depend on the speed of observer relative to rod and lamp. And, as it seemed, all these (and other) changes in physics can be reduced to the invariance of the new fundamental value of the interval between two events P and P’:

(4) δ(P,P’)² = (δx1)² + (δx2)² + (δx3)² + -c²(δt)²

with differences in the Cartesian coordinates and time interval δt. The invariance of intervals means that any observers, who calculated intervals on the basis of their dimensions value (4) for one and the same pair of events P and P', will obtain one and the same value, although the δ-values will be different.

An Event, or world point, in this case is characterized by four numbers (for each observer of three space coordinates and the moment of the time. Value (4) is differenct from (3) only by one minus sign before c²δt². But this small difference marked the rise of new physics and the theory of relativity, and a new interrelation between the metric and topological properties, which will be discussed below.

The critically thinking reader, who for the first time becomes acquainted with the theory of relativity in the geometric appearance and who knows about this theory only by hearsay, will feel an increasing distrust in proportion to his acquaintance with geometry. The reader will possibly transfer this distrust also to theory of relativity. The author must recognize that the few previous pages barely prevent the appearance of this distrust. In the books, intended, as now sometimes called, for “the pedestrians” and specially dedicated to the theory of relativity, there is the possibility, examining concrete physical experiments, to convincingly show the degree of the validity of the special theory of relativity,and to show, why physics are assured in the theory of relativity not less than, let us say, in the sphericity of the Earth.

In this book we can only attempt to understand, why the theory of relativity is so incomprehensible for “the pedestrians” and why, in particular, some “pedestrians”, until now, undertake desperate efforts to refute this theory.

For this let us continue the comparison of the special theory of relativity with the theory of the sphericity of the Earth, which once caused stormy disputes. To most “pedestrians” it is frequently incomprehensibe because the predicted effects are unknown in every day’s experience. To each of the compared theories correspond certain characteristical values: the radius of the Earth (6400 km) and speed of light (300.000 km/h).

Both values are enormous compared to the human scales values, this determines the incomprehensibility. The physical scales, which characterize the life experience “of pedestrians” are about 5 km/h and 2 m. It is possible to say that incomprehensibility of

relativity more incomprehensible than sphericity of Earth by one hundred times.

In our time these theories become more intelligible. The fact is that now the life experience “of pedestrians” includes round-the-world journeys, straight telecasts from space and much other. There is a theory, saying the theories become intelligible, if the ratio of its characteristic parameter T to the value I, which corresponds to the life experience of man, becomes the order of one or less than one. In history the life experience of very few people the distances close to the radius of Earth occured, among them were astronomers, travellers, geographers, and to these people the theory of sphericity was intelligible

Analogously now the life experience of physicists includes such phenomena as the motion of elementary particles in the synchrotron, the penetrating power of the muons, which reach the earth's surface, although according to “the rules” (according to the Newtonian rules) they must be decomposed in the upper air. These and many other phenomena, which are obeyed the law of special relativity (SR), make this theory intelligible to physicists.

It is difficult to say, when SR will become intelligible all people. Possibly, they will begin to find the universal understanding when interplanetary telephone conversations are normal and it is necessary to wait several minutes or even hours for the answers of the telephone partner. But so far it is only the physicists, whose experience (in the region of physical phenomena) exceeds the experience “of pedestrians”.

The paragraphs, which preceded this small retreat, of course, cannot give authentic idea about this immense upheaval created by special theory of relativity, in which participated the following physics and mathematicians: Lorenz, Poincare, Einstein and Minkowski [2].

In particular, the geometric four-dimensional point of view, which is most convenient for our purposes (and without which it is not possible to present the general theory of relativity), it arose in essence in the work of Minkowski. The mathematical model of space-time in the special theory of relativity is called Minkowski's space-time and is designated M3+1. As it is already clear from previous, M3+1 is this the totality of all possible quadruple of numbers, called events, or world points, and for any pair of world points according to formula (4) the number, called the interval between these points, is assigned.

In classical physics and in SR there is the possibility to determine the dimensionality of space and space-time, without the aid of topology. There are known even several such methods. The construction of the Cartesian coordinate system in classical physics and the inertial Cartesian coordinate systems in SR are well-defined and feasible tasks. Therefore it is possible to name the quantity of such coordinates dimensionality, necessary and sufficient for the task of specifying the position of point in the space or the event in space-time of SR. This is the same number, talked about by Galileo and Leibnitz, when they defined dimensionality as the maximum number of the mutually perpendicular straight lines, passing through one point of space. Eddington, as we remember, doubted that this determination can explain, why dimensionality must be equal to three, but it is possible to learn empirically, what dimensionality space posesses, with the aid of this definition.

Another way to speak about the dimensionality, other than topology, is with the concept of vector. In GR, as in classical physics, it is possible to speak about the vectors, which connect one point of space (or space-time) with another, it is possible to speak about the vector addition and about the multiplication of vector by the number, moreover these operations are subordinated to all usual rules of vector algebra. Then we can identify dimensionality with the maximum number of independent vectors, in any set of vectors, one of which cannot be represented as the sum of remaining vectors with some coefficients.

Existence of the class of inertial reference systems is a fundamental physical fact for SR. This fact in the geometric language is described by the fact that expression (4) in SR takes

one and the same form independent of the differences of the time-spatial coordinates for any two world points and independent of the choice of origin. Expression for distance (1) in classical physics possesses the same property. This circumstance made it possible to speak about the 3-dimensional nature of space in the classical mechanics and about 3+1-dimensions of space-time in SR, without using topological concepts.

The form of formula (1) expresses the 3-dimensional nature (three terms), and formula (4) 3+1-dimensionality. Certainly, the form of formulas (1) and (4) depends on the use of Cartesian coordinates. It is possible to use many others (spherical, cylindrical and, etc), but since the structure of space-time is considered Euclidean, it is possible to switch over to the Cartesian coordinate system, or into M3+1 Cartesian inertial reference system.

Only ten years after the article of Einstein 1905 creation of general theory of relativity was completed., which was derived from SR as physical theory. However, for the concept of the dimensionality of the space these ten years brought more changes, than two and one-half centuries, which separate Newtonian mechanics from the special theory of relativity.

It will not be described, how Einstein began to work at the construction of the theory of gravity, coordinated with SR; as he saw in the equality of gravitational and inert masses the key principle of equivalence; as Einstein recognized the geometric nature of gravitational field and understood that it must be described not by one value of the Newtonian potential, but ten values, which simultaneously describe the metric structure of space-time [2]. There are numerous good books, which present the basic ideas of general theory of relativity.

For us in GR the metric properties of space-time are important, as to SR, they are described by the interval, whose value is determined by differences in the coordinates of two world points:

(5) δs² ~ ∑ gik(x) xi xk

Difference between (5) and (4), i.e. from SR, consists, in the first place, in the fact that the formula (5) gives the value of the interval between two world points with differences in the coordinates only approximately (~), and that it is the more precise, the nearer the coordinates of these points are to each other. In the second place, the value of the interval is determined not only by values of the distances, but also by the coefficients which depend on the four coordinates of the world point, near which formula (5) is examined. Values gik are symmetrical on their indices, therefore in reality there are only ten (and not 16=4X4) different values. The fact that formula (5) will be precise only in the case of infinitely close points, is written symbolically thus:

(6) ds² = ∑ gik(x) xi xk

with the aid of the equations obtained by Einstein

(7) E(gik) = G/c² T

which includes the values gik=metric tensor, and G=Newtonian gravitational constant, c=speed of light.

Let us talk about the innovation of Einstein. In the geometry of multidimensional spaces and generally in those the fields of mathematics, where it is necessary to use many uniform values, expressions, similar (6) are frequently encountered (being sums of equally arranged terms). For the record of such sums of mathematics since old times was used the symbol ∑. The summing up conditions were written under and above this symbol. If we look at the mathematical texts, written up to 1916, the symbol ∑ was repeated in one formula so many times, it hurts in the eyes. Remarkable mathematicians (being always worried about convenience in the designations) were not bothered by this. But here physicist Einstein, mastering new for himself and for physics the apparatus of Riemann geometry, found an ingenious way out. However, in the first

systematic account of GR he preliminary lay down expressions of the type (6) to write simply:

(6’) ds² = gik(x) xi xk

in this case for the indices, which are repeated in the formula twice, the summing up is implied, and indices pass all possible values, for example, in the formula (6) i,k=1, 2, 3, 4. This “summing convention of Einstein” was very rapidly accepted - not only by physicists, but also mathematicians, even workers in the fields, not working directly with Riemann geometry. Facilitating perception and “the carrying out” of physico mathematical constructions, this convention, dividing the fate of other salient inventions in the region of designations, rapidly became conventional and anonymous. There are almost no cases, when Einstein's convention proves to be inconvenient. This book being one of these very few exceptions, since in the record (6') in contrast to (6) it is not indicated clearly, how many values assume indices i and k.

But now let us find out, how in the general theory of relativity it is possible to understand the dimensionality of space. It would seem, the answer is to call dimensionality a quantity of coordinates, utilized in the mathematical model of space, i.e. a quantity of of indices in (6). However, what coordinates? In the general case there is no established concept of inertial-frame of reference. It is not possible to establish coordinates so that the expression for the interval would take on the standard form, independent from the choice of the origin, similar to (4) – because the gik depend on x. The special features of the mathematical model of space-time in GR is called Riemann manifold and designated R3+1 is radically different from the previous models E3 and M3+1. In GR it is not the model of space-time itself, but only the principle of the construction of this model depending on how space-time “is filled”. Therefore the geometric properties of space-time in GR at different world points are different. Let us say, the sum of the angles of triangle in one place can be more than 180, in other it is less, etc. This means that the space-time in GR is bent, the nature of bend in each place depending on the physical processes. On other hand, the curvature of space-time influences the physical processes. According to the descriptive expression of well-known physicist John Archibald Wheeler: “Space tells substance how to move, and substance tells space how to be bent”. Thus, the number of indices are not of the same quality as in E3 and M3+1.

Since the space-time in GR can be bent in an arbitrary manner, among the coordinate systems there are no preferred ones. Therefore if we identify dimensionality with the number of coordinates, then this number must exist in any possible coordinate system. But what are these “any coordinate system”? In SR and even more in classical physics it was possible to indicate the physically well-defined methods for the establishment of the coordinate system, e.g. with the aid of the solid rods with chronometers and light beams. In GR there is no such possibility.

The ideas of Einstein about the dimensionality of space are tightly connected with the introduction of coordinates, but his attitude toward the role of coordinates considerably changed upon transfer from SR to GR.

In the special theory of relativity, it is possible to say, it began with giving real physical meaning to time coordinates (from the explanation of the concept “absolute simultaneity” being meaningless) and refining the physical sense of space coordinates with the definition of the procedure of the measurement: “Separate coordinates relate to the measurement of the position of solid bodies” [3].

In the general theory of relativity the final success, in the opinion of Einstein, became possible only after the acknowledgment of the equivalence of “arbitrary” coordinate systems: “The postulate of relativity in its most common format, which deprives the time-spatial coordinates of physical sense, lead with to the well-defined theory of gravity, which explains the perihelion motions of mercury” [4] .

This alone shows, what difficult way it was, to go for Einstein: success arrived after the refinement of the concrete physical sense of coordinates by comparison with the specific procedure of measurement.

This is how Einstein he wrote about the coordinates in GR: “For describing the time-spatial regions of final extension the arbitrary coordinates of the four-dimensional manifold are necessary, which ensure the single-valued designation of each of the points of space-time with numbers”, or: “Each point of continuum we arbitrarily place in the correspondence of the numbers, which are called “coordinates”. Adjacent points correspond to the adjacent values of coordinates” [6]. Thus before introducing coordinates, it is necessary to know “the continuity” of time-spatial manifold, it is necessary to know, what points are adjacent, i.e. to know the topological structure of space-time. According to Einstein, “coordinates express only order or degree “of proximity” and, consequently, also the dimensionality of space” [7].

Therefore in GR topological dimensionality of space for the first time proved to be inevitable.

Metric and topological properties of space-time.

Ideas of Einstein about the dimensionality of the space:

In the special theory of relativity the metric properties of space-time were described by formula (4), which gives the value of the interval for any pair of world points. In GR the parent element of the metric description become the purely local values of g ik(x), which give the value of interval only for the infinitely close world points. However, if time-spatial model isthe result of solving the equation of Einstein (7). The values gik(x) must be known for each point X, then it is possible in GR to introduce the concept of interval for remote points. It is necessary to examine the shortest, so called geodetic lines, which connect points P and P'. Geodetic lines in R3+1 determine the motion of material point in GR. It is more conveniently not to use not the interval itself, the square of it, because that is always a real value, which wecall for the brevity also the interval between the world points P and P'.

(8) I(P,P’) = (δx1) ² + (δx2)² + (δx3)² - (δt) ²

The characteristic properties of the interrelation of metric and topological properties are manifested already in the case of flat space-time. We get the picture, after limiting to a 1+1-dimensinal space-time. This space-time can be presented graphically on the two-dimensional sheet of paper. In this case the interval

(9) I(P,P’) = (δx)² - (δt)²

(The units of the measurement of length and time they are selected in a way, that the speed of light it is equal to one). This two-dimensional space-time we will compare with the usual two-dimensional Euclidean space E2, i.e. with the plane geometry, the square of distance on which is given by the formula

(10) K(P,P’) = (r(P,P’))² = (δx)² + (δy)²

In order to graphically represent what the values k(P, P') and I(P,P') in the general case of the bent space would be, considering that between two points P, P' on the bent surface (let us say, on the sphere) a “rope” stretches so that each of ist points fits closely to the surface; the minimum length of this “rope” will give the unknown value. It is necessary to only remember that in the case of the interval I the “rope” is the time-spatial trajectory of the motion of a point.

I can be said, that function I(P,P') and function k(P,P') determine a metric structure in a certain

meaning, since in both cases to each pair of points is a certain number is assigned. However, these two types of metric structures are of different kind. One difference strikes immediately: before (δt)² in (8) and (9) stands a minus sign. Interval I(P,P') can take on positve and negative values in contrast to the formula of type (10). Therefore we will say: ±metric and +metric. Usual intuitive ideas about the distance correspond precisely to the +metric. However the + metric has the important advantage that it reflects the most important physical laws governing the theory of relativity.

It is possible to show that interval (8) is compatible with the geometry of Minkowski's space-time and the relativistic law of addition of velocities, and a change in the values of distance and time interval for the moving observer, and even laws of the electromagnetism of of Maxwell's equation. But we will be concentrated on the relationship of the metric and topological properties of space-time.

Einstein spoke about “Gaussian” generalization of the geometry (this generalization is more frequently called “Riemann”) as sufficient for GR: “Gauss proposed the method of the mathematical description of any continuum, in which metric relationships are defined (“distance” between the adjacent points). To each point of the continuum as many the numbers (Gaussian coordinates) are assigned, as the dimensions of that continuum demand. The method of distance-calculation is selected in a way, that it would be single-valued and that to adjacent points would correspond numbers, which are also „adjacent“. The Gaussian coordinate system is the logical generalization of Cartesian. It is applicable also to non-Euclidean continua, but if and only if the part of the continuum in question is small [8].

However, Gauss and Riemann considered only +metric, similar to formula (10). This formula generates a topological structure and point p adjoins the set M, if it is located at zero distance from this set, i.e. if within M for each value of distance ε there are points, which are located on a distance smaller than ε from p. A completely definite meaning acquire words like“small parts of continuum”: the size of a region of space can be characterized with the greatest distance between the points, which belong to this region. Passage to ±metric of space-time in the theories of relativity principally changes situation. Now the proximity of two world points cannot be connected directly with the value of the interval between them, since the equality to zero intervals indicates no longer the agreement of points, but only the possibility to connect these two point events with a light signal.

The difference between the two types of the formulas can be demonstrated graphically in the two-dimensional case.

Fig. 2. Difference in the two types of formulas

In the case of ±metric (Fig. 2, b) the set of points, removed from the point 0 less or equal to ε are limited by a hyperbola. With the decrease of ε that hyperbola increasingly is more forcing against the straight line I (formed by the world points, which they can be connected with the point 0 by light signals). This difference is a reason that +metric does not assign topological structure in the manner that the +metric does. Therefore with general theory of relativity the topological structure of space-time is considered to be given before the introduction of the interval, i.e., metric structure. Such a situation is never completely satisfactory, since to physics is nearer to the metric structure (which can be reduced to dimensions) than to topological structure, which is mathematical.

Perhaps, a very clear examples of this situation is absolute space and absolute time in the mechanic Newton. These concepts by no means interfered with the development of physics for two centuries, despite the fact that Leibniz (contemporary of Newton) criticized deficiencies in these concepts, and despite the fact that the realization of these deficiencies stimulated Einstein’s revolutionary conversion of physical ideas about the space and the time.

The words “It stimulated”, of course, does not mean “completely determined”. The appearance of the special theory of relativity is obliged to the first of all in-depth analysis of the laws of the electromagnetism of Maxwell and experiments, most known of which was Michaelson's experience. The fact of the equality of gravitational and inert masses was extremely important for generating the general theory of relativity. However, understanding the logical imperfection of the classical concepts of absolute space and time facilitated for Einstein the critical revision of these concepts.

Returning to the topological structure of space-time in SR, it should be noted that Einstein himself did not discuss the relationship of the metric and topological properties of space-time in his theory of relativity and it was, apparently, limited to intuitive idea about space relations existing in the proximity of points and the like - for this was an intelligible reason.

The fact is that, although the relationship of metric structure and topology in GR are sufficiently complex, the mathematical apparatus of the theory, which describes the local curvature of space-time with the aid of Einsteins equation, does not depend essentially on the nature of the formula. The theory assumes that in the space-time the coordinates are already assigned. At the same time, according to Einstein, the sole limitation to the introduction of coordinates of is the compatibility of coordinates with the topological structure of space-time.

And nevertheless it cannot be said that Einstein did not have doubts about the relationship of the metric and topological properties of space-time, about the role of interval in the structure of theory. The geometrization of gravitational field was inseparably connected with the idea about the fundamental role of interval. However, in the first work, dedicated to geometric approach to the gravity (1913), Einstein it proceeded from the fact that “the interval must be the absolute invariant”, which “should be understood as the invariant measure for the distance between two adjacent time-spatial points. Therefore the interval must also make physical sense independent from the selected frame of reference

Thus, “ds is the distance between two time-spatial points” [9]. From the positions of the developed Einstein's theory he emphasized the fundamental position, which occupies interval in the theory of relativity, and he indicated that “there are physical objects, which… measure the invariant ds” [10]. Finally, in 1952 Einstein wrote that “the functions gik(x) describe not only a field, but at the same time topological and metric structural properties of space: „The concrete methods of the generation of the topological structure of space-time were found with its formula”[11]. For example, in the so-called L topology (called on the name of well-known Soviet mathematician Aleksandrov) it is not necessary to know the entire metric structure, it suffices to know for what pairs of points the interval is equal to zero.

But if in GR the formula (let us recall that the discussion does not deal with ±metric, but with the interval) is capable of generating topological structure and thus: Can we speak about the dimensionality directly in the metric language, without reference to the topological level?

Obviously this is possible. Any definition of dimensionality must “calibrate” with the aid of the selected standard of 4-dimensionality, i.e. it must be selected a certain mathematical object, whose dimensionality is equal to the general definition of dimensionality. From the point of view of physics this standard is Minkowski's space.

Since it would be desirable to find the definition of dimensionality, based only on the concept of distance between two points, it is necessary to lean on any property of Minkowski’s space, which would separate the number n by calculations. It is not difficult to find this property. Let us examine for the beginning the two-dimensional case. If we choose a certain point on the plane E2 (or M2), then the distance (interval) determines the position of the infinite number of points, generating a circle with the center at point P (hyperbola in the case M2). But if we chose two points P1 and P2, then the calculation of two distances to these reference points will determine the position ot only a finite number of points, not greater than two, since two circles can not be crossed not more than at two points. It is not difficult to see that the three-dimensional space possesses a similar property, if we select three points. The same property possess the n-dimensional spaces En and Mn. In each of them it is possible to fix n of basic points so that the distances (or intervals) to all n of these points of would determine the position of any point of space. In this case a smaller number of basic points is, generally speaking, insufficient, and larger is excess. This can be transferred also to the case of the bent space Rn.

Thus, in the general case of space-time dimensionality can be understood as the minimum number of world points, such, that the values of intervals to these points can assign the position of any event. This means parametric representation of the dimensionality (point of space are parametrized by their distances to the reference points). The method of parametrization has been assumed to be arbitrary, being based on the metric structure of space itself. Thus, the dimensionality of space-time can be expressed in the metric concepts.

Einstein's approach to the dimensionality can be described thus: there is a standard of the 4-dimensionality of En and, establishing the defined connection with this standard, it is possible to speak about the dimensionality of space as “a quantity of coordinates”. This is the most general common approach to the dimensionality of space, but it is necessary to remember that even in mathematics the more general common approach to toplogical dimenion was created by Uryson and Menger in 1922—1923., seven years after completion of GR and several years afterof its experimental confirmation.

But the fact that the creation of the relativistic theory of gravity did not require the more common mathematical models of space, and no other ideas about the dimensionality, than coordinate. Even the subsequent development of physics did not advance past coordinate description.

Einstein considered 4-dimensionality the fundamental property, inseparably belonging to the continuity of space, i.e. the fact that space-time is “continuum”. Einstein explains „continuum“ as follows: “I can switch over from any point of a manifold to any other point, passing from one point to the next without “the jumps” (only passing adjacent point). The reader hopefully clearly understands, what the concepts “adjacent” and “jumps” mean. We express the same thought, asserting that the surface is continuum” [12]. But it is difficult to understand this description precisely. It is not clear, for example, is it possible to consider as continuum the set, which consists of all rational points on the straight line represented by E1. Tor Einstein the concepts of “continuity” of space and “continuum” did not cause the need to deeply analyze them, although these concepts are not more physically trivial, than, for example, the concept “absolute simultaneity”, so important for SR.

Just as Poincare, Einstein connected the concept of dimensionality with the continuity. The examination of quantum phenomena gave rise for doubts about the continuity of space-time. Einstein asks: “How can QM be reconciled with continuity?” [18].

From these words it is possible to understand the fundamental value of 4-dimensionality, which he connected with the continuity because “he could not devise” another concept of dimensionality.

There is no evidence, that Einstein attempted to comprehend 4-dimensionality of space-time at the physical level. Dimensionality for Einstein is the most important physical property of material world. In the preface to the book “the concepts of space” Einstein considers the history of the concept of space as development and interaction of two substantially different concepts: absolute space and “space as the properties of material objects”. He connects this victory of the second concept above the concept of absolute space with the introduction of the concept of field. Einstein concludes with the words, which refer straight to the problem of the dimensionality of the space: “… entire physical reality can be represented in the form of the field, whose components depend on four time-spatial parameters. The case of “empty” spaces, i.e., of space without the field, does not exist” [14]. Einstein attempts to present dimensionality as the property of the physical field.

Is there besides the four dimensons of physical space-time a fifth dimension? Several works introduce the fifth dimension in connection with the electromagnetic field.

Other works, beginning from the articles of Klein and Kaluza 1926, tried to describe GR with the aid of the five-dimensional classical geometry quantum phenomena. Paul Bergmann wrote that “the description of a five-dimensional world with the aid of the four-dimensional formalism is incomplete”, and “… quantum phenomena finally can be explained by field theory” [15].

The uncertainty, which appears with the four-dimensional description of five-dimensional wold, can be explained as follows. Let us visualize the modernized version of the Platonian cave - the name of the movie “the Platonian physics” [16], but the «movie“ is not demonstrated on the illuminated screen, but are moved shadows of spheres, which move in the hall in the rays of a projector. Let us assume that for physicists it is possible to observe all events, happening in (three-dimensional) hall, only the shadows on (two-dimensional) screen. Thus, to describe the events they can only use the values, which characterize arrangement and displacement of shadows on the screen. Let us assume further that the spheres interact with each other (in the three-dimensional hall) according to the laws of elastic impact. Then as a result their two-dimensional studies the physicists unavoidably encounter with the fact that “the same” (two-dimensional) conditions lead to substantially different results: in some cases (two-dimensional) collision of spheres leads to a change in their motion, and in other cases spheres without difficulty pass through each other. The uncertainty of the result of interaction of spheres could be described with the three-dimensional point of view.

The ratio of the diameter of spheres to the depth can not be observed by our physicists. This same type of uncertainty, the probabilistic nature of laws it would create, if it turned out that our world is 5-dimensional - and we see only the space-time on a 4- dimensional screen,

Five-dimensional theories, as they were customary to assume until recently, did not leave any track in contemporary physics. It is not possible to publish this opinion in recent yearss opinon without massice criticism. The intensive work on the development of unified theories in physics of elementary particles led to the unexpected revival of Kaluza-Klein ideas. Certainly, this revival occurs at an entirely new level: excess dimensions no longer limited to one, but several. The purpose is not obtaining quantum laws on classical geometric basis, but general idea. Any final results in this direction could not yet be obtained, but A. Salam, the laureate of the first Nobel Prize for the achievements in the region of unified theories (1979), mentions “Kaluza-Klein” as a hope for the construction of the unified theory of fundamental,

interactions [17].

However, in any case for the historian of physics, five-dimensional theories present an extremely interesting object. Beginning with the work of Kaluza 1921, for a period of several decades large efforts of theoretical physicists were applied in this direction. Kaluza succeeded in part to embed electromagnetism in the geometric structure of five-dimensional space, equivalent to usual Maxwellian electrodynamics within the framework of SR.

Passage to the five-dimensional space was a radical step, comparable with passage from space and time of nonrelativistic physics to space-time of GR.

Bergmann wrote: “Kaluza introduced the fifth dimension exclusively with the purpose of an increase in the number of components of metric tensor, assigning to it no real meaning”. The efforts of theorists subsequently were directed toward giving physical sense to the fifth measurement. The vulnerable point of five-dimensional theories was the fifth dimension could not be observed. On the other hand in the work of Einstein according to the five-dimensional theory ithe following problem exists: “To explain, why the continuum is limited to four dimensions” [18].

Bergmann mentioned about the similarity of the incompleteness (and hopefully probabilistic nature) of the description of five-dimensional world on the one side, and the quantum description on the other side. This incompleteness was the manifestation of the so-called idea of the concealed parameters. The supporters of the hypothesis of the concealed parameters strove for the classical interpretation of quantum theory, assuming that the physical reality is governed by classical laws, but it is characterized by a large quantity of dynamic variables how it is usually considered (coordinate, pulses, the tension of electromagnetic field). “Excess” variables (concealed parameters) are not observed by themselves, and therefore the results of physical experiments “are spread” throughout the entire region of the possible values of the concealed parameters (as in the picture of movie auditorium in the example given above). But for the idea of the concealed parameters there were no other clues, except nostalgia related to the loss of customary classical description. In the case of five-dimensional theory it was not clear, why it is necessary to limit precisely to one concealed parameter, and not to examine 4+k-dimensional theory.

In the beginning of the article of Einstein 1944, for the first time he declared the failure of the project (three years after the last work of Einstein, dedicated to a five-dimensional theory). This was a welcome argument “for others”. Rather should be accepted another explanation of Einstein, which declares that the five-dimensional theory could not explain, why continuum is in an obvious manner limited by four by dimensions.

However, it is possible that the ideas, which lie at the basis of five-dimensional approach, some day will play their role in physics. This would confirm the widely known “theorem” that any beautiful mathematical idea sooner or later finds physical use.

Quantization and the four-dimensional space-time continuum

“The concepts, which prove to be useful… easily conquer such authority, that we forget their terrestrial origin and accept them as something invariable. They call this “logically necessary”, “a priori given”, etc…. the analysis of the concepts long ago utilized by us and the development of the circumstances, on which depend their validity and how they are derived from experimental data. The superfluously great authority of these concepts makes it possible to blow up this analysis. They will be rejected, if they cannot be legalized properly, they are corrected, if they do not completely correspond to given things, they are substituted with others, if it is necessary to create any new more preferable system ". This statement of Einstein summed up his methodology and skepticism toward the steady ideas, his tendency toward the analysis of their real physical status. The value of Einstein’s methodology for the science of 20th century is

enormous. With his creation of the theory of relativity this methodological position proved to be decisive. (Although, of course, precisely appearance SR and GR as the result of this position made possible its mastering, in particular, by the creators of quantum mechanics, for whom this position was even more fruitful.

With respect to the three-dimensional concepts most vividly Einstein expressed his position as follows: “What is a priori undoubted, or necessary, respectively in the geometry (doctrine of space) or in its bases? Before we thought - everything, now we think - nothing. Already the concept “section” is logically arbitrary: things are not obligated to exist, which correspond to it even approximately. Analogous observation can be made about the concepts of straight line, plane, about the 3-dimensional nature of space and about the validity of the Pythagorean theorem. Even the doctrine of continuum is by no means given to us in the nature of human thinking, so that from the point of view of the theory of knowledge it is not possible to attach great significance to purely topological relationships more than to other relationships“ [19].

Other statements of Einstein are known, expressing doubts about the absolute applicability of the usual three-dimensional ideas; moreover these doubts are clearly connected with the special features of the quantum phenomena: “The proposed physical interpretation of geometry cannot be directly applied to the regions of the space of sub-molecular sizes… it can only serve as the justification for the attempt to assign physical reality to the basic concepts of Riemannian geometry out of the field of their physical background. " [20].

For a physical theory it is insufficient to have some doubts, and in connection with this Einstein writes: “… the introduction of space-time continuum can be considered unnatural in the microcosm. They assert that the success of Heisenberg's method can be attributed to the purely algebraic method of describing nature, i.e. to eliminating from physics all continuous functions. But then it will be necessary in principle to abandon the space-time continuum. It is possible to think that the human resourcefulness will finally find the methods, which will make it possible to follow this way. But at present this program whisks away to the attempt to breathe in the vacuum” [21].

In his last works Einstein recognized the victory of the quantum ideas: “It is possible to convincingly prove that the reality can not be continuous field. It follows from the quantum phenomena that the final system with the final energy can be described completely by the final collection of the numbers (quantum numbers). This cannot be combined with the theory of continuum. It requires for ist description purely algebraic theory. But today no one knows how to find basis for this theory” [22].

Perhaps most vividly Einstein's position is visible in the answer to well-known mathematician Menger, who expressed doubts about the physical space as a continuum. Einstein writes:

“For constructing the contemporary theory of relativity:

1. Physical objects are described by the continuous functions and fields, which depend on four coordinates. If topological connectedness remains, then the selection of these coordinates is arbitrary.

2. Variable fields are the components of tensor. Among these tensors there is a symmetrical tensor of gravitational field.

… Examining quantum phenomena, we begin to suspect, that doubts can appear about the final expedience of the program, briefly characterized...

As yet there are no new concepts, which possess a sufficient creative force. Such, is unfortunately, my position. I adhere to ideas about the continuum not because I proceed from a certain prejudice, but because I cannot devise anything, which could for replace these ideas (of

continuity)“. [23].

He places the well-defined problem before mathematics and theoretical physics: to learn to describe the discrete structures, would it be possible to give four-dimensional some definite meaning. (Let us note that Menger one of the creators of the topological dimensional theory did not propose in his article any concrete construction). We still wait for the future synthesis of relativistic quantum theory and Einstein theory of gravity.

Chapter III

THE 3-DIMENSIONAL NATURE OF PHYSICS“How in the fundamental laws of physics it is manifested, that the space has three dimensions? “

The article of Paul Ehrenfest (1880—1933) with this name was printed in 1917 in “the works of Amsterdam academy”[1]. Only after this article the real foundations for considering the dimensionality of space as physical concept was established, and the 3-dimensional nature a physical fact.

Let us begin from the brief account of the work itself.

Introduction to the article consists in several phrases, in which Ehrenfest emphasizes the unusualness of the question, carried out into the title, and refines it:

„Why has our space three dimensions? or, in other words, what special features distinguish geometry and physics in the (Euclidean) space R3 from the geometry and physics in Rn?. In this form these questions, possibly, have no sense. Probably, they are subjected to the justified criticism. Is it really clear, whether the space of physical reality is three-dimensional? Why not R4 or R7?

I will not attempt to find the best form for these questions. Possibly, others will succeed in the indication of some more singular properties of R3, and then it will become clear, such are „the correct " questions, for which our examinations are the suitable answers ".

This last phrase speaks, that itself Ehrenfest not only did not consider it a question depleted, but even he did not consider as the his completely satisfactory means set. And actually, from a contemporary point of view the question: “Why space has three dimensions?” — can be understood in two substantially different senses.

First, it is possible to attempt to explain the 3-dimensional nature of space on the basis of the deeper properties of the material world from within the framework as a certain fundamental theory; in this case the discussion must deal with the theory, incomparable with the existing physical theories, since in them the 3-dimensional nature of space is taken as the initial assumption, the postulate. The answer, for example, to the question: “Why the electromagnetic radiation of atoms is characterized by discrete frequencies?” — became the construction of the fundamental theory (quantum mechanics), which explained, on the basis of one and the same fundamental principles, not only the nature of atomic spectra, but also the set of other phenomena.

The second sense, which can be put in the question: “Why space does have three dimensions?” — no longer is connected directly with such immense concepts. A question can be refined thus: “Why are physicists assured in the fact that the space has three dimensions?” or: “What bases determine the confidence of physicists in the 3-dimensional nature of space?” This question can seemtrivial with respect to the area of the macroscopic phenomena, when 3-dimensional nature is received “directly” by sensory organs.

But the same question loses any triviality, if one considers that the range of the phenomena, studied by contemporary physics, left far beyond the limits of macroscopic scales and that the study of phenomena, for example, in physics of elementary particles and in cosmology is only possible by indirect methods. Therefore a question about the dimensionality of space in these regions is the completely justified and nontrivial task. In this case it should not be supposed, that the dimensionality of the space is only one of many properties, whose extrapolation to the substantially new phenomena requires special substantiation. The fact is that the dimensionality is the most general quantitatively expressed property of space-time; at present any physical theory, which claims to the time-spatial description of reality, takes dimensionality value as the initial postulate. (For a while in physics of elementary particles the ideal of theory was popular, which is circumvented

without the concepts of space and time; however, at present in connection with the successes of the theory of calibration pour on hopes these concepts are connected in essence with the field-theory description [2], which essentially assumes the time-spatial picture of phenomena.)

Let us return now to the work of Ehrenfest, which corresponds exactly to the second sense of a question about the dimensionality of space, and let us look, how in its work 3-dimensional nature was substantiated in the range physical phenomena from the atomic to the astronomical scales.

Ehrenfest examines “physics” m in Euclidean space En. In this case he derives the law of interaction with the point center (analogously with the three-dimensional case) from differential Poisson's equation in En for the potential, which determines this interaction. Poisson's equation is equivalent to the law of Gauss, which asserts that the flow of the field strength through an arbitrary closed surface is equal to the summary charge (or mass in the case of gravity), which is located inside this surface. As we remember, Poincare acted completely differently: he proceeded from the law of interaction in the particular case of two point particles. However, Ehrenfest proceeds from the invariability of the general law of interaction, from which it is possible to obtain the law of interaction not only for two point particles, but also any system of bodies of arbitrary form and distribution of density.

In order to have the capability to pose the locked physical problems, Ehrenfest also subordinates motion to the Newtonian laws of dynamics, more precisely telling, their natural generalization to the case of En.

On the basis of such laws of interaction and motion, Ehrenfest examines, in particular, the following concrete consequences of these laws: closure and the stability of orbits into the field of the gravitating center (“planetary system”), and the Bohr spectrum of hydrogen atom. It turned out that: only in the space E3 both steady finite (in this case always with the locked trajectories) and infinite motion are possible;

In the space E2 finite motion is possible but is restricted to circular paths;

In the space En n>3 finite motion corresponds only to circular paths, also, moreover always unstably, i.e. any slight disturbance leads either to the drop in the center or to the removal into infinity.

In the spherically symmetric case in En from Poisson's equation for the potential or from the law of Gauss for the tension follows the expression for the potential energy (in the designations, Ehrenfest)

(1) Vn( r ) = - k Mm/((n-2)rn-2) für n >= 3

V2( r ) = k Mm ln r and V1( r ) = k Mm r

where the constants M and m are the mass of star and planet (or nuclear charges and electron), and k is the coupling constant.

Expression for the force of interaction corresponds to these expressions for the potential energy

(2) Fn ( r ) = k Mm r1-n

(3) F = ma

which are the equations of motion (n dimensional second Newton's law). In the case of central field it leads to two preserved values: energy and moment of momentum. It is not difficult to understand that in the central field in En the motion is always flat (two-dimensional). Actually, the plane, determined by velocity vector and by radius-vector, which connects the moving point with the center of field, does not change its position, since velocity change occurs only

along a radius, i.e. in the same plane.

By the he properties of the electric field, which is subordinated to Laplace's equation in En the special features of the spectrum “of hydrogen” in this space are determined also. Ehrenfest obtains this spectrum with the aid of the quantization of Bohr.

But at first let us recall briefly what is Bohr atom model is: One of the most mysterious facts for classical physics was the fact that the atoms of this substance emit the light not with any wavelengths, but well-defined, always one and the same. Let us say, sodium-atoms (for example in the usual common salt), at a high temperature emit yellow light. Without knowing the explanation of this fact, physics and chemists nevertheless very successfully and fruitfully used this property for the recognition and identifications of substances (spectral analysis). For physicists it was necessary to be reconciled with the fact that they they did not know for nearly a half century, why determined spectral “are the passport” of different substances.

But after the famous experiences of Rutherford and after appearence in 1911 of the Rutherford planetary atom model the position became entirely unbearable. Indeed according to the classical theory of electromagnetic field the electron, which revolves around the atom, in the first place, can, and even it must, emit the light of all wavelengths, and, in the second place, an electron, constantly emitting energy, it must within very short time fall into the nucleus.

Both these difficulties were solved by Bohr in 1913 with his atom model. According to this model the electron motion along the orbit around the nucleus is determined by the laws of classical mechanics, but orbits themselves can not be any, but only such, for which the following condition is obeyed:

Mvr = nh/2π

Where M and v are the electron mass and the electron velocity; r is the radius of the orbit (by assumed for simplicity circular); n is any positive integer: n==1, 2, 3, … and h is the constant, which was introduced in 1899 by M. Planck (and with the aid of which hesolved not less serious difficulty of the explanation of thermal electromagnetic radiation) “according to Bohr's model, being found in one and the same orbit, electron does not emit, but passage from the orbit, in which the energy of electron is equal to εk, into an orbit with the energy εn it corresponds to the emission frequency νkn=(εk-εn)/h analogously Ehrenfest obtains the spectrum of hydrogen in the n-dimensional space En “with the circular electron-motion with the charge e and a mass m around the fixed proton with the charge e and second Newton's law together with the law for the force, analogous (2).

The radiated frequencies are defined from the condition with νkn=(εk-εn)/h.

For n=4 the singular result (“in particular remarkable with respect to the quantum theory”, as Ehrenfest mentions) mr²ω=em1/2 i.e. the moment of momentum, can have only one specific value. The condition of quantization leads to the stiffening joint em1/2=kh/2π

With n>4 we obtain “the series in the spectrum, which with the constant/contain lines into ultraviolet, ever more and more distant from each other”. But more importantly there can be the fact that with n>4 the electron must pass into ever more distant orbits, which correspond to smaller energy, i.e. the atom spontaneously ionizes.

The results of the analysis of the properties of the n-dimensional “atom” make it possible to draw the conclusion that the 3-dimensional nature of space in the atomic phenomena is completely substantiated, since the difference from the 3-dimensional nature would bring, as showed Ehrenfest, a radical difference in the spectrum from that observed.

Quantum-mechanical solution of the problem about the spectrum of n-dimensional atom [3](on the basis of the Schroedinger equation), although it is distinguished substantially (with n=3)

from the results of the Bohr quantization, used by Ehrenfest, also it leads to “the super-instability” of atom with n>=4 (the electron spontaneously must fall on the nucleus) and to “the super-stability” of the atom with n<=2.

Besides the properties of planetary system and atom of hydrogen in En Ehrenfest examines also the properties of wave process and some geometric properties of the space En. As a result he comes to the conclusion about a qualitative difference in the case of n=3 from other values of different dimensionality [4].

Two years after his article in 1917 Ehrenfest directs into the periodical “Annalen der Physik” a somewhat changed version of this article [5]. As the occasion for this publication he mentions Weyl's observation about space-time. Weyl in connection with his version of the unified field theory, in which the vector potential of electromagnetic field obtained geometric interpretation, and only in the four-dimensional world the equation for the electromagnetic field possess the special property of conformal invariance, which in Weyl's theory substituted usual covariance of SR, i.e. the independence of the form of fundamental equations from the transformation of coordinates.

In the final observations to the article of 1920 Ehrenfest emphasizes that the dimensionality of space is manifested practically in entire physics.

What is physics in n-dimensional space?

However, what did Ehrenfest really do? With the eyes of physics he looked to the totality of the spaces En, which are characterized by one parameter of dimensionality. Certainly, this totality did not exhaust all possible geometric structure, claiming the name „space“ to certain mathematical models,describing the properties of real physical space. Sufficient are Riemann spaces, without speaking about the general metric and topological spaces.

The equations, which are used for the writing of basic physical laws, easily allow generalization from E3 to En. For this it suffices in the appropriate sums the number of terms to replace 3 by n (for example, in the expression for the radius-vector, etc). However, the form of fundamental equations which fall into this category are the equation of Poisson (law of Gauss), the wave equation, Newton's law, the quantum postulate of Bohr, the equation of Schroedinger, etc.

The fundamental physical laws of interactions are now assigned usually in the so-called variation form. In this case it proves to be sufficient to indicate one function from the values, which characterize this field (this function it is called Lagrangian), in order to obtain all equations, which describe the laws of this field. For example, The Lagrangian for the simplest case of scalar mass-free field takes the form

(7) L =∂ϕ/∂t ∑ ∂ϕ∂xk

or in the briefer record L = ϕkϕk

This Lagrangian reduces to Poisson's equation and, therefore, to the field of the point i.e. to expressions (1). The dimensionality of space is considered in the expression (7) only in the form of condition to the set of the values, which can assume indexes. in 3+1- dimensional case of k=0, 1, 2, 3. Thus, the expression for Lagrangian (7) given above makes it possible to obtain the counterpart of physics (physics of scalar mass-free field) in the space of any dimensionality, more precisely saying, in the Euclidean and in the Riemann the spaces of any dimensionality.

Poisson's equation is mathematically equivalent to the Lagrangian. The simplest extrapolation of (7) is:

(7’) L = (ϕkϕk)N-3

(dg. of — the diensionality of space-time), also passing into usual Lagrangian in the case of 4- dimensional space-time.

In response to this it must be noted, that Ehrenfest did not simply devise some physics in some space - such a physics would be completely arbitrary. He placed the problem otherwise, explaining, on what was based the confidence of physicists in the fact that the real physical space is really three-dimensional, i.e. that entirethe set of phenomena (including the phenomena known to physics, which far fall outside the limits of the ordinary, macroscopic experience, in which the 3-dimensional nature was obvious) in the best way is described with the aid of the model of the space E3. But once speech goes about the totality of known phenomena, the possibility of the selection of Lagrangian is, it goes without saying, severely limited. Such limitations are, in particular, the properties of physical phenomena, which do not require for their formulation the specific value of the dimensionality of space. Thus, for instance, th principle of superposition, known in the theory of electromagnetism testifies about the linearity of equations (and, that means without the squareness of Lagrangian). But the fact that for the task of the motion of system it is sufficient to know only initial position and speed (this it relates not only to the mechanics, but also to the field theory), he says, that the differential equations, which describe this system, they must be not higher than the second order (i.e. into Lagrangian they must enter derived not higher than the first order). These limitations lead exactly to usual Lagrangian (7).

True, with these limitations it would be possible to say that they cannot be the complete and absolutely strict substantiation of the form of Lagrangian, since they are connected with the specific range of phenomena, limited, in particular, to the characteristic time-spatial scales. But the limitedness of empirical basis is at the given moment the usual situation for the science.

It is mportant that in this usual situation with the aid of the work of Ehrenfest proved to be implicated the fundamental concept of the dimensionality of space, before it had been in the state of apriority. In Ehrenfest's article there is no analysis of epistemological status of the concept of the dimensionality of space, there are no methodological observations whatever about this concept. However, if physicists wanted to answer a question, which physically indicates the 3-dimensional nature of real space, then they must begin with an analysis, similar to that of Ehrenfest.

A difference in the approaches of Ehrenfest and Poincare to the physical understanding of dimensionality is completely obvious. In Poincare the tendency toward convetionalism made him show the physical equivalence of the spaces of any dimensionality and three-dimensional space. As a result the simplistic and physically unfounded analysis of the law of universal gravitation on the assumption, that the dimensionality of space is different from three. The initial and constant fact for Poincare is the inverse proportionality for the square of distance in the law of gravity, and passage to the space of another dimensionality only substitutes the expression of scattering between the bodies through the coordinates. He does not see that the law of F∼r-2 in En with n≠3 is incompatible with entire remaining (even only gravitational) physics: with the principle of superposition, with the equivalence of the field of the spherical mass distribution and field of material point with the mass, equal to the total mass of distribution and which is located in the distribution center, etc.

Poincare did not consider that physics cannot supplement any geometry so that their combination would correctly describe real world. Although his observation about the fact that on the experience is checked only the sum “of physics and geometry”, correctly and, therefore, certain conventional element always exists in physics, the comparison of positions and results of Ehrenfest and Poincare very clearly demonstrates, that there is no complete conventionalism in reality. Still more graphic this would be evidently, if Poincare examined not 4-, but 2- or 1- dimensional space as the permissible model of reality. What physics in the sum with the one-dimensional geometry would be equal to the sum of usual physics with the three-dimensional geometry?

Ehrenfest's analysis is not based on this preconceived position. And if Ehrenfest revealed that the spectrum of hydrogen in the space of another dimensionality more precisely will agree with the experimental data than three-dimensional spectrum, then this could give the foundation for assuming another dimensionality of space on atomic scales. Certainly, this assumption would require first of all an answer to the question: how to coordinate other “this” dimensionality with the undoubted 3-dimensional nature of space on macroscopic scales? A study of Ehrenfest does not affect this question, which is equivalent to the implicit assumption: and without the concrete realization of this agreement has sense to examine the specific, internal properties of the physical systems (similar to the spectrum of atom and stability of planetary system) under the assumption of a difference in the dimensionality from three. But the validity of a similar kind of studies assumes the idea about how a three-dimensional observer could obtain information about the physical phenomena, characterized by the dimensionality of space, different from three.

The historic importance of the work of Ehrenfest lies in the fact that, after raising the question about the sense of 3-dimensional nature and after correlating it with the concrete physical phenomena, Ehrenfest established the boundaries, in which the 3-dimensional nature has real physical substantiation and out of which it is only assumption. In Ehrenfest's analysis these boundaries were on top determined by the scales of the solar system, and from below by atomic scales. Out of these boundaries a question remained open. Thus, the dimensionality of space because of the work of Ehrenfest became physical concept.

At the same time it should be noted that the work of Ehrenfest unconditionally outdistanced his time until now, in physics practically it is not perceived the hardness of the framework of absolute 3-dimensional nature. However, if physicists will come up with the question about empirical status of the fact of the 3-dimensional nature of space, then they will have to recall that for the first time way to the establishment of this empirical status, he saw the correct way to the interpretation, the transfer of the concept of the 3-dimensional nature of space into the language physicists.

Ehrenfest's analysis was disseminated subsequently, in the first place, to Riemann geometry (motion “of planet” m dimensional generalization GR) [7], and, in the second place, as has already been said, Bohr quantization was substituted with the analysis of the n-dimensional Schroedinger equation.

Prerequisites of the work of Ehrenfest

About the 3-dimensional nature of space the historians of the science, until now, did not pay adequate consideration to Ehrenfest's article. And this not by chance. At first glance it seems that to the work 1917 can be attributed of Ehrenfest's word from his letters, addressed to a. F. Joffe in 1913: “… everything, that I, until now, knew how to make, was based on the love for the puzzles, the interest in any paradoxes, but not on the tendency to make anything „significant”; … in contrast to you, Ritz, Einstein and even Debye, I do not have main and solid ideological directions, there are no „ problems of my own”, and so, only „amusing problems”…” [8]. In these words, repeatedly expressed by Ehrenfest (and not substantiated) dissatisfaction with himself and insufficient fundamentality of his studies [9].

However, the work of 1917. is not nevertheless by chance; it is not by chance both for the general state of physics of that time and in the individual creation of Ehrenfest.

It goes without saying, the work of Ehrenfest on very posing of problems considerably outdistanced his time. However, it should not be supposed, that this work was isolated from the contemporary state of physics.

One of the most important features of the development of physico mathematical sciences at the beginning of 20th century. became the destruction of the ruled ideas about the Euclidean three-dimensional space as the only possible mathematical description of the properties of real physical space. By the manifestation of these previous ideas was the apriority of geometric concepts and identification of physical space with its mathematical model of Euclidean three-dimensional space. However, in connection with the creation of non-Euclidean geometry as a result of the work of Lobachevsky, Boljai, Gauss and Riemann mathematics of 19th century. the Euclidean geometry would be only one of the logically possible geometric systems. This great reaching of mathematics became known to the physicists as a result of popularization as well by mathematicians, as by physicists, especially sensitive to the problem of the empirical substantiation of the conceptual apparatus for physics [10].

This atmosphere, the sensation of the need to select and to physically base geometric description, undoubtedly, facilitated Einstein's way to the creation of special and in particular general theory of relativity, although, of course, the creation of these theories would be completely unthinkably without the physical causes. One of the starting points for Einstein became the analysis of empirical, physical status of Newtonian three-dimensional and temporary concepts.

In turn, revolutionary changes in the ideas about the space and the time, connected with the theory of relativity, only a short time after “the first defense line” of classical ideas was destroyed, they brought to the extremely large readiness of physicists new, even more profound alterations in the time-spatial ideas. It suffices to recall the extraordinary activity in this direction of those days: Weyl's theory, which geometrically unites gravity and electromagnetism; five-dimensional theory; the searches for unified field theories. Somewhat later this disposition appeared, for example, under the assumptions that the problem of nucleus can be solved only by radical changes in the geometry (discrete space) [11]. The relativistic quantum field theory cannot exist without radical changes in the concept of space-time [12].

In this geometric “fermentation of minds” the dimensionality of space did not remain untouched. It has already been discussed the five-dimensional direction. It is possible to note also the curious observation of Planck in his answer to the opening academic address of Einstein in 1914: “We most all see in this circumstance (second degree in the law of universal gravitation) the natural consequence of 3-dimensionality of our space, which we assume as fact and being reasonable physicists we must not be worried why space does not possess four or even larger number of dimensions” [13]. It is possible not to agree with Planck that any reasonable physicist must receive 3-dimensional nature as fact, without attempting to analyze, what bases force physicists to consider space three-dimensional as the only one. There do not exist boundaries in the region of the validity of this assertion. However, it is interesting that a question about the dimensionality of space proved to be significant even for Planck, in general not inclined to radical changes in the physical picture of the world.

Thus, the trends of the work of Ehrenfest can be considered in a certain sense characteristic for physics of that time.

The work of Ehrenfest was not random, also, in his own creation. It completely corresponded to its tastes and aspirations in theoretical physics. Furthermore, in the scientific biography of Ehrenfest it is possible to find such episodes, which make his turning to a question about the dimensionality of space more intelligible.

The concept of the dimensionality of space even is more fundamental than idea about the metric structure of space-time. The basic result of the theory of relativity consisted of the establishment of precisely metric structure; in this case fact 3+1-dimensionality of space-time was considered as the initial, subject to analysis. The fundamentality of the problem raised by Ehrenfest completely corresponded to his aspirations, which, however, more often can be revealed in his letters, than in his laconic, clear works, devoid of all methodological

observations, intended for others. One observation of this type nevertheless can be faound, and it occurs in a connected curious manner with the problem of the dimensionality of space.

In 1911 in the abstract “Magniton” Ehrenfest very clearly expresses his taste, calling true physics the introduction of order into chaos of phenomena with the aid of the fundamental, “illuminating” ideas.

In the “physical dictionary” of Jesuit Poliana (1761): “PHYSICS. This science has by object a body in its natural state, i.e. substance long, wide and deep. To examine, can being omnipotent The body, deprived of its three dimensions and the requirement of extent only, would be rather the object of metaphysics, than physics” [15]. As we see, in this uncommon determination the 3-dimensional nature it is considered as the most important sign of reality. Not this whether arbitrariness, can be subconsious, it did lead to the fact that Ehrenfest did think above a question about the dimensionality of space?

But there can be someone who did influence more actually to the appearance of a article about the 3-dimensional nature of space? Ehrenfest writes nothing either about the reasons, which impelled him to study or which works ideologically influenced it. The works, to which refers Ehrenfest, could not lead to the posing of problems.

Of the histories of science is known only one preceding to the work of Ehrenfest the attempt to connect the 3-dimensional nature of space with physics. These is Kant's hypothesis, which has already been discussed in the introductory chapter: “3-dimensional nature occurs, apparently, because substance in the existing world they act on each other in such a way that the effective force is inversely proportional to the square of distance”.

Apparently, this assumption of Kant in no way directly acted on Ehrenfest. Besides the fact that in the work of Ehrenfest there are no indications of this influence, should be noted the sufficiently careless attitude of Ehrenfest to the classical philosophy generally. In its correspondence and articles the names of philosophers or any references to their works barely are encountered. Exception is made for Mach. In Mach’s books, which attentively read Ehrenfest [16], in connection with the critical analysis of basic physical ideas actively popularized the achievements of mathematicians in the region of non-Euclidean and multidimensional geometry [17]. A question about the dimensionality of space was not central in the work of Mach, but nevertheless in it appeared, in particular, the question: “Why space is three-dimensional?”. Mach even assumed that in atomic physics the turning to multidimensional spaces is completely possible. It would be perhaps tempting consider the work of Ehrenfest as concrete (negative) answer to this prejudice, but there are no straight bases for this hypothesis. It should be noted that general philosophical position of Mach did not cause Ehrenfest's sympathy, who wrote to Joffe: “Especially interested me the criticism of idealistic philosophy (which, from my point of view, personifies Mach)” [18].

Was there not some more direct occasion for the reflections of Ehrenfest above a question about the dimensionality of space? But how to search for this occasion? The origin of thought, as a rule, occurs not in the official situation, and often the scientist himself does not realize completely, which occurred. Only after it nurtures, rears this thought to the viable state, the corresponding publication can appear as the evidence about the generation. In no article of Ehrenfest in 1917 there are no signs of his reflections about the dimensionality of space. Therefore it is necessary to turn to the letters. Among the published correspondence of Ehrenfest Ehrenfest's correspondence is of greatest interest with Joffe, first of all because of its volume and, in the second place, because Ehrenfest considered Joffe one of the nearest friends. However, in this correspondence there was interruption from 1914 through 1920 – but one phrase in a letter from 1912. makes it necessary to be on guard.

This letter Ehrenfest wrote on 4 November 1912 several weeks after he arrived into Holland

at Lorenz's invitation, in order to engage his department at the Leyden university [19]. The first weeks of Ehrenfest in Holland were occupied besides device at the new place with official visits and acquaintances with the Dutch scientists. Among others: “Friday of visit with the expression of thanks to Van der Waals (trustee of university!) — old man (far beyond 70), and is it seems more than nothing. Then Waals low-order — of approximately 40 — of 45 years. Gm… yes. With it it dashed to Brouwer assistant professor of mathematics in Amsterdam of — now to only mathematician in Holland, — is entirely young — pupil Getting- on — axiomatics, the theory of functions, set theory. On the whole this of men, to whom I should thoroughly be introduced. Everything, including Lorenz, speak with special ponderability about his salient endowment” [20].

Thus, in Ehrenfest's letter the discussion deals with l. Brouwer, one of the outstanding mathematicians XX v., and, which for us is especially important, about one of the creators of topology, the author of the salient works, connected with the topological concept of dimensionality. Moreover for years its greatest activity in this direction they were exactly of 1911-1913!

Brouwer was located under the effect of a. Poincare. This influence was manifested also with respect to the philosophy of mathematics and, which is especially substantial, in connection with the problem of the topological determination of dimensionality. In 1911 Brouwer proved the topological nonequivalence of En with different n in 1913 he found the correct mathematical definition belonging to Poincare’s idea of the inductive topological definition of dimensionality.

And here in the period between these two remarkable results of Brouwer in the topological dimensional theory he becomes acquainted with Ehrenfest. We assume that the object of their conversation was, in particular, occupying then Brouwer the problem of the dimensionality of space. This assumption is even more plausible, if Ehrenfest succeeded in carrying out the intention thoroughly to be introduced to Brouwer. But taking into account Ehrenfest's sociability, his greediness to the new, it is difficult to doubt the realization of this intention.

Thus, pulse for the reflections of Ehrenfest above the concept of the dimensionality of space could arrive from the topology with the aid of l. of Brouwer (and thus and a. Poincare).

But are five years not a too large interval between the initial pulse and the publication of work? The record, which Ehrenfest made in 1912 g. in the notebook, can be answer to this question: “Any question, which you are encountered during the reflection, either during reading or finally in the conversation, it is must be brief and clear to fix. From time to time similar questions it follows to examine, trying to somehow systematize them. It is first of all, pleasant to realize that you knew how to solve some of them, although initially hardly only could formulate them, — to say nothing of that, in order to develop! In the second place, we thus learn something misty and obscure to gradually convert into the clearly formulated questions" [21]. ]These statements illuminate his style of work. The long period becomes thus intellegible.

Measure of the hardness of the mathematical model of space-time. Value of works about physics of the dimensionality of the space

Assertion about the 3-dimensional nature of space with the aid of the conceptual apparatus of contemporary science is trivial in the sense that already Aristotle realized the fact of the 3-dimensional nature of space and expressed it correspondingly. But practically this fundamental property of space was known to the distant ancestors of man.

The millenia, which separate the twentieth century from the time of Aristotle, by no means made fact of 3-dimensional nature less known and clear. In this sense the proof of the 3-dimensional nature of space is superfluous. However, one should think that the 3-dimensional

nature is obvious for the man on the basis of his macro-experience, i.e. the totality of the phenomena, with which it it is necessary to contend in the daily life.

The sense of the work of Ehrenfest consists of the indication of the need for rechecking the fact of the 3-dimensional nature of space with the expansion of the region of the phenomena being investigated on “superhuman” scales. Atomic and astronomical phenomena exceed the limits of macro-experience, and the result of the investigations of Ehrenfest is the confirmation of the fact of the 3-dimensional nature of space in these phenomena. It could seem that the assumption about another dimensionality would lead only to quantitative changes, and then it would be necessary to compare the more thoroughly different cases. However, Ehrenfest established that difference in the cases in question are qualitative, and this facilitated the selection between the models of the space of different dimensionality.

The discussion deals with the measure of the hardness of the mathematical model of space-time, utilized in physics. Adoption of the model of three-dimensional Euclidean space by Newtonian physics is the result of macro-experience. This 3-dimensional nature practically without change moved to 3+1-dimensional pseudo-Euclidean model of the space in SR and quantum theory and into 3+1- dimensional pseudo-Riemann model in GR. Question naturally arises, will the assumption about the 3-dimensional nature (or the generally specific dimensionality) with a sufficient “removal” from the region of macroscopic phenomena prove to bee too rigid for physics?

After putting off to the chapter 5 the more detailed examination of possible answers to this question in contemporary physics, let us note now that some symptoms of this superfluous hardness are located. Of course the attempts was made to make bended time-spatial model appear from the concrete problematic (or paradoxical) situations. With this it should not be supposed, that all these problems arose recently. One of them was realized in the beginning of 20th century during the combination of classical electrodynamics and SR. These are the problems “of the self-energy of electron”, or, in the language of field theory, the problem of ultraviolet divergences. Other problems arose very recently in connection with the description of strong interactions at high energies, with “the retention” of quarks and the like although with respect to the problems of cosmology (or physics in the mega-scales) there is no such flow of experimental information as in physics of elementary particles, also here the main problem of cosmology is the problem of the initial state of the universe (beginning of expansion, initial singularity) which leads to an attempt to doubt the absolute nature of property 3+1-dimensionality and to realize these doubts about the concrete models.

It is interesting to compare the attitude of physicists toward 3-dimensional nature as the fundamental property of space, which is manifested, as showed Ehrenfest, in the fundamental physical laws, with the laws of conservation - two of the most effective tools of theoretical physics. Dimensionality is in a sense more fundamental than the laws of conservation. In the latter is “placed” the determinate structure of space-time, in particular, its symmetry and dimensionality. It is known that in physicists repeatedly appeared the assumptions about the disturbance of the law of conservation of energy-momentum (for example, in Bohr), to say nothing of the less fundamental laws of conservation, whose disturbance (under specific conditions) is already included in theory, for example parity nonconservation.

In physics there is for a long time a direction, connected with the ideas “discrete space” and “fundamental lengths” [22]. This direction attempts to find such time-spatial description of reality, which would not assume the continuous structure of space-time (more precisely saying, the local structure of E4). This direction is, generally speaking, not equivalent to failure of the absolute nature of the number of dimensions as one parameter, characteristic for all physical phenomena, and the concept of size. It is usually implicitly assumed that model, the locally Euclidean 3+1-dimensional model of space, are completely discrete, point, zero-dimensional, or, in the more physical language,

the completely quantized model of space. However, it is possible that in proportion to deepening into the microcosm of the dimensionality of space will “step back” gradually and that for some physical situations will prove to be 2+1- dimensional or 1+1-dimensional model of space-time to be appropriate. This change (decrease) in the dimensionality of space upon transfer to the micro- scales could become the consequence of the realization of the general idea of fundamental length and discrete space.

However, one main difficulty remains - the absence of the mathematical model of space-time, which possesses, from one side, a sufficient flexibility in order to describe possible changes and, from the other side that it would be possible to include the extensive apparatus for contemporary physics in the framework of this model (i.e. the region of the studied phenomena) .

With this difficulty is connected that Ehrenfest could not examine all possible p dimensional spaces, but he examined only simplest Euclidean spaces. He could not examine the mathematical model of space-time, in which the dimensionality would depend on the scales of phenomena. However, all this does not at all decreases the value of the Ehrenfestian analysis, after which physics was obtained the possibility to look at the dimensionality of space as a physical concept.

Chapter IV

APPEARANCE OF TOPOLOGICAL DIMENSIONAL THEORY AND PHYSICS

Prehistory of topological dimensional theory (Poincare, Brouwer, Lebesgue)

In the history of topological dimensional theory usually is separated the initial period, to which they carry the published into 1911-1913 of the work of Poincare, Brouwer and Lebesgue. In these works the topological concept of the dimensionality was formed: the idea of the inductive definition of Poincare's dimensionality, converted into the mathematical definition by Brouwer; the remarkable idea of Lebesgue — the prototype of the definition of dimensionality with the aid of the tiling; proof by Brouwer the topological N-dimensionality of the space of En.

However, the chronological framework of the first period.(1911—1913) one should move to the beginning of century in order to include the first appearance of the idea of the inductive topological definition of dimensionality and the physico psychological generation of the very concept of topological dimensionality (1901 — of 1902[gg].). Actually, if we consider as usually [1], the beginning of dimensional theory the article of Poincare 1912 “why space has three dimensions?” (in this work is contained the inductive definition of dimensionality, expressed in the mathematical language), then a question about the origin of basic idea, remains without an answer, since in this work definition of dimensionality is given immediately in the finished form, while in the work 1902 this generation occurs almost in front of the eyes of the reader.

It was in the second place, thus far considered that the idea of the topological definition of dimensionality was unknown to Poincare 1912 (year of his death), the question did not arise, why this idea did not convert into the correct mathematical definition - he simply did not have time to make this. However, the same question is not so easy to answer, if one considers that Poincare worked on the appropriate geometric idea of at least 10 years. The possible answer to this question is connected with the absence of the vital mathematical need for a similar definition. The mathematical objects (geometric figures, spaces) in question, as a rule, could be sufficiently simply assigned. This noted Poincare's himself, indicating that the topological approach to the dimensionality is dictated in essence by philosophical and physico psychological needs.

Before passing to the analysis of further development of Poincare's ideas, most clearly presented in his article 1912, one should return a little back, to the extremely important work of Brouwer and Lebesgue for the history of topological dimensional theory, published in 1911.

In Brouwer's article “proof of the invariance of the number of dimensions” was for the first time proven the general case of the topological nonequivalence of the Euclidean spaces of different dimensionality. This work, in the first place, completed a whole series of the studies, which contain either special cases (En with n<3), or only approaches to the solution of problem. Brouwer in this connection mentions the work of several mathematicians.

In the second place, Brouwer's article, dedicated to a completely special question, was strong point not only for constructing the topological dimensional theory, but in a certain sense and for the topology as a whole. If it turned out that during the topological mapping the dimensionality can not remain (understood, for example, as linear dimensionality) of geometric objects in En, then this would indicate that the class of topological (homeomorphous) conversions is too wide in order to have meaningful geometric applications. Indeed En and the figures in it are some of the most usual objects in mathematics.

In this connection it is possible to recall the remarkable result of Cantor - the proof of the set-

theoretical equivalence of the set of points of square (or generally cube in En) and set of points of a section. This result meant that it is not possible to determine the concept of dimensionality only in the set-theoretical language.

The work of Lebesgue was published in the same number of the same periodical as Brouwer's article, and immediately after it. A fragment from Lebesgue's letter to one of the publishers of this periodical. This three-page note begins with the words: “Recently, when you told me about the proof of the impossibility to establish single-valued and the discontinuous correspondence between the points of two spaces of dimensions n and n+p, the proof, which belongs to Brouwer, I indicated to you the principle of some proofs of this theorem. I will demonstrate to you simplest of these proofs”. This theorem, as Lebesgue writes, it follows from the generalization of one proposal of Jordan and formulates the remarkable theorem: “If each point of n-dimensional region D belongs at least to one of the closed sets E1, …, Ep and if these sets are sufficiently low, then common points are had at least n+1 from these sets”.

True, the initial proof of Lebesgue proved to be unsatisfactory, and for the first time this theorem was completely proven in the work of Brouwer 1913 “about the natural concept of dimensionality”. In this work, which completes the first period of the history of topological dimensional theory, Brouwer turns himself to definition of the dimensionality created by Poincare: “Continuum is called n-dimensional, if with the aid of one or several n-1-dimensional continua it is possible to decompose it into separate parts”. He shows the impossibility to use this definition directly in the given form and refines it in two directions. First, he refines the concept “continuum” and, in the second place, refines the words “with the aid of one or several”. In this case he gives a simple example (double cone), when a set can be divided by one point, but it is clearly not one-dimensional.

The definition of dimensionality in Brouwer is such: Space is n-dimensional if it can be separated by a n.1-dimensional subset, but not by any subset of lower dimension. Brouwer calls the space, which contains no continuum, zero-dimensional. This is the initial point of inductive chain. Then Brouwer sollves the deficiencies in Lebesgue's proof for the theorems about the multiplicity of tiling of cube in En, and proves this theorem, also.

The first period in the history of topological dimensional theory was completed by this work of Brouwer 1913. In connection with this the questions arise: why Brouwer did not pose the problem of constructing the general dimensional theory after solving the problem of dimensionality in En? Why did the following essential advance occur in 1921? Why did Uryson and Menger carry out this advance? Discussion of these questions is naturally connected with the examination of the role of Uryson and Menger in the creation of principles of dimensional theory.

Uryson & Menger - the creators of phonological dimensional theory

Uryson (1898-1924) and Menger (1902-1985) - the creators of the first mathematical theory, based on the concept of the dimensionality of space. They revealed the possibility to extend the concept of dimensionality to the broad class of geometric objects and “they justified the new concept, after making its cornerstone of the extremely beautiful and fruitful theory, which introduced unity and order into the large region of geometry” [3].

The definition of dimensionality, whose idea arose in Poincare and which acquired present mathematical form from Brouwer, is called at present the definition of the large inductive dimensionality. Once in this name there is a word large, it is reasonable to assume that there is also small inductive dimensionality, and since into name enters also word inductive, we assume that there is a not-inductive dimensionality. So it is in reality. To “noninductive” dimensionality we were introduced in the first chapter. This the dimensionality, which ascends to

Lebesgue's theorem and is defined as minimum multiplicity of sufficiently small tiling.

However, Uryson and Menger as the basis of topological dimensional theory placed such definition, which is now called the definition of small inductive dimensionality: the space X is called n-dimensional at point p, if the point p has small environments, whose boundaries have a dimensionality, not more than n-1, but there are no small environments, whose boundaries have a dimensionality less than n-1. The initial point of inductive chain forms the empty set, to which is assigned the dimensionality –1.

Uryson and Menger at first did not consider that there are three different definitions of dimensionality. It was considered that there is a definition of dimensionality and its two properties, whose existence was proven for the sufficiently broad class of topological spaces. Actually this indicated (and it was one of the most important first results of topological dimensional theory) that for the broad class of the topological spaces X coincide three invariants. [razmernostnykh] invariants: IndX=indX=dimX. Only subsequently these three values were considered as equal; bases for this appeared at the end of the fourties, when the first examples were built with the noncoincident values of the dimensionality.

The surprising situation arose: to one, it would seem, the concept of dimensionality could be defined by three different, generally speaking, values. Already this alone it could cause doubt about the fitness of topological language for describing the physical concept of dimensionality. However, we already met with a similar situation of – one concept of the continuity of space is described by mathematically different methods.

However, in the twenties during the construction by Uryson and Menger in topological dimensional theory this ambiguity was not yet known, and the topological concept of dimensionality could seem to be the natural mathematical refinement of the physical idea about the dimensionality. This observation is appropriate because when you become acquainted with the biographies of the remarkable mathematicians Uryson and Menger, special attention draws their completely uncommon interest in physics.

The first scientific publication of outstanding mathematician Uryson was dedicated to physics, and, which is especially interesting, to experimental physics. In 1915 in “the periodical of Russian physico chemical society” was published summer Uryson's article 17- “X-ray radiation of Coolidge tube” [6]. In the article on seven pages the results of a experimental study of the dependence of the frequency of X-radiation on the stress, applied to the electrodes of Coolidge tube are presented and are discussed (type of X-ray tube). In the beginning article the author mentions several work of different authors, then in detail he describes the experiment itself and presents the obtained results (with the aid of figures and the tables). That he will be an outstanding mathematician in the future is almost impossibly; we would rather predict the future of thorough, accurate experimenter, so thoroughly he described the components of experiment.

However, one should nevertheless note that there is no evidence about any strictly mathematical interests of Uryson before his entering in 1915 to physico mathematical department of Moscow University. Moreover, according to the evidence of Aleksandrov, Uryson at first intended to become physicist and only under the effect of the lectures of Yegorov and Luzina occurred the turning of his scientific interests to the side of mathematics. Uryson entered into the graduate study under his management in 1919 at the proposal of Luzina. From the diary of Uryson [7] it is evident, he mastered the new mathematical material. He mastered actively, revealing errors and weak places in the work of venerable authors. The exceptionally fruitful period of the mathematical creation of Uryson began from this. This period, one of highest achievements of which became the dimensional theory constructed into 1921-1922 , continued only four years and was broken on 17 August of 1924 of as a result of an accident Uryson perished.

The not completed by Uryson works, many of which existed only in the form rough drafts or even only in the oral form, were prepared to the press by Aleksandrov. Subsequently works of Uryson with the extensive notes were assembled and published in 1951 “works on the topology and other the fields of mathematics”.

l. Brouwer participated in the preparation of Uryson's manuscripts for the publication by, who, in spite of skeptical relation to the point-set topology generally, very highly valued Uryson (this is, however, not surprising, since the work of Uryson were pierced by remarkable geometric ideas and they are by no means reduced only to the axiomatic side of general topology). In connection with the tragic loss p. s. Uryson Brouwer wrote his father: “I fell in love with your highly talented and amiable son, that easily I can understand your loss is heavy. For the mathematicians his death is a unique loss. He would be the most outstanding mathematician of our time. I survive this loss together with you”.

The first work of Menger he deposited in the Viennese academy of sciences. It relates to the autumn of 1921, when its author was 19 years old.

What could stimulate searches by Uryson and Menger dimensional theory? What did help them to see naturalness and need for the stated goal— to determine the concept of dimensionality for the broadest possible class of the mathematical models of space? To these questions it is possible to propose the answer, connected with the examination of interaction of physics and mathematics.

Role of physics in the mathematical creation of Uryson and Menger

It is important to note that in winter 1922/23 Uryson thoroughly and with the large enthusiasm studied the theory of relativity. He obtains means for the mission by Aleksandrov abroad together with after reading a cycle of four public lectures by the name “about the mathematical knowledge of the world in light of the theory of relativity” during January 1923.

In the autumnal semester the course was called “the mathematical bases of the law of relativity”, in the the spring of — “the mathematical bases of the theory of relativity” [9]. The large volume of this course (on 4 hours every week) gives grounds to think that in the course the general theory of relativity was included. This confirms also the well-known Soviet mathematician Menshov.

As is known, the theory of relativity examines the structure of physical space-time in connection with the physical processes. It is possible to think that for Uryson word space in his works on the topology and the word space in the physical theory of relativity they were not altogether only homonyms. Although in the work of Uryson according to the dimensional theory there are no direct indications of the possible connection of this theory with physics, his separate observations, apparently, testify about this. Let us examine several such places in the basic work of Uryson according to the dimensional theory “memoirs about the Cantorian manifolds. Dimensionality”.

Aleksandrov writes that Uryson's way to the dimensional theory began from the fact that Yegorov in 1921 placed before him the task of the internal definition of the total sets, which it would be possible to name lines or surfaces.

Uryson calls the n-dimensional jordan manifold the set, topologically equivalent to n- dimensional cube in En. At the very beginning memoirs Uryson formulates the problem: “To give the purely geometric definition of n-dimensional jordan manifolds”, i.e. the definition, which is not reduced simply to the possibility of a certain mapping. He writes that “the purely mathematical interest of these problems is considerably less than their philosophical interest (which is completely significant)…”. It is possible to think that here the sense of word “philosophical” is close to the sense which was packed into this word by Poincare, speaking about the dissatisfaction of philosophers with “arithmetical” definitions of dimensionality. The

dimensionality of a manifold is reduced to the need for explaining the internal reasons for the possibility of corresponding topological mapping onto En.

Then Uryson approaches the problem Kn(“to indicate the total sets, which can still be named lines, surfaces, etc”), noting that in this case there is no “poor” definition. His entire memoirs is dedicated to problem Kn. One should emphasize that to contemporary physics (both in 1921 and in 1982 ) much larger relation has exactly the problem Jn, or more precisely J3 and J4, since Riemann the manifold of the general theory of relativity one of the deepest today theory of the physical space-time is locally is J4.

Summing up the sum to the description of the obtained by the results in our introductory chapter, Uryson writes: “… the obtained results make it possible for the first time to spill a certain light to the not explained question, what is this number of dimensions. It seems to me that the answer, which I propose, is completely suitable to the concept of Poincare”. Uryson refers to the work of Poincare 1912, in which the problem of the number of dimensions is placed not as purely mathematical problem, but as physico mathematical.

Uryson called the n-dimensional Kantorian manifold n-dimensional continuum, which remains connected after the removal from it of any set of a dimensionality less than n-2, i.e. to a high degree topologically uniform space.

In the introductory chapter Uryson indicates “there is a difference between the integral definitions (being concerned with the integral properties, i.e., of the properties of set as a whole) and the local definitions (relating to the properties of a set in environment of one of its points)”and we give preference to the local definitions: “. This escapes from the following obvious observation: the concept (until now, definition) behind the number of dimensions is the jordan manifold (i.e. the topological means of cube into En), whose definition would make it possible to recognize lines, surfaces, bodies, etc. is the integral concept of local origin.

Words “obvious” and “as easily to consider” do not have, of course, evidential effect, and for the more general common class of topological spaces, then examined by Uryson, primary meaning has the nonlocal definition of dimensionality with the aid of the tiling (not coinciding, generally speaking, with the inductive dimensionality). Thus, the discussion deals with the subjective limitation, whose origin can be connected the with the fact that fundamental language of physics up to the present time, and already in any case and the language of classical physics, is the language of differential equations.

In the first works of Menger according to the dimensional theory distinct physical undercurrent does not succeed in revealing. Possibly, partly this is connected with the fact that these works are small notes in contrast to the main work of Uryson, in which there is the extensive introductory part, which contains observations, in particular, extra-mathematical and autobiographical nature. However, at least in the later work of Menger one cannot fail to note sufficiently the distinctly expressed interest in physics.

In this sense the article of Menger is characteristic: “theory of relativity and the geometry” in the volume, dedicated to the 70-anniversary of Einstein. In it Menger discusses, in particular, the possibility of use in physics instead of is Riemann the manifold of more general common spaces; in this case special doubt causes the properties of continuity. In the article is evident if not the profound knowledge by Menger of the vital problems of physics, then the completely physical motivation of his mathematical creation. As an example it is possible Menger's acquaintance with the famous article of Minkowski 1908 : “… I dare to develop Minkowski's idea about the fact that the laws of nature can find their most perfect expression in the establishment of the relations between the world lines” [10].

The concept of statistical metric space Menger introduced in 1942 so he named the set, for any pair of elements of which was assigned the distribution function, which generates “the distance” of the random variable satisfying the conditions, which generalize the usual postulates of the metric

space: nonnegativity, symmetry, nondegeneracy and triangular inequality. As one of the possible applications of this concept Menger directly indicates physical microcosm. Physics noted work on the statistical metric spaces, although no real physical theories were based on this idea, until now.

Why was ist possible for Uryson and Menger to build the first dimensional theory? Without diminishing the value strictly of the mathematical stimuli of creation the, it can be assumed that natural-science, physical interests of Uryson and Menger could help them during the construction of dimensional theory. This aid as the realization of naturalness and need for the stated goal of the construction of the general dimensional theory could be achieved, for example, as follows.

Knowing much the more common mathematical models of space, than Euclidean, it was easy to see the specific arbitrariness of assumption about the Euclidean three-dimensional (or even pseudo-Riemann four-dimensional) structure of physical space (space-time). This insufficiency of the Euclidean model of physical space was more evident after the failure of Einstein’s SR with its Euclidean structure of physical space (true, only in the global, but not on a local scale). It was possible to consider the also general fact of the limitedness of physical information the properties of space, obtained from the [ogragshchennogo] (in particular, to the three-dimensional scales) range of physical phenomena. Therefore natural could be the tendency to build a maximally common (or at least more general common than E3 or R3+1) mathematical model of real physical space. The task hence emerges: to build the sufficiently common mathematical model of space, in which the concept of dimensionality preserves sense.

The perhaps, proposed hypothesis is formulated too rectilinearly. But, on the other side, it is untypical for the mathematicians of 20th century, who are occupied by this abstract field of mathematics as topology, natural-science interests of Uryson and Menger.

In this connection one cannot fail to recall also mathematical creation of one of the greatest mathematicians of 19th century: Riemann's, which he received to the, apparently, subordinate physical purposes [11]. The action of the physical attitude of Riemann on his mathematical creation did not become less fruitful because he could not build united physical theory.

It is known that the stimulating idea is not entirely “obligated” to be correct in narrow the practical, immediate sense of word. In this case, in spite of half century intensive development of most topological dimensional theory, it is necessary to establish that the physicist of space-time these theories directly in no way proved useful, until now. This brings us close to the more general common problem, examined in the following chapter.

Chapter V

DIMENSIONALITY AND CONTEMPORARY PHYSICS

In what sense is isolated dimensionality of space-time, equal to 3+1?

The fact 3+1-dimensionality of physical space-time was established after Ehrenfest with relation to the concrete physical phenomena, it arose in one row with many other physical facts according to the degree of validity, but, of course, not according to the degree of fundamentality. However, dimensionality of space proved to be substantiated by different phenomena: the spectrum of atom, the propagation of electromagnetic waves, motion.

It is not surprising therefore the striving to reduce dimensionality to the deeper properties of real world, i.e. to find such a property, which (within the framework specific theory) would give 3+1-dimensionality and other fundamental physical positions, which are considered now independent variables.

A similar tendency “to derive 3-dimensional nature”, as we saw in the introductory chapter, was even found in Kant’s work. Usually the attention is not turned to this attempt; however with the young Kant began his analysis. He attempted, as we remember, to connect the 3-dimensional nature of space with the fact that 3 is the smallest number, which precedes composite 4=2*2.

After the theory of relativity brought idea about 4-dimensional space-time, number 4 became the object of analogous reflections. In particular, about this number reflected English astrophysicist Eddington (1882-1944), to him belong not only the first experimental confirmation of GR in 1919, but also numbers of fundamental theoretical results. Furthermore, he wrote several remarkable scientific popular books.

In one of them Eddington attempted to establish fact of 4-dimensionality with the following observation (of humorous nature). He focused attention [1] to the fact that any measurement of length assumes the presence of two extensive objects, for example two measuring rods. But since, as is known, each stick has two ends, we obtain, that the total number of ends is equal to number 4. Of no more intelligible “dimensional theory” Eddington succeeded in devising, although this theory it greatly required.

Another example gives the work of the contemporary American physicist D. Finkelstein [2], in which the attempt is made to build the model of quantum space, which passes within classical limit into the space-time of GR. In this work a certain binary code and dimensionality of space-time are somehow connected. If this approach led to the viable physical theory, then the trivial relationship of 2X2 = 4 would prove to be the connected with the fundamental property of the material world and dimensionality of space-time, brought to the quantum properties of microcosm. One should emphasize that the work Finkelstein is not aimed at obtaining the value of dimensionality. Here 3+1-dimensionality is only one of the consequences of the assumed theory. And this is, apparently, the only possible approach to “the conclusion” of dimensionality, what cannot be said about the concrete specific realization of this approach. This cannot be said at least because there are known many other hypotheses and preliminary constructions, which have analogous purposes and also did not come to the state, which would make it possible to raise the question about their experimental check.

However, regardless of the fact, if it will be possible to realize the conclusion of dimensionality or not, the attempts are undertaken to comprehend isolation 3+1-dimensionality. It is clear that the substantiation of this isolation by purely mathematical means is completely hopeless. And not because there are no mathematical facts, which release the number 3 (or 3+1). There are too many such facts on the contrary. The mathematical property, to which Kant focused attention (that 4 is the first composite number) is not the only, and not most interesting

property of this type. There is a entire collection of mathematical assertions, which release not only number 3, but even a dimensionality of space, equal to three.

But it is possible to indicate such mathematical assertions, which separate other values of dimensionality. Therefore up to the appearance of the physical theory, in which some one of these mathematical facts will become the physically intelligible assertions, it will not be, apparently, the foundations for replacing the straight statement of the 3-dimensional nature of space by some mathematical circumstance, which releases the number 3. But also this, inevitable now postulative nature of assertion about the 3-dimensional nature of space leaves possibility for the analysis.

In this case the dimensionality of space appears in the form of specific parameter, which characterizes (together with other cosmological parameters and fundamental physical constants) our metagalaxy. The fact of the 3-dimensional nature of space is connected with the fundamental properties of the material world. However, these connections remain implicit to those times, where three-dimensional, physics is examined itself, but not against the background of others n-dimensional “physics”. Only examination of 3-dimensional nature as one of the possible assumptions about the dimensionality of space makes it possible to reveal the deep connections of 3-dimensional nature with other most important structures of physical world [3].

Understanding dimensionality as a certain cosmological parameter leads also to the known problem of the uniqueness of the model of the universe. This problem, not solved within the framework of SR, can be formulated as the problem of the selection of initial conditions for Einstein's equations in the cosmological task. The usual formulation of the physical problems assumes the possibility of the task of arbitrary initial conditions, and these different initial conditions can be realized in actuality. However, the universe exists “in one copy”, and therefore the arbitrariness of initial conditions in cosmology loses usual meaning.

This situation led, in particular, to the assumption that the set of the parameters (including the dimensionality of space and value of fundamental constants), which can be considered as initial conditions, does not characterize the entire universe as a whole, but only one of its many parts – a metagalaxy; in this case the set of the parameters (and, in particular, 3-dimensional nature) is declared random[4].

Attempts were made to solve the problem of cosmological initial conditions (but also of today's cosmological parameters) with the aid of the joint calculation of SR and the quantum regularities (“the principle of ignorance" [5]). By this method it is intended to obtain as the most probable real cosmological parameters, in particular the proximity of the critical value and the thermal nature of cosmological background-radiation. It is possible to attempt even 3+1-dimensionality of space-time to also obtain as the most probable value.

Since here the discussion deals only with the most probable (but not unambiguously specific) value, the question seems legitimate: why nevertheless our space-time is 3+1- dimensional? The most attractive answer to this question now is, apparently, such: space-time is 3+1-dimensional because this same question can naturally arise only in this space-time. In this case those giving this answer assume that the system, compared with the complexity of man, who poses to himself similar questions, can arise only in 3+1-dimensional world. 1- and 2nd dimensionality of space are excluded as insufficient for any complicatedly arranged nervous system, and more than 3-dimensional leaves no room for the biological evolution on a planet, which stably moves around the star [6]. As showed already Ehrenfest, the stable motion of planet is possible only in the three-dimensional space.

In other words, it is proposed to consider that the searches for answer to a question, why our space is three-dimensional, not are more than (but also not less) [osmyslenny] than the searches

for answer to a question, why our sun of — precisely such type star (but not white dwarf or the red giant, for example). Answer to the second question, obviously, consists in the fact that the white dwarf or the red giant would not ensure the conditions, necessary for the appearance of the civilization, in which could arise question itself. This substantiation is advanced apropos and other cosmological parameters and relationships [7].

However, as far as dimensionality is concerned, it is difficult to recognize not only the answer, but also the formulation of a question as completely satisfactory. In particular, the assertions about the fact that in the space of another dimensionality the atoms would be unstable and the appearance of life and reason is impossible, acquire some sense only during the fixation of the nature of physical laws (principle of superposition, the form of dynamic equations, etc). But indeed if the discussion does not deal with the real observed physical space, but only with the possible, but it is not possible to say anything about “possible” physics. It is this qualitative difference from the Eherenfestian posing of the question about the dimensionality of the observed real physical space under different physical conditions.

Physical space-time and the topology

Let us turn now to the question, which in explicit or implicit form appeared in the previous chapters.

Why did topological dimensional theory not "prove useful” to physics? Indeed the greatest generalization of the ideas about the dimensionality of space, utilized in physics, gives topology. This, it would seem, must lead automatically to the fact that a question about the dimensionality of space must be assigned to nature in the topological language. This is assumed in some works, dedicated to this problem [8]. In this case the topological properties of physical space are discussed, the topological definition of dimensionality, etc.

This position simplifies and distorts real situation, since at present no methods are known for constructing physics, which preserves (in the sense of the correspondence principle) at least the elements of usual physics in the space more general common than the so-called manifold. But manifold is locally arranged exactly as Euclidean space. Therefore from the point of view of contemporary physics it is not possible even to formulate a question about the dimensionality of space, concerning the generality being differed significantly from Ehrenfest approach. In this approach, let us recall, were used only Euclidean spaces of different dimensionality, but not more general topological spaces.

Another, more essential circumstance lies in the fact that the concept of topological space, which was appeared (in practice in contemporary form) in 1914, is not used for describing the real physical space. It will poorly agree with the relativistic quantum theory. The concept of topological space implies as the primary concept „point“. But the concept of point, which ascends even to the Euclidean “that what does not have length”, seems incompatible with the language of the relativistic quantum theory, in which the uncertainty principle together with the relativistic possibility of particle production limits the absolute accuracy of the definition of time-spatial sizes.

From similar considerations arose the hypothesis of minimum, or fundamental, length [9]. This hypothesis caused the appearance of an entire direction in theoretical physics, which includes the following pproaches and some concrete models: “discrete space”, “quantized space”, “length element”. The first works of this direction appeared in the thirties, through several years after the construction of quantum mechanics, and they had a purpose to solve the problem of infinity of the electromagnetic mass of electron and even problem of the structure of atomic nucleus. The problem of nucleus was solved, as is known, by more traditional (from a present point of view) means; however, the idea of discrete space repeatedly was renewed in the different forms up to the last time [10,11].

This direction is most naturally called the problem of fundamental length, since the

introduction of certain fundamental value with the dimensionality of length into physics was the general element of all attempts. At the same time concrete methods of the introduction of this value are essentially different and each is individually vulnerable for criticism.

It is easy to understand, even without examining concrete models, that assumption about existence of a fundamental length will not be coordinated with the idea about the fundamentality of the topological model of space. First, the usual concept of the point proves physically to be senseless - and, in the second place, the introduction of fundamental length is possible only in the metric language, which in the usual mathematical hierarchy of structures is less fundamental than topological.

But metric language is not obligated to be reduced to the usual concept of the metric space (this, however, follows already from the fact that the time-spatial structure, examined by the theory of relativity, is not metric space, since in SR naturally appears only distance as two-point function). It is possible to assume that this new metric language must somehow generalize the concept of interval in SR. In the spirit of quantum ideas metric function can become an operator, and then, possibly, topology will arise after the operation of the corresponding averaging or it will arise immediately from the spectrum of topologies.

The indirect illustration of these statistical metric spaces, is considerably later introduced into mathematics than the usual metric spaces (Menger, 1942). Topology in this space can be introduced by several nonequivalent ways, i.e. with one and the same metric structure (understood already in the generalized sense) it is possible to coordinate nonequivalent topologies. Thus, the traditional hierarchy of topological and metric properties is disrupted in mathematics. So why should one insist on it in physics?

There are doubt about the feasability to solve the problem of the dimensionality of real physical space by the analysis of its topological properties. When they speak about the topological structure of real space, they do not identify physical space and certain (inaccurate) connections with its model?

In connection with this one should remember that the known hierarchy of “spacelike” properties (in which in a specific manner the described metric properties are less fundamental than topological) it arose in the beginning century without knowledge for the domain of quantum phenomena.

Is it possible to be confident that the mathematical concepts, which are not influenced by quantum physics, can satisfy all needs of a new theory? To hope for this corresponding attempts were made possible,(the general theory of relativity, in which the concept of Riemann space is used, an example of the success of a similar attempt), but it cannot be considered that mathematics deliberately contains all concepts, necessary for contemporary physics.

Nonconformity between the topological ideas about the dimensionality of space and the needs of physics can be illustrated as follows. In physics there are foundations for considering the following assertion for the phenomena of sufficiently large time-spatial scales space-time is 3+1-dimensional and at the same time for sufficiently small scales it is discrete [11]. Another discussed possibility lies in the fact that at short distances it is not possible to generally introduce the specific dimensionality [12].

However, within the framework of topological ideas about the dimensionality of space it is not possible even to simulate a similar situation, i.e. to agree on 3+1-dimensional space (“in the large”) and possible discrete (“in the small”) structure of space and to consider the possibility of dimensionality changes with a change in the scale of phenomenon.

In order to explain the aforesaid, let us examine a simple example. Let the set D03(a) of infinite

cubic lattice with the step a in the three-dimensional Euclidean space E3, i.e. the point set with

the coordinates (k1a, k2a, k3a) where the k’s ar integers. It would be possible to expect that “the space”, with the sufficiently small a would approach sufficiently well to the properties (and, in particular, on the dimensionality) E3. However, a similar assertion is meaningless from the point of view of topology, since in view of topological definitions the dimensionality “of the space” D0

3(a) is equal to zero - independently from the values a. the topological definition of dimensionality is local, and cannot “note” the passage through some chosen scale a moreover, even space, which consists of all rational points of E3 (i.e. the points, which have all coordinates with rational numbers), it is topologically zero-dimensional, although this space seems generally physically indistinguishable from E3.

The example examined testifies about the impossibility to give sense to topological ideas like the assertion about the dependence of dimensionality on the scales of phenomenon from within the framework.

But perhaps not only topological, but also any other mathematical formulation of the concept of dimensionality generally (as the saying goes, by nature) cannot realize the above-indicated wish of the common sense? Let us examine ideas about the number of dimensions which are correlated with the topological approach to the dimensionality.

The creation of the general theory of relativity - one of the deepest physical theory of the space-time today - did not require the mathematical models of space, more general than manifold, and ideas about the dimensionality, more general than a quantity of coordinates. Even in contemporary physics there are no real alternatives to the coordinate description, which makes it possible to examine the behavior of any physical system. Although the concept of the dimensionality of manifold has topological nature, there is no need for the general topological definitions of dimensionality in the case of manifold, since the very structure of manifold automatically assumes the specific dimensionality.

Topological approach to the dimensionality would be promising, if the development of physics (and first of all realizing of complete synthesis of SR an quantum theory) led to the failure of the examination of the metric structure of space and adoption as the basic object the study of its topological properties; in this case the turning to the topological definitions of dimensionality would be unavoidably. However, at present this course of events seems improbable; in particular, evidently the values, which could play the role of the dynamic variables, which describe state must somehow resemble space-time. Therefore for the development the theory of space-time it seems desirable the generalization of the concept of manifold, the generalization of metric, the coordinate structure of space (concept of interval and metric tensor gik) with the failure of the the local structure of space, identical to the structure of E4, then the concept of dimensionality could remain connected with the concept of a quantity of coordinates, since only coordinates have been understood in the generalized sense.

Physics actually ties us to parametric representation about the dimensionality of space. “Working” physical concepts of the number of degrees of freedom, a quantity of dynamic variables in the equations of motion, a quantity of components of physical field, prompt the need for the realization of parametric representation of dimensionality, or, it is better to say, about the number of dimensions of space. This idea would be actually connected with “the number of dimensions” - necessary for the description states of physical system.

As Cantor already showed, if we limit the method of parametrization in no way, then to realize parametric approach to the dimensionality is impossible. Topology (in connection with manifold) limits the method of parametrization by the requirements of one-to-one correspondence and continuity. But this is the only possible type of limitation. If we consider the metric structure of space as the given one, then the method of the parametrization of the points of space can be limited by requirement so that parameters themselves would make metric sense.

In order to specifically formulate this limitation, let us focus attention on the following property of the n-dimensional Euclidean space En: in order to assign the position of point into En with the aid of its distances from the collection (basis) of some fixed points, it is necessary that in this collection there would be exactly n points. A smaller number of points in the basis is insufficient, since the set of the corresponding distances separates in the general case an infinite quantity of points (for example, into E3 any set of two distances gives a circle – which is the intersection of two spheres); any larger number of points in the basis is excess. The n-dimensional pseudo-Euclidean space (Minkowski's space-time) possesses the same property; in this case the role of distance plays interval.

As a result we obtain the possibility to limit the method of parametrization to the fact that it is permitted to use its distances from other points only as the parameters of point. Generalizing this construction, it is possible to introduce the following concept of dimensionality: I will name a region in the space n-dimensional, if for its parametrization in the above-indicated sense are necessary bases of n points.

Thus, the calculation of the metric structure of space gives the possibility to introduce the concept of dimensionality, more flexible than topological. But, it goes without saying, mathematical realizability alone of the complex structure of space does not make it physically meaningful. Similar attempts are physically justified, if the signs of the superfluous hardness of the usual mathematical model of physical space would be revealed.

3+1-dimensionality of the space of time is reliably substantiated not only in the region of macro- scopic phenomena (“ordinary” experience“), but accrording to the Ehrenfestian analysis and in the much broader band of the physical phenomena: from the atomic to the astronomical scales. Therefore deviation from 3+1-dimensionality can be discussed only with respect to the phenomena out of this range, to the extreme states of substance. Such states are examined in physics of elementary particles and in cosmology. Attention deserve only such outputs “beyond the framework”, which are connected with the unresolved physical problems.

Before we examine from this point of view the situation in physics of elementary particles and in cosmology, it is worth to contemplate the question: are there limits of the geometric description of physical reality?

Physics and the geometry

Dimensionality is a geometric concept. If it is permitted to doubt the absoluteness of the fact of the 3-dimensional nature of space with the expansion of the range of the phenomena studied by physics, then indeed it is possible to doubt also the applicability of geometric concepts generally for describing the physical reality with a sufficient removal from the region of macroscopic phenomena. What is possible to say about the interrelations of physics and geometry?

Examining the history of these interrelations, it is possible to come to two conclusions. There is no doubt that in geometric description always limits were revealed. And, in the second place, just as there is no doubt that the geometric description did not have no limit; always after exclamation “the king died!” it was heard “yes the king is in good health!”, and to the throne was raised new geometry.

The geometric description of Newtonian physics, in which it is possible to work with E3, stumbled on the limits during the sufficiently deep penetration into the region of electromagnetic phenomena. Moreover, despite the fact that the theory of electromagnetic field was created actually in the sixties of 19th century, the realization of the need for replacing Newtonian ideas about the space and the time was achieved only in the first decade of 20th century. But even after the creation of the theory of relativity and the concept of Minkowski space-time M3+1 the revolution was not universally recognized. Even Einstein at

first underestimated Minkowski's geometry, considering it only as a formal method. Only when Einstein began to work at the construction of the theory of gravity, matched with the theory of relativity, he revealed that the point of view of Minkowski makes advance possible. True, this advance led to the detection of the limits of geometric description by the force of its own logic soon. As it proved to be, sufficiently deep penetration into the region of gravitational phenomena requires the replacement of Minkowski's geometry M3+1 to Riemann geometry K3+1.

Who could see in 1865 anything about the future geometric revolution, connected with the special theory of relativity (implicitly hidden) in Maxwell's equations? Who could see, that the fact, known already by Galileo, that in the gravitational field the motion of bodies does not depend on their properties, created the need for passage to Riemann geometry of the bent space-time? No one could see in the ambiguity of the value of electromagnetic potential the hint to the change in the geometric component of physical theory.

As it was explained relatively recently, all four fundamental interactions (strong, weak, electromagnetic and gravitational) possess the general property of the so-called calibration symmetry; the first manifestation of this property was the possibility to describe electromagnetic potentials, without changing the observed characteristics of field (tension). This generality of all fundamental interactions greatly strengthened the hope in physicists for the construction of the unified theory of all interactions.

The Story about the theory of calibration would take us away us from our main theme. Therefore let us limit to the most necessary. The geometry of space stratification, which composes the mathematical language of the theory of calibration, takes for the basis Minkowski space-time M3+1, but the concept of world point, or event changes. Each event now is characterized not only three-dimensional and time coordinates, but also by additional N values. (N can be different in different variations of the theory). Point now no longer is “that that it does not have parts”, but the entire world, arranged by the well-defined means; it is possible to say that the role of each point now fulfills certain N- dimensional (internal) space S(N). Exactly as passage from one coordinate system to another in the Euclidean space does not change the geometric interrelations of different figures, so a change in the coordinate system, or gauge transformation, in each internal space there must not change physical situation.

The geometry of space stratification we will designate by the symbol M3+1ΧS(N); the sign of work here makes much the same sense as in the designation of Newtonian space-time E3ΧT.

On the basis of the geometry of space stratification it was possible to build the satisfactory and even honoured by the Nobel Prize (Weinberg, Salam, 1979) theory, which unites electromagnetic and weak interactions. The calibration theory is developed from the strong interaction of so-called quantum-chromodynamics. And finally even now the outlines of a new theory are planned, which unites all four interaction modes. As the basis of this theory, called super-gravity, there is a geometric description, which unites the features of geometry GR i.e. R3+1 and the geometry of space statifications.

The discussion deals with ideas, which are in the stage of development. The discussion deals not only with the super-space, but also with the super-immense purposes. Indeed together with the construction of the unified theory of all interactions must be solved the problem of the quantization of gravity, since association must be achieved on the quantum basis.

From the numerous physical constants (including, in of particular, the of mass of of numerous of elementary of particles, the of value of of elementary of charge, etc) we see only three constants: the of speed of of light, the gravitational constant, and planck's constant. These the three constants which occupy special position in physics.

Then there is so-called “cube of theories”, along the axes of which of there are plotted the coordinates of the three constants.

Let us give examples: theory with the coordinates (0,0,h) these are quantum mechanics, (0, G, 0) — the Newtonian theory of gravity, (c-1, 0, 0)=SR, (c-1, G, 0) GR, etc, or, somewhat otherwise, quantum mechanics can be called h-theory, the Newtonian theory of gravity G- theory, SR c-theory, GR cG-theory, the relativistic quantum theory of gravity cGh-theory. In this three-dimensional cube of theories the idea of the space of physical theories it is not difficult to see the interesting schematic of the changed geometric models; they represent the edges of cube.

Let us contemplate again the question about the limits of geometric description. The discussion deals with the incompatibility of the Riemann structure of space-time and quantum ideas, i.e. about the quantum limits of the applicability of the general theory of relativity, or relativistic theory of gravity, and at the same time with the gravitational limits of the applicability of relativistic quantum theory. Existence of such boundaries, which correspond, as we will see, to the specific values of characteristic physical quantities, it undermines the hopes for reaching the synthesis of general relativistic and quantum ideas with the aid of the already known theoretical constructions only.

For the first time a question about the compatability of SR and quantum theory arose in Einstein himself. To him physics is obliged, as is known, not only for the general theory of relativity, but also for the fundamental development of quantum theory. In 1916, in several months after creation of GR, Einstein examined a question about the gravity waves, whose existence followed from the equations of GR. He derived the formula for the intensity of gravitational radiation and unexpectedly (in any case, for the reader) noted the following: “… atom, as a result of the intra-atomic electron motion, must emit not only electromagnetic, but also gravitational energy, although in the minute quantity. Then, apparently, quantum theory must modify not only Maxwellian electrodynamics, but also new theory of gravity " [15].

Einstein has formulated the insoluble problem of the stability of atom. As is known, it consisted in the fact that according to the laws of classical (Maxwellian) electrodynamics the electron motion around the nucleus must be accompanied by continuous emission. As a result of energy losses the electron must approach a nucleus in the spiral trajectory. If we calculate the emission of electromagnetic energy by the electron from the same formulas, as the emission of the radio station (there are no other formulas in the classical electrodynamics), then we will obtain the absurd result: the electron must fall into the nucleus in the time of the order of 10-10 s! But the atom for the important reasons was the most stable physical structure. Classical physics could not manage this difficulty. The problem “of the luminiscence” of atom was solved, as has already been said, only in the atom model of Bohr, in which central place occupied the constant h.

However, the analogy of gravity and electromagnetism, which had been formulated by Einstein, is far from obvious. If we use Einstein's formula for the intensity of gravitational radiation, then it will seem that “the luminiscence” of energy of the atom of in the form of gravity waves occurs in characteristic time of 1029 years. So that there is no direct contradiction with empirical data. And nevertheless the effect of the gravitational instability of atoms has consequences. This is, apparently, connected with its cosmological ideas. In the work of Einstein 1917, in which was born relativistic cosmology, the static character of the picture of the universe was assumed. But in the static universe, which exists eternally, the effect of the instability of atoms is not admitted independent of the magnitude of effect itself. It is interesting to compare this position, completely justified for the state of astronomy of that time, with the fact that in our time the possible instability of proton (characterized, by the way, with the close value of the lifetime) is mentioned in the Nobel lectures 1979 even as preferable [16].

The fates of quantum gravity and cosmology thus interlaced. But in actuality, we will soon see these fates are interlaced still closer, and the deepest questions of cosmology cannot be answered, without knowing that it is located in the apex cGh of our cube, outermost from the origin of coordinates, i.e. in the apex with the coordinates (c-1,G,h).

But the way to this apex (they approach both senses of word) is barred, as already is known to us, a question about the compatability of SR and the quantum theory. For the first time in the precise physical language this question was examined by remarkable Soviet physicist Bronstein (1906-1938) [17]. In 1935 Bronstein carried out the first in-depth research, dedicated to the quantization of gravitational field. In essence his work was dedicated to the case of weak gravitational field, which gives the possibility not to consider the geometric nature of gravity, i.e. the curvature of space-time.

However, this work contains also the very important analysis, which showed that if we are not limited to the condition for weakness and gravitational field, then it will be revealed that the usual approach to the quantization and the concepts of Riemann geometry are insufficient for the creation of the complete quantum gravitational-field theory.

Simultaneously were revealed the boundaries of the region of the substantially quantum-gravitational phenomena. As a result Bronstein came to the fundamental conclusion that the authentic quantization of gravity “requires radical reconstruction of theory and, in particular, failure of Riemann geometry, which operates, as write Bronstein, with principally not observed values”.

Despite the fact that from the time, when these words were written, almost half a century ago (from 1982), they not only did not lose force, but are even more urgent for contemporary physics of elementary particles and cosmology.

To reproduce Bronstein's reasonings without the attraction of complex physical concepts is sufficiently difficult. Therefore we will select the lighter way, which will lead to the boundaries of the region of the quantum gravitational phenomena. In order to pass this way, it is not compulsory to know these complex Einstein's equations. Everything that will be necessary, the Newtonian law of gravity; the position of the theory of relativity about the fact that the speed of the light is the maximum possible speed of body; the atom model of Bohr (in examination they are thus included respectively constants c, G, h. In 1913 appeared Bohr’s quantum atom model. For eight years was the maximality of the speed of light was already known and during several centuries Newtonian’s law of gravity.

But thus far we will visit 1899 in those May days, when at the session of the academy of sciences in Berlin report of M. Planck “ was delivered about the irreversible processes of emission”. In this report it was for the first time the existence of a new universal physical constant was prolaimed, which was subsequently named Planck's constant i. in the same report appeared the values, which in 1957 American physicist Wheeler named Planck’s values and which refer most direct to quantum gravity.

“Planckian values” and their relation to the the quantum gravity

Planck himself did not introduced these values in relation with quantization of gravity. Moreover, in the report 1899 even constant h did not have an even quantum sense (there were no ideas about the quanta of energy E=hν). The introduction of this constant allowed Planck at first only to write the formula for the spectrum of thermal radiation to agree with the experimental data. To Planck, for five years was attempting himself to solve the problem of thermal radiation, was the fundamentality of the new constant clear. Apparently the universality of new constant impelled him in the same report of 1899 to turn to the question, not in general

connected with the basic theme, about the natural units of measurement. At the end of his report Planck focused attention on the fact that the selection of normally utilized systems of units “was not made on the basis of the common point of sight, it is compulsory acceptable for all places and times, but it is exclusive on the basis of the needs of our terrestrial culture”, and, as he writes, “it is not difficult to visualize that in another time, with the confined external circumstances, any of the systems of units, used up to now would partially or completely lose its initial natural value ".

In connection with this Planck notes that, relying on the new constant and, and also to the speed of light in the vacuum and the gravitational constant G, “we obtain the possibility to establish the units of length, mass, time and temperature, which would not depend on the selection of any bodies or substances and they would compulsorily preserve their value for all times and for all cultures, including extraterrestrial superhuman and therefore it would be possible to introduce them as “natural units of dimensions”.

New (natural) units are selected so that in the new system of units each of the constants indicated would be turned into one. Thus Planck obtains the units of length, mass, time:

l=(hG/c³)1/2=1.6 10-33 cm

m=(hc/G)1/2=2 10-5 eV

t=(hG/c5)1/2=5 10-44 s

(here they are used contemporary designations and values of constants). It is not difficult to understand that analogously it is possible to obtain “natural unit”, also, for any other physical quantity. The purpose to introduce the natural units of measurement, suitable “for all times and peoples”, did not find support. In particular because the collection did not have the essential advantages (at least in 1899) over other collections. Furthermore, one view on the values of Planckian units repulses any desire to call them natural. If the values of l and t are located monstrously far from the field of the reach of contemporary physics then the values are entirely awkward.

And even Planck himself ceased to recall about his natural units after a certain time.

The deep physical sense of those values became clear only after many decades, when it was explained that the Planckian values define the quantum limits of the applicability of general theory of relativity and at the same time gravitational boundaries of relativistic quantum theory [18].

The generation of quantum theory was caused by the fact that in the classical theory of electromagnetic field there were serious difficulties, one of which was even called a catastrophe (moreover of ultraviolet). And quantum theory overcame these difficulties, it is possible to say, it became soon clear with the aid of the new constant h. that a similar “aid” will be required also the gravitational-field theory. To this circumstance, Einstein focused attention in 1916, as has already been said. Long time the association of gravity and quantum theory did not draw the attention of physicists (partly, possibly, because of the abundance of more vital tasks, connected with nuclear physics).

The work of Bronstein in 1936 was only published, after it became clear that the authentic association of the theory of gravity and quantum theory is a complex problem, which cannot be solved, using simple analogies with the electrodynamics.

And Planckian values indicate that the field of physical phenomena, in which known methods cannot give even the approximate description. The difficulty of constructing the theory of

quantum gravity is already visible from the fact that this theory, in spite of numerous ingenious attempts, is not created, until now, 65 years after Einstein realized the need for such theory and 45 years after Bronstein explained essential features, which it must possess. Thus, at present in physics there are two general theories of GR, which can be named cG-theory and ch-theory, since in it these constants are considered, and the quantum theory, but no as yet a cGh-theory, from which would follow both the cG-theory and the ch-theory.

How it is possible to understand the fact that the field of the physical phenomena, for describing which is necessary the cGh-theory, it is limited by the Planckian parameters?

Let us examine the simple system, which consist of two identical point particles of the mass of m, connected with gravitational interaction and which revolve along the circular orbit of a radiu r with the spee v (this system would be possible to name “the molecule of gravitation”). How do we describe the state of this system? It is clear that the parameter m and r can be selected so that the dual stellar system will come out, for which Newtonian law of gravity is sufficient. However, in the general case one must take into account the existence of constant h and; it is most simple to make this with the aid of the main element of the Bohr atom model - the requirement that the moment of momentum of system would be integral multiple of h.

Let us thus, subordinate our system to classical mechanics and Newtonian law of gravity

Ma=mv²/r=Gm²/(2r)² (2)

and also to the quantum postulate of Bohr

2mvr=nh (n=1,2,3,..) (3)

With the aid of equation (2) it is possible to express v through m and r. Therefore the nature of system is completely determined by two parameters m and r. the field of all possible values is divided by the parameters m and r on the parts, in which the system can be described only with the Newtonian theory of gravity plus the quantum theory and also the theory of relativity. We obtain some inequalities.After some calculations we come to the Planckian values. Of course we cannot relate completely seriously to the obtained results: with the aid of the concepts and laws of classical physics we came into the region, where they are deliberately not applied. But to now we easily believe, that the more serious analysis also leads to the the Planckian values as the boundaries of the region of the quantum-gravitational phenomena.

Why is cGh-theory necessary?

We said much about future of cGh-theory, or about the theory of quantum gravity, But is it necessary? Are now any phenomena, processes etc. known, which cannot be managed without the cGh-theory? Or does the monstrous difference in Planckian values (1) from the values, accessible to contemporary experimental physics, mean that this theory will be required not earlier than several centuries ahead?

No, such phenomena are known even now. One of the interesting phenomena relates to cosmology and consists of the following. As is known, th universe e surrounding us consists of the galaxies (more precisely saying, from the clusters of galaxies), one of which the Milky Way or the galaxy (from the capital letter) enters together with hundred of billions of other stars and our sun. One of the most important values in cosmology is the average density of substance in the universe ρ. This value is obtained, if the mass, which is contained in the volume, whose dimensions are much greater the average distance between the galaxies, are divided into this volume. Fundamental observant fact is that wherever we select this volume,

the value of density is approximately one and the same (this property indicates the uniformity of the universe). The dependence of ρ on the time is determined by the equations of the general theory of relativity of Einstein.

According to observations the average distance between the galaxies in the course of time increases. This means that in the past the density was more than the present, moreover according to GR a while back (appr. 10 billion years ago) density was equal to infinity. Now it is time to recall that GR this altogether only the cG-theory, which does not consider the existence of constant h. Thus, developing the history of the universe according to the laws GR, or cG-theory, we unavoidably come to such a stage, which may be described by cGh- theory.

But perhaps if the discussion does deal with the events of billion years remoteness, is our interest in the cGh-theory only idle curiosity? No, not only. The fact is that the theory of gravity of Einstein, which has been the basis of contemporary cosmology, makes it possible to predict the course of events when a situation is known at a certain moment. Therefore it is important to know, what inheritance the universe obtained from the epoch of its development. In this epoch the answers to the questions are hidden, which are concerned with the formation of galaxies, the values of their mass and other physical parameters [19]. But indeed without the answers to such questions the most important questions, which are concerned the origin of the solar system, lives and finally man remain obscure.

In light of the cGh-theory can explain not only cosmological problems, but also astrophysical, the final stage of the evolution of the stars, the so-called gravitational collapse. Furthermore, one should not forget that the new theory always not only answers old questions, but it also leads to the numerous, frequently unexpected consequences.

From the point of view of contemporary physics, as has already been said, most the construction of cGh-theoryis probable within the framework of the unified theory of fundamental interactions on the basis of the synthesis of the ideas of the Einstein theory of gravity and principle of calibration symmetry, which from a mathematical point of view is expressed by the geometry of space stratifications.

Let us return now to a question about the limits of geometric description generally and to a question about the dimensionality of space-time in particular. The examination of the geometric descriptions and the physical space-time makes it necessary to believe, that the physical geometry, given by general theory of relativity, is not the truth in the last instance (this becomes especially clear, if we look at the history of physics not through the prism of experiment, but through the cube of theories). But, besides this, the surprising vitality of the geometry produces impression: the detection of the limits of one geometric description was overcome in other, more general, or the deeper geometric description of physical reality. This makes it possible to hope that also the cGh-theory will be built on the basis of geometry.

How is this power of geometry explained? This question is, of course, only the part of a wider question about “incomprehensibility of the effectiveness mathematics” in physics [20].

It is difficult not to come to the conclusion that the power of geometry in physics reflects the certain objective special feature of the device of material world, the time-spatial nature of its structure. However, the interrelation of geometry and physics, is seen now with the examination “the space of the ideas” of theoretical physics proves not to be simple.

Is it possible to assert that the role of geometry in physics grows constantly? Yes and no. Yes, because in history we find the use of ever more complex geometric models. No, because this assertion inadequately describes the real situation, and with a broad base it is possible to say that the role of physics in the geometry constantly grows (in the form of physical geometry).

Actually, physical geometry ever more "absorbs” physics. Minkowski's geometry, which absorbed into itself the purely physical fact of the constancy of the speed of light, already could be examined at least as the geometric part of physical reality.

Passage to Riemann geometry in GR, that absorbed into itself one additional physical fact (in the form the principle of equivalence), deprived the possibility to consider geometry the individual part of physical reality, geometry rather was the specific projection of the physical reality. In this case geometric concepts themselves could not even be physically comprehended separately from the corresponding component of theory.

Passage from Minkowski's space to the space stratification of calibration theories replaced the concept of the point of space-time (“not having parts according to Euclid”) with the internal space, which describes the state of the physical field, whose dynamics against the background of M3+1 must give physics.

And finally if super-gravity actually is able to become a cGh-theory, then, most likely, the concept of time-spatial coordinates and physical add-on will cease to be characterized by such fundamental means, and then it would be possible to say that the geometric description has physical limits only because the physical description has geometric limits.

This forecast of the development of interrelations between physics and geometry possesses only one deficiency it is too single-valued. But indeed the history of physics with the greatest available accuracy attests to the fact that in the science the smallest chances possess the single-valued forecasts.

What role can play the concept of dimensionality in the forthcoming immense changes in the physical picture of world? About the importance of this concept speaks already the fact that 3-dimensionality survived all radical reconstructions of physical geometry. However, this does not mean that dimensionality is to remain at the unattainable height. Many signs in the contemporary physical situation speak, that the dimensionality can play the constructive role in the development of physical theory. Even with the quick survey of the indexes of contemporary physical periodicals one cannot fail to note, how more frequently than recently, there is the word the dimensionality encountered: the model variations of the theory of field the space of different dimensionality and the dependence of the properties of these models on dimensionality value, the concept of strings (i.e. 1-dimensional objects) in the theory of strong interactions, dimensional regularization, etc exists.

The start into the geometry of internal space (thus far well-defined dimensionality), the indissoluble connection of four time-spatial coordinates and N coordinates of the space of the internal of symmetry in the super-gravity not only raises the question about the connection of the numbers N and 3+1. The difference between the time-spatial degrees of freedom and the degrees of freedom of field is erased; thus it is possible to expect the dynamically nontrivial behavior of the dimensionality of physical space-time. Indeed for the calibration theories this nontrivial behavior is characteristic: spontaneous asymmetry can supply originally mass-free fields with mass, the intensity of interaction can diminish with the rapprochement of particles, etc. but will not attempt to guess those answers, which can be obtained only by heavy “physical” labor, and let us look to the situation from another point of view. What kind of new possibilities could deliver the nontrivial structure of space-time for physics of elementary particles and cosmology?

Dimensionality of space-time and physics of the elementary particles

As has already been said in chapter 3, the problem of the dimensionality of physical space, connected with the concrete regions of physical phenomena, unavoidably leads to a question

about the measure of the hardness of the mathematical model of real physical space.

From the point of view of the deepest physical theory of space-time GR about the local structure of space, up to the arbitrarily small distances, everything is known. And this structure coincides with the structure of pseudo-Euclidean 4-dimensional space. However, in this description it is assumed too detailed information, incompatible even with the idealized experiment, and therefore is physically senseless [21]. Assumption about the fact that the local structure of space coincides with the structure of Euclidean space, can be compared with the requirement to describe particle spin elementary as the usual rotation of ball. In the latter case in physics it was necessary to foreget about the excess (and incorrect) information. The failure of some elements of a theory never solves any problem. The design replacement of old concepts by new ones is necessary.

At the basis of one of the deepest problems of the relativistic quantum field theory of lies the problem of the divergences, because of too distant extrapolation of the properties of the existing mathematical model of physical space [22].

For the first time the physicists encountered the problem of divergence even in the classical theory of the electromagnetic field, when they attempted to consider from the theory of relativity the pointlike nature of the electron in the task to calculate its self-energy, or about the electromagnetic nature of its mass. This led to the divergence of the self-energy of point electron. In the quantum electrodynamics this divergence assumed the form of the divergence of the mass of electron, but quantitatively, in other words, it did not change; all this relates not only to the electron, but also to other particles.

In the forties the methods “of fight” with these divergences were proposed (methods of renormalization), which make it possible to obtain the specific quantitative predictions in the theory. Despite the fact that these predictions could be coordinated with the experiment with unpreceded accuracy, the procedure of renormalization had more the nature of a prescription or trick, than that of theory, It did not satisfy even its authors. Subsequently the method of renormalization was improved; it was also explained that besides electrodynamics there exist other renormalizable field theories.

At present in connection with the successes of the theory of calibration the opinion is popular that for all physically add-ons the method of renormalization is applicable. One way or another, but the problem of divergence is extremely important for physics of the elementary particles.

There are different relations to the method of renormalization. Some physics assert that this is simply one of the methods of derivation of the empirically checked effects of the theory, is not better and not worse than others. Others consider the real success of the method of renormalization evidence only of the fact that this is the indication of some deep property of true theory (perhaps, calibration invariance), which will prove to be understood properly and mathematically in the future theory.

How could the problem of divergence refer to the nontrivial structure of space? Within the framework of classical field theory it is possible to give the illustration, connected with the divergence which demonstrates the consequences of passage to the smaller dimensionality of space at short distances. As has already been said, in the classical electrodynamics point electron possesses infinite self-energy [81).

In contemporary theoretical physics the concept of dimensionality is by no means central; and nevertheless one cannot fail to note the large number of works, in which the number of dimensions of space undergoes active and even completely free rotation. In these works physico mathematical constructions are examined in the space-time with the dimensionality, not equal to four, in essence with two purposes.

First, for construction and analysis of the model versions of the theories in question some characteristic of 4-dimensional case the difficulties (in particular, the mentioned problem of divergence) are solved more easily if we assume smaller dimensionality.

In the second place, different dimensionality is considerd for giving sense to the results, senseless in the usual mathematical understanding. Even non-integer values of dimensionality are used. They propose, for example, in the general formulas (obtained, naturally, for En, i.e. for the integer values of dimensionality) that the dimensionality of 3 is considered as too low a value; |

On the other side, revived at the new level the attempts to use models with the space of the large number of dimensions (>3+1); in this case the agreement with the macroscopic 3-dimensional nature is achieved by the fact that “excess” dimensions are compactified. These attempts have large historical tradition, going back even to the five-dimensional field theories, in development of which participated, in particular Einstein and Pauli. It is not possible to completely exclude the possibility of the fact that attempts at the solution of some problems in microphysics on the basis of geometric models with the dimensionality, not equal to three, are not simply systematic exercises, but they make physical sense. With these ideas, in particular, are connected the hopes for the solution of the problem of electric charge [24].

If the theoretical models, which use space with the dimensionality, not equal to three, they do actually possess remarkable properties. As is known, Planck, introducing his constant was using the purely technical method, with the aid of which it is possible to describe the spectrum of thermal radiation. Subsequently, when it was explained that the quantum postulate is a new law of nature, for physicists it was necessary to include this law in physical theory,

Relativistic strings (1-dimensional objects) in the theory of strong interactions [25], the model of the field theory and retention of quarks in the space with the dimensionality, not equal to three, are examined now, as a rule, as the technical methods, which do not have physical sense [26]. But perhaps these are not simply technical methods, but hints to the complex structure of space-time? If the problems of divergences are removed in the case of smaller dimensionality and it is possible to isolate quarks, then perhaps it is necessary to consider as evidence the dimensionality of space at short distances smaller than 3? Indeed in exactly the same manner the stability of planetary orbits and the spectrum of hydrogen were connected by Ehrenfest with the 3-dimensional nature of space in the appropriate range of distances.

Opinion about the prematurity of cardinal changes in the time-spatial description is supported also by the following considerations. Whatever form the generalization of time-spatial description would have, a certain characteristic value, which corresponds to boundary, must exist. (It is convenient this characteristic value to consider length; any other value of mass, energy can be correlated with a certain length.) at present the most natural length of this type does seem the quantum-gravitational Planckian length,

We see, since gravity and geometry of space-time are described by the general theory of relativity, the value corresponds to the region, where the geometry must acquire substantially quantum properties and, in particular, the classical concept of of manifold will become unfit.

Problem of the dimensionality of space and cosmology of the early universe

The validity of fact 3+1- of dimensionality as any physical fact, is connected with the specific (and by necessity for those limited at the given moment) range of physical phenomena. The at the same time deepest problems of contemporary cosmology one way or another deals with the limits of the applicability of the existing theories. Such problems include, in particular, the problem of initial cosmological singularity, initial cosmological conditions, the problem of the

description of the physical processes, which occurred in the initial expansion period of the universe.

The fact of cosmological expansion speaks, that the in the past average density of substance in the universe was extremely (but formally, even unlimited) large. This, in turn, means that in the initial period of cosmological expansion the quantum-gravitational phenomena were determining, i.e. the phenomena, for describing which were essential the values and by Planck's constant h (quantum mechanics) and the gravitational constant G (gravity), and the speed of light (relativity). Density in the expanding universe was in the past exceeding any limit. The Planckian parameters differed by many orders from the values, accessible to the experimental study of contemporary physics. And nevertheless there are serious foundations for considering that the authentic solution of such problems of cosmology as large-scale isotropy and the galactic structure of the universe, the origin of cosmic backgraound-radiation and specific enthropy, can be achieved only upon consideration of initial cosmological conditions. At present there is no theory, in which would be realized the complete synthesis of the theory of gravity and quantum theory, but from the general considerations it is clear that the state of material, space and time in the qunatum-gravitaional significantly differed from usual.

However, under the conditions of quantum-gravitational cosmology gravitational field and together with it space-time must acquire quantum properties and therefore the normal description of the structure of space-time becomes unfit.

A similar attempt undertook English astrophysicist Saslo, who proposed the uncommon solution of the known problem of the horizon, connected with the interpretation of the observed large-scale uniformity of the universe. (This attempt at least clearly illustrates the importance of a question about the dimensionality of space for cosmology of the early universe; therefore let us examine it in more detail).

First about the problem itself, which appears, when they attempt to agree on two most important observant facts of cosmology; the uniformity of the universe and the extremity of the time of its expansion.

The uniformity of the universe is understood as the identity of its properties in the different places of space. It is clear that the discussion can deal only with the large-scale uniformity, i.e. it is necessary to compare the regions of the universe, whose sizes are much more than the average distance between the galaxies. Before the discovery of cosmic electromagnetic radiation about the uniformity could testify only the calculations of the number of galaxies in different sections of the firmament; this was a labor-consuming and inaccurate method. The discovery of relict cosmic background radiation in 1965 principally enlarged the observant possibilities of cosmology.

The name “relict” speaks, that composing this emission photons were preserved in the protoplastic form from the very old time from the so-called moment of separation of emission from the substance. According to contemporary cosmology a sufficient removal into the past of our universe must lead not only to an increase in the average density of substance (about which speaks the fact of expansion of universe), but also to a increase in the temperature. Once the entire universe was filled with the hot plasma of the mixture of the actively interacting elementary particles: protons, electrons, photons and the like. In proportion to expansion of universe its temperature fell, until the finally medium energy of photons became less than the typical excitation energy of hydrogen atoms, which by that time could stably exist. This moment is called the moment of separation of emission from the substance, since from those burrows the photons of cosmological origin ceased to interact with the substance. According to contemporary data, the detachment of emission occurred only approximately ten thousand years after the beginning of expansion and for many billion years “to our era”. Thus, the relict photons, which are recorded by instruments now, passed way into billions light years and are a stored memory

about the state, in which the universe was billions years ago.

And here, that is most surprising, the photons, which came from the opposite ends of the observed universe, testify about the identity of the properties of the regions of the universe, by the divided billion light years. But perhaps in this high uniformity there is nothing surprising? Even without examining those processes of the mixing, which could occur in the early universe, it is possible to assert that the theory of relativity allows the establishment of uniformity in the regions, whose sizes not are greater than several ten thousand light years.

The previous reasonings contain a essential inaccuracy they do not consider the fact of expansion of universe. This expansion is described by the so-called scale factor that depend on the time and showing the change with the time of the distance between the galaxies. Scale factor enters into the formula, which describes cosmological model, and it is determined by Einstein's equations, which connects the geometric properties of space-time with the properties of substance.

During the construction of cosmological models they use several types of assumptions about the substance of the universe. In the simplest case each such assumption is characterized by only one number, which connects energy density with the pressure in the so-called equation of state of the substance.

Special importance in cosmology have two equations of state. The first applicably in our era, when separate galaxies are located at large distances from each other and practically they do not interact as separate specks; the corresponding equation of state so is called the equation of state of dust, it is reduced to the equality to zero pressures.

The second, so-called ultrarelativistic, equation of state is applicable to that epoch, when (if we look cosmological film to the reverse side) galaxies so drew together and their substance was mixed and was heated so, that it became the very hot plasma of the mixture of all possible particles, which move with near-light speeds.

However, English astrophysicist [Saslo] [29] approached the problem of the horizon entirely otherwise. He turned to the fact that if the universe in the initial stage of expansion was 2+1-dimensional and substance was subordinated to the equation of state of dust then the dependence of scale factor word be proportional to time. this law which follows from Einstein's equations in 2+1 dimensional space-time, it means that problem of the horizon disappears.

The idea of Saslo is not placed into the framework of contemporary theoretical physics. There is no answer to the main question appearing in connection with it: how it would be possible to understand and to describe passage from 2. to 3-dimensional space in the process of the evolution of the universe? The result of Saslo about the two-dimensional character of the early universe rests on the vulnerable assumption about dust-picture of substance in the early universe. This assumption seems the incompatible with the model of a hot universe. More natural is ultrarelativistic equation of state in it gives a 2-dimensional space no longer sufficiently rapid initial expansion.

The nontrivial behavior of the dimensionality of space-time in the course of cosmological evolution would refer, it goes without saying, also to other cosmological problems, in particular to the problem of the initial disturbances, which lead to the observed large-scale structure of the universe.

Certainly, given considerations are not more than the illustration, which indicates the importance of the concept of dimensionality for cosmology, and they are located thus far beyond the framework of physics. But once we went beyond the framework of contemporary physics, it is possible to attempt to make several scientific fantastic steps.

In the cosmological task, first of all, it should be explained, how it would be possible to coordinate the change of dimensionality with Einstein's equations on the basis of relativistic cosmology. In this case a vital difference in Einstein's equations from the equations of field in the space of Minkowski should be considered (for example, Maxwell's equations), in the space-time with the fixed (flat) geometry. In Maxwell's equations, for example, the solution of a problem with the flat (two-dimensional) symmetry is exactly equivalent to the solution of the two-dimensional equations of Maxwell. For Einstein's equations, which define geometry itself, this is not so, and the behavior of scale factor in anisotropic three-dimensional cosmology is not the same as in two-dimensional isotropic cosmology.

However, in Einstein's equations the dimensionality of space-time is considered by the number of coordinates. Therefore one should look at the concept of coordinates in SR again. Lying at the basis GR is the geometric model of the arbitrarily bent space-time . This includes assumption the permissibility of the arbitrary coordinate systems. In SR this understanding of the word arbitrary is impossible, and with this are connected several complex problems of GR, in particular the description of frames of reference, and the laws of conservation.

The role of the laws of conservation of energy and momentum in physics, and also the concepts of energy and pulse is well known. There are other laws of conservation of momentum of pulse and speed of the center of masses. All these laws of conservation, as it occurs, are tightly connected with the properties of space-time. Such laws are only ten (energy and three components of the pulse, moment of momentum and speed of the center of masses).

Number 10 is very simple it is connected with dimensionality value of the space-time. In the general theory of relativity the space-time can be bent in an arbitrary manner, so that in the general case there is neither uniformity nor isotropism of space-time. The observercan be moved in the space-time as before “in different directions”. If we use in this case not “arbitrary” coordinates, but the frame of references of coordinates, built with the aid of the interval [34].

The nonphysical nature “of the arbitrariness of coordinates” did not completely satisfy Einstein himself. He attempted to reduce the three-dimensional structure of physical reality to the properties of physical field[30]. A similar thought expressed e. Wigner [31]. Thus, the role of coordinates charges thus to the fields, which enter into the right side of the equations of Einstein (equation 7 chapter 2) and the geometry of space-time.

Now came time to recall the remarkable innovation in the contemporary quantum field theory, with which to the high degree was connected the success of calibration field theories. The discussion deals with the phenomena, which very recently were considered relating only to macroscopic physics, about the spontaneous asymmetry and about the phase transitions in the field theory [32]. These phenomena in the contemporary unified theories are critical for the fact that the particles, mass-free in their essence, at a sufficiently low temperature can acquire mass and become similar to those actually observed. This acquisition of mass occurs because the field, “acting like” mass, at a sufficiently low temperature can “freeze” and become a classical field. The phase transformations of this type already found their place and in cosmology of the early universe must be built upon consideration of the theory of the elementary particles [33], where, in particular, is explained the concept of spontaneous asymmetry.

But what relation can have similar phase transformations to our field- coordinates? Since the scale factor of the macroscopic (more precise, even megascopic) parameter and the equations of Einstein have classical, macroscopic nature, the time-spatial coordinates, which participate in these equations, must be classical values. Such coordinates must be no longer v (quantum) fields. Now let us assume that the right side of Einstein's equations (or the structure of interaction) is

arranged so that at a sufficiently low temperature as a result of spontaneous asymmetry occurs.s. This phase transition would indicate the decrease of the number of classical (macroscopic) degrees of freedom or the decrease of dimensionality. Passage from the 3-dimensional nature to the two-dimensional character, for example would be possible to understand that one measurement no longer can be described classical, macroscopically, i.e. with the aid of the usual equations of Einstein, and “it is turned off”. A similar passage would appear as phase transition in the structure of space.

The simplest model of this passage of mechanical oscillatory system with two degrees of freedom of with two springs, characterized by two coefficients, one of which goes to infinity with a temperature decrease to a certain critical value (one spring “it freezes”).

It can seem that it is thoughtless to tell about low temperatures with respect to the early (and very hot, as it is now known) universe. However, one should consider that the critical temperature, which separates sufficiently low temperatures from the sufficiently high, depends on the density of the substance and other factors. Therefore the early universe could “freeze”, also, at very high (from a terrestrial point of view) temperatures.

It is possible to differently relate to the possibility of changing the dimensionality of space in the extreme physical situations. Similar studies can, of course, lead to the expansion of the range of phenomena, in which the 3-dimensional nature of space will be substantiated, and this will be not bad replacement of thoughtless extrapolation (by macroscopically undoubted) 3-dimensionality immediately to all phenomena. Convincing bases for the assumption about the variability of the dimensionality of space can also be revealed. Then the construction of the corresponding theory would become answer to the question, similar to Einstein's question: “How it is possible to preserve essential features 3+1- of dimensionality, if we forego ourselves the usual ideas about the dimensionality?”