It is the first paragraph of my article for you to be able to judge about the theme of the article.
© P.V.Poluyan

Numbers in Space

P. V. Poluyan

1. Transformation of four-dimensional space-time into quaternion time-space.

2. Rotation as a kind of motion, non-reduced to rectilinear one.

3. Non-interrupted continuum and varieties of numbers.

Author’s notes: During the International Mathematical Conference “Multidimensional Complex Analysis” (Krasnoyarsk, Russia, August 5-10, 2002) I made a poster report “Do the Hyperreal Numbers Exist in the Quantum-Relative Universe?” The report was devoted to the extensive theme “Non-Standard Analysis of Non-Classical Motion”,the main subject was mathematical and methodological problems, connected with basing of non-interrupted analysis model of A. Robinson and real number field expansion. This work is addressed, first of all, to physicists, because mathematical aspects are mostly excluded, and physical problems are paid attention to. The author recommends the readers who have got interested in this work to visit his internet site,,

and read the electronic Russian and English versions of “Do the Hyperreal Numbers Exist in Quantum Relative Universe?” and “Time and Chronometrics. Areal Multitudes”. The author expresses his gratitude towards the mathematicians and physicians, who gave personally or by e-mail their critical and constructive comments to the stated problem.


One of the Wolfgang Pauli’s scientific texts begins with a remarkable phrase: “Let us introduce, as usual, material co-ordinates Xk for space and imaginary co-ordinate X4= iCt for time and consider Lawrence’s transformations…” (W. Pauli. Works on Quantum Theory. M. “Nauka”, 1977, see article “About Mathematical Matrix Theory Of Dirak”, p. 5, “Lawrence’s Transformations of Dirak’s Wave Functions”, p. 233). The phrase “as usual” can be considered here as a kind of a witty intellectual provocation, which means that the above-mentioned procedure can be performed not “as usual”, but in “an unusual way”. But how? It is not difficult to say: we try to maintain the material co-ordinate for time and consider 3 spatial co-ordinates imaginary. Then Minkowsky’s four-dimensional pseudoeuclidean continuum will transform into some unusual variety, which we shall call “Quaternion time-space”.

The appearance of the term “quaternion” here is evident: it is easier to present 4 numbers, expressing co-ordinates (one material, three – imaginary) as quaternion. But quaternion is algebraic numbers, and four-dimensional space-time is continuum. If it is so, are there enough reasoning to make them correspondent? We shall try to answer this question later and for the present we shall consider quaternion time-space as some pure logical construction, which can be seen as a whole and analyzed in particular. It is also important to mention that the term “space” in modern science is not connected any more with distance measuring, and nothing disturbs us to make a four-dimensional space, where a measure in [t] is put on the axis. But as time is of physical character, which reflects the important aspect of reality, not formal mathematical qualities of the made-up construction, but its physical interpretation will be of greatest interest to us in this article.

The fact that the algebra of quaternions is not commutative leads us to the idea that an abstract object, made-up this way, is directly connected with quantum-mechanical peculiarities of the physical world. But let us consider quaternion time-space as if we do not know anything about quantum mechanics. In other words, we shall try to preserve the classical notions of time and space.

Thus we have a four-dimensional variety, where the material axis is pure time, and the rest three ones are spatial co-ordinates transformed into imaginary temporal axes. While building Minkowsky’s four-dimensional pseudoeuclidean continuum, all the co-ordinates were measured in [x] as a result of multiplication of a temporal co-ordinate and co-efficient C which is velocity of light [m/s]. That is why in our quaternion time-space a ‘one-measurement’ is achieved in analogical way: Multiplication of imaginary spatial co-ordinates and some co-efficient S, measured in [s/m]. One can say that it is ‘the reverse velocity of light’, but it is not. The reverse velocity of light 1/c, as real physical quantity cannot be an unknown co-efficient, while the scale of reverse velocities is irregular. In classical notion velocity is a ratio, where the numerator is the distance segment, and the denominator is time period, time as independent variable quantity. Then dealing with ‘reverse velocity’, where the numerator and the denominator exchange their places, there appears not only new, but also irregular measuring scale: 1[m/s] = 1[s/m], 2[m/s] = 1/2 [s/m], 3 [m/s] = 1/3 [s/m], 4 [m/s] = 1/4 [s/m], etc. It seems that due to this reason quaternion time-space cannot be analogue of the four-dimensional continuum. But it easy to find the way out, if we do not consider S to be ‘reverse velocity’, but some co-efficient measured in [s/m].

Let us turn from mathematics to physics. If co-efficient C in Minkowsky’s pseudoeuclidean continuum is a concrete physical quantity – velocity of light, which has in different measurement-system concrete numerical realization, in our quaternion time-space co-efficient S must be some physical constant quantity, different in itself from velocity of light, but having a measurement [s/m]- a reverse one to the measurement of velocity. We can offer a combination of constant h/e2 to suit this new constant, where h is Plank’s constant, and e is the charge of an electron. It is well known that this combination as well as C is included in the expression of the non-measured constant of thin structure 1/α = ħC/e2 = 137. 0306… (ħ is Plank’s constant divided into 2 “π” – h/2π). I believe that is true, that quaternion time-space is a mathematical expression of the real aspect of microphysical reality, where the constant S = h/e2 measured in [s/m] is as important as velocity of light for Minkowsky’s four-dimensional continuum.

Making such an interpretation we are binding quantum physics and relative physics logically, discovering at the same time a great, but still formal mathematical connection between the global spatial–temporal picture of the universe and microphysical quantum reality. Thus, the logical sense of non-measured constant of thin structure can be seen in the fact that it shows the correspondence between Minkowsky’s continuum and quaternion time-space. I believe Wolfgang Pauli, who insisted on the theoretical grounding of physical status of this mysterious number 137. 0306… meant something of that kind.

But formal arguments are not enough here. We must show the physical essence to discover correspondence, that is to discover the connections between the velocity of rectilinear forward movement C and constant S, the meaning of which is not quite clear yet. S=h/e2 is a combination of empirical constants measured in [s/m], we include it in some mathematical structure, but that has not cleared up its meaning.

In classical physics velocity is a quantitative measure of forward movement, which binds spatial and temporal characteristics of motion as rectilinear forward movement. If constant S is included in quarter time-space, it means that it must be also understood as an expression of some aspect of motion, where spatial and temporal characteristics are bound somehow. Moreover, one of the most important qualities of Minkowsky’s continuum is Lawrence’s transformations, which lead to that law of adding velocities while leaving one-measure system for the other gives us maximum meaning for the rectilinear forward movement.

It would be logical to suppose that in quaternion time-space there is also an analogue of Lawrence’s transformations, which will let us interpret constant s as an invariant and limit in adding some quantities. Thus, the matter should look like a case of using 2 measures, where on the complex plane by means of pseudoeuclidean way one temporal and one spatial axes are being bound. For Minkowsky’s continuum an imaginary axis will be iCt – a temporal axis, for quaternion time-space – a spatial axis iSx. While dealing with 2-measured case the matter does not seem difficult, as we do not consider non-commutability (on the other hand, it is discovered that non-commutability is directly connected with the presence of two more imaginary spatial co-ordinates).

While velocity of light C is non-classical limitation of maximum velocity (velocity of signal spreading along some distance cannot be endless), correspondently, constant S also does not let the ratio Δx/Δt take endless meanings. But S is a limit for “reverse velocity”, and increasing of Δt/Δx means at the same time decreasing of ratio Δx/Δt. That leads us to the thought: “zero velocity” is as unattainable as endless velocity.

Nevertheless, in case of a simplified 2-measured complex notion of quaternion time-space, it is still not clear what measures they should be, what is the physical meaning of “measure system” in this case? We are expected to answer these questions.

While S is some co-efficient of proportionality between time-measurement t[s] and space-measurement x[m], constant S as the independent parameter expresses some aspect of motion. But while the quantity measurement for forward rectilinear movement is the classical notion of velocity V[m/s] and its non-classical limit C, this new constant must be a non-classical limit of some classical movement measurement, which is a forward movement, nevertheless. We suppose that the form we need is rotation.*

*There are some microphysical premises to connect the mentioned quantity with nothing else, but rotation. For example, in physics of elementary particles the existence of the so-called isotope- transformations, which are completely the same as ordinary rotations, is experimentally discovered. Werner Geisenberg , accounting basic symmetry groups, places some special group next to Lawrence’s group, it is ”the group explored by Pauli and Gucci, which corresponds according to its structure to the group of three-dimensional spatial rotations, it is isomorphous to this group and reveals itself in appearance of the quantum number, which was discovered empirically and which characterizes elementary particles, it is called “isospin” . (“Quantum Theory and Material Structure” in W.Geisenberg’s book “Physics and Philosophy. Part and the Whole” M., “Nauka”, 1990, p. 103) Ratios, which are the result of isotope-in-variety, are observed to calculate to within amendments, the quantity of which is determined by the ratio e2/hC. It is noted in the textbook “that isotope-in-variety means a special symmetry of great interactions, which is not connected with general qualities of space and time. Though isotope- in-variety is discovered quite well experimentally, the qualities of symmetry, connected with it, do not follow from this theory, and the nature of these qualities is not discovered yet”(“Isotope Spin “ in the book “Physics Encyclopedia”, M., 1962, v2, p.143)

From the author: The enclosure is the text of the same article in Russian. I decided not to offer you English translation, as I was afraid that it would misrepresent the meaning. If you are interested in this article, I hope, at your disposal there will appear qualified translators.

Pavel Poluyan