P. V. Poluyan
Krasnoyarsk, Russia
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 by 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 http://res.krasu.ru/non-standard, http://geocities.com/quntum_math_poluyan, http://sciteclibrary.ru/eng/catalog/pages/3773.html and read the electronic Russian and English versions of "Do the Hyperreal Numbers Exist in Quantum Relative Universe?", "Time and Chronometrics. Areal Multitudes" and "Incident Circle Starts to Rotate". 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. Also the author wants to thank his friends who help him to popularize his works in the World Web.
I
One of the Wolfgang Pauli's scientific texts begins with a remarkable phrase: "Let us introduce, as usual, material coordinates Xk for space and imaginary coordinate 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 coordinate for time and consider 3 spatial coordinates imaginary. Then Minkowsky's four-dimensional pseudo-euclidean continuum will transform into some unusual variety, which we shall call "Quaternized time-space".
The appearance of the term "quaternion" here is evident: it is easier to present 4 numbers, expressing coordinates (one material, three - imaginary) as quaternion. But quaternion is algebraic numbers, and four-dimensional space-time is continuum. If it is so, is 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 particulars. 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 coordinates transformed into imaginary temporal axes. While building Minkowsky's four-dimensional pseudo-euclidean continuum, all the coordinates were measured in [x] as a result of multiplication of a temporal coordinate and coefficient 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 coordinates and some coefficient S, measured in [s/m].Sometimes it is considered for the interpretation of Minkowski's continuum that the shift of t into x with the help of the co-efficient C is of no importance- this is a strange illusion, because time cannot be physically equal to the temporal extension. If we take C for a unity, measure [t] and [x] will not disappear because of that. The same is with the statements like "only spatial-temporal interval is truly important", "space and time are united in their nature" and so on, which are more philosophic statements than physical. That is why it is extremely important that we choose a different one-measure system in our quaternion time-space: imaginary spatial co-ordinates must be multiplied by some co-efficient S, measured in [s/m]. And again it may seem that nothing special happens because it is just "the reverse velocity of light". But the changing of the co-efficient, which is not important in mathematical sense, leads to great changes in physical sense.
The reverse velocity of light 1/c, as real physical quantity cannot be an unknown coefficient, 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 being 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.Standard mathematical analysis and pseudo-euclidian space do not contradict each other just because space does not possesses inner metrics (as it was underlined by Rieman), in other words a unity can be as big as possible, it is not set as some inner measure unity of distance. In our case, velocity of light C, which plays the part of the co-efficient by the imaginary unity, is quite a concrete physical quantity, the velocity of electromagnet waves. We can imagine it as some unity only relatively. For mathematical characteristics of Minkowski's space-time it is not essential, but in the real world this unity C is used to characterize the unique physical process, that is the change, which is mathematically harmless, can be approved physically.
It seems that due to this reason quaternion time-space cannot be an 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 coefficient measured in [s/m].Let us turn from mathematics to physics. If coefficient C in Minkowsky's pseudo-euclidean continuum is a concrete physical quantity - velocity of light, which has in different measurement system concrete numerical realization, in our quaternion time-space coefficient S must be some physical constant quantity, different in its nature from velocity of light, but having a measurement [s/m] - a reverse one to the measurement unit 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/a = ħC/e2 = 137.0306... (ħ is Plank's constant divided into 2p - h/2p ). 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.
Of course, the author can be reproached for a kind of arbitrariness because it is possible to construct the measure [s/m] from the constants in some other ways (e.g., by using the gravitational constant). The only reason, by which the author is motivated here is the desire to find the logical connexion between the quantum and relative physics, discovering at the moment only formal deep mathematical connexion between the global spatial-temporal picture of the world and microphysical quantum reality, while these constants are accepted to be used to express a size-less constant of a thin structure. 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 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 Quaternized 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 pseudoe-uclidean 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 two-dimensional 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 coordinates).
While velocity of light C is non-classical limitation of maximum velocity (velocity of signal expansion over some distance cannot be endless), correspondently, constant S also does not let the ratio D x/D t take endless meanings. But S is a limit for "reverse velocity", and increasing of D t/D x means at the same time decreasing of ratio D x/D t. 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 coefficient 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 microphysical and mathematical premises to connect the mentioned quantity with nothing else, but rotation.
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 according to its structure corresponds to the group of three-dimensional special 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" (W. Geisenberg, "Quantum Theory and Material Structure" Physics and Philosophy. Part and the Whole. Moscow: 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" Physics Encyclopaedia, Moscow, 1962, Vol.2, p.143).
Mathematics-educated readers have, probably, understood that that object which is known here as quaternion time-space is quite well-known Klifford's algebra of the four-dimensional vector space. Its applicability in physics has been shown several times, as well as for isospins. But the usual attitude towards such applicability of vector algebra in non-classical physics is quite skeptical. What has been done in that respect in France (works by G. Ñusanova, C.R. Acad and others) is regarded usually as the result of specific interrelation of quantum physics.
Thus, the real aim of the work the author proposes to discus here is grounding of the fundamental importance of vector algebra for studying the Universe. The author believes that quaternion time-space is a logically necessary element of the four dimensional space-time (Clifford's algebra determined in the field of hyperreal numbers for the space with time measure system), which closes spatial-temporal structural of the universe, and the division of the size-less quantity into two measurable constants determines that fragment of the universe where physical processes take place in time. The author attempts to show that vector algebra is not just a specific mathematical language to reform the well-known physical data, but, on the contrary, it appears so logically and naturally on the basis of the classical Decartes space pseudo-euclidian Minkowski's continuum is built.The applicability of the mentioned approaches is doubted by many thanks to the standard notions of limits and infinitesimal. The author thinks that while the logical non-contradictoriness of the non-standard analysis has already been proved by Abraham Robinson's works, nothing disturbs us to re-realize the standard notions of interrelations infinitely big numbers and infinitesimal. This is what happens, when the four-dimensional space-time is closed with quaternion time-space into one whole unity. And this really happens.
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From the author:
It is the first paragraph of my article for you to be able to judge about the theme of the article.
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,
E-mail: poluyan@fromru.com, polyan2002@mail.ru