ЯДЕРНАЯ ФИЗИКА, 2009, том 72, № 5, с. 856-870
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
ALGEBRODYNAMICS OVER COMPLEX SPACE AND PHASE EXTENSION OF THE MINKOWSKI GEOMETRY
© 2009 V. V. Kassandrov*
Institute of Gravitation and Cosmology, Peoples' Friendship University, Moscow, Russia
Received October 14, 2008
First principles should predetermine physical geometry and dynamics both together. In the "algebrody-namics" they follow solely from the properties of biquaternion algebra B and the analysis over B. We briefly present the algebrodynamics over Minkowski background based on a nonlinear generalization to B of the Cauchi—Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of B multiplication and found it to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex B space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements ("duplicons"), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at macrolevel, etc. In partucular, the concept of "dimerous electron" naturally arises in the framework of complex algebrodynamics and, together with the above-mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to recently accepted wave—particle dualism paradigm.
PACS:02.30.-f, 03.30.+p, 03.50.-z, 03.65.Vf
1. STATUS OF MINKOWSKI GEOMETRY AND THE ALGEBRODYNAMICAL PARADIGM
A whole century after German Minkowski introduced his famous conception of the 4D space—time continuum, we come to realize the restricted nature of this conception and the necessity of its revision, supplement, and derivation from some general and fundamental principle.
Indeed, formalism of the 4D space—time geometry was indispensable to ultimately formulate the Special Theory of Relativity (STR), to ascertain basic symmetries of fundamental physical equations and related conservation laws. It was also the Minkowski geometry that served as a base for formulation of the concept of curved space—time in the framework of the Einstein's General Theory of Relativity (GTR).
Subsequently, Minkowski geometry and its pseudo-Riemannian analog have been generalized via introduction of effective geometries related to correspondent field dynamics (in the formalism of fiber bundles), or via exchange of Riemannnian manifold for spaces with torsion, nonmetricity or additional "hidden" dimensions (in the Kaluza— Klein formalism). There have been considered also the models of discrete space—time, the challenging scheme of causal sets [1] among them.
E-mail: vkassan@sci.pfu.edu.ru
However, none of modified space—time geometries has become generally accepted and able to replace the Minkowski geometry. Indeed, especial significance and reliability of the latter is stipulated by its origination from trustworthy physical principles of STR and, particularly, from the structure of experimentally verificated Maxwell equations. None of its subsequent modifications can boast of such a firm and uniquely interpreted experimental ground.
From the epoch of Minkowski we did not get better comprehension of the true geometry of our World, its hidden structure and origination. In fact, we are not even aware whether physical geometry is Riemannian or flat, has four dimensions or more, etc. Essentially, we can say nothing definite about the topology of space (both global and at microscale). And, of course, we still have no satisfactory answer to sacramental question: "Why is the space three dimensional (at least, at macrolevel)?" Finally, an "eternal" question about the sense and origin of physical time stands as before on the agenda.
Meanwhile, the Minkowski geometry suffers itself from grave shortcomings, both from phenomenologi-cal and generic viewpoints. To be concrete, complex structure of field equations accepted in quantum theory results, generally, in string-like structure of field singularities (perhaps, it was first noticed by Dirac [2]) and, moreover, these strings are unstable
and, as a rule, radiate themselves to infinity (see, e.g., [3] and the example in Section 2).
Another drawback (exactly, insufficiency) of the Minkowski geometry is the absence of fundamental distinction of temporal and spatial coordinates within its framework. Time enters the Minkowski metrical form on an equal footing with ordinary coordinates though with opposite sign. In other words, in the framework of the STR geometry time does not reveal itself as an evolution parameter as it was even in the antecedent Newton's picture of the World. At a pragmatic level this results, in particular, in the difficulty to coordinate "times" of various interacting (entangled) particles in an ensemble, in impossibility to introduce universal global time and to adjust the latter to proper times of different observers, or in the absence of clear comprehension of the passage of local time and dependence of its rate on matter. All these problems are widely discussed in physical literature (see, e.g., [4]) but are still far from resolution.
However, the main discontent with generally accepted Minkowski geometry is related to the fact that this geometry does not follow from some deep logical premises or exceptional numerical structures. This is still more valid with respect to generalizations of space—time structure arising, in particular, in the superstring theories (11D spaces) and in other approaches for purely phenomenological, "technical" reasons which in no way can replace the transparent and general physical principles of STR, of relativity and of universal velocity of interaction propagation.
At present, physics and mathematics are mature enough for construction of multidimensional geometries with different number of spatial and temporal dimensions. Moreover, they aim to create a general unified conception from which it would follow definite conclusions on the true geometry of physical space and on the properties and meaning of physical time, on the dynamics of Time itself!
In most of approaches of such kind the Minkowski space does not reproduce itself in its canonical form but is either deformed through some parameter (say, fundamental length and mass in the paradigm of Kadyshevsky [5]) under correspondence with canonical scheme, or changes its structure in a radical way. The latter takes place, in particular, in the theory of Euclidean time developed by Pestov [6] (in this connection, see also [7]), in the concept of Clifford space—time of Hestenes—Pavsic (see, e.g., [8, 9]), in the framework of 6D geometry proposed by Urusovskii [10], etc.
At a still more fundamental level of consideration, one assumes to derive the geometry of physical space—time from some primordial principle encoding it (perhaps, together with physical dynamics). One can try to relate such an elementary
Code of Nature with some exceptional symmetry (theory of physical structures of Kulakov [11] and binary geometrophysics of Vladimirov [12]), group or algebra (quaternionic theory of relativity of Yefremov [13] and algebrodynamics of Kas-sandrov [14, 15]), with algebraically distinguished geometry (Finslerian anisotropic geometry of Bo-goslovsky [16] and geometry of polynumbers of Pavlov [17]) as well as with some special "World function" (metrical geometry of Rylov [18]).
Generally, all the above-mentioned and similar approaches affecting the very foundations of physics differ essentially one from another in the character of the first principle (being either purely physical or abstract in nature), in the degree of confidence of their authors to recently predominant paradigms (Lorentz invariance, Standard model, etc.) and in their attitude towards the necessity to reproduce, in the framework of the original approach, the principal notions and mathematical insrumentation of canonical schemes (of Lagrangian formalism, quantization procedure, Minkowski space itself, etc.). In this respect the neo-Pythagorean philosophical paradigm professing by the author [19—21] seems most consistent and promising, though difficult in realization.
Accordingly, under construction of an algebraic (logical, numerical) "Theory of Everything" one should forget all of the known physical theories and even experimental facts and to unprejudicely read out the laws of physical World in the internal properties of some exceptional abstract primordial structure, adding and changing nothing in the course of this for "better correspondence with experiment". In this connection, one should be ready that physical picture of the World arising at the output could have little in common with recently accepted one and that the real language of Nature might be quite different from that we have thought out for better description of observable phenomena. In this situation none principle of correspondence with former theories could be applied.
We have no opportunity to go into details of the neo-Pythagorean philosophy, quite novel and radical for modern science, sending the reader to [19— 21]. Instead, in Section 2 we briefly present its realization in the framework of the "old" version of algebrodynamics developed during the period 1980— 2005 [14, 15]. Therein an attempt has been undertaken to obtain the principal equations of physical fields and the properties of particle-like formations as the only consequence of the properties of exceptional quaternion-like algebras, exactly, of the algebra of biquaternions B.
We have forcible arguments to regard this attempt successful. From the sole conditions of B analitivity (generalization of the Cauchy—Riemann equations,
see Section 2) we were able to obtain an unexpectedly rich and rather realistic field theory. In particular, as a principal element of the arising picture of the World there tu
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