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ISSN 1063-7788, of Atomic Nuclei, 2009, Vol. 72, No. 5, pp. 813–827. c Pleiades Publishing, Ltd., 2009.

ELEMENTARY PARTICLES AND FIELDS Theory

Algebrodynamics Over Complex and Phase Extension of the Minkowski Geometry*

V. V. Kassandrov ** Institute of and Cosmology, Peoples’ Friendship University, Moscow, Russia Received October 14, 2008

Abstract—First principles should predetermine physical geometry and dynamics both together. In the “algebrodynamics” they follow solely from the properties of algebra B and the analysis over B. We briefly 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 . Then we pass to study the primordial dynamics that takes place in the complex B space and brings into consideration a of remarkable structures: an ensemble of identical correlated matter pre-elements (“duplicons”), caustic-like signals (interaction carriers), a concept of random complex 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 num b e r s : 02.30.-f, 03.30.+p, 03.50.-z, 03.65.Vf DOI: 10.1134/S106377880905010X

1. STATUS OF MINKOWSKI GEOMETRY additional “hidden” (in the Kaluza– AND THE ALGEBRODYNAMICAL Klein formalism). There have been considered also PA RA D I G M the models of discrete space–time, the challenging scheme of [1] among them. A whole century after German Minkowski intro- However, none of modified space–time geometries duced his famous conception of the 4D space–time has become generally accepted and able to replace the continuum, we come to realize the restricted nature Minkowski geometry. Indeed, especial significance of this conception and the necessity of its revision, and reliability of the latter is stipulated by its orig- supplement and derivation from some general and ination from trustworthy physical principles of STR fundamental principle. and, particularly, from the structure of experimentally verificated Maxwell equations. None of its subsequent Indeed, formalism of the 4D space–time geometry modifications can boast of such a firm and uniquely was indispensable to ultimately formulate the Spe- interpreted experimental ground. cial (STR), to ascertain basic From the epoch of Minkowski we did not get better symmetries of fundamental physical equations and comprehension of the true geometry of our , its related conservation laws. It was also the Minkowski hidden structure and origination. In fact, we are not geometry that served as a base for formulation of the even aware whether physical geometry is Riemannian concept of –time in the framework of the or flat, has four dimensions or more, etc. Essentially, Einstein’s General Theory of Relativity (GTR). we can say nothing definite about the of Subsequently, Minkowski geometry and its space (both global and at microscale). And, of course, pseudo-Riemannian analog have been generalized we still have no satisfactory answer to sacramental via introduction of effective geometries related to question: “Why is the space three dimensional (at correspondent field dynamics (in the formalism of least, at macrolevel)?” Finally, an “eternal” question about the sense and origin of physical time stands as fiber bundles), or via exchange of Riemannnian before on the agenda. for with torsion, nonmetricity or Meanwhile, the Minkowski geometry suffers itself ∗The text was submitted by the author in English. from grave shortcomings, both from phenomenolog- **E-mail: [email protected] ical and generic viewpoints. To be concrete, complex

813 814 KASSANDROV structure of field equations accepted in quantum the- At a still more fundamental level of consid- ory results, generally, in string-like structure of field eration, one assumes to derive the geometry of singularities (perhaps, it was first noticed by Dirac [2]) physical space–time from some primordial principle and, moreover, these strings are unstable and, as a encoding it (perhaps, together with physical dy- rule, radiate themselves to infinity (see, e.g., [3] and namics). One can try to relate such an elementary the example in Section 2). Code of Nature with some exceptional symme- Another drawback (exactly, insufficiency) of the try (theory of physical structures of Kulakov [11] Minkowski geometry is the absence of fundamental and binary geometrophysics of Vladimirov [12]), distinction of temporal and spatial coordinates within or algebra (quaternionic theory of relativ- its framework. Time enters the Minkowski metrical ity of Yefremov [13] and algebrodynamics of Kas- form on an equal footing with ordinary coordinates sandrov [14, 15]), with algebraically distinguished though with opposite sign. In other words, in the geometry (Finslerian anisotropic geometry of Bo- framework of the STR geometry time does not reveal goslovsky [16] and geometry of polynumbers of itself as an evolution parameter as it was even in Pavlov [17]) as well as with some special “World the antecedent Newton’s picture of the World. At function” (metrical geometry of Rylov [18]). a pragmatic level this results, in particular, in the Generally, all the above-mentioned and similar difficulty to coordinate “” of various interacting approaches affecting the very foundations of physics (entangled) particles in an ensemble, in impossibility differ essentially one from another in the charac- to introduce universal global time and to adjust the ter of the first principle (being either purely physical latter to proper times of different observers, or in or abstract in nature), in the degree of confidence the absence of clear comprehension of the passage of of their authors to recently predominant paradigms local time and dependence of its rate on matter. All (Lorentz invariance, Standard model, etc.) and in these problems are widely discussed in physical liter- their attitude towards the necessity to reproduce, in ature (see, e.g., [4]) but are still far from resolution. the framework of the original approach, the principal notions and mathematical insrumentation of canon- However, the main discontent with generally ac- ical schemes (of Lagrangian formalism, quantization cepted Minkowski geometry is related to the fact that procedure, itself, etc.). In this re- this geometry does not follow from some deep log- spect the neo-Pythagorean philosophical paradigm ical premises or exceptional numerical structures. professing by the author [19–21] seems most consis- This is still more valid with respect to generaliza- tent and promising, though difficult in realization. tions of space–time structure arising, in particular, in the superstring theories (11D spaces) and in other Accordingly, under construction of an algebraic approaches for purely phenomenological, “technical” (logical, numerical) “Theory of Everything” one reasons which in no way can replace the transparent should forget all of the known physical theories and general physical principles of STR, of relativity and even experimental facts and to unprejudicely and of universal of interaction propagation. read out the laws of physical World in the internal properties of some exceptional abstract primordial At present, physics and mathematics are mature structure, adding and changing nothing in the course enough for construction of multidimensional geome- of this for “better correspondence with experiment”. tries with different number of spatial and temporal In this connection, one should be ready that physical dimensions. Moreover, they aim to create a general picture of the World arising at the output could have unified conception from which it would follow definite little in common with recently accepted one and that conclusions on the true geometry of physical space the real language of Nature might be quite different and on the properties and meaning of physical time, from that we have thought out for better description on the dynamics of Time itself! of observable phenomena. In this situation none In most of approaches of such kind the Minkowski principle of correspondence with former theories space does not reproduce itself in its canonical form could be applied. but is either deformed through some parameter (say, We have no opportunity to go into details of the fundamental length and in the paradigm of neo-Pythagorean philosophy, quite novel and radi- Kadyshevsky [5]) under correspondence with canon- cal for modern science, sending the reader to [19– ical scheme, or changes its structure in a radical 21]. Instead, in Section 2 we briefly present its re- way. The latter takes place, in particular, in the the- alization in the framework of the “old” version of ory of Euclidean time developed by Pestov [6] (in algebrodynamics developed during the period 1980– this connection, see also [7]), in the concept of Clif- 2005 [14, 15]. Therein an attempt has been under- ford space–time of Hestenes–Pavsic (see, e.g., [8, taken to obtain the principal equations of physical 9]), in the framework of 6D geometry proposed by fields and the properties of particle-like formations as Urusovskii [10], etc. the only consequence of the properties of exceptional

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 ALGEBRODYNAMICS OVER COMPLEX SPACE 815 -like algebras, exactly, of the algebra of a completed version of algebrodynamics [23, 24]. B. Inparticular,itturnsoutthatjust(andonly!)in We have forcible arguments to regard this attempt the primordial complex space there may be realized successful. From the sole conditions of B analitivity one of the most interesting and original ideas of (generalization of the Cauchy–Riemann equations, Wheeler–Feynman about “all identical electrons as see Section 2) we were able to obtain an unexpectedly oneandthesameelectron”, in its distinct positions rich and rather realistic field theory. In particular, on a unique . In [23] the set of copies of as a principal element of the arising picture of the the sole “generating charge” correlated in dynamics World there turned to be a flow of light-like rays has been called the ensemble of “duplicons.” We densely filling the space and giving rise to a sort of consider geometrodynamical properties of duplicons particle-like formations at caustics and focal points. and related particle-like formations in Section 4. Such a primordial, matter generating structure has In Section 5 a naturally arising concept of com- been called the Flow of Prelight. From mathemat- ical point of view, this flow is defined by the twistor plex time is presented. Indeed, already in the previous structure of the first equations for biquaternionic field, version of algebrodynamics (on the real Minkows- whereas geometrically it represents itself a congru- ki background) the temporal coordinate is distin- ence of null rays of a special type (namely, shear- guished in a natural way as an evolution parame- free), below—the generating congruence. ter of the primordial biquaternionic (and associated twistor) field: the generating Prelight Flow is identi- Meanwhile, the “geometrical scene” on which the algebraic dynamics displays itself has been, in fact fied with the flow of time [19, 21]. Now, in the complex “by hands,” restricted to a subspace with canon- pre-space such a parameter unavoidably turns to be ical Minkowski metric, to ensure the Lorentz in- two-dimensional, and the related order of sequence of variance of the scheme. Such a procedure was in events—indefinite. Thus, in the framework of initially evident contradiction with the whole philosophy of deterministic “classical” theory there arises inevitable algebrodynamics, since corresponding subspace does uncertainty of evolution of states related to effec- not even form a subalgebra of B and is thus in no way tively stochastic alteration of the evolution parameter distinguished in the structure of B algebra. From a itself on the complex ; we are led, therefore, to more general viewpoint, neither in our old works nor accept the concept of complex random time.Onthe in those of other authors there has been found any other hand, existence of geometrical phase makes it space–time algebra, that is, ascertained an alge- possible to suggest a novel treatment of the phenom- braic (“numerical”) structure which could naturally ena of quantum interference, alternative to gener- induce the Minkowski geometry (or include the latter ally accepted concept of wave–particle dualism.In as its part).1) particular, such a treatment relates the notion of the However, in 2005 in [22] we have demonstrated phase of wave function to the classical of a that, under a thorough consideration, the primordial particle quite in the spirit of Feynman’s version of complex geometry of B algebra unavoidably induces quantum theory. Consideratons of these issues con- a real geometry just of the Minkowski type. In this clude the paper. scheme, the additional coordinates of (8D in reals) of B are naturally compactified and be- have like a geometrical phase suggesting thus a 2. ALGEBRODYNAMICS geometrical explanation of the wave properties of OVER MINKOWSKI SPACE matter in general. In the following, this geometry has Biquaternionic (B) algebrodynamics is completely been called the phase extension of the Minkowski based on the (proposed by the author in 1980) version space. Its derivation and simplest properties are pre- of noncommutative, including biquaternionic, analy- sented in Section 3. sis, that is, on generalization of the theory of functions Discovery of the novel geometry induced by the of complex variable to the case of noncommutative primordial algebraic struture of complex quater- 2) algebras of quaternionic type. This version is exposed nions , opened wide perspective for construction of in detail in the monograph [14] (where one can find references to preceding works) and in the recent re- 1)Hestenes was one of the first to consider the concept of space–time algebra [8]. We think, however, that his favourite view [15]. 16D Dirac algebra cannot in fact be considered in this role Essentially, the whole structure of the theory of since the additional dimensions have no natural physical B ∈ B interpretation. functions of variable Z follows from invariant 2)The algebra B is distinguished as a unification of two excep- definition of a differential dF of such, differentiable tional (associative with norm and division) algebras, namely in B,functionF : Z → Z (a direct analog of an an- of complex and of Hamilton’s quaternioins. alytical function in complex analysis). Specifically, in

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 816 KASSANDROV account of associativity yet non-commutativity of the Realization of this program requires, meanwhile, algebra one has to resolve the problem of relationship of the 4D com- Z B dF =Φ∗ dZ ∗ Ψ, (1) plex coordinate space ,vectorspaceof algebra, to the Minkowski physical space–time. As it was where Φ(Z), Ψ(Z) are some two auxiliary functions already mentioned, correct correspondence between formerly called (left and right) semi-derivatives of these spaces has been ascertained not long ago in F (Z). our works and leads to a principally novel view on the geometry of space–time (see below). As to this Relation (1) explicitly generalizes the well-known section, we shall expose only the former version of Cauchy–Riemann conditions. Indeed, in the case of algebodynamics in which the coordinate space Z is commutative algebra of complex numbers it acquires forcibly restricted onto the subspace with Minkowski a familiar form metric, in order to guarantee the Lorentz invariance dF = F ∗ dZ, (2) of the scheme and to avoid the problem to prescribe a ∗ particular meaning to additional “imaginary” coordi- with F := Φ Ψ being an ordinary derivative of nates. an analytical function F (Z) of complex variable Z. Specifically, it is well known that the biquaternion Writing (2) down in components one comes to the algebra B is isomorphic to the full 2 × 2 matrix al- standard Cauchy–Riemann system of equations. gebra over C. Further on we shall use the following Thus, requirement of invariance of the differential (2) two equivalent matrix representations of an element represents itself one of a number of equivalent ver- Z ∈ B: sions of complex analysis suitable, moreover, for ⎛ ⎞ ⎛ ⎞ its generalization to a noncommutative case in the − ⎝uw⎠ ⎝ z0 + z3 z1 iz2⎠ form (1). Note that such version was, perhaps, first Z = = (3) proposed by Sheffers [25] for construction of the pv z1 + iz2 z0 − z3 analysis over an arbitrary commutative–associative { } algebra and nearily after a century used by Vladimirov through four complex “null” u, w, p, v or “Carte- { } and Volovich [26] for generalization to superalgebras. sian” zµ , µ =0, 1, 2, 3, coordinates of a biquater- nion Z, respectively. Therefore, the 4D complex Remarkably, in the case of real Hamilton quater- B Q vector space of possesses a natural complex nions the proposed conditions (1) reproduce an- (quasi)metrical form correspondent to the determi- other exceptional property of complex analysis, nant of representative matrix (3), namely, the conformity of correspondent mapping 2 − 2 − 2 − 2 implemented by any analytical function [27, 28]. D = z0 z1 z2 z3. (4) However, since the of Euclidean This form turns into the Minkowski pseudo- Ek under k ≥ 3 is finite (exactly, 15-parametrical metric if only one considers the coordinates zµ → for k =4), quaternionic analysis built on the base xµ ∈ R as reals. This corresponds to restriction of of relation (1), turns to be unattractive in respect of a generic matrix (3) to a Hermitean one Z → X = physical applications. X+. However, we do not intend to restrict, in a When, however, one passes to the case of similar way, the “field” matrices associated with B algebra (i.e., under complexification of Q) the class functions F (X) of the space–time coordinates X = of differentiable (in the sense of (1)) functions {xµ}, since the principal physical fields (especially in essentially expands due to special elements of B— quantum theory) are considered as complex-valued. null divizors (see details in [14, 15]; corresponding Thus, we come to the second interpretational prin- mappings have been called degenerate conformal). ciple of the considered version of algebrodynamics: In this way we naturally come to formulate the first B “interpretational” principle: In algebrodynamics the coordinate physical space–time is represented by a subspace of the B In the paradigm of algebrodynamics there 4C vector space of B correspondent to Hermitean exists a unique fundamental physical field. This is 2 × 2 matrices X = X+ with determinant being a (essentially complicated and even multivalued) just the Minkowski metric. After such a restriction B function of variable obeying the conditions of the whole algebrodynamical scheme becomes B differentiability (1)—the only primordial “field Lorentz invariant by itself. equations”.Alloftheother“fields” arising in On a coordinate “cut” correspondent to the the scheme are secondary and can be defined Minkowski space first conditions of B differentiability through (semi)derivatives, contractions, etc. of take the form the input B field. Their equations also follow from ∗ ∗ the “master equations” (1). dF =Φ dX Ψ (5)

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 ALGEBRODYNAMICS OVER COMPLEX SPACE 817 and represent themselves a sort of “master equations” 2. Primary conditions (5) can be reduced to a for some algebraical field theory which uniquely simpler system of equations of the form determines all its derivable properties. It is also dξ =ΦdXξ (7) noteworthy that none Lagrangians, commutation relations or other additional structures are used in for effectively “interacting” 2- field ξ(X)= the theory under consideration. Moreover: system of {ξA},A=1, 2, and potentials Φ(X)={ΦAA } of a equations (5) turns to be overdetermined and does complex gauge-like field (see for details [15]. not allow for any generating Lagrangian structure. 3. Integrability conditions of reduced over- Consequently, overdetermined character of the determined system (7) are just the self-duality primordial algebrodynamical relations (5), together conditions with nonlinearity of the arising field equations (see ∗ below), makes it possible to consider B algebro- F = iF (8) dynamics as a theory of interacting fields (and for the field strengths of gauge potentials ΦAA .Con- “particles” with rigidly fixed, “self-quantized” char- sequently, complexified Maxwell and SL(2, C) Yang– acteristics, see below). Mills free equations are both satisfied on the solutions As for particles, in the classical theory (like alge- of master system (7). brodynamics in its original form) in the capacity of µ 4. A field of a null 4-vector kµ: k kµ =0 can those one can obviously take either regular (soliton- be constructed from fundamental 2-spinor ξ(X) as like) or singular field formations localized in 3-space. follows: Now we are ready to formulate the last (third) inter- + pretational principle completing the set of first state- kµ = ξ σµξ, (9) ments of algebrodynamical scheme: where σµ = {I,σa}, a =1, 2, 3, is the canonical ba- In the framework of B algebrodynamics, “par- sis of 2 × 2 matrices. As a consequence of master ticles” (particle-like formations) correspond to system (7), the null congruence of rays tangent to (point or extended but bounded in 3-space) sin- kµ is rectilinear (geodetic) and shear-free. This con- gularities of the biquaternionic field or its deriva- gruence of rays plays an extremely important role in tives. The latters may be put into correspondence algebrodynamics; below we shall call it generating with singularities of secondary (Maxwell, Yang– congruence3). In the context of algebrodynamics it is Mills, and other) fields associated with any dis- important that an effective Riemannian metric g of B µν tribution of the primary field. Shape, spatial aspecialform arrangement, characteristics, and temporal dy- namics of these particle-like formations are again gµν = ηµν + h(X)kµkν (10) completely determined by the properties of the (the so-called Kerr–Schild metric [32]) may be put master algebrodynamical system of equations for B B in correspondence with any field or associated field (5). generating congruence. This is a deformation of the It should be noted that symmetries (relativistic and flat Minkowski metric ηµν preserving all the defining conformal among them) of system (5) are considered properties of generating congruence. Note that a self- in detail in the review [15]. Gauge and twistor struc- consistent algebrodynamical scheme over a curved tures specific for system (5) are also described therein space–time background has been developed in [33]. (below we shall return to some of them). Let us now 5. In contrast to ordinary nonlinear field mod- briefly review principal properties and consequences els, in the algebrodynamicsitturnstobepossible of the B algebrodynamics (based solely on the condi- B to obtain general solution of the master system of tions of differentiability (5)). equations (7) or (5) in an implicit algebraic form. The 1. Each matrix component S(x, y, z, t) of a procedure is based upon the (well-known in GTR) B-differentiable function F (X) satisfies the complex Kerr theorem [32, 34] that gives full discription of null eikonal equation shear-free congruences on the Minkowski or Kerr– Schild background, and makes also use of a natu- ∂S 2 ∂S 2 ∂S 2 ∂S 2 − − − =0. (6) ral generalization of this theorem [35, 36], namely, c∂t ∂x ∂y ∂z of general solution of the complex eikonal equation obtained therein. Briefly, the procedure of searching This nonlinear, Lorentz and conformal invariant equation substitutes the Laplace equation in complex 3)Congruences like these naturally arise in the framework of analysis and form the of the algebraic field GTR and were widely studied, in particular, by Newman [29], theory. Kerr [30], and Burinskii [31].

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 818 KASSANDROV the solutions of eikonal equation and associated con- 6. It has been demonstrated in [39, 40] that the gruence can be desribed as follows (for details see [15, complexified electromagnetic field associated with 36]). fundamental spinor ξ (it identically satisfied self- Using gauge (projective) , one reduces duality conditions (8) and, thus, the homogeneous fundamental spinor ξ(X) to the ratio of its two com- Maxwell equations) can be directly expressed through ponents choosing, say, the function g (obtained as a solution of algebraic { } T constraint (15)) and its derivatives ΠA, ΠAB with ξ =(1,g(X)); (11) respect to the twistor arguments {τ A},A=1, 2. then any solution of the algebrodynamical field Specfically, for spintensor of electromagnetic field strength ϕ one gets theory is defined via the only complex function AB g(x, y, z, t)—component of the projective 2-spinor ξ. 1 d Π Π ϕ = Π − A B , (16) In turn, any solution for g(X) is obtained in the fol- AB P AB dg P lowing way. Consider an arbitrary (almost everywhere smooth) surface in the 3D complex projective space with P := dΠ/dg. Strengths of the associated Yang– CP 3; it may be set by an algebraic constraint of the Mills field can also be represented algebraically form via (16) and the spinor g itself. Π(g, τ1,τ2)=0, (12) 7. It can be seen from representation (16) that the electromagnetic field strength turns to infinity at the where Π(...) is an arbitrary (holomorphic) function points defined by the condition of three complex arguments. Let now these latters be dΠ ∂Π linearily linked with the points of Minkowski space P = ≡ + wΠ1 + vΠ2 =0. (17) through the so-called incidence relation [34] dg ∂g τ = Xξ (13) Similar situation takes place for singularities of as- sociated Yang–Mills field and the field of or, in components, effective Kerr–Schild metric (10) (see [31, 32, 37]). B τ 1 = wg + u, τ 2 = vg + p, (14) Therefore, in the context of algebrodynamics one is brought to identify particles with locus of common where in the considered case of real Minkowski space singularities of the curvature and gauge fields.It the coordinates (3) u, v = ct ± z are real and p, w = is also reasonable to assume under this identification x ± iy complex conjugated. It is known that two that, instantaneously, particle-like singularities are ξ, τ related with points X via the incidence bounded in 3-space.4) relation (13) or (14) form the so-called projective 8. With respect to the primary C field g(X) ob- twistor of the Minkowski space [34]. tained from constraint (15), condition (17) defines its After substitution of (14) into equation of gener- branching points. Geometrically, this corresponds ating surface (12) the latter acquires the form of an to caustics of the light-like rays of generating con- algebraic equation gruence. Generally speaking, instead of the primary Π(g, wg + u, vg + p)=0 (15) B field and correspondent multivalued field g(X) one can equivalently consider the fundamental congru- with respect to the only unknown g, whereas the ence consisting, generically, of a (great) number of in- coordinates {u, v, p, w} play the role of parameters. dividual branches (“subcongruences” [36]) and form- Resolving the equation above at each point of the ing caustics-particles at the points of merging of rays Minkowski space X, one obtains some (generally from some two of them, i.e., at the envelope. This all- multivalued) field distribution g(X). matter-generating primordial structure in [19, 21, 36] In a rather puzzling way (the proof may be found, has been called the prelight flow,orthe“Prelight.” say, in [37, 38]), for any generating function Π 9. At the same time, existence of the Prelight flow and any continious branch of the solution under immediately distinguishes the temporal structure5). consideration the field g(X) identically satisfies Indeed, incidence relation (13) preserves its form un- both fundamental relativistic equations—linear der a one-parametrical coordinate transformations of wave equation g =0and nonlinear equation of the form complex eikonal (6). Correspondent spinor ξ (in x → x + n s, t → t + s, (18) the gauge (11)) satisfies meanwhile the equations of a a a shear-free null congruences and, according to the 4)For string-like singularities expanding to infinity and found, above-mentioned Kerr theorem, all such congruences e.g., in [37] another interpretation is needed (cosmic strings, can be obtained with the help of the exposed algebraic etc.). procedure. 5)Actually, this is true for any twistor structure in general.

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nana =1,s∈ R, fixed in absolute value (in the accepted normaliza- tion q = ±1/4) [14, 28]. Correspondent effective met- corresponding to a in 3-space along each ric (10) is just the Reissner–Nordstrem¨ solution of of rectilinear rays of the congruence, i.e., along spatial the Einstein–Maxwell equations. directions specified by the vector n defined as In a general case a =0 solution (21) leads to ξ+σξ n = (19) the field and metric exactly correspondent (under ad- ξ+ξ ditional requirement on electric charge to be unit!) 1 to the above-mentioned Kerr–Newman solution (in = {g + g∗,i(g − g∗), 1 − gg∗}. 1+gg∗ the regime of a naked singularity free of horizon). Carter [42] was the first to notice that correspondent Under such transformations physically correspon- gyromagnetic ratio for this field distribution is ex- dent to the process of propagation of the principal actly equal to its anomalous value for Dirac fermion. field with universal velocity V = c =1, all of the three This stimulated subsequent studies (of Lopez, Is- components of the projective twistor are preserved in rael, Burinskii, Newman, et al.) in which the Kerr value, as well as the direction vector n itself. Then, in singular , with associated set of fields, has been accord with representation (18), one can regard these regarded as a model of electron. Note that in the transformation as a prototype of the course of time algebrodynamical scheme this consideration is still and the Prelight Flow itself as the Time Flow.Inmore more justified since the electric charge therein is nec- details these issues were considered in [21, 36], and essarily fixed in modulus and may be identified as the in Section 5 we shall see in what an interesting way elementary one. Thus, they are refracted under introduction of complex pre- In the framework of B algebrodynamics over space. Minkowski space the electron can be represented 10. Particles identified with common singularities by the Kerr singular ring (of Compton size) of gauge and curvature fields exhibit a number of related to a unique static axisymmetrical so- remarkable properties specific for real matter con- lution (21) of equations (7), or of the con- stituents. The most interesting is, perhaps, that of straint (15). self-quantization of electric charge. This property 11. A number of other exact solutions of the initial follows from over-determinance of master system (7) algebrodynamical equations and of related eikonal, and self-duality of associated field strength (8) and Maxwell, and Yang–Mills equations have been ob- is, partially, of topological origin. According to the tained in [36, 41], among them a bisingular solution quantization theorem proved in [41]), for any isolated and its toroidal modification [38]. They correspond to and bounded (i.e., particle-like) singularity of elec- generating function Π in (15) quadratic in g.More tromagnetic field (16) electric charge is either null complicated solutions demand the computer assis- or necessarily multiple to some minimal, ele- tance for solving the algebraic relation (15). However, mentary value, namely, to the charge of fundamental B the (most interesting) structure of singular loci of static solution to equations (7). The latter is a direct these distributions can be determined through elimi- analog of the well-known Kerr–Newman solution in nation of the unknown g from the set of two algebraic GTR. It follows from twistor constraint (12) with 6) generating function Π of the form equations (15) and (17) . The complex equation aris- ing under the procedure Π=gτ − τ +2iag = wg2 1 2 (20) Π(x, y, z, t)=0 (22) +2(z + ia)g − p =0,z:= (u − v)/2, represents itself the equation of of particles- resolving which one obtains the two-valued solution singularities and, moreover, at a fixed instant fixes p x + iy their spatial distribution and shape. In this way, we g = ≡ , (21) have examined the structure of singularities for com- z ± r ± 2 2 2 z + ia x + y +(z + ia) plicated solutions of master equations and associated a ∈ R. biquaternionic and electromagnetic fields. As for the latter, there has been obtained a peculiar solution With the above solution one can associate the fa- to free Maxwell equations (!) describing the process mous Kerr congruence with caustic of the form of of annihilation of two unlike (and necessarily unit) a singular ring of radius a correspondent to the lo- charges, with accompaning radiation of a singular cus of branching points of function (21). Particularly, in degenerate case a =0of a point-like singularity 6)In the case of polinomial form of generating function Π the the associated via (16) electric field is the Coulomb procedure reduces to determination of the resultant of two one but electric charge q of singularity is strictly polinoms and can be easily algorithmized.

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 820 KASSANDROV wave front [36], a class of the wave-like singular other hand, the structure of string-like singularities- solutions [40], etc. particles arising on M under this procedure turns 12. If one restricts itself by generic solutions to be unstable and, perhaps, diffuses with time. To- to master equations (7) or to associated Maxwell gether, these considerations suggest the necessity of equations7), then their singular locus will (instan- a more successive analysis of the geometry “hidden” in the algebraic structure of , taneously) represent itself as a number of one- C dimensional —“strings.” Generally, these and of probable links of its 4 vector space with the strings (though neutral or carrying unit charges) are true physical geometry. On this way we immediately unstable in shape and size with respect to a small discover a completely novel geometry of (extended) variation of parameters of the generating function space–time presented in the next section. Π. As an example, consider a special deformation of the Kerr solution and congruence [3] defined 3. BIQUATERNION GEOMETRY AND PHASE by the following modification of the Kerr generating EXTENSION OF THE MINKOWSKI SPACE function (20): Let us return to matrix representation (3) of the Π=gτ1(1 − ih) − τ 2(1 + ih)+2iag, (23) elements Z ∈ B of biquaternion algebra. Restriction Z → U : U + = U −1 ∗ det U ∈ R to unitary matrices re- in which the parameter h enters in addition to duces the algebra B to that of real Hamilton quater- ∈ R the standard Kerr parameter a . As a result, from nions Q. Remind that Q is one of the two exceptional the constraint Π=0 one obtains a novel solution associative division algebras, together with com- for function g that defines still axisymmetrical but plex algebra. Transformations preserving, together now time-dependent generating congruence of rays. with unitarity, the structure of multiplication in Q Its caustic defined by the branching points of g is (inner automorphisms) are of the form represented by a uniformly collapsing into a point and, −1 −1 + afterwards, expanding to infinity singular ring: U → S ∗ U ∗ S ,S = S , (25) 2 2 S ∈ SU(2). z =0,ρ:= x + y = v(t − t0), (24) √ Under these, the diagonal (real) component of a ma- where t = a/ 1+h2 and velocity of collapse/ex- 0 √ trix U is invariant whereas the other three {x1,x2,x3} pansion v = h/ 1+h2 is always less than the light behave as components of a rotating 3-vector (note one c =1.Thus, that both ±S correspond to the same : spinor The Kerr congruence is unstable with respect to structure). So the of quaternion ∼ a small perturbation of controlling parameters algebra Aut(Q)=SU(2) = SO(3) is 2:1 isomor- of the generating function. This lets one expect phic to the group of 3D rotations, with the main also the instability of the Kerr (Kerr–Newman) invariant solution of (electro) Einstein equations, 2 2 2 l = x1 + x2 + x3, (26) since the latter is defined, to a considerable de- gree, by the structure of null congruence of the defining Euclidean structure of geometry induced by above-presented type. the algebra Q. In this sense, from the times of Hamil- Note in addition that the deformed ring still car- ton, exceptional algebra of real is ries a fixed elementary charge but cannot escape, considered as the algebra of physical background nonetheless, being radiated to infinity. space and, in the algebrodynamical paradigm, pre- determines its dimensionality and observable Eu- At this point we complete our brief review of clidean structure. the “old” algebrodynamics on the Minkowski back- We can then apply the same “Hamilton logic” to ground by the following remarks. In fact, from a B ∈ B the algebra of biquaternions .NowtheelementsZ single initial condition of differentiability we were B are represented by complex matrices of generic able to develop a self-consistent theory of fields and type (3), and multiplication in B is preserved under particle-like formations possessing a whole set of transformations unique and physically realistic properties. The only → ∗ ∗ −1 ad hoc assumption made during the construction Z M Z M , det M =1, (27) of algebrodynamical theory, in order to ensure its M ∈ SL(2, C). Lorentz invariance, was a rather artificial restriction of the coordinate 4C vector space of B algebra onto In full analogy with the real case, the diagonal com- the subspace with Minkowski metrical form. On the ponent z0 in (3) remains invariant, and the three others z = {z1,z2,z3} manifest themselves as a 3D 7) complex vector under complex rotations. Thus, one That is, by solutions free of any symmetry, in particular, ∼ nonstatic and nonaxisymmetric. has: Aut(B)=SL(2, C) = SO(3, C).

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 ALGEBRODYNAMICS OVER COMPLEX SPACE 821 Some explanations must be presented at this aquire, respectively, the meaning of temporal and spa- point. It is well known that the 6D (in reals) group tial coordinates of some effective 4D space with a SL(2, C) is a covering of the realizing Minkowski-type metric. Note also that such an iden- its spinorial representation; the same is true for tification is quite unformal since under the B auto- the 2:1isomorphic group of 3D complex rotations morphisms acting as 3D complex rotations the quan- SO(3, C).Specifically, Lorentz transformations can tities T,R transform one through the others just be represented in the form analogous to (27), as the temporal and spatial coordinates do under 8) X → M ∗ X ∗ M +, (28) Lorentz transformations. Thus, the main real invariant of biquater- but act on the subspace Z → X of Hermitean ma- + nion algebra, being positive definite, induces trices X = X with determinant representing the nonetheless the structure of causal domain of the Minkowski metric. It is just this restriction that we Minkowski space correspondent to the interior of considered in the previous section. Now, however, we the (together with its light-like boundary). are interested in natural geometry induced by the full In this scheme, the events that are not causally Z B structure of 8D (in reals) vector space of algebra, connected as if do not exist at all (just as this should in its hypothetical relations to the Minkowski space be from a successive viewpoint of STR). We come, and in physical meaning of four additional coordi- therefore, to a paradoxical but much interesting, both nates. It should be noted that, surprisingly, this ge- from physical and mathematical viewpoints, concept ometry has not been discovered until now. As we shall of the physical space–time with positive definite see, corresponding construction is rather transparent metric. and successive. We have seen that the structure of B multiplica- Consider now the phase part of complex invari- tion is preserved under 3D complex rotations forming ant (29). The latter can be represented in the form the SO(3, C) group. The main complex invariant of σ = S expiα, (34) these transformations, the analog of Euclidean in- variant (26) of the real algebra Q, is represented by with absolute value S correspondent to the Minkows- a holomorphic (quasi)metrical ki interval and the phase α also invariant under 3D 2 2 2 ≡| |2 complex rotations (that is, in fact, under Lorentz σ = z1 + z2 + z3 z , (29) transformations). In this connection, the noncom- the (squared) “complex length” of a vector z.Itshould pact (corresponding to modulus) part of the initial be emphasized that all other metrical forms, the Her- invariant is responsible for macrogeometry explicitly mitean metric among them, that could be canonically fixed by an observer: remarkably, it turns to be exactly defined on the vector space C4 itself (or on its sub- of a Minkowski type. At the same time, its phase, space C3) are, in fact, meaningless in the framework compact part determines geometry of the “fiber” and, of the algebrodynamics since they do not preserve perhaps, reveals itself at a microlevel being, in par- their structure under B automorphisms. ticular, related to universal wave properties of matter On the other hand, from complex invariant (29) (see Section 5). In the other respect, invariant α has one can naturally extract a positive definite (exactly, the meaning of the phase of proper complex time as non-negative) Finslerian metrical form of the fourth this can be seen from (34) and will be discussed below. degree taking the square of complex modulus of the Thus, we accept a novel concept of the back- considered invariant ground space–time geometry as of the phase exten- ∗ ∗ S2 := σσ = |z|2|z |2. (30) sion of (a causal part) of the Minkowski space predetemined by the initial complex-quaternionic As the next step, one can make use of the following structure, with coordinates bilinear in those of the remarkable identity (see, e.g., [43]): primordial and “actually existing” C3 space. 2 ∗ 2 ∗ 2 ∗ 2 |z| |z | ≡ (z · z ) −|iz × z | (31) Note that in literature dealing with different ver- that can be explicitly verified. Taking it into account, sions of complex extensions of space–time (see, one can represent the positive-definite invariant (30) e.g., [44, 45]) one usually encounters the procedure of in the form of a Minkowski-like interval [22]: separation of complex coordinates into real “physical” 2 2 2 and imaginary “unphysical” parts (alternative to their S = T −|R| ≥ 0, (32) separation into “modulus” and “phase” parts in our in which the quantities T and R defined through approach). This procedure is, actually, inconsistent, · × the scalar ( ) and vector ( ) products of complex 3- 8) vectors as In fact, these transformations have some peculiarities in ∗ ∗ comparison with canonical Lorentz transformations, see [22] T := z · z , R := i z × z , (33) for details.

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 822 KASSANDROV since both parts are completely equivalent in inter- 4. COMPLEX ALGEBRODYNAMICS nal properties and should equally contribute to the AND THE ENSEMBLE OF “DUPLICONS” induced real geometry one constructs. Accordingtothefirst principles of algebrodynam- Nonetheless, the above-mentioned linear separa- ical approach, the true dynamics takes place just in 4 tion of complex coordinates is rather demonstrative. the biquaternionic “pre-space” C . In fact, we are Specifically, consider acoupleof3Drealvectors able to explicitly observe only a “shadow” of this pri- {p, q} associated with a complex vector z: mordial dynamics on the induced (via mapping (33)) real Minkowski-like space with additional phase and z = p + iq. (35) causal structures. As another ground under construction of complex In this representation the principal invariant (29) algebrodynamics there serves a distinguished role takes the form of the “complex null cone”, a direct analog of the σ =(|p|2 −|q|2)+i(2p · q), (36) real Minkowski light cone. Specifically, consider two points P, P (0) ∈ C4 with coordinates Z connected and corresponds to a pair of invariants in which one through algebraic relation of the form easily recognizes the two well-known invariants of − (0) 2 − (0) 2 − (0) 2 electromagnetic field (with vectors p, q identified [z1 z1 ] +[z2 z2 ] +[z3 z3 ] (39) as the field strengths of electric and magnetic field, − (0) 2 respectively). The noticed analogy of complex coor- =[z0 z0 ] . dinates and electromagnetic field seems much sug- Then it is easy to demonstrate [23] using the inci- gesting and requires thorough analysis. dence relation τ = Zξ (comp. with real case (13)) that the twistor field (as well as the principal spinor Express now through vectors p, q the effective field g and the initial biquaternionic field) takes equal temporal and spatial coordinates (33): values in all the points of the complex null line T = |p|2 + |q|2, R =2p × q (37) connecting these points, that is, along an element of the null cone. In this respect the position and and note that the temporal coordinate is positive def- displacement of such points are correlated. It is inite (in Section 5 we shall relate this property with noteworthy that in both sides of nullconeequa- time irreversibility). As to three spatial coordinates, tion (39) there stands one of the two fundamen- they form an axial vector so that the choice of sign tal invariants of B algebra so that the primordial corresponds to reference frame of definite chirality. complex geometry dynamically reduces to the geometry of smaller space C3 with holomorphic Finally, let us write down a remarkable rela- (quasi)metrical form (29). As we are already aware tion [22] that links the module V of the velocity of, this gives rise to real effective geometry of the V = δR/δT of motion of a material point in the Minkowski type. Note also that for a fixed value of (the induced Minkowski space with characteristics of two equal) invariants equation (39) defines a complex initial complex space C3, namely, with invariant phase 2-. The latter manifold is SO(3, C)-invariant α and the θ between vectors p and q: and closely related to unitary representations of the Lorentz group [46]. 1 − V 2 cos2 θ = . (38) Finally, in full analogy with the “old” version of 2 2 1+V coth α algebrodynamics on M,letusidentify particles In a limited case of motion with fundamental velocity with singularities of the biquaternionic and as- V = c =1 one gets θ = π/2,sothatvectorsp, q sociated fields, geometrically—with caustics of are orthogonal to each other and to the direction of generating congruence. In this connection, recall motion V (in analogy with electromagnetic wave). that generic singularities on M have the structure of From (37) one obtains also that in this “light-like” one-dimensional curves—“strings”. However, on the C3 case the two vectors are equal in modulus, invariant complexified background singularities manifest σ turns to zero, and the phase α becomes indefinite. much richer structure. Let us also emphasize from the beginning that In the opposite case of a “particle” at rest V =0 the dynamical principles of complex algebrodynamics one gets θ =0,π, so that two different (“para” and are completely the same as of its old version on M. “ortho”) relative orientations of vectors p, q are pos- Specifically, we only make use of biquaternionic fields sible. This remarkable property might be related to obeying the B-analyticity conditions or, equivalently, two admissible projections of the spin vector onto an of twistor fields defining a generating congruence of abitrary direction in 3-space. complex null rays with zero shear. In particular, all

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 ALGEBRODYNAMICS OVER COMPLEX SPACE 823 the rules of definition of the set of relativistic fields conjecture, in the arising picture “all of the electrons” (Section 2) do not require any modification in the are, essentially, “oneandthesameelectron” in vari- complex case. ous locations on a unique world line. In fact, however, Consider now generating congruences (and, cor- the arising structure of singularities suggests a more respondingly, biquaternionic, twistor, and associated natural, though exotic, interpretation. gauge fields) of a special and physically interesting Indeed, let us consider a primordial “generating” type. These are congruences with a focal line—a duplicon in the capacity of an “elementary observer” world line of some virtual point charge “moving” O.12) All other duplicons on its null cone (40), though in complex extension C4 of the Minkowski space.9) dynamically correlated with O, are in fact “invisible” Structures like this have been first considered in the and not perceived by the elementary observer: any framework of GTR by Newman [29, 47]; further on, “signal” is absent! It is thus natural to conjecture congruences with a focal line will be called New- that the act of “perception” (actually—of interac- man’s congruences.10) tion) takes place when only a null complex line (an In this connection, consider a point-like singula- element of the complex null cone) connecting O with rity-particle “moving” in complex space C4 along a some duplicon becomes “material”,thatis,—a caus- tic of the generating congruence. “” zµ = zµ(τ), τ ∈ C, µ =0, 1, 2, 3.Points, where the primordial twistor (spinor) field takes the It is easy to determine the caustic locus of the same values as in the vicinity of the “particle,” are congruence from null cone equation (39). Similar to defined by null cone equation (39). However, let these the case of general solution (15) of the B-analiticity points belong themselves to the considered world line equations, in the case of a Newman’s congruence and represent thus other “particles”.Thennullcone caustics coincide with the branching points of the equation (39) acquires the form principal spinor field g or, equivalently, of the field of local time of a duplicon τ(Z).13) At these points one L := [z (λ) − z (τ)]2 +[z (λ) − z (τ)]2 1 1 2 2 (40) observes the amplification of the principal twistor- 2 2 +[z3(λ) − z3(τ)] − [z0(λ) − z0(τ)] =0 biquaternionic field (preserved along the elements of the complex null cone) that can be regarded as the and for any τ has, in general, a great (or even infinite) process of propagation of a “signal” to (from) the (n) number of roots λ = λ (τ) defining an ensemble of observer O, see below. (n) correspondent “particles” zµ . These are arranged in In turn, branching points correspond to multiple various points of the same complex world line and roots of the null cone equation defined by the condi- dynamically correlated (“interact”) with the initial tion (generating) particle. 1 dL L := − = z (z (λ) − z (τ)) (41) Such a set of “copies” of a sole point-like particle 2 dλ a a a “observing itself” (both in its past and , see − z (z (λ) − z (τ)) = 0 Section 5) was first considered in our works [23, 24] 0 0 0 and was called therein the ensemble of duplicons. (summation over a =1, 2, 3 is assumed and prime It is noteworthy that on real Minkowski background denotes differentiation by λ). Together with initial equation (40) (which on M turnstobeanordinary defining relation for duplicons (40), the above con- retardation equation), in the case when the point dition specifies a disrete set of positions of the of observation belongs itself to the world line of a observer (via its local times τ = τ (k))andofa particle, has a unique solution independently of the pair of duplicons joining at a correspondent in- form of a trajectory, namely, the trivial solution λ = τ. stant (defined as one of mupliple roots λ = λ(k)). Thus, the concept of duplicons cannot be realized on Thus, elementary interaction act can be regarded as the background of ordinary Minkowski space M. a fusion of some two duplicons (a, b)(withλ(a)= In [23, 24] we were guided by the old idea of λ(b)) considered with respect to the observer O at 11) R. Feynman and J.A. Wheeler and considered each some of its positions (with τ = τ (k)). At such instants duplicon in the capacity of an electron model.In- deed, in full correspondence with Feynman–Wheeler 12)To model a real macroscopic observer, instead of a trajectory of an individual duplicon zµ(τ), one should introduce some 9)Exact definition and specification of such congruences is averaged trajectory simplest of which is represented by a presented, say, in [23]. null complex line and, under its mapping into real Minkowski 10)At present, this approach is intensively developed by New- space, corresponds to uniform rectilinear motion of an inertial man himself with collaborators [48] as well as by Burin- observer. skii [49]. 13)This field satisfies the complex eikonal equation [23]. On the 11)Their below presented construction, by virtue of the above- real Minkowski background this is analogous to the field of mentioned reason, cannot be realized on real M. the “retarded” time.

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 824 KASSANDROV a process of the field amplification occurs along a α of the principal complex invariant (34). Since, how- null complex line—a caustic—connecting the ob- ever, the complex coordinates of both “twins” at the server and the two coinciding duplicons. As it was instant of fusion should be equal, for the acquired already mentioned, under its mapping into M this phase lag one has line corresponds to some rectilinear path of a field ∆α =2πN, N =0, ±1, ±2,... (42) perturbation moving in uniform with velocity V ≤ c. We are now in a position to naturally distinguish Thus, there exists only a certain set of points at particles-singularities in the scheme under consid- which a microobject might be once more observed eration as matter constituents and interaction after some its primary “registration”. This stronly carriers, in full analogy with generally accepted theo- resembles the well-known procedure of preparation retical classification. First of them form the ensemble of a quantum-mechanical state and of the following of identical duplicons and can move along a very QM measurement, respectively. However, in the complicated and mutually concordant , above presented picture we do not encounter any geometrically—along the focal of generating sort of the wave–particle dualism, of the de Broglie congruence. As to the second, they always move wave, etc. Each matter pre-element—duplicon— along rectilinear line elements of the complex null manifests itself as a typical point-like corpuscular, cone connecting a pair of “interacting” duplicons. In whereas phase relations are of a completely geomet- this process, the two merging duplicons represent an rical nature and relate to some internal space of a entire particle (see below) and stand for an emitter, 15) whereas the observer—for a detector of propagating “fiber” over M . Below we shall once more return to discuss the interference phenomenon. “signal”; the problem of temporal ordering arising in this connection will be discussed in Section 5. Thus, we are led to the conclusion that any 5. RANDOM COMPLEX TIME elementary object (electron?) may be fixed by an AND QUANTUM UNCERTAINTY observer only at some particular instants and repre- sents itself a pair of pre-elements—duplicons— Essentially, it is meaningless to discuss the prob- emmiting a signal towards the observer when and lem of dynamics in complex space before one specifies only when their positions coincide in (complex) the notion of complex time. In fact, we have already space. At all the rest time these pre-elements– seen that the evolution parameter τ ∈ C of an “el- duplicons–are separated in space, do not radiate, and, ementary observer” O, that is, the parameter of the consequently, can be detected by none observer.14) “world line” zµ(τ) of a generating point singularity Conjecture about dupliconsashalvesofthe is now complex-valued. This means that subsequent position of the observer on its “trajectory” (under electron revealing themselves solely at the instants → of pairwise fusion strongly correlates with modern τ τ + dτ)isindefinite by virtue of arbitrariness of concept and observations of fractional charge (see, alteration of the phase of parameter τ. e.g., the review [50]) and, on the other hand, makes On the one hand, any value of τ one-to-one corre- it possible to offer an alternative explanation for the sponds to a certain position of the observer O (and of wave properties of microobjects, particularly, for the associated set of duplicons correlated with O through quantum interference phenomena. thenullconeconstraint) and, therefore, to adefinite Indeed, let a pair of duplicons be identified as “state of the ” with respect to a given ob- an electron at an instant of the first fusion, via the server. caustic-signal emitted towards an observer. In the On the other hand, a particular realization of following, these “twins” diverge in space and, in par- those or other continuations of the trajectory is am- ticular, can pass through different “slots” in an ide- biguous being ruled by an unknown law of “walk” of alized interference experiment. As a result, they can the evolution parameter τ across the , again reveal themselves only at an instant of the that is, by the form of a curve τ = τ(t) with mono- next fusion accompanied by a new act of emission tonically increasing real-valued parameter t ∈ R.16) of a signal-caustic in the direction of the observer. In [23, 24] this was called the evolution curve. Between the two fusions, each of the “twins” acquires aparticularphase lag, namely, of geometrical phase 15)In our scheme, the fiber itself defines the structure of effective Minkowski base. Such situation is unique and, in paticular, 14)In the complex algebrodynamics there exists also another can find application in the theory of Calabi–Yau class of singularities representing themselves as a sort of with a 3C fiber structure. “three-element” formations. One can speculate about their 16)One can evidently represent this parameter by the length of probable relation to the quark content of matter. the curve.

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 ALGEBRODYNAMICS OVER COMPLEX SPACE 825 Note that only after specification of the form of Now under averaging the mixed term dT = evolution curve one can ascertain the order of suc- 2(p · dp + q · dq) vanishes, and the time interval cession of events and even distinguish past from fu- δT = dp · dp + dq · dq ≡ ∆T (44) ture. It is just this form that defines the time arrow and predetermines, in particular, which of the (com- at a “physically infinitesimal” scale behaves as a full pletely identical in dynamics) duplicons is “younger” differential, an actually holonomic entity. The same is and which “older” than a certain “elementary ob- true for increments of the averaged spatial coordi- server” O. nates δR ≡ ∆R. In the framework of neo-Pythagorean ideology of Moreover, property of increment of the time coor- ≥ algebrodynamics, the form of the universal evolution dinate δT 0 be positive definite “in average” leads curve should follow from some general mathematical immediately to a natural kinematical explanation considerations and be exceptional with respect to its of the irreversibility of physical time. Actually, any internal properties; unfortunately, at present the form “macroscopic” alteration of the particles’ posi- is unknown. Up to now it only seems natural to tions (of the state of a system of particles) in expect that this “Time Curve” is extremely compli- the primary complex space necessarily results in cated and entangled (being, probably, of a fractal-like an increase of the value of time coordinate of the nature). Where this is the case, for us the character effective MInkowski space. Thus, in the algebrody- of alteration of the evolution parameter on its com- namical approach irreversibility of time seems to plex plane would effectively represent itself a random be of kinematical and statistical nature,andinthe latter respect time resembles the entropy-like quan- walk. Moreover, one may conjecture that this walk is tity in the orthodox scheme (if the latter is understood discrete whereas the generating worldline z (τ) itself µ as a probability measure). remains complex analytical: these two are completely independent. Then in the scheme there arises the To conclude, in the context of initially determinis- time quanta—“”. We shall see below that tic “classical” dynamics there arises an unremovable it has to be of order of the Compton size, not of uncertainty and, effectively, randomness of evolu- the Plank one. From different viewpoints the latter tion of an observable ensemble of micro-objects. This concept has been advocated in a number of works uncertainty is of a global and universal character and is related to conjectural stochastic type of al- (see, e.g., [51] and references therein). teration of the complex time parameter, to complex It is noteworthy that despite a probable random and effectively random nature of physical time character of the Time Curve it gives rise to mutually itself. It is noteworthy that numerous problems and correlated alterations of the locations of different par- perspectives arising under introduction of the notion ticles or, more generally,—to global synchroniza- of two-dimensional time have been considered by tion of random processes of various nature.Ata Sakharov [53];Kechkin and Asadov [45] studied the microlevel this may be related to quantum nonlocality with complex time parameter and entanglement, at macrolevel—to universal corre- and introduced, in this connection, the notion of dif- lations already observed in the experiments of Shnoll ferent alteration regimes of this parameter similar to (see, e.g., the review [52]). the above-introduced notion of “evolution curve.” Remarkably, in the capacity of rather unspecified Conjecture about random nature of the Time dy- parameter τ of the generating world line zµ(τ) one namics and resulting randomness of the motion of can (should!) use the principal invariant σ of complex microobjects makes it possible to solve also the prob- (34), with its modulus S correspond- lem of concordance between increments δT, δR of ing to ordinary Minkowski proper time, and phase the effective space–time coordinates (33) and differ- α responsible for uncertainty of evolution. This is ences of their final and initial values ∆T = T − T , the only parameter ensuring preservation of both the ∆R = R − R17). Indeed, for, say, the time coordinate primary twistor field and the caustic structure (along one gets straight null rays of the generating congruence) (see the proof in [23]). In this sense (despite its accepted T T − T |p dp|2 −|p|2 ∆ := =( + ) (43) name) complex “proper” time acquires the meaning +(|q + dq|2 −|q|2)=2(p · dp + q · dq) of universal global time governing the concordant +(dp · dp + dq · dq). dynamics of the Universe. We are now ready to return back to the analysis 17)Generally, these are not necessarily equal due to bilinearity of quantum interference experiment started in the of the induced space–time coordinates with respect to the previous section. Recall that we have undertaken an primary complex “holonomic” coordinates zµ. attempt to relate the wave properties of matter to the

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 5 2009 826 KASSANDROV conjecture of dimerous electron (formed by two pre- matter—duplicons—constituents of observable par- elements—duplicons—at the instants of their fusion) ticles (electrons?) as well as uniformly propagating and to the geometrical phase (phase of the complex interaction carriers (caustics) and random complex time α) “attached” at each point of the generating time that predetermines kinematical irreversibility of world line and lagging along the latter. In the simplest physical time at macrolevel. However, after 25 years case, assuming linear proportionality of (physically) of development of the algebrodynamical field theory infinitesimal increments of the module dS and phase on ordinary Minkowski background, “new” complex dα of complex time, algebrodynamics is just at the very beginning of its dα = const · dS, (45) march. We expect that the properties of biquaternion algebra and of associated mathematical structures and choosing as the scale factor one half of the in- are rich enough to encode in themselves the most fun- verse Compton length of electron const = Mc/2 damental laws of dynamics and geometry of physical (this corresponds to the above-mentioned assump- World. tion about the quanta of complex time —“chronon”), one obtains from the fusion condition (42) ACKNOWLEDGMENTS Mc ∆A ∆α = ∆ dS = =2πN. (46) 2 The author acknowledges S.I. Vinitsky and I.B. Pestov for fruitful discussions, A.V. Goosev for Essentially, the above formula represents condition assistance in computer visualization of the duplicons’ for maxima of classical interference in the rela- dynamics. tivistic case. According to it, the phase lag for two “halves”—duplicons under interference are propor- tional to the path difference, with the Minkowski REFERENCES interval as invariant measure. On the one hand, 1. R. D. Sorkin, in Relativity and Gravitation: Classi- this is in remarkable correspondence with famous cal and Quantum, Ed. by J. C. D. Olivo et al. (World Feynmann representation of the wave function Ψ= Sci., Singapore, 1991); Int. J. Theor. Phys. 39, 1731 R exp(iA/) whose phase is proportional to the (2000). classical action A (for free particle—to the proper 2. P.A. M. Dirac, Phys. Rev. 74, 817 (1948). time interval). On the other hand, in nonrelativistic 3 . V. V. K a s s a n d r o v, i n Proc. of the Intern. School approximation decomposing the interval dS over the on Geometry and Analysis (Rostov-na-Donu Univ. powers of velocity V/c and taking into account the Press, 2004), p. 65; gr-qc/0602046. integrability of zero power term, one obtains as the 4. L. P.Horwitz, hep-ph/9606330. condition of quantum interference thedeBroglie 5. V.G. Kadyshevsky, Phys. Part. Nucl. 29, 227 (1998). relation 6. I. B. Pestov, in Space–Time Physics Research Trends. Horizons in World Physics, Vol. 248, Ed. by dL h ∆ = N, λ := , (47) A. Reimer (Nova Science, New York, 2005), p. 1; gr- λ Mv qc/0308073. with integer path difference of two duplicons in frac- 7. J. M. C. Montanus, Hadronic J. 22, 625 (1999); tions of the de Broglie wave length λ. J. B. Almeida, physics/0406026; arXiv:0807.3668 [physics.gen-ph]; www.euclideanrelativity.com Thus, the phase invariant α seems to be of funda- 8. D. Hestenes, Space–Time Algebra (Gordon and mental physical importance being at the same time a Breach, New York, 1966). measure of uncertainty of the evolution of microob- 9. M. Pavsic, The Landscape of : jects and the measure of their wave properties. The AGlobalView(Kluwer, Dordrecht, 2001). latters have their origin in the peculiarities of pri- 10. I. A. Urusovskii, Zarubezh. Radioelectron., No. 3, 3 mordial complex geometry and do not appeal to the (1996); No. 4, 6, 66 (2000); www.chronos.msu.ru paradigm of wave–particle dualism. 11. Yu. I. Kulakov, Theory of Physical Structures (Univers Contract, Moscow, 2004) [in Russian]. To conclude, we have endeavored to demonstrate 12.Yu.S.Vladimirov,Foundations of Physics (BI- the following. Exceptional complex geometry based NOM, Moscow, 2008) [in Russian]. on the properties of remarkable algebraic structure 13. A. P. Efremov, Quaternionic Spaces, Reference (biquaternions) when introduced into foundations Frames and Fields (Peopl.Friend.Univ.Press, of physics as the primordial “hidden” geometry Moscow, 2005) [in Russian]; Gravit. Cosmol. 2, 77, of space–time (instead of the habitual Minkows- 335 (1996). ki geometry) results in a quite novel and unex- 1 4 . V. V. K a s s a n d r o v, Algebraic Structure of Space- pected picture of the World. As its principal ele- Time and Algebrodynamics (Peopl. Friend. Univ. ments one can distinguish identical pre-elements of Press, Moscow, 1992) [in Russian].

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