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Gravitation & Cosmology, Vol.1 (1995), No.3, pp.216–222 c 1995 Russian Gravitational Society

BIQUATERNION ELECTRODYNAMICS AND WEYL-CARTAN GEOMETRY OF - V.V.Kassandrov

Russian People’s Friendship University, Department of General Physics, 3 Ordjonikidze Str., Moscow 117302, Russia

Received 5 July 1995

The generalized Cauchy-Riemann equations (GCRE) in algebra appear to be Lorentz-. The Laplace equation is in this case replaced by a nonlinear C -eikonal equation. GCRE contain a 2- and a C - gauge structures, and their integrability conditions take the form of Maxwell and Yang-Mills equations. For the value of electric charge from GCRE only the quantization rule follows, as well as the treatment of Coulomb law as a stereographic map. The equivalent geometrodynamics in a Weyl-Cartan affine space and the conjecture of a complex- structure of space-time are discussed.

1. Introduction mine physical dynamics as well. Indeed, if we con- sider the physical fields as algebra-valued functions of In the frames of the geometrodynamic approach all fun- an algebraic variable, the generalized Cauchy-Riemann damental physical quantities and above all the equa- equations (GCRE), i.e., the differentiability conditions tions of physical dynamics should be of a purely geo- in the STA, become fundamental equations of field dy- metric nature. The twistor program, the Kaluza-Klein namics. Wonderfully, the generally accepted physical theories and string dynamics give representative ex- equations (in particular, the Maxwell or Yang-Mills amples of this concept, perhaps the most general ones equations) become a direct consequences of GCRE, up to now. In essence, any physical interaction may namely their integrability conditions (see below). be regarded as a manifestation of geometry (by using From an epistemological point of view, the alge- multi-dimensional , fiber bundles, etc.). brodynamic (AD) concept returns us to the ideas of However, the diversity of admissible geometries and Pythagoras, Hamilton and Eddington on a crucial their invariants makes the “kinematic” part of this role of in the structure of the Uni- procedure (selection of space and geometric identifi- verse. At the modern stage we deal with the pri- cation of physical quantities), as well as the dynamic mary structure of multidimensional ST arithmetics, one (choice of a Lagrangian) quite ambiguous. Even completely different from the classical arithmetics of for the electromagnetic (EM) field one has a lot of dif- the Macroworld, or the of reversible processes ferent geometric interpretations (Weyl’s conformal fac- and weakly interacting objects. tor, bundle connection, the Kaluza metric field, torsion A genuine ST arithmetics ought to be non- [1] or nonholonomic [2] structures of space-time (ST), commutative and even non-associative! Indeed, etc.). these properties are just algebraic equivalents of causal Alternatively, within the algebrodynamic paradigm and interactive structures of the physical World (en- [3] the ST is regarded as a supplied with a suring the dependence of “out-state” on the order and basic algebraic structure, the structure of linear alge- composition of reactions). For such reasons the most bra in the simplest case. But it is well known that suitable STA candidate is the algebra,the the exceptional algebras — algebras with division and unique exceptional non-associative algebra. However, positive norm — exist in the d = 4 (Hamil- the difficulties of “intercourse” with are well- ton ) and d = 8 (Cayley octonions). So it known (see, nevertheless, [6 7]). would be natural to suppose that the ST algebra (STA) Meanwhile, the non-commutativity of algebraic [4] should be exceptional in its internal mathematical structures is closely connected with the non-linearity of properties. If it is the case, the of automor- the corresponding dynamic equations (this is the case, phisms (Aut) of STA would generate the ST geometry, in particular, for the Yang-Mills fields). We will see for example, by operating as an . later that the GCRE in non-commutative algebras also Moreover, the STA structure can completely deter- possess a nonlinear structure and are therefore capa- Biquaternion Electrodynamics and Weyl-Cartan Geometry of Space-Time 217 ble to describe both quantum phenomena and physical 2. B -algebra and B -differentiability field interactions. Let z M (4, C), z = zµ,µ=0, 1, 2, 3 be an element In this paper we choose for a STA the algebra of bi- of the∈ complex vector{ space M (4, C) of} d = quaternions B , the extension of real Hamilton quater- 4. The function nions H to the field of complex numbers C. The H F(z)= F µ(z) = F µ(z0,z1,z2,z3) (1) algebra is known to have Aut (H) = SO (3) and is { } in perfect correspondence with the structure of the 3- F M, maps an open domain O M to the domain ∈ µ ⊂ dimensional space. We are unaware of a similar O0 M ; let its components F (z) be complex and an- algebra for the case of Minkowski 4-space! For alytic.⊂ obvious reasons one often considers the Clifford-Dirac Then a structure B of associative algebra of com- algebra C(1, 3) to be the STA [4, 5]. However, a re- plex quaternions () M M Mmaybe duction from the 16-dimensional total of introduced on M . According to the × → B = C(1, 3) to a 4-dimensional physical ST is a completely L(2, C), L being the full 2 2 complex matrix algebra, “voluntaristic” procedure; even the we shall use the matrix representation× of B of the basic generator space may be chosen in different uw ways [8]. z M:z = zµσ = , (2) ∀ ∈ µ pv

The B-algebra, isomorphic to the Clifford alge- σµ = e, σa , e being the 2 2 matrix and bra C(3,0) of smaller dimension d = 8, is preferable σ ,a=1{ , 2}, 3 the ; ×u, v = z0 z3 ; from this point of view. On the other hand, the B - a p,{ w = z1 iz2 are} the DeWitt coordinates on M .± Now dynamics, based on GCRE, appears to be Lorentz the multiplication± ( ) in B is equivalent to the usual invariant, so the B -algebra may be treated as a matrix one; the function∗ (1) becomes a matrix-valued, minimal STA. This choice leads to the conjectures or B -valued function of a B -variable. Let for some on a fundamental role of null divisors as a subspace z O of STA and on complex-valued structure of ST; these ∈ questions will be discussed below. dF = F(z + dz) F(z)(3) − Now we are ready to the contents of the be an infinitesimal increment (differential) of F(z), cor- paper. In Sec. 2 we begin with the basic definitions responding to a differential of a B -variable dz and ac- of the B -algebra and B -differentiability. The general cording to the usual Euclidean metric ρ2 = zµ 2 . µ | | problems of (bi-)quaternionic analysis are also briefly Then we come to the following definition. P discussed. Then, in Sec. 3, after preliminary physical The function (1) F(z)issaidtobeB-differentiable identifications, we demonstrate the 2-spinor structure in some domain O Miffor z O there are some ⊂ ∀ ∈ of the basic GCRE and obtain a complexified eikonal G(z), H(z) such that the differential (3) may be equation for each component of the B -field. Global presented in the invariant form symmetries of the model are studied as well. dF = G(z) dz H(z), (4) ∗ ∗ Sections 4 and 5 are devoted to B -electrodynamics i.e. only through the operation of multiplication in B . as the basic case of B -differentiability. Firstly (Sec. 4) For the commutative algebra of complex numbers, the self-duality conditions are obtained from GCRE, from (4) the Cauchy-Riemann (CR) equations follow whence follow the Maxwell equations. Gauge invari- in the coordinate representation, F0 = G H being ance of a model of special type is demonstrated in Sec. a derivative of F(z). So the relation (4)∗ naturally 5. From the eikonal equation, a geometrical origin of generalizes the CR equations to the case of a non- the Coulomb law as a becomes commutative associative B -algebra. Eqs. (4) will be evident, and we get for the admissible values of an elec- further designated as GCRE (in the invariant form). tric charge q = 1, i.e., a quantization rule! ± A detailed study of B -differentiability and analyt- In Sec. 6 we demonstrate the equivalence of the icity, based on GCRE (4), may be found in [3], and a theory to geometrodynamics in a complexified Weyl- review of other approaches in [9]. The most profound Cartan space. A reduction to iden- is perhaps Fueter’s work [10]; G¨ursey et al. [11] applied tifies the magnetic monopole field as that of torsion it within the d = 4 gauge and chiral theories (see also and the Coloumb electric one as the ST Weyl non- [12]). metricity. We conclude in Sec. 7 by the establish- ment of complex-valued Yang-Mills equations as the 3. Spinor splitting and the eikonal eq- integrability conditions of GRCE and a discussion of uation general consequences of a complex-quaternionic struc- ture of physical space. Finally, we discuss the relation Let us turn now to the construction of field theory, of the AD approach to binary geometrophysics. based on the concept of B -differentiability. Consider a 218 V.V. Kassandrov

subspace M+ M of the points with real coordinates Z-transformations in (10) define a 6C -parameter x = xµ = ⊂zµ :Im(zµ)=0 , or else the subspace group of rotations SO (4, C); the restriction of this { } { } + 1 + of Hermitian matrices with elements z = z.TheB- group to M+ (with n− = m ) leads to the Lorentz 2 norm N (z)=Det(z) then generates on M+ the real transformations for x. Now, if we put in (10) k = n, 1 Minkowski metric l = m− , the functions F, G, H manifest their nature as 4-vectors (F n F m 1 etc.). However, when 2 − N (x)=Det(x)=uv pw k = n, l =Ident→., G∗ transforms∗ as a 4-vector, for F 2 2− 2 2 = x0 x1 x2 x3 (5) and H we have F n F, H n H, preserving − − − → ∗ → ∗     the structure of the spinor splitting (8). Moreover, a so M+ may be identified with the physical ST. Solu- double row-column splitting of (6) corresponds to the tions to (4) on M may be obtained by analytic contin- case k = l =Ident.,whenF(x) has to be considered uation from M+ . We will return to a detailed study as a scalar, while G and H transform as conjugated 1 of the relation between M and M+ in Sec. 7. G G m− , H n H. It is now evident that the B -differentiable functions Let us→ return∗ now to the→ dynamical∗ consequences F(x), realizing the mappings F:M+ M, should be of GCRE (6) and (9). Using the Fiertz identity and → considered as a fundamental physical field; its spinor the double row-column splitting of (8), for every matrix nature will be seen below. We will assume the dy- component ψ = FAB ; A, B =1, 2ofF-field we get [13] namics of a basic F-field to be completely de- 2 2 2 2 termined by the GCRE (4) with z = x M+ , (∂0ψ) (∂1ψ) (∂2ψ) (∂3ψ) =0. (11) i.e. ∈ − − − Hence, every matrix component of a B -differ- dF = G(x) dx H(x)(6)entiable function satisfies the nonlinear, Loren- ∗ ∗ tz invariant, complexified eikonal equation (11). Except direct physical identifications of the abstract For the B -algebra it plays a role similar to that of the variables, in what follows no other assumptions will be Laplace equation in complex analysis; as for physics, its necessary. fundamental properties (for ψ R) were emphasized Let us rewrite now the matrices F, H in (6) in the by V.A. Fock [14]. ∈ form F = ψ(x),η(x) , H = α(x),γ(x) (7) 4. B -electrodynamics. Self-duality co- k k k k each of ψ,η and α, γ being a matrix-column with nditions and Maxwell equations two components; the columns transform independently We shall further restrict ourselves to the case of the through left multiplication. Then (6) splits into a pair spinor equality α(x)=ψ(x) in (9), i.e. to the funda- of equations: mental equation dψ = G dx α, dη = G dx γ. (8) ∗ ∗ ∗ ∗ dψ = G(x) dx ψ(x). (12) ∗ ∗ From (7) and (8) it follows that each solution to (6) For (12) the global continuous symmetries (10) are re- may be presented in the form F(x)= ψ0(x),ψ00(x) k k duced to the transformations of the where ψ0,ψ00 are two arbitrary solutions of the unique irreducible equation x m x m+,ψ sψ, G m G m+, (13) → ∗ ∗ → → ∗ ∗ dψ = G(x) dx α(x). (9) + 1 ∗ ∗ where s =(m )− , G is a B -conjugated field: G 2 ∗ The functions ψ(x),α(x)belongtotheleft-side G =(DetG) . ideal of the Clifford algebra B = C(3, 0) and are there- So relativistic invariance is ensured, and the conju- fore obviously 2-spinors. A conjugated spinor reduc- gated field G(x) forms a 4-vector. Later on G(x) will tion of (6) is also possible, if the row splitting of G(x) be regarded as a C -valued matrix of electromagnetic is used; a double reduction may be realized as well. (EM-) 4-potential A(x). Precisely, we set These properties stand side by side with the widest A (x)=2G (x) 2Gµ(x). (14) group of Eqs. (4) or (6), including the trans- µ µ ≡ formations Such an identification will be justified further by its dy- 1 z m z n− , F k F l namic and geometric consequences, as well as by the → ∗ ∗ 1 → ∗ ∗ , (10) establishment of gauge invariance of (12). Therefore, G k G m− , H n H l  → ∗ ∗ → ∗ ∗ the latter can be considered as the basic equa- m, n, k, l being arbitrary constant biquaternions of tions of B -electrodynamics, i.e., some type of clas- unit norm (neglecting the dilatations z λz, λ C), sical spinor electrodynamics, generated by solely the 1 1 → ∈ m− , n− are the inverse ones. GCRE-structure. Biquaternion Electrodynamics and Weyl-Cartan Geometry of Space-Time 219

Written in components, Eqs. (12) form the set of The R-valued vectors E~ , B~ satisfy the linear Maxwell differential equations equations as well. However, they are mutually in- dependent (contrary to (19)) and create a non-zero ∂ f =Guf, ∂ f =Gwf, ∂ f =Guh, ∂ f =Gwh u p w v energy- density (W , P~ ) of the usual form p v p v . (15) ∂uh=G f, ∂ph=G f, ∂wh=G h, ∂vh=G h  W E~ 2 + B~ 2 , P~ [E~ B~ ]. (23) Here f(x)andh(x) are the components of a 2-spinor ∼ | | | |  ∼ × field ψ(x), and ∂ denotes a partial derivative with Moreover, an infinite series of conservation laws can be respect to the corresponding DeWitt coordinate. obtained for (15) using routine procedures (see [15] for The equations for the EM field follow from the over- an example). determined system (15) as its integrability (compatibil- ity) conditions 5. Coulomb field as a stereographic map. ∂µ(∂ν ψ) ∂ν (∂µψ)=0,ψ= f(x),h(x) . Electric charge quantization − { } Assuming then both f(x),h(x) 0 (otherwise we 6≡ Let us search now for solutions to the B -electrodyna- would have obtained the same final results), we obtain mic equations (15). Each of the two components f(x), after derivation h(x) of the spinor field in (15) satisfies the C -eikonal 1 equation (11). Starting from one of its solutions, all ∂uAw ∂pAu =0,∂wAw ∂vAu = 2 Det A − − 1 , (16) the other quantities, including the EM potentials (14), ∂vAp ∂wAv =0,∂pAp ∂uAv = Det A − − 2 should be derived. In particular, the wave-like solu- Aµ(x) being the EM potentials (14) and Det A = tions of (11) lead to EM fields, identical to the usual AuAv ApAw . Going back to the Cartesian coordi- EM waves [3, 13]. nates,− we observe that Eqs. (16) are equivalent to the Notice now that the eikonal equation (11) pos- self-duality conditions (SDC) sesses a wonderful invariance property under the transformations ~ ~ + i ~ = 0 (17) P≡E B ψ(x) Φ(ψ(x)) (24) → for the C -valued electric ~ = a and magnetic ~ = components of the EME field{E } strength B with an arbitrary (C -differentiable) function Φ(f). {Ba} Accordingly, one can easily verify the gauge invariance µν = ∂µAν ∂ν Aµ; (18) of the basic system (12) (and, therefore, (15)) of a F − special type: here ψ(x) ψ(x)α(ψ),Aµ(x) Aµ(x)+∂µ ln α(ψ), (25) 1 → → a = 0a, a = 2 εabc bc; a,b,c,...=1, 2, 3. (19) E F B F α(ψ) being an arbitrary scalar function of the ψ com- In addition to (17), from (16) we have ponents f(x), h(x). The C -structure of the eikonal equation (11) essen- ∂ Aµ +2A Aµ =0, (20) D≡ µ µ tially enlarges the spectrum of its solutions. The most i.e. an inhomogeneous Lorentz condition. important are certainly two static solutions found in Combined with the definitions (18) and (19), the [3]: SDC (17) lead then to the Maxwell equations x1 + ix2 in free space f + = =tan θ exp (iϕ) r + x3 2   , (26) µν 1 µνρλ x1 ix2  ∂ν = ∂ν ε ρλ =0. (21) θ  2i f − = − =cot exp ( iϕ) F F r x3 2 −  −   So the Maxwell equations represent nothing but the  where r, θ, ϕ are usual spherical coordinates on R3 . consistency conditions of a basic GCRE-system and { } are satisfied identically for each solution to the From a geometric point of view, the expression (26) latter. The inverse statement generally does not take corresponds to the stereographic projection S2 C of a unit 2- onto the C - (from the place! → Now, it is easy to see that, according to the SDC south (+) or north (-) poles, respectively). Substitut- (17), the energy-momentum density of a complex- ing (26) into (15), (14), we get after trivial integration: valued EM-field turns to zero. Therefore, we ought 2 f(x)=f (θ, ϕ),h(x)=[f (θ, ϕ)] to define the physical fields E~ , B~ through the real ± ± 1 2  (Re) or imaginary (Im) parts of (19). For geometric  Au = ,Av =  (27) reasons (see part 6), we prefer ∓ r ± r  iϕ θ 1 iϕ θ 1  e− tan ∓ 2e tan ± ~ ~ ~ ~ A = 2 ,A = 2 E =2Re( ), B =2Re( ). (22) p r  w − r   E B   220 V.V. Kassandrov

µ or, for spherical components of the C -valued 4-potential where δν , ηµν , εµνρλ are the Kronecker, Minkowski 1 1 and Levi-Civita , respectively, and Aµ(x) are A = ,A= ,A= iA = C -valued potentials (14). 0 ±2r r −2r θ ∓ ϕ 1 3 Thus in the basic electrodynamic case the initial cot θ . (28) GCRE system (6) is equivalent to the defining equa- − 2r ± 2r sin θ tions (32) of the covariantly constant vector fields Now, a transition to the physical vectors of EM- F µ(x) on a B -manifold with a “dynamically cre- field strengths (22) shows that the magnetic monopole {ated” effective} geometry of Weyl-Cartan type, and gradient-like terms in (28) disappear and we get represented by the affine connection (34). Note that 1 the C -vector Aµ(x) completely determines both the E = E = B = B = B =0,E= , (29) θ ϕ r θ ϕ r ± r Weyl part of (34) and its torsion structure. A general- ization by introduction of a Riemann metric structure i.e. the Coulomb law with a fixed value of elec- is natural as well. tric charge q = 1. To obtain the ST geometry induced by (34), let us Whereas the stereographic± projection (26) and the pass from F(x) to the unitary field U(x)=F F+ . transformations (24) realize the conformal mappings Using (32), we get ∗ S2 C, C C respectively, EM fields behave by (25) → → µ µ ρ in a gauge invariant manner, and the electric charge ∂ν U =∆νρ (x) U (x), (35) remains quantized. An exceptional role of conformal with the R-valued connection mappings in algebrodynamics has been clarified in [3] µ µ µ µ µ α (chapter 1). ∆νρ (x)=2(aν δρ + aρδν a ηνρ ε νρα b ), (36) q -quantization is a crucial point for B -electrody- − − · where a (x)andb (x) are real and imaginary parts namics; a fundamental significance of this problem has µ µ of the potentials A (x). been evident to Dirac, Eddington, Wheeler and other µ A connection similar to (36) has been introduced grands. In orthodox field theory the q -quantization in Ref. [17] from physical considerations; in [18] it was is postulated rather than explained. The most ele- shown to be the only ST connection compatible gant approach to this problem is produced, perhaps, by with a spinor bundle structure with the conven- multidimensional ST theories [16]; B -electrodynamics tional notion of a covariant spinor derivative. In our presents another possibility. approach these results follow from the GCRE structure In our approach the algebraic and purely classical alone. origin of q -quantization becomes evident. The fact However, the torsion field b (x) in (36) satisfies the is that the initial GCRE are not invariant under the µ Maxwell equations, as well as the non-metricity field scaling A λA, contrary to the linear Maxwell equa- a (x). By the key Ansatz (28), precisely the Weyl part tions. We→ suppose, however, that the phenomenon µ a (x) corresponds to the ordinary Coloumb electric of “algebraic q -quantization” should have deeper µ field, justifying the previous identification of the EM topological reasons; we hope to discuss them in the field with the real part of the C -field. [20]. As for the imaginary part bµ(x), for (28) it has the magnetic monopole form 6. Spinor connection and Weyl-Cartan 1 3 b = b = b =0,b= cot θ ; (37) geometry of ST 0 r θ ϕ ∓2r − 2r sin θ The fundamental equation of B -electrodynamics (12) we thus come to an exotic geometric interpretation may be presented in the form of magnetic monopoles as a ST torsion (with a totally antisymmetric tensor structure). Accord- ∂ν ψ = Γν (x) ψ(x), (30) ingly, the field bµ(x) cannot appear in the equations of geodesics. If we assume the latter to present the with laws of test particle , then monopoles should Γν (x)=G(x) σν (31) have no effect on it and therefore be entirely unob- ∗ servable! being a 2-spinor connection of special type. The initial Let us now return to the study of the primary C - GCRE, corresponding to (30), have the matrix form geometry of the B -space. The integrability conditions (6) with H( )=F( ), i.e. x x for the irreducible spinor equation (30) may be written dF = G(x) dx F(x), (32) in the form ∗ ∗ or, in a 4-vector representation [3] Rµν ψ(x)=0, (38) µ µ ρ with ∂ν F =Γνρ (x) F , (33) µ µ µ µ µ α Γνρ (x)=2(Aν δρ + Aρδν A ηνρ iε νρα A ) (34) Rµν = ∂[µΓν] [Γµ, Γν ] (39) − − · − Biquaternion Electrodynamics and Weyl-Cartan Geometry of Space-Time 221 being the tensor in the matrix representa- corresponds to the traceless part of the curvature ten- tion. For its self-dual components sor (39) and may be written as usual:

i a (R~ ) = R + ε R (40) Lµν = (x) σa = ∂ N [Nµ, Nν ]. (46) a 0a 2 abc bc Lµν [µ ν] − with the connection of the form (31), we get We see now that the self-dual part of (46) coincides with the traceless part of the tensor (41) and, in view R~ = ~ + ~σ i[ ~ ~σ ]. (41) of (17) and (20), we have again P D − P× Here the quantities i ρλ Lµν + 2 εµνρλL =0. (47) µ µ ~ = ~ + i ~, = ∂µA +2AµA From (47) and the Bianchi identity, the YM equations P E B D follow immediately in a usual way: coincide with (17) and (20), respectively and therefore µν µν vanish along with the entire self-dual tensor (40). ∂ν L =[Nν , L ]. (48) It is easy to see that is proportional to the curva- So we can indeed consider the field N (x)asaC- ture invariant =6ηµνDRα (= 0). So the B -space ν µαν valued YM field of a special structure (45). From (38) appears to beD an self- with zero scalar for any non-trivial ( ) we obtain, in addition, curvature. Now, if there are two linearly indepen- ψ x dent spinor solutions of (30), then, as follows from Det Rµν = 0; (49) (38), written in components, this condition leads to an ex- Rµν (x)=0, (42) pression of the EM field strength in terms of the YM ones (for each [µν] separately): i.e. a trivial case of flat geometry and zero field strengths. To avoid that, the primary B -field F(x) in (32) a a =( )2. (50) Lµν Lµν Fµν should split into two spinors ψ0(x), ψ00(x) (see (9)), proportional to each other; therefore, we have So the EM field may be regarded as a modulus of the YM triplet field in the isotopic complex- Det F(x)=0, (43) ified 3-space. Contrary to EM fields, the YM ones cannot be split and the field F(x) takes the values on the subspace of into real and imaginary parts (due to the nonlinearity null divisors of the B -algebra, or, physically, on the of the YM equations) and therefore are essentially C - complex “”. valued. This seems quite natural in connection with The null B -fields are the most fundamental objects the pseudo-Euclidean structure of the ST (the dual- throughout the AD approach as a whole. However, ity operator is known to have imaginary eigenvalues they can exist only on with an indefinite in the Lorentz signature). Since the self-duality condi- metric signature. So the pseudo-Euclidean struc- tions play a crucial role both in orthodox field theory ture of the World should not be postulated within the and in AD, we come again to the conjecture on a AD approach, but is just a necessary condition of C -analytic structure of real physical ST. This nontrivial dynamics (and effective geometry). possibility, discussed repeatedly within the frames of GRT, the twistor and string programs, as well as within 7. Yang-Mills fields and the C -structure the binary geometrophysics approach [19], seems to be of space-time inevitable in AD in view of the non-existence of a R-valued STA with an Aut group isomorphic to the Now we will demonstrate that the Yang-Mills (YM) Lorentz one. Thus, we suppose that close connec- gauge fields also appear in theory in rather a natural tions between the field equations nonlinearity and the way. To see that, let us separate the trace-free part in C structure of ST do exist, as well as its noncommuta- the basic spinor connection (31) tive quaternion structure (see Sections 1 and 3 for the latter). Γ (x)=G(x) σ = 1 A (x)+N (x) . (44) ν ∗ ν 2 ν ν Moreover, we may think of the C structure  as some natural way of ST dimension enlarging Then the zero component A (x) coincides with the C - µ (namely, doubling), just in the sense of Kaluza-Klein potentials (14) of the EM field, and the trace-free part theories. As for physics, such an effect should be essen- N (x) can be expressed in its terms in a linear way: µ tial at high energies; asymptotically, in the linear ap- a a proximation, the ST C structure should split into the Nµ(x)=Nµ (x) σa; N0 = Aa(x), a Minkowski space observed plus a conjugated one. The N = δabA0(x) iεabcAc(x). (45) b − same is done by the field C -structure: it exhibits a re- The quantities Nµ(x) can be regarded as the matrix duction to a linear R-valued EM-field (doubled through potentials of some C -valued gauge field; its strength the SDC (17), too). 222 V.V. Kassandrov

Generally, we assume the existence of a biquater- [6] F. G¨ursey and H.G. Tze, Phys. Lett. B 127 (1983), nion (i.e. complex-quaternion) algebraic structure of 191. ST and field manifolds consistent with each other. [7] “Quazigroups and Nonassociative Algebras The non-commutativity of such a B -algebra re- in Physics”, (J. L¨ohmus and P. Kuusk, eds.), Proc. sults in the nonlinearity of fundamental dy- Inst. Phys. Estonia Ac. Sci., vol.66, Tartu, 1990. namics: the GCRE (6), as well as its P- and even [8] N. Salingaros, J. Math. Phys. 23 (1982), 1. T-noninvariance (the connections similar to (35), [9] V.V. Vishnewsky, A.P. Shirokov and V.V. Shurygin, (36) are efficiently employed by V.G. Krechet for a “Spaces over Algebras”, Kasan Univ. Press, 1985 (in 5-geometrical description of electroweak interaction). Russian). Nevertheless, here the usual reversible dynamics of gauge fields has been obtained in Sections 4 and 7; [10] R. Fueter, Commun. Math. Helv. 4 (1931–32), 9. this latter should be regarded as nothing more but [11] F. G¨ursey and H.G. Tze, Ann. Phys. 128 (1980), 29. some “trace” of a primary B -structure, responsible for [12] M. Evans, F. G¨ursey and V. Ogievetsky, Phys. Rev. interactions, the “time arrow” and the left-right pref- D47 (1993), 3496. erence on the Minkowski ST. We expect an extensive [13] V.V. Kassandrov, Vestnik Peopl. Fried. Univ., Fizika, presentation of our views of these problems as well as 1993, No.1, 60 (in Russian). numerous generalizations of the AD approach. [14] V.A. Fock., “Theory of Space, Time and ”, IIL, In conlusion, peculiar correlations between AD and Moscow, 1955 (in Russian). binary geometrophysics (BG) [19] should be noted. Both of the approaches start from some abstract ex- [15] M.K. Prasad, Phys. Lett. B87(1979), 237. ceptional algebraic structures and deal with either [16] Yu.S. Vladimirov, “Physical Space-Time Dimension basic relations (in BG), or special mappings (in AD). and Unification of Interactions”, Moscow Univ. Press, In both theories zero determinant structures (see 1987 (in Russian). (43), (49)) are of particular importance. Finally, [17] Yu.N. Obukhov, V.G. Krechet and V.N. Ponomariev, the ideas of multipoint geometries [19, 20] originate in: “ and Relativity Theory”, Kazan, from purely algebraic considerations and should find 1978, No.14–15, 121. their place in AD as well. It seems plausible that other [18] V.E. Stepanov, Izvestiya Vuzov, Mathematica, 1987, deep interrelations will be found out in future. No.1, 72. We see that the simplest AD model, based on the [19] Yu.I. Kulakov, Yu.S. Vladimirov and A.V. Kar- conditions of B -differentiality alone, naturally con- naukhov, Introduction to Physical Structures The- tains the geometric, spinor-gauge and discrete struc- ory and Binary Geometrophysics, Moscow, Arkhimed tures, capable of solving the charge quantization and Press, 1992 (in Russian). monopole problems. Within this model, the Coulomb [20] V.Ya. Skorobogat’ko, G.N. Feshin and V.A. Pielykh, law gains an exotic geometrical meaning, and the C - in: “Math. Methods and Physico-Mechanical Fields”, eikonal equation becomes a fundamental equation of Kiev, Naukova Dumka, (1975), No.1, 5. field dynamics. Related problems (in particular, the problem of motion law and many-sources distributions) are yet to be solved.

Acknowledgement I am grateful to D.V. Alexeevsky, B.V. Medvedev and especially to Yu.S. Vladimirov for helpful advice and (Yu.S. Vladimirov) for organizational support.

References

[1] V.I. Rodichev, Izvestiya Vuzov, Fizika, 1963, No.2, 122 (in Russian). [2] S. Mandelstam, Ann. Phys. 19 (1962), 25. [3] V.V. Kassandrov, “Algebraic Structure of Space-Time and Algebrodynamics”, Peoples’ Friend. Univ. Press, Moscow, 1992 (in Russian). [4] D. Hestenes, “Space-Time Algebra”, N.Y., Gordon & Breach, 1966. [5] G. Casanova, “”, Presses Univers. France, 1976.