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BIQUATERNION ELECTRODYNAMICS and WEYL-CARTAN GEOMETRY of SPACE-TIME V.V.Kassandrov

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BIQUATERNION ELECTRODYNAMICS and WEYL-CARTAN GEOMETRY of SPACE-TIME V.V.Kassandrov View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Gravitation & Cosmology, Vol.1 (1995), No.3, pp.216–222 c 1995 Russian Gravitational Society BIQUATERNION ELECTRODYNAMICS AND WEYL-CARTAN GEOMETRY OF SPACE-TIME 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 biquaternion algebra appear to be Lorentz-invariant. The Laplace equation is in this case replaced by a nonlinear C -eikonal equation. GCRE contain a 2-spinor 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-quaternion 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 spaces, 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 Numbers 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 world 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 manifold 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 octonion algebra,the the exceptional algebras — algebras with division and unique exceptional non-associative algebra. However, positive norm — exist in the dimensions d = 4 (Hamil- the difficulties of “intercourse” with octonions are well- ton quaternions) 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 group 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 isometry group. 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} dimension 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 (biquaternions) M M Mmaybe duction from the 16-dimensional total vector space of introduced on M . According to the isomorphism× → 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 metric signature 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 unit 2 2 matrix and bra C(3,0) of smaller dimension d = 8, is preferable σ ,a=1{ , 2}, 3 the Pauli matrices; ×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 present 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 stereographic projection 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 Minkowski space 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.
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