Relativistic algebra of space-time and algebrodynamics

Abstract

We consider a manifestly Lorentz invariant form L of the biquaternion algebra and its generalization to the case of curved manifold. The conditions of L-differentiability of L-functions are formulated and considered as the primary equations for fundamental fields modeled with such functions. The exact form of the effective affine connection induced by L-differentiability equations is obtained for the flat and curved cases. In the flat case, the integrability conditions of the latter leads to the self-duality of the corresponding curvature, thus ensuring that the source-free Maxwell and SL(2, C) Yang-Mills equations hold on the solutions of the L-differentiability equations.

Maxwell, Yang–Mills, Weyl and eikonal fields defined by any null shear-free congruence

“…Note also that from the skew-symmetric part of (44) follows an additional "inhomogeneous Lorentz condition" [18,19] for the C-valued electromagnetic potentials Φ A ′ A = A µ :…”

“…Finally, we observe that the last system (6) admits an invariant Pfaffian (matrix) form [17][18][19] dξ = ΦdXξ (7) for a 2-spinor ξ(x) = {ξ A ′ } and a M at(2, C)-valued vector field Φ(x) = {Φ A ′ A }. In the case of flat spacetime, we have dX = {dX AA ′ } and the symbol d in the left-hand side of (7) denotes the ordinary operator of external differentiation.…”

Twistor structures and boost-invariant solutions to field equations

“…(now Π (C) are two arbitrary functions of four twistor variables ξ, τ = Zξ) arises, in the framework of so-called algebrodynamical (AD) approach [4,5,6]. In fact, (3) represents the general solution of the conditions of biquaternionic B-differentiability [4,5] which are, in a sense, a natural generalization of the Cauchy-Riemann conditions in complex analysis.…”

Self-dual connections and the equations of fundamental fields in a Weyl–Cartan space

“…where the gauge parameter α = α(ξ, Xξ) depends on X only through the components of the transformed twistor W = {ξ, Xξ} (the so-called "weak gauge invariance" [4]). Moreover, the compatibility conditions of the overdetermined GSE (3) ddξ = 0 = Rξ, R := (ΦdXΦ) ∧ dX imply the self-duality of the curvature 2form of the effective connection Ω := ΦdX on the GSE solutions (the so-called weak self-duality [4]). Even on a flat space-time background, the connection Ω already possesses nonmetricity (of the Weyl type) and torsion of a specific form [4].…”

“…Moreover, the compatibility conditions of the overdetermined GSE (3) ddξ = 0 = Rξ, R := (ΦdXΦ) ∧ dX imply the self-duality of the curvature 2form of the effective connection Ω := ΦdX on the GSE solutions (the so-called weak self-duality [4]). Even on a flat space-time background, the connection Ω already possesses nonmetricity (of the Weyl type) and torsion of a specific form [4]. The very GSE equations (in form (2)) can be considered as the equations of covariantly constant fields in the corresponding space with the Weyl-Cartan connection [2], dF = ΩF .…”