Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space <I>R<sup>1,3</sup></I> by Clifford Algebra
Аннотация
This paper presents the unification of Einstein's equations, Maxwell’s equation systems, and Dirac's equation for three generations of particles in the <I>R<sup>1,3 </sup></I>pseudo-Riemannian space with torsion. The use of Dirac matrices as an orthonormal basis (in general, as a canonical basis) on the tangent plane permits the replacement of vectors with second-rank tensors. The symmetric component of the differential form <I>DA</I> (<I>D</I>•<I>A</I>, where <I>D</I> is the Dirac operator, <I>A</I> is the tensor field, and • is the inner product) represents the deformation of the field, while the antisymmetric one (<I>D</I>Ʌ<I>A</I>, Ʌ is the outer product) denotes the torsion. The differentiation of <I>DA</I> (i.e., <I>DDA</I>) yields an equation from which both Einstein's equation and the two independent Maxwell systems can be derived. The differentiation of the field deformation <I>D</I>(<I>D</I>•<I>A</I>), that is, the gradient of the field divergence, yields a four-dimensional current. This four-current formulation results in nonlinearity in the inhomogeneous Maxwell's equations. In particular, the four-current <I>J</I> is not a constant in the inhomogeneous system of Maxwell's equations, <I>D</I>•<I>F</I> = <I>J</I>. In accordance with this definition, a field singularity is defined as a source of current, or alternatively described as a "hole," which is a necessary component for the existence of the field. The description of field inhomogeneity (<I>DA</I>) in the form of biquaternions through complex hyperbolic functions in <I>R<sup>1,3 </sup></I>permits the decomposition of <I>DA</I> into three pairs of spinors–antispinors (spinor bundle). The differentiation of spinors and the subsequent determination of eigenvalues and eigenfunctions yield three pairs of Dirac-type equations that are applicable to both bosons and fermions, which describe the fundamental particles of the three generations. The solution of Dirac-type equations in pseudo-Euclidean space for massless particles (eigenvalues <i>m</i> = 0) unifies the photon and three generations of neutrinos (γ, ν<sub>e</sub>, ν<sub>μ</sub>, ν<sub>τ</sub>) into a single entity, namely, a singlet (photon) + a triplet (three generations of neutrinos).