The generalized Clifford algebra is a universal tool for describing and unifying fundamental fields

This article presents an overview of the construction of Clifford algebra in pseudo-Riemannian space R ^1,3 for describing and unifying fundamental fields: gravitational, electromagnetic, and Dirac fields. The quadratic differential form ∇ A = ∇• A + ∇∧ A was defined as a measure of the local inhomogeneity of a vector field in R ^1,3 . Instead of the classical scalar and vector products in ∇ A , the Clifford product was used: the inner ∇• A and outer ∇∧ A ones. In addition, instead of the orthonormal basis in the Minkowski space ( E ^1,3 ) { t,x,y.z }, the so-called canonical basis, composed of the Dirac matrices {γ ⁰ , γ ¹ ,γ ² ,γ ³ }, was used, which significantly expanded the meaning and capabilities of the classical basis. The associativity property of the product ∇∇ A = (∇∇ A ) = ∇(∇ A ) made it possible to obtain the following: a) the Einstein equation from (∇∇ A ); and b) the Maxwell equation from ∇(∇ A ). Simultaneously, two independent Maxwell systems were combined into a single equation too. Transforming ∇ A into a sum of three independent biquaternions ∇ A = Σ _α ℬ _α (α = 1,2,3) allowed us to obtain bispinors (antibispinors), rotations on the spatial and temporal planes in R ^1,3 , and representation functions and generators of the Lorentz group in R ^1,3 . The gradient of the bispinors allowed us to derive three pairs of Dirac-type equations for the three generations of fermions and bosons. Transformations of vectors using generalized biquaternions ( x ′ = ℬ _α • x •ℬ̃̃ _α ), or more precisely, Lorentz transformations in curvilinear coordinates, led to the universal form of the Doppler law. The approximation function of the proposed nonlinear Hubble law was calculated from experimental data on the dependence of the redshift on the distance to the observed stars. This nonlinearity of the Hubble law explains the mechanism for the accelerated increase in redshift for the steady-state model without the Big Bang.