@article{10.11648/j.ajmp.20241306.13,
  author = {Alimzhan Kholmuratovich Babaev},
  title = {Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra
},
  journal = {American Journal of Modern Physics},
  volume = {13},
  number = {6},
  pages = {102-111},
  doi = {10.11648/j.ajmp.20241306.13},
  url = {https://doi.org/10.11648/j.ajmp.20241306.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20241306.13},
  abstract = {This paper presents the unification of Einstein's equations, Maxwell’s equation systems, and Dirac's equation for three generations of particles in the R1,3 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 DA (D•A, where D is the Dirac operator, A is the tensor field, and • is the inner product) represents the deformation of the field, while the antisymmetric one (DɅA, Ʌ is the outer product) denotes the torsion. The differentiation of DA (i.e., DDA) yields an equation from which both Einstein's equation and the two independent Maxwell systems can be derived. The differentiation of the field deformation D(D•A), 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 J is not a constant in the inhomogeneous system of Maxwell's equations, D•F = J. 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 (DA) in the form of biquaternions through complex hyperbolic functions in R1,3 permits the decomposition of DA 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 m = 0) unifies the photon and three generations of neutrinos (γ, νe, νμ, ντ) into a single entity, namely, a singlet (photon) + a triplet (three generations of neutrinos).
},
 year = {2024}
}

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TY  - JOUR
T1  - Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra

AU  - Alimzhan Kholmuratovich Babaev
Y1  - 2024/12/25
PY  - 2024
N1  - https://doi.org/10.11648/j.ajmp.20241306.13
DO  - 10.11648/j.ajmp.20241306.13
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 102
EP  - 111
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20241306.13
AB  - This paper presents the unification of Einstein's equations, Maxwell’s equation systems, and Dirac's equation for three generations of particles in the R1,3 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 DA (D•A, where D is the Dirac operator, A is the tensor field, and • is the inner product) represents the deformation of the field, while the antisymmetric one (DɅA, Ʌ is the outer product) denotes the torsion. The differentiation of DA (i.e., DDA) yields an equation from which both Einstein's equation and the two independent Maxwell systems can be derived. The differentiation of the field deformation D(D•A), 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 J is not a constant in the inhomogeneous system of Maxwell's equations, D•F = J. 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 (DA) in the form of biquaternions through complex hyperbolic functions in R1,3 permits the decomposition of DA 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 m = 0) unifies the photon and three generations of neutrinos (γ, νe, νμ, ντ) into a single entity, namely, a singlet (photon) + a triplet (three generations of neutrinos).

VL  - 13
IS  - 6
ER  - 

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