Papers by Alimzhan K H . Babaev
Description of Lorentz Transformations, the Doppler Effect, Hubble's Law, and Related Phenomena in Curvilinear Coordinates by Generalized Biquaternions, 2025
This paper presents the derivation of Lorentz transformations in curvilinear coordinates utilizin... more This paper presents the derivation of Lorentz transformations in curvilinear coordinates utilizing generalized biquaternions. Generalized biquaternions are rotations in curvilinear coordinates, including on the tx, ty, and tz planes. These space-time rotations are precisely the Lorentz transformations in curvilinear coordinates. The orbital rotation of the source and/or receiver, which mathematically represents the Lorentz transformation in spherical coordinates, is identified as the cause of the transverse Doppler effect. The change in wave frequency, specifically the "redshift," results in nonlinearities of Hubble's law manifesting as phenomena such as accelerated and anisotropic expansion of the universe, aberration, and wave polarization. Apparently, redshift exists even without radial expansion of the universe, i.e., without the "Big Bang". The reasons for the accelerated expansion of the universe, the anisotropic (angular) distribution of relic radiation, and the polarization of light from distant stars become clear in this approach. This greatly simplifies the mathematical description and understanding of the supposedly complex processes occurring in the universe.
Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra, 2024
This paper presents the unification of Einstein's equations, Maxwell's equation systems, and Dira... more This paper presents the unification of Einstein's equations, Maxwell's equation systems, and Dirac's equation for three generations of particles in the R 1,3 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 R 1,3 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, ν e , ν μ , ν τ) into a single entity, namely, a singlet (photon) + a triplet (three generations of neutrinos).
This paper presents the unification of Einstein's equations, Maxwell's equation systems, and Dira... more This paper presents the unification of Einstein's equations, Maxwell's equation systems, and Dirac's equation for three generations of particles in the R 1,3 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 R 1,3 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, ν e , ν μ , ν τ ) into a single entity, namely, a singlet (photon) + a triplet (three generations of neutrinos).
Аннотация: В статье рассматривается взаимосвязь между неоднородностью векторного поля и квантовой... more Аннотация: В статье рассматривается взаимосвязь между неоднородностью векторного поля и квантовой статистикой частиц. Доказывается, что причиной существования (наличия) только двух типов частиц (фермионов и бозонов) являются деформация и вращение неоднородности поля A.
The article presents the unification of two Maxwell's systems equations (homogeneous and inho... more The article presents the unification of two Maxwell's systems equations (homogeneous and inhomogeneous) within the generalized Clifford algebra. In this new formalism, an electromagnetic current and certain gauges acquire a geometric meaning associated with the properties of space.
В статье представлен альтернативный формализм на основе обобщенной алгебры Клиффорда (на случай к... more В статье представлен альтернативный формализм на основе обобщенной алгебры Клиффорда (на случай криволинейных координат). Однородная (∇∧F=0) и неоднородная (∇•F=J) системы уравнений Максвелла объединены в единое уравнение. Связанная с явными свойствами пространства трактовка дана для 4-х мерного электромагнитного тока и некоторых калибровок. The article presents the formalism based on generalized Clifford algebra (curvilinear coordinates’ case). The homogeneous (∇∧F = 0) and inhomogeneous (∇ • F = j) Maxwell's equations have been combined into the single equation. The interpretation for 4-dimensional electromagnetic current and some gauges associated with explicit spaces properties was given.
The paper presents the relation between line and surface integrals in Clifford algebra (ℰ4) and, ... more The paper presents the relation between line and surface integrals in Clifford algebra (ℰ4) and, in particular, in Cartesian space (ℰ3). The bijection between hypercomplex numbers and elements of space ℰ4, in particular ℰ3, has been set. The generalized Stokes theorem and Cauchy's integral theorem are generalized and combined into one. The physical interpretation of the formulas is in accord with the laws of the circulation of the electromagnetic field and gives some nontrivial results.
In 4-dimensional curved space, the article presents the relations between the vector field inhomo... more In 4-dimensional curved space, the article presents the relations between the vector field inhomogeneity, biquaternions, rotations, and spinors. As a mathematical tool, the generalized Clifford algebra has been employed. The electromagnetic field inhomogeneity is proven to be made up of three independent rotations, biquaternions, and three pairs of spinors-antispinors.
В статье получены уравнения непрерывности и закон сохранения вихревого 4-х тока в новом формализм... more В статье получены уравнения непрерывности и закон сохранения вихревого 4-х тока в новом формализме на основе обобщенной алгебры Клиффорда (на случай криволинейных координат). . The article presents the derivation of the continuity equation and the conservation law of the eddy 4-current on the generalized Clifford algebra (curvilinear coordinates’ case) based formalism.
Curvilinear integrals in the concept of hypercomplex numbers of Clifford algebra
The relationship between curvilinear and surface integrals in Clifford's algebra (ℰ4 - 4-dimensio... more The relationship between curvilinear and surface integrals in Clifford's algebra (ℰ4 - 4-dimensional pseudo-Euclidean space), and in particular, in Cartesian space (2nd and 3rd dimensional) is considered in the article. A one-to-one correspondence (bijection) between hypercomplex numbers and elements of the space ℰ4 (γi - Dirac matrices and their combinations, in particular, σα - Pauli matrices) was established. The Green's theorem, the Stokes' theorem and the Cauchy's integral formula and integral theorem were generalized. The physical interpretation (electromagnetism) of the formulas corresponded well with Maxwell's laws and gives some non-trivial results.
В статье рассматривается взаимосвязь между криволинейными и поверхностными интегралами в алгебре Клиффорда (ℰ4 - 4-х мерное псевдоевклидово пространство) и, в частности, в декартовом (2-х и 3-х мерное) пространстве. Установлено взаимно однозначное соответствие (биекция) между гиперкомплексными числами и элементами пространства ℰ4 (γi – гамма - матрицы и их комбинации, в частности, σα – матрицы Паули). Обобщены формулы Грина, Стокса, интегральная теорема и формула Коши. Физическая интерпретация (электромагнетизм) полученных формул хорошо согласуется с законами Максвелла и дает некоторые нетривиальные результаты.
Формализм на основе модели неоднородности векторного поля в обобщенной алгебре Клиффорда , 2017
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Papers by Alimzhan K H . Babaev
В статье рассматривается взаимосвязь между криволинейными и поверхностными интегралами в алгебре Клиффорда (ℰ4 - 4-х мерное псевдоевклидово пространство) и, в частности, в декартовом (2-х и 3-х мерное) пространстве. Установлено взаимно однозначное соответствие (биекция) между гиперкомплексными числами и элементами пространства ℰ4 (γi – гамма - матрицы и их комбинации, в частности, σα – матрицы Паули). Обобщены формулы Грина, Стокса, интегральная теорема и формула Коши. Физическая интерпретация (электромагнетизм) полученных формул хорошо согласуется с законами Максвелла и дает некоторые нетривиальные результаты.
В статье рассматривается взаимосвязь между криволинейными и поверхностными интегралами в алгебре Клиффорда (ℰ4 - 4-х мерное псевдоевклидово пространство) и, в частности, в декартовом (2-х и 3-х мерное) пространстве. Установлено взаимно однозначное соответствие (биекция) между гиперкомплексными числами и элементами пространства ℰ4 (γi – гамма - матрицы и их комбинации, в частности, σα – матрицы Паули). Обобщены формулы Грина, Стокса, интегральная теорема и формула Коши. Физическая интерпретация (электромагнетизм) полученных формул хорошо согласуется с законами Максвелла и дает некоторые нетривиальные результаты.