Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space <I>R<sup>1,3</sup></I> by Clifford Algebra

Alimzhan Kholmuratovich Babaev

doi:doi:10.11648/j.ajmp.20241306.13

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Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R^1,3 by Clifford Algebra

Alimzhan Kholmuratovich Babaev^*

Published in American Journal of Modern Physics (Volume 13, Issue 6)

Received: 23 November 2024 Accepted: 6 December 2024 Published: 25 December 2024

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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 R^1,3pseudo-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^1,3permits 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).

Published in	American Journal of Modern Physics (Volume 13, Issue 6)
DOI	10.11648/j.ajmp.20241306.13
Page(s)	102-111
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Einstein Equations, Maxwell Equations, Dirac Equations, Unified field theory, Pseudo-Riemannian Manifold R^1,3, Biquaternions

1. Introduction

The use of new applications of algebra complements and expands the possibilities for describing physical processes. For example, the methods of Clifford algebra

[1]

unite fundamental fields into a single field in pseudo-Riemannian space.

In this research, we will combine the homogeneous and inhomogeneous systems of equations of Maxwell, Einstein’s

[2]

and Dirac's equations

[3]

for three generations of particles using the generalized Clifford algebra method in R^1,3 with torsion. In other words, we will create a single theoretical framework by combining the vector (electromagnetism), tensor (gravity), and spinor (fermions) fields into one field.

This study extends and generalizes previous research on Clifford algebra applications in field theory.

In contrast to the classical Clifford algebra, which is applicable to pseudo-Euclidean space (Minkowski

[4]

), the generalized Clifford algebra describes objects in non-Euclidean space, R^1,3

[5]

. The incorporation of torsion in the Riemannian manifold R^1,3 is a necessary condition for the unification of Maxwell's and Einstein's equations.

The representation of the differential form DA (D is the Dirac operator; A is a tensor field) in the form of generalized biquaternions

[6]

permits the expression of DA (inhomogeneity) as the sum of three pairs of independent spinors - antispinors

[6]

. Subsequently, Dirac-type equations are derived by differentiating the spinors-antispinors.

Using the Clifford product ab = a•b + a∧b (outer a∧b = (ab – ba)/2 and inner a•b = (ab + ba)/2) instead of the usual scalar and vector products of vectors allows us to reconsider many previous concepts. For example, the divergence gradient D(D•A) is interpreted as a four-dimensional electromagnetic current, thereby expanding its physical and geometric meaning.

2. Results

2.1. Theoretical Basis

Let be given a vector field A in a pseudo-Riemannian space R^1,3:

A = \sum_{i = 0}^{3} e^{i} A_{i}

(1)

Formula (1) is the expansion of A in the basis {eⁱ}.

Clarification. In classical Clifford algebra, eⁱ are called vectors, but in fact, eⁱare 4x4 matrices related to the Dirac matrices (γ⁰, γ¹, γ², γ³) through the transition functions X = X_i (q^j) by the formula:

e_{i} = \sum_{j = 0}^{3} γ^{j} \frac{\partial X_{j}}{\partial q^{i}} \equiv \sum_{j = 0}^{3} γ^{j} \partial_{i} X_{j}

(2)

The basis {γⁱ} of Dirac matrices is referred to as canonical:

γ_iγ_j+γ_jγ_i= ±2Iδ_ij(3)

δ_ij is the Kronecker delta and I is the identity matrix. If i= 0 and/or j = 0, then the sign “+” is used in (3). In other cases, the sign “-” is used.

Mathematically, e_i are second-rank tensors. The terms "second-rank tensor" and "vector" are equivalent because Dirac matrices are employed as the orthonormal basis, rather than ordinary vectors (unit vectors).

Basis (3) greatly expands the concept of an orthonormal basis because it also includes ordinary (scalar and vector) vector products. This allows us to consider space not only with curvature (deformation) but also with torsion.

The terms “vector, bivector, trivector” accepted in Clifford algebra are quite acceptable, since the internal structure of the Dirac matrices and their products γⁱ, γⁱγ^j, γⁱγ^jγ^k does not change, and therefore the name has no principled importance. The main objective is to satisfy Condition (3).

The movable (local) basis {eⁱ} is connected to the canonical basis, which is drawn at the point of contact of the local tangent plane with the pseudo-Riemann surface. The basis vectors eⁱ depend on the coordinates, that is, they change when moving “parallel to themselves” (in R^1,3, along geodesic lines). Thus, ordinary differentiation must be replaced by covariant differentiation.

eⁱe^j = eⁱ•e^j + eⁱ∧e^j is the Clifford vector product

[1]

eⁱ•e^j= (eⁱe^j + eeⁱ)/2 is the inner product of vectors.

eⁱ∧e^j= (eⁱe^j - eeⁱ)/2 is the outer product of vectors or a bivector.

In general, eⁱ•e^j is a symmetric second-rank tensor obtained by the contraction of the dyadic product of two second-rank tensors

[7]

, and then by symmetrization:

eⁱ•e^j= (e⊗e

In particular, eⁱ•e^j = I g_ij, where g_ij is a metric tensor. In other words, eⁱ•e^j is the curvature of the metric, that is, a measure of the difference from flat space, a measure of deformation.

In general, eⁱ∧e^j is an antisymmetric tensor of the second rank, obtained by the contraction of the dyadic product of two tensors

[7]

and then by their antisymmetrization:

eⁱ∧e^j= (e⊗e

eⁱ∧e^j denotes the torsion tensor

[8]

. Module |eⁱ∧e^j| is the area of the parallelogram constructed from vectors eⁱ and e^j. It is clear that eⁱ∧e^j = – e^j∧eⁱ.

In 3-dimensional space, the geometric meaning of the torsion tensor eⁱ∧e^j is more obvious: eⁱ∧e^j corresponds to the associated accompanying axial vector (dual, i.e., pseudovector) e^k, which, like a screw, changes its direction depending on the replacement of eⁱ and e^j.

2.2. The Inhomogeneity of a Vector Field

We consider the gradient from A to:

DA=eⁱe^jD_iA_j(4)

In accordance with Clifford's product, Equation (4) can be separated into symmetric and antisymmetric components:

DA=ID•A+D∧A=Ieⁱ•e^jD_iA_j+eⁱ∧e^jD_iA_j(5)

where D = eⁱD_i is the Dirac operator

[9]

and D_i is the symbol of covariant differentiation

[10]

We refer to this (5) as the field inhomogeneity (DA), which

comprises the following:

1) field deformations:

I D•A = Ieⁱ•e^{j I}D_iA_j= Ig^ijD_iA_j= ID_iAⁱ,

where D_iA_j= D_iAⁱ= Div A is a 4-dimensional divergence, a true scalar, and the first invariant of the DA

[11]

, because g^ijD_iA_j is the contraction of the tensor DA.

2) field torsion:

D∧A = eⁱ∧e^jD_iA_j = eⁱ∧e^j(D_iA_j – D_jA_i)

D∧A is an antisymmetric second rank tensor. F = D∧A is the electromagnetic field tensor.

We again differentiate Equation (5). The DDA can then be written in two equivalent versions:

D(DA)= D(D•A)+D•(D∧A)+D∧(D∧A)(6)

(DD)A =(D•D)A+ (D∧D)•A +(D∧D)∧A(7)

Next, we will verify that (6) gives the system of Maxwell's equations and (7) gives Einstein's equation.

2.3. Homogeneous System of Maxwell's Equations

The last term in Equation (6) is equal to zero

[12]

D∧(D∧A) =D∧F= 0(8)

(8) is a system of homogeneous Maxwell’s equations. We write (8) in coordinate form as

D∧F=E^ijknD_kF_ij= 0(9)

where E^ijkn is a contravariant antisymmetric tensor of fourth rank or Levi-Civita symbol

[13]

2.4. Four-dimensional Electromagnetic Current

In Equation (6), the first term, D(D•A), is identified using the density of the four-dimensional electromagnetic current:

j=D(D•A)(10)

Then, the electric charge q and 3-dimensional current

q=e⁰e⁰j₀=g⁰⁰D₀(D•A)(11)

In a flat space, the charge and current have the following forms:

q=γ⁰γ⁰j₀=σ₀j₀,

since γ⁰γⁱ = σ_i. where σ_i is a Pauli matrix.

Formula (11) has a clear geometric interpretation: the electric charge can be understood as the time derivative of the divergence of the field D₀(D•A). Therefore, an electric charge has only two signs: “plus” or source, if D₀(D•A) > 0 and “minus” or drain if D₀(D•A) < 0.

The electric charge (±) can be represented as a "hole" in the field. Gauges, such as Lorentz and Coulomb gauges.

[14]

can be understood as a method of removing singularities or sources of the field (current) with the aim of simplifying calculations.

2.5. Inhomogeneous System of Maxwell's Equations

The third-rank tensor D(DA) or trivector, or more precisely its part D∧(D∧A), is dual to the pseudovector

[12]

, which is equal to zero, i.e. D∧(D∧A) is equivalent to the homogeneous system of Maxwell's equations (9)

[12]

The remaining part is equivalent to the vector:

D(DA) =μT•A(13)

The inner product T•A gives a vector, as does the right side of equation (6). T is a second-rank tensor; μ is the proportionality coefficient. The properties of T and μ will be defined later.

Considering (10), (13), and F = D∧A, we obtain an inhomogeneous system of Maxwell's equations:

D•F=μT•A–j=J(14)

When μT•A = 0, (14) has the classical form. For values of μT•A ≠ 0, the inhomogeneous system of Maxwell's equations is nonlinear, that is, the effective charge J (right-hand side of (14)) is not constant; it increases or decreases depending on the growth of μT•A (charge screening).

2.6. Continuity Equation

The differentiation of (14) gives the continuity equation:

μT• (DA) –D•j= 0,(15)

because D•(D•F) = 0 and (D•T) •A = 0

[15]

. The inner product of the vectors is used in Eq. (15).

Formula (15) in general is a differential form of the law of conservation of energy-momentum in an elementary volume, i.e., the law of conservation of 4-current.

We write Equation (15) in the coordinate form and transform the left side.

μT^k_nAⁿ_k= 0.5μ(g^nkg^ijT_ki▽_nA_j+g^nkg^ijT_kiD_jA_n)=

= 0.5μg^nkg^ijT_ki(D_nA_j+ D_jA_n) = 0.5μT^njε_nj

Then, we obtain the continuity equation in the form

[15]

0.5μT^njε_nj–D_kj^k= 0(16)

ε_nj = 0.5 (D_n A_j + D_j A_n) is the deformation tensor.

The four-current j_k consists of the sum of the positive and negative currents:

j_k=Σj⁺_k+Σj^-_k(17)

Equation (16) represents the law of conservation of four-currents during field deformation. The deformation of electromagnetic and/or gravitational fields gives rise to the creation of pairs of positive and negative electric charges (particles + antiparticles), resulting in a constant total charge.

From (16), it is clear that tensor T (13) is the energy -momentum tensor.

2.7. Law of the Conservation of Eddy Currents or Electromagnetic Induction

By differentiating (14), we obtain the law of conservation of the eddy current

[15]

D∧J= 0(18)

D∧(D• F) = 0, because D• F is “scalar”.

Equation (18) represents the law of conservation of eddy currents, also known as Faraday’s law of electromagnetic induction (Faraday's law)

[16]

. Note that in (18), the outer product of the vectors is employed.

Formula (18) in the usual 3-dimensional form looks like

[15]

▽-ρ+ ∂_t

where ρ is the electric charge density; ∂_t

Let the current carriers be positive charges and ρ₁ < ρ₂. For simplicity, we let ▽ρ = const. Then, the change in current over time (in general, -▽ρ + ∂_t

Within the framework of generalized Clifford algebra, we combined two independent Maxwell system equations into a single equation: In addition, we derived conservation laws for the four-current.

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Figure 1. Eddy currents conservation.

2.8. Einstein's Equations

We now obtain Einstein's equation from Equation (7). Because

(D∧D)∧A = 0, (DD)A = μT•A, (D•D)A = D²A = □A, where □ is the D’Alembert operator,

(D∧D)•A = - Ric•A,

where Ric is the Ricci tensor, we can write Equation (7) in the following form:

μT•A=D•DA–Ric•A(20)

According to the Lichnerowicz formula

[17, 18]

□A =□_BA+ 0.5ℛic(A),(21)

where □A = eⁱ•e^je^k D_iD_j A_kis the action of the Lichnerowicz operator on vector A;

□_BA ≡ – tr (□A) = – eⁱ•eⁱe^k D_iD_iA_kis the action of the Bochner Laplacian

[18]

on A_k (on a scalar). This is the trace of □A, or the “rough” D’Alembertian;

λ_k are the eigenvalues of each eigenfunction (scalar!) A_k of the operator ▽•▽:

ℛic(A) = eⁱ•e^je^kR_ij A_k = g^ije^kR_ij A_k = e^kA_k R;

R is the scalar curvature.

Then, considering (21), from (20), we obtain the Einstein equation:

Ric– 0.5R+λ= –μT(22)

λ_k represent the eigenvalues of the Bochner operator:

–eⁱ•eⁱD_iD_iA^k=δ_kⁿλ_nA^k

From (22), we can determine the coefficient μ. μ = – 8 π k/c⁴

[2]

Furthermore, it is assumed that all components of the vector A^k (without e_k) are a scalar function, i.e., each scalar function A^k corresponds to its own number λ_k (there are four of them).

Einstein's equation (22) can be derived by differentiating Eq. (20)

[19]

. The proof is provided in Appendix 1.

Obtaining Einstein equation (22) by direct calculation from the inhomogeneous system of Maxwell equations in curvilinear coordinates proves the validity of the Lichnerowicz formula (21) for the case of a 4-dimensional pseudo -Riemannian space. Thus, from an inhomogeneous system of Maxwell equations, we obtain the Einstein equation.

2.9. Biquaternions, Bispinors

We square the field inhomogeneity (5), which consists of symmetric and antisymmetric parts, as follows:

(DA)²= (D•A+D∧A)²

(DA)²= (D•A)²+2 (D•A)D∧A+(D∧A)²(23)

whereD∧A=

γ = γ⁰γ¹γ²γ³ is the matrix analogue of the imaginary unit: (γ)² = (γ⁰γ¹γ²γ³)² = – I.

Substituting

(DA)²=SR + SP + VR + VP(25)

where

SR = (D•A)²I is scalar,

SP = – 0.25 γ E_ijkn F^ijF^knis pseudoscalar;

VR = e_α∧e₀ F_α0 (D•A) is bivector;

VP = γ (e_α∧e₀) E^βλα0 F_βλ (D•A) is pseudobivector.

Greek letters take the values 1, 2, 3.

SR, SP, VR and VP are expressed using hyperbolic functions:

SR = |τ_α0| |τ_β0| cosh((η_α + η_β)/2) cosh(γ (φ_α + φ_β)/2);

SP = |τ_α0||τ_β0| sinh((η_α + η_β)/2) sinh(γ (φ_α + φ_β)/2);

VR = τ_α0 |τ_β0|(sinh((η_α + η_β)/2) cosh(γ (φ_α + φ_β)/2) – sinh((η_α – η_β)/2) cosh(γ (φ_α – φ_β)/2));

VP = τ_α0 |τ_β0|(cosh((η_α + η_β)/2) sinh(γ (φ_α + φ_β)/2) – cosh((η_α – η_β)/2) sinh(γ (φ_α – φ_β)/2))

where

τ_α0= e_α∧e₀, |τ_α0| = (g_0α g_0α – g₀₀ g_αα)^0.5,

η_α is rapidity or angle of rotation on a hyperplane x_α0t,

φ_α is angle of spatial rotation around an axis x_α.

Substituting the new notations SR, SP, VR, and VP into (25), and simplifying and extracting the square root, we obtain:

DA = \sum_{α = 1}^{3} R_{α} = \sum_{α = 1}^{3} (|τ_{α 0}| \cosh \frac{z_{α}}{2} + τ_{α 0} \sin \frac{z_{α}}{2})

(26)

where z_α = Iη_α + γ φ_α.

In (25), we change the order of multiplication of bivectors e_α∧e₀ = – e₀∧e_α, in general, for all even Clifford numbers: e_i•e_j, e_i∧e_j, e_i∧e_j∧e_k∧e_n

[6]

. By repeating the procedure that we have already performed, we obtain:

\tilde{DA} = \sum_{α = 1}^{3} {\tilde{R}}_{α} = \sum_{α = 1}^{3} (|τ_{0 α}| \cosh \frac{z_{α}}{2} + τ_{0 α} \sin \frac{z_{α}}{2})

(27)

R_{α}

and

{\tilde{R}}_{α}

are referred to as generalized biquaternions and antibiquaternions, respectively. They described rotations on Riemann surfaces (on tangent planes x0y, z0y, y0z, t0x, t0y, t0z).

Formula (27) means that the field inhomogeneity DA is the sum of three biquaternions (antibiquaternions). Mathematically, biquaternions are second-rank tensors.

In the Minkowski space, where eⁱ are replaced by Dirac matrices γⁱ, biquaternions assume the form of a matrix exponential

[21]

R_{α} ({\tilde{R}}_{α}) = Icosh \frac{z_{α}}{2} \pm γ^{α} γ^{0} \sin \frac{z_{α}}{2} = \exp (\pm γ^{α} γ^{0} \frac{z_{α}}{2})

(28)

Thus, it is difficult to overestimate the role of biquaternions in physics. Any transformations of a vector, including those in curvilinear coordinates, are performed by biquaternions

[22]

x^{'} = R_{α} x {\tilde{R}}_{α} / {|τ_{0 α}|}^{2}

(29)

Using Euler's formulas, we write the biquaternion R_α as:

R_{α} = |τ_{α 0}| \frac{\exp (\frac{z_{α}}{2}) + \exp (- \frac{z_{α}}{2})}{2} + τ_{α 0} \frac{\exp (\frac{z_{α}}{2}) - \exp (- \frac{z_{α}}{2})}{2}

R_{α} = (|τ_{α 0}| + τ_{α 0}) \frac{\exp (\frac{z_{α}}{2})}{2} + (|τ_{α 0}| - τ_{α 0}) \frac{\exp (- \frac{z_{α}}{2})}{2}

(30)

where

Ψ_{α} = (|τ_{α 0}| + τ_{α 0}) \frac{\exp (\frac{z_{α}}{2})}{2} = (|τ_{α 0}| + τ_{α 0}) Φ_{α}

(31)

{\tilde{Ψ}}_{α} = (|τ_{α 0}| - τ_{α 0}) \frac{\exp (- \frac{z_{α}}{2})}{2} = (|τ_{α 0}| - τ_{α 0}) {\tilde{Φ}}_{α}

(32)

In general

Φ_{α} ({\tilde{Φ}}_{α}) = \frac{1}{2} \exp (\pm \frac{z_{α}}{2}) = \frac{1}{2} \exp (\pm \frac{I X_{α} + γ Y_{α}}{2})

(33)

X_{α} (q^{i})

and

Y_{α} (q^{i})

are scalar functions.

Ψ_{α}

and

{\tilde{Ψ}}_{α}

will be called generalized bispinor - antibispinor

[6]

According to the strict terminology of algebra, an ideal of a ring K is a subring k for

\forall b \in K

and

\forall S \in k

the equality holds

[23]

bS = cS

where c is a real number. If c > 0, then S is a positive ideal, if c < 0, then S is a negative ideal. In physics, ideals are associated with spinors: if c > 0, then

S = Ψ_{α}

is a bispinor, if c < 0, then

S = {\tilde{Ψ}}_{α}

is an antibispinor.

Really,

τ_{α 0} • Ψ_{α} = ((e_{α} ⋀ e_{0}) |e_{α} ⋀ e_{0}| + {(e_{α} ⋀ e_{0})}^{2}) Φ_{α} =

= |τ_{α 0}| (τ_{α 0} + |τ_{α 0}| Φ_{α}) = |τ_{α 0}| Ψ_{α}

Since

c = |τ_{α 0}| > 0

, then

Ψ_{α}

is a bispinor. The anti-bispinor can be similarly checked:

τ_{α 0} • {\tilde{Ψ}}_{α} = ((e_{α} ⋀ e_{0}) |e_{α} ⋀ e_{0}| - {(e_{α} ⋀ e_{0})}^{2}) {\tilde{Φ}}_{α} =

= |τ_{α 0}| (τ_{α 0} - |τ_{α 0}| {\tilde{Φ}}_{α}) = - |τ_{α 0}| {\tilde{Ψ}}_{α}

It can be verified that all three ideals (α = 1, 2, 3) are two-sided (left- or right-hand multiplication of

τ_{α 0}

is equal), since

τ_{α 0}

commutes with

Φ_{α}

and

{\tilde{Φ}}_{α}

Spinors and antispinors are independent

[6]

. The equality

\sum_{α = 1}^{3} (μ_{α} Ψ_{α} + {\tilde{μ}}_{α} {\tilde{Ψ}}_{α}) = 0

is satisfied if all real numbers (

μ_{α}, {\tilde{μ}}_{α}

) are equal to zero.

From (30), it is evident that a biquaternion is the sum of a spinors and antispinors. In the general case, the field inhomogeneity DA (26) consists of three generalized biquaternions R_α, that is, three pairs of independent spinors and antispinors

[6]

. Because DA is a second-rank tensor, we can say that biquaternions, bispinors, and antibispinors are also second-rank tensors (

Ψ_{α 0}

). In simple terms, a spinor field is a spinor bundle

[24]

of a tensor, that is, a DA inhomogeneity.

2.10. Dirac Type Equations

By differentiating bispinors (31) and (32), and finding the eigenvalues and eigenbispinors, we obtain Dirac-type equations in covariant form:

D (Ψ_{α} ({\tilde{Ψ}}_{α})) = m Ψ_{α} ({\tilde{Ψ}}_{α})

(34)

We write Equation (34) in the coordinate form as:

e^{i} (I |τ^{α 0}| + e^{α} ⋀ e^{0}) D_{i} Φ_{α 0} = I g^{jα} m_{j} Ψ_{α 0}

(|τ^{α 0}| e^{i} + e^{i} • τ^{α 0} + e^{i} ⋀ τ^{α 0}) D_{i} Φ_{α 0} = I g^{jα} m_{j} Ψ_{α 0}

(35)

|τ^{α 0}| = |e^{α} ⋀ e^{0}|

Equation (35) is a Dirac-type equation expressed in the coordinate form. No summation for α, and 0.

In Minkowski space, equation (34) assumes the form of classical Dirac equations:

γ^{i} \partial_{i} Ψ_{α} ({\tilde{Ψ}}_{α}) = - m_{α} Ψ_{α} ({\tilde{Ψ}}_{α})

(36)

where

Ψ_{α} ({\tilde{Ψ}}_{α}) = (I \pm γ^{α} γ^{0}) Φ_{α} ({\tilde{Φ}}_{α}) = \frac{I \pm γ^{α} γ^{0}}{2} e^{\pm \frac{I X_{α} + γ Y_{α}}{2}}

(37)

3. Calculations

Equation (36) was solved for massless particles

[25]

. Below, we discuss several important aspects.

Let α=3.

3.1. Solutions for Neutrinos

We consider (37) with only field torsion (without deformation) as follows:

γ^{i} γ^{3} γ^{0} U^{T} \partial_{i} \exp ((I + γ) \frac{p • x}{2}) = 0

,(38)

where

U^{T} = {(u_{0}, u_{1}, u_{2}, u_{3})}^{T}

is a constant column vector;

γ^{i}

are the Dirac matrices in the Weyl representation;

p • x = I p_{i} x^{i}

p_{i}

denotes the four-vector energy momentum.

In two-component (block) spinors, the solutions of (38) are given by:

ψ_{31, 2} = (|\begin{matrix} - p_{3} \\ p_{1} + i p_{2} \end{matrix}| + |\begin{matrix} p_{1} - i p_{2} \\ p_{3} \end{matrix}|) \frac{e^{(I + γ) \frac{p • x}{2}}}{p_{0}}

(39)

ψ_{33, 4} = - (|\begin{matrix} - p_{3} \\ p_{1} + i p_{2} \end{matrix}| + |\begin{matrix} p_{1} - i p_{2} \\ p_{3} \end{matrix}|) \frac{e^{(I + γ) \frac{p • x}{2}}}{p_{0}}

(40)

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Figure 2. Neutrino.

It is evident that the solutions

ψ_{31, 2}

and

ψ_{33, 4}

differ only in sign. Simply put,

ψ_{31, 2}

and

ψ_{33, 4}

describe the same particle. A change in helicity is observed when a particle transitions from state

ψ_{31}

ψ_{32}

as well as from

ψ_{33}

ψ_{34}

. This phenomenon is illustrated in Figure 2. If the helicity was left-handed, then it would be right-handed in the opposite direction. Helicity is not a P-parity-invariant quantity.

The particle is its own antiparticle, that is, solution (38) does not change if we change the

\exp ((I + γ) p • x / 2)

to the

\exp (- (I + γ) p • x / 2)

However, each state,

ψ_{α 1}

and

ψ_{α 2}

(also

ψ_{α 3}

and

ψ_{α 4}

), behaves as independent, separate particles.

Solutions (39) and (40) can be written in compact form as follows:

ψ_{3} = (σ_{1} p_{1} + σ_{2} (\pm p_{2}) + σ_{3} (\mp p_{3})) \frac{\exp ((I + γ) \frac{p • x}{2})}{p_{0}}

(41)

The upper signs before the impulses refer to the solution

ψ_{31, 2}

, and the lower signs refer to

ψ_{33, 4}

ψ_{α}

and

{\tilde{ψ}}_{α}

are obtained from the torsion of the field (without deformation); thus, they must be antisymmetric functions. To verify this, we considered the wave function of a system comprising two identical non-interacting particles and performed a swap operation on them:

ψ_{α, β} = \frac{p \exp ((I + γ) \frac{p • x}{2})}{p_{0}} ⋀ \frac{q \exp ((I + γ) \frac{q • x}{2})}{q} =

= - \frac{q \exp ((I + γ) \frac{q • x}{2})}{q} ⋀ \frac{p \exp ((I + γ) \frac{p • x}{2})}{p_{0}} = - ψ_{β, α}

The wave function of the superposition of two identical, non-interacting particles changes sign when the particles are permuted. The particles described by the function

ψ_{α}

are fermions. Equations (36) and (37) are equivalent to Dirac equations for fermions.

Because

ψ_{3}

is a spinor and the mass of this fermion is zero, the particle itself is one of the three generations of neutrinos. The equations for the remaining generations can be solved similarly.

3.2. Solutions for Photon

Now, we consider (37) without field torsion, that is, only with deformation:

γ^{i} U^{T} \partial_{i} \exp ((I + γ) \frac{p • x}{2}) = 0

(42)

In two-component spinors, the solutions of (42) are as follows:

χ_{31, 2} = (|\begin{matrix} p_{3} \\ p_{1} + i p_{2} \end{matrix}| + |\begin{matrix} p_{1} - i p_{2} \\ - p_{3} \end{matrix}|) \frac{\exp ((I + γ) \frac{p • x}{2})}{p_{0}}

(43)

χ_{33, 4} = - (|\begin{matrix} p_{3} \\ p_{1} + i p_{2} \end{matrix}| + |\begin{matrix} p_{1} - i p_{2} \\ - p_{3} \end{matrix}|) \frac{\exp ((I + γ) \frac{p • x}{2})}{p_{0}}

(44)

Solutions (43) and (44), that is,

χ_{31, 2}

and

χ_{33, 4}

, differ only in terms of sign.

Functions

χ_{31, 2}

and

χ_{33, 4}

describe the same particle. Moreover, the particle is its own antiparticle with a mass of zero.

Upon transitioning from state

χ_{31}

χ_{32}

(and similarly from

χ_{33}

χ_{34}

), the helicity of the particle remained unaltered, maintaining a right-handed polarization (Figure 3). There is no difference between particles and antiparticles.

Solutions (43) and (44) can be written in compact form as follows:

χ_{3} = (σ_{1} p_{1} + σ_{2} (\pm p_{2}) + σ_{3} (\pm p_{3})) \frac{\exp ((I + γ) \frac{p • x}{2})}{p_{0}}

(45)

The upper signs before the impulses refer to the solution

χ_{31, 2}

, and the lower signs refer to

χ_{33, 4}

It is clear that

χ_{α}

and

{\tilde{χ}}_{α}

are derived from field deformation (without torsion), and they must be symmetric functions. To verify this, we consider the wave function of a system of two identical noninteracting particles and swap them as follows:

χ_{α, β} = \frac{p \exp ((I + γ) \frac{p • x}{2})}{p_{0}} • \frac{q \exp ((I + γ) \frac{q • x}{2})}{q} =

= \frac{q \exp ((I + γ) \frac{q • x}{2})}{q} • \frac{p \exp ((I + γ) \frac{p • x}{2})}{p_{0}} = χ_{β, α}

The wave function of the superposition of two identical noninteracting particles remained unchanged when the particles were rearranged. The particles described by the function

χ_{α}

are bosons. Equations (43) and (44) are equivalent to Dirac equations for bosons.

Given that function (45) is an function

χ_{3}

for a boson, it can be inferred that the particle corresponding to it is a boson. The mass of this boson is zero, and there exist only two possible states. Based on these observations, it was concluded that the particle in question is a photon.

We have considered a photon in a flat isotropic space. In such a space, all three "generations" of photons are identical. In a Riemannian, and even more so in the anisotropic space

|τ_{α 0}|

, the photon (boson) will be different for each "direction" of α0.

Download: Download full-size image

Figure 3. Photon.

4. Conclusions

1) Clifford algebra in the pseudo-Riemannian space R^1,3 with torsion and spinor bundles can be employed to unify gravitational, electromagnetic, and spinor fields into a single structure. Field deformation is a prerequisite for gravity, and torsion is a prerequisite for electromagnetism. The existence of a spinor field is contingent on the presence of deformation, torsion, and the spinor bundle.

2) The tensor basis method and Clifford product of vectors permit the consideration of both deformation and torsion within the field, thereby facilitating the unification of the Einstein equation and Maxwell system of equations. Einstein and Maxwell's equations are equivalent. Einstein's equations describe spatial quantities including the Ricci tensor and scalar curvature. Maxwell's equations describe field quantities such as the electromagnetic field tensor, 4-current, etc.

3) The inhomogeneity of the DA field can be represented by the sum of the three biquaternions, each of which splits into a pair of bispinors. Differentiation of the bispinors yields three Dirac-type equations for the bosons and fermions. So the solution of Dirac-type equations with deformation describes a boson, whereas the solution with torsion describes a fermion.

In a four-dimensional space, there are only three biquaternions: R₁₀, R₂₀, and R₃₀. Consequently, there could only be three particle generations.

Massless particles, photon, and three generations of neutrinos (γ, ν_e, ν_μ, ν_τ) were combined into a single structure: singlet (photon) + triplet (three neutrinos).

4) The divergence gradient (D(D•A)) is a 4-dimensional electromagnetic current. This formulation of the four-current gives the nonlinearity of the inhomogeneous Maxwell system and the laws of conservation of the four-current and eddy currents. It also clearly shows the geometric meaning of singularities – field "holes," i.e., current sources and the need for calibrations such as Lorentz, Coulomb, and others.

Acknowledgments

It would be unfair to not thank my alliance, my wife Lyubov Gomazkova, for her selfless and colossal help, without which this work would hardly have come into being. Furthermore, I would like to express my gratitude to all those who contributed to this work.

Author Contributions

Alimzhan Kholmuratovich Babaev is the sole author. The author read and approved the final manuscript.

Conflicts of Interest

The author declares no conflicts of interest.

Appendix

D•(□A–Ric•A) =D•(μT•A)

In coordinate form:

eⁿ•e^kD_n(□A_k–A_pR^p_k) =eⁿ•e^kD_n(μA_pT^p_k)

By simplifying we get:

g^nkD_n(g^ijD_iD_jA_k) –g^nkD_n(A_pR^p_k) =μg^nkD_n(A_pT^p_k)

D_k(g^ijD_iD_jA^k) –D_k(AⁱR^k_i) =μD_k(AⁱT^k_i)

We open brackets under differentiation:

D_kDⁱD_jA^k– D_kAⁱR^k_i– AⁱD_kR^k_i=μD_kAⁱT^k_i+μAⁱD_kT^k_i(A-1)

It is known that

[2]

D_k R^k_i = 0.5 D_i R

and according to formula (16) μ D_k Aⁱ T^k_i = D_k j^k.

Then, (A-1) takes the form:

D_kDⁱD_jA^k– D_kAⁱR^k_i– 0.5AⁱD_iR = D_kj^k+μAⁱD_kT^k_i(A-2)

In the first term we change the differentiation order (k with i).

D_nD_iD_jA^k = D_iD_nD_jA^k,

because

g^ij (D_kD_iD_jA^k – D_iD_kD_jA^k) =

=g^ij (D_pA^k R^p_jik – D_jA^p R^k_pik) = – D_pA^k R^p_k + D_jA^p Rⁱ_p = 0.

Now in D_iD_kD_jA^k we change the order of differentiation (k with j):

g^ijD_i(D_kD_jA^k– D_jD_kA^k) =g^ijD_i(–A^pR^k_pjk) =

=D_i(A^pR^j_p) =D_iA^pR^j_p+A^pD_iR^j_p

org^ijD_iD_kD_jA^k=g^ijD_iD_jD_kA^k+ D_iA^pR^j_p+ A^pD_iR^j_p

Considering that according to formula (10)

e^jD_jD_kA^k = j is a four-dimensional electromagnetic current, from the last equality, we obtain:

g^ijD_kD_iD_jA^k= D_ijⁱ+ D_iA^pR^j_p+ A^pD_iR^j_p(A-3)

(A-3) substitute into (A-2):

D_kj^k+ D_kA^pR^k_p+ A^pD_kR^k_p– D_kA^pR^k_p– 0.5 AⁱD_iR=

=D_kj^k+μA^pD_kT^k_p

Now, we reduce the four currents and simplify:

D_kR^k_p– 0.5δ^k_iD_kR =μD_kT^k_p

“Getting rid” of differentiation, we obtain Einstein’s equation:

R^k_p– 0.5δ^k_iR+δ^k_iλ_i=μT^k_p(A-4)

λ_i are the cosmological constants or eigenvalues of the Bochner operator.

References

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[2]	L. D. Landau and E. V. Lipschitz, The Classical Theory of Fields, vol. 2. pp. 70 – 80, 295 – 301. https://haidinh89.wordpress.com/wp-content/uploads/2015/08/landau-l-d-lifshitz-e-m-course-of-theoretical-physics-vol-02-the-classical-theory-of-fields-3305.pdf
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[13]	Wikipedia. Levi - Civita epsilon. Available from: https://en.wikipedia.org/wiki/Levi-Civita_symbol
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[15]	A. Kh. Babaev. Four-dimensional current conservation law in a Clifford algebra-based formalism. sci-article, №42 (February) 2017. pp. 27 – 33. Preprint https://doi.org/10.24108/preprints-3112478 (In Russian).
[16]	Faraday’s law. https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction
[17]	Peter Petersen. Demystifying the curvature term in Lichnerowicz Laplacians. https://doi.org/10.3842/SIGMA.2020.064
[18]	Bochner Laplacian. https://en.wikipedia.org/wiki/Laplace_operators_in_differential_geometry
[19]	A. Kh. Babaev. Equivalence of the inhomogeneous system of Maxwell's equations and Einstein's equations. sci-article, №43 (March) 2017. https://doi.org/10.13140/RG.2.2.33648.47362 (In English)
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[21]	Wikipedia. Matrix exponential. Available from: https://en.wikipedia.org/wiki/Matrix_exponential
[22]	A. Kh. Babaev. Description of Lorentz transformations, the Doppler effect, Hubble's law, and related phenomena in curvilinear coordinates by generalized biquaternions. December 2024, https://www.researchgate.net/publication/386334104_Description_of_Lorentz_transformations_the_Doppler_effect_Hubble's_law_and_related_phenomena_in_curvilinear_coordinates_by_generalized_biquaternions.
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[25]	A. Kh. Babaev. Derivation of the Dirac equation from the inhomogeneity of space and solution for neutrino generations. sci-article, №52 (December) 2017. https://sci-article.ru/number/12_2017.pdf pp. 237- 244 (In Russian)

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Babaev, A. K. (2024). Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra. American Journal of Modern Physics, 13(6), 102-111. https://doi.org/10.11648/j.ajmp.20241306.13

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Babaev, A. K. Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra. Am. J. Mod. Phys. 2024, 13(6), 102-111. doi: 10.11648/j.ajmp.20241306.13

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AMA Style

Babaev AK. Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra. Am J Mod Phys. 2024;13(6):102-111. doi: 10.11648/j.ajmp.20241306.13

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@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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Author Information

Alimzhan Kholmuratovich Babaev

Department Physics, National University of Uzbekistan, Tashkent, Republic of Uzbekistan; Department of Applied Mathematics and Computer Science, Novosibirsk State Technical University, Novosibirsk, Russian Federation

Biography: Alimzhan Kh. Babaev defended his thesis for a PhD in physical and math sciences on the topic “Multiple collisions of particles and fragmentations of 22Ne nuclei in a photoemulsion at P/A=4.1 GeV/c” in 1989 at the Institute of Nuclear Physics of the Academy of Sciences of Uzbekistan (Tashkent). He worked at the Department of Nuclear Physics and Cosmic Rays at the National University (Tashkent, Uzbekistan) as an associate professor. Since 2000, he has worked as a lecturer at the Novosibirsk State Technical University (Novosibirsk, Russian Federation) in the Department of Higher Mathematics. At present, he is working as an independent researcher in the field of applications of methods of abstract algebra to classical and quantum field physics. He is the author (co-author) of more than 30 scientific papers published in peer-reviewed journals.

Research Fields: Clifford algebra, Gravity, Electromagnetism, Biquaternions and Spinor fields, Partial differential equations, Unified field theory.

Contact Email

http://orcid.org/0000-0002-4071-5524

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Babaev, A. K. (2024). Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra. American Journal of Modern Physics, 13(6), 102-111. https://doi.org/10.11648/j.ajmp.20241306.13

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Babaev, A. K. Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra. Am. J. Mod. Phys. 2024, 13(6), 102-111. doi: 10.11648/j.ajmp.20241306.13

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AMA Style

Babaev AK. Unification of Maxwell Systems, Einstein, and Dirac Equations in Pseudo-Riemannian Space R1,3 by Clifford Algebra. Am J Mod Phys. 2024;13(6):102-111. doi: 10.11648/j.ajmp.20241306.13

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@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}
}

Copy | Download

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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