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Similarity Between Dirac and Maxwell Equations Open Access Journal of Physics Volume 3, Issue 3, 2019, PP 15-20 ISSN 2637-5826 Similarity between Dirac and Maxwell Equations Doron Kwiat* 17/2 Erez Street, Mazkeret Batya, Israel *Corresponding Author: Doron Kwiat, 17/2 Erez Street, Mazkeret Batya, Israel, Email: [email protected] ABSTRACT Assuming a charged electron, with a ring current distribution, the resulting Maxwell equations for the charged electron resembles Dirac equation. Thus, a connection is made between Dirac's electron field and Maxwell's electromagnetic field. Furthermore, based on the assumption of a ring shaped electron, estimation is made on the electron's radius. Keywords: Dirac equation; Maxwell equation; Ring-like Electrons; Electron radius INTRODUCTION Which is impossible, as it is much larger than Ever since its discovery, the structure of the the proton radius, which was recently 3, based electron has been discussed by many. Many on experimental electron-proton scattering, have tried to reach an explanation of its internal estimated to be? structure and its radius. −15 푅푃 ≈ 0.87 푥10 [m] It is believed today that the electron as a From the close agreement of experimental and Electron is a one of the elementary particles theoretical g-values4 a new value for the which cannot be divided into smaller particles. electron radius R < 10−23 [m] may be Thus the electron and the positron are the e extracted smallest massive Electrons. They have a unit charge e and spin-half and mass m. It was discussed earlier and stated there5, that Maxwell’s equation cannot be put into a spinor It was Dirac1 who described the fermion in the form that is equivalent to Dirac’s equation. First relativistic Dirac equation. of all, the spinor ψ in the representation of the It is now clear that all massive elementary electromagnetic field bivector depends on only particles must be charged fermions with half- three independent complex components whereas spin multiplications. the Dirac spinor depends on four. Second, Dirac’s equation implies a complex structure Much earlier, the electromagnetic field was specific to spin 1/2 particles that has no described by Maxwell and the two fields, the counterpart in Maxwell’s equation. This fermionic field and the electromagnetic field complex structure makes fermions essentially were considered separately. different from bosons and therefore insures, that The internal structure of the electron have been there is no physically meaningful way to the subject of many articles, from Max Born2 transform Maxwell’s and Dirac’s equations into who wrote in 1933: "The attempts to combine each other. Maxwell's equations with the quantum theory More ecently6a matrix formulation of the (Pauli, Heisenberg, Dirac) have not succeeded. Maxwell field equations was presented One can see that the failure does not lie on the employing an 8-by-8 matrix operator, referred to side of the quantum theory, but on the side of then as the spacetime operator. The primary the field equations, which do not account for the purpose of that was to employ the methods of existence of a radius of the electron (or its finite matrix calculus to determine electromagnetic energy=mass) ". fields for arbitrary charge and current The classical electron radius is given by distributions. Four-dimensional Fourier 1 푒2 transforms, transfer functions, and the −15 푅푒 = 2 ≈ 2.8 푥10 [m] convolution theorem were employed. New 4휋휀0 푚푐 Open Access Journal of Physics V3 ● 13 ● 2019 15 Similarity between Dirac and Maxwell Equations 0 1 3 2 matrix formulations of the classical electromagnetic 푖ℏ 훾 휕푡 + 훾 휕푥 + 훾 휕푧 + 푖훾′ 휕푦 − (EM) Maxwell field equations and the quantum 푚푐 훹푟 = 0 (3) mechanical (QM) relativistic Dirac equations 푖훹 푖 were presented and referred to as the Maxwell or, Which leads (after separation of real and and Dirac spacetime matrix equations, respectively. imaginary parts) to: From the Dirac spacetime matrix equation they 휕푡 훹푖 + 휍′푦 휕푦 훹푟 = 0 (4) were able to form four relativistic vector 휕푡 훹푟 − 휍′푦 휕푦 훹푖 = 0 (5) 푚푐 equations which resemble the four Maxwell 휍 휕 + 휍 휕 훹 = − 훹 (6) vector equations. 푥 푥 푧 푧 푖 ℏ 푟 푚푐 휍 휕 + 휍 휕 훹 = − 훹 (7) An extended model of the electron in general 푥 푥 푧 푧 푟 ℏ 푖 relativity was described7, where a classical Where by definition, σy = iσ′y withσ′y = model of the spinning electron was proposed in 0 −1 which the electron is regarded as a charged 1 0 rotating shell endowed with surface tension. The Since both Ψr and Ψi are made of 2 real shell is the surface of an oblate ellipsoid of components each, there are apparently 4 revolution having a minor axis equal to the constituents of the Dirac particle, denoted by, classical electron radius and a focal distance of ΨA , ΨB, ΨC and ΨD. the order of the corresponding Compton ΨA These 4 components are: Ψr = andΨi = wavelength. This surface is undergoing rigid ΨB Ψ rotation with its equator at a velocity almost C Ψ equal to the velocity of light. D Eqs.4-5 can be then formulated as follows: Recently8, a formulation of quantum mechanics without use of complex numbers is shown to be 휕푡 훹퐴 + 휕푦 훹퐷 = 0 (8) equivalent to the complex formulation. With 휕푡 훹퐵 − 휕푦 훹퐶 = 0 (9) real wave functions formulation, the Dirac 휕푡 훹퐶 − 휕푦 훹퐵 = 0 (10) equation is presented as 4 real wave functions, 휕푡 훹퐷 + 휕푦 훹퐴 = 0 (11) solving 4 KG equations of massive point-like particles. These 4 solutions represent 4 states of While Eqs. 6-7 yield: 푚푐 a fermion. 휕 훹 − 휕 훹 = − 훹 (12) 푥 퐷 푧 퐶 ℏ 퐴 푚푐 In a recent work9, the internal structure of a 휕 훹 − 휕 훹 = − 훹 (13) 푥 퐶 푧 퐷 ℏ 퐵 charged finite size electron was examined. 푚푐 휕 훹 + 휕 훹 = − 훹 (14) 푥 퐵 푧 퐴 ℏ 퐶 푚푐 Based on Maxwell's equations, the 4-vector 휕 훹 − 휕 훹 = − 훹 (15) internal electromagnetic field Aμ of the charged 푥 퐴 푧 퐵 ℏ 퐷 electron was investigated. The solution gives 4 These equations hint to interactions between the non-homogeneous KG equations. possible states of the electron. Under the assumption of a ring-shaped current Applying ∂xand ∂tto Eqs.8-15 results in 2 2 distribution, the Maxwell equations for the 휕푡 + 휕푦 훹푖 = 0 for i= A,C (16) electron internal vector field have the same form 휕2 − 휕2 훹 = 0 for i= B,D (17) as the Dirac equations. 푡 푦 푖 This can be combined by subtraction/addition to Dirac Equation in a Real Vector Space give: The relativistic Dirac equation, describing a free 2 2 휕 2 푚푐 − 훻 + 훹푖 = 0 for i= A,C (18) Fermion of mass m is: 푐 2휕푡 2 ℏ 휇 2 2 푖ℏ 훾 휕휇 − 푚푐 훹 = 0 휕 2 푚푐 + 훻 − 훹푖 = 0 for i= B,D (19) (1) 푐 2휕푡 2 ℏ Separate the complex wave function Ψ into its We thus see that Dirac equation becomes 4 real real and imaginary parts8 waves’ equations. Each component satisfies 훹 = 훹푟 (2) Klein-Gordon equation. Since the solutions to 푖훹 푖 the Klein-Gordon equations are well known, we and insert them into the Dirac equation, can conclude and say, that the Dirac relativistic decomposing Ψ intoΨr + iΨi where Ψr and Ψi Fermion is actually a 4entities equation. These are real, the Dirac equation becomes 4entities are of same mass m. 16 Open Access Journal of Physics V3 ● 13 ● 2019 Similarity between Dirac and Maxwell Equations 휕2휓 푚푐 2 The electromagnetic field vector 퐴 − 훻2휓 = − 휓 (20) 푐 2휕푡 2 퐴 ℏ 퐴 휇 휕2휓 푚푐 2 퐴 = (ϕ, A ) 퐶 − 훻2휓 = − 휓 (21) 푐 2휕푡 2 퐶 ℏ 퐶 2 2 Satisfies Maxwell's equations: 휕 휓퐵 2 푚푐 2 2 + 훻 휓퐵 = + 휓퐵 (22) 2 푐 휕푡 ℏ 휕 휙 2 휌 휕2휓 푚푐 2 2 2 − 훻 휙 = (28) 퐷 + 훻2휓 = + 휓 (23) 푐 휕푡 휀0 푐 2휕푡 2 퐷 ℏ 퐷 휕2퐴 2 2 2 − 훻 퐴 = 휇0퐽 (29) 3-DIMENSIONAL MIRROR FUNCTIONS 푐 휕푡 Assume the Dirac charged electron has an Two functions, f x, y, z and f x, y, z are said 9 1 2 internal current density jμand hence an internal to be mirror functions if f x, y, z = -f x, y, z 2 1 vector potential Aμ. for all x,y,z. This particle has an internal electric charge A surface of a sphere for instance represents a 3- distribution휌, which is responsible for an dimensional mirror function. internal current J. Assume now ψ B and ψ D, to be the mirror functions of ψ and ψ , respectively. Because of current conservation and the B D definition of the vector potentials one has 2 2 2 2 Then ∇ ψ = − ∇ ψ , and ∇ ψ = − ∇ ψ . 1 휕휙 B B D D + 훻 ∙ 퐴 = 0 (30) In such case, 푐 2 휕푡 휕휌 휕2휓 푚푐 2 + 훻 ∙ 퐽 = 0 (31) 퐴 − 훻2휓 = − 휓 (24) 휕푡 푐 2휕푡 2 퐴 ℏ 퐴 휕2휓 푚푐 2 Suppose now that the current density is 퐶 − 훻2휓 = − 휓 (25) 푐 2휕푡 2 퐶 ℏ 퐶 proportional to the vector potential: 2 2 휕 휓 퐵 2 푚푐 − 훻 휓 = + 휓 (26) 휌 = 푘1휙 (32) 푐 2휕푡 2 퐵 ℏ 퐵 2 2 퐽 = 푘 퐴 (33) 휕 휓 퐷 2 푚푐 2 − 훻 휓 퐷 = + 휓 퐷 (27) 푐 2휕푡 2 ℏ (Which, for example, is the case of the vector In other words, we have four possible KG potential and current density for a current in a solutions.
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