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 , with a ring current distribution, the resulting Maxwell equations for the charged electron resembles . Thus, a connection is made between Dirac's electron 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 ; Electron radius

INTRODUCTION Which is impossible, as it is much larger than Ever since its discovery, the structure of the the 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 -half and mass m. It was discussed earlier and stated there5, that Maxwell’s equation cannot be put into a It was Dirac1 who described the 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 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

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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.

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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 2 2 2 2 − 훻 휙 = (28) 휕 휓퐷 2 푚푐 푐 휕푡 휀0 2 2 + 훻 휓퐷 = + 휓퐷 (23) 2 푐 휕푡 ℏ 휕 퐴 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. Two solutions (A, C) with positive ring)? energy term and two solutions (B, D) with Then negative energy term 휕휌 휕휙 푘1 휕휙 + 훻 ∙ 퐽 = 푘1 + 푘2훻 ∙ 퐴 = 푘2 + 훻 ∙ Maxwell Equations for a Ring Current 휕푡 휕푡 푘2 휕푡 퐴=0 (34) Maxwell's equations and the Lorentz force law are extraordinarily successful at explaining and 2 Provided that푘1/푘2 = 1/푐 , then if Eq. 30 is predicting a variety of phenomena; however true, then so is Eq. 31 and vice versa, Eq. 32 they are not exact, but a classical limit will lead to Eq. 30. Then, the conservation of (QED). equations (Eqs. 30, 31) are always satisfied. Some observed electromagnetic phenomena are Thus, the proportionality conditions (Eqs. incompatible with Maxwell's equations. These 32,33), are enough to make the charge-current include photon–photon scattering and many conservation will force the gauge condition Eq. other phenomena related to photons or virtual 30, and the gauge condition Eq. 30, will force photons, "non classical light" and quantum charge current conservation, Eq. 31. entanglement of electromagnetic fields. Then, from Eqs. 28, 29one obtains: 휕2휙 − 훻2휙 = 푘 휙 (35) 푐 2휕푡 2 1 휕2퐴 − 훻2퐴 = 푘 퐴 (36) 푐 2휕푡 2 2 2 And 푘1/푘2 = 휀0휇0 = 1/푐 (37)

Now, based on the fact that the units of 푘1 = 퐦퐜 푚푒푡푒푟−1 , it will be assumed ( has ℏ -1 dimensions of meter ) that 푚푐 2 Figure1. A ring-shaped electron of radius Re with 휌 = 휀 휙 (38) current I and potential field휙 0 ℏ

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1 푚푐 2 ϕ, A , A , A provided that the current 퐽 = 퐴 (39) x y z 휇 ℏ 0 densities ρ, Jx, Jy , Jzare related to the mc 2 1 mc 2 electromagnetic fieldsϕ, A , A , A by Eqs. Thus, k = ε and k = x y z 1 0 ℏ 2 μ ℏ mc 2 0 37,38 through the constant . ℏ When inserted into Maxwell Eqs. 28, 29, they become of same form as the Dirac 4- So, if one identifies the four Dirac fields (ψ , ψ , ψ , ψ ) with the vector componentsEqs. 24-27: A B C D potential (ϕ, Ax , Ay , Az ), Eqs. 39-42, become 휕2휙 푚푐 2 − 훻2휙 = 휙 (40) identical with Eqs. 24-27. 푐 2휕푡 2 ℏ 2 2 휕 퐴1 2 푚푐 Note though, that in order to keep with signs, as 2 2 − 훻 퐴푥 = 퐴푥 (41) 푐 휕푡 ℏ required byEqs24 -27, the currents J x and J z 2 2 휕 퐴2 2 푚푐 must also be symmetric with J and J about their − 훻 퐴푦 = 퐴푦 (42) x z 푐 2휕푡 2 ℏ spatial coordinates (mirrors), so that their spatial 휕2퐴 푚푐 2 3 − 훻2퐴 = 퐴 (43) derivatives change signs accordingly. 푐 2휕푡 2 푧 ℏ 푧 In other words, the four Dirac fields There are no external electromagnetic fields, so these are confined electromagnetic fields which ψA , ψ B , ψC , ψ Dare equivalent to the electromagnetic vector potential fields, create the electron and give it its mass m.

Maxwell Eqs for a ring current Dirac Eqs 2 2 2 2 휕 휙 푚푐 휕 휓퐴 푚푐 − 훻2휙 = + 휙 − 훻2휓 = − 휓 푐2휕푡2 ℏ 푐2휕푡2 퐴 ℏ 퐴 2 2 2 2 휕 퐴 푚푐 휕 휓퐶 푚푐 1 − 훻2퐴 = + 퐴 − 훻2휓 = − 휓 푐2휕푡2 푥 ℏ 푥 푐2휕푡2 퐶 ℏ 퐶 휕2퐴 푚푐 2 휕2휓 푚푐 2 2 − 훻2퐴 = + 퐴 퐵 − 훻2휓 = + 휓 푐2휕푡2 푦 ℏ 푦 푐2휕푡2 퐵 ℏ 퐵 2 2 2 2 휕 퐴3 푚푐 휕 휓 푚푐 − 훻2퐴 = + 퐴 퐷 − 훻2휓 = + 휓 푐2휕푡2 푧 ℏ 푧 푐2휕푡2 퐷 ℏ 퐷

A more realistic approach would be to take the Notice the difference in sign of the energy terms ratio of proton/electron mass, then divide the between theψ , ψ and the ϕ, A equations. A 퐶 x proton’s radius by the cube root of this number One possible explanation could be that the (because mass increases according to the cube electron may have a positive charge and a of radius). This ratio, 1836, would set the negative charge, together with two possible electron’s radius at approximately 12 times opposing current directions (spins). Therefore, smaller than a proton, namelyRe ≈ Maxwell equations will describe opposite spins, 9.1 x10−17 m. only if we assume opposite currrents. Dirac However, the proton is believed to consist of equations contains opposite spins as a result of . If these quarks occupy only a small mirror functions. volume within a proton, this would greatly Electron Radius lower its density and the electron’s radius would calculate to be far smaller again, perhaps by an Many attempts are known to evaluate the additional factor of 10 or 100. electron's radius7-10. A classical model of the spinning electron was The classical radius of the electron is calculated proposed10 in which the electron is regarded as a by equating the electrostatic potential energy of charged rotating shell endowed with surface a sphere of charge e and radius Re with the rest tension. The shell is the surface of an oblate -15 energy of the electron. It is Re =2.8179 x 10 m ellipsoid of revolution having a minor axis equal If we compare this classical radius, with the to the classical electron radius and a focal distance of the order of the corresponding measured radius of a proton, which is3R ≈ 푃 Compton wavelength. This surface is −15 1.11x10 m, an electron has a radius 2.5 undergoing rigid rotation with its equator at a times larger than a proton. Given that a proton is velocity almost equal to the velocity of light. around 2000 heavier, one should not take this The arrangement of charges gives rise to a ‘classical radius’ seriously. quadrupole electric moment, in addition to the

18 Open Access Journal of Physics V3 ● 13 ● 2019 Similarity between Dirac and Maxwell Equations magnetic dipole moment of the current electron will align its z-direction according to distribution. and with the direction of the applied magnetic field. In this work, the electron is assumed to have a doughnut shape, of radius Re with a ring- CONCLUSIONS current. Based on Eq. 37, the charge e of this Based on real wave functions description of electron will be given by: Dirac's equation, similarity of Dirac with the 2휋 푚푐 2 2휋 푒 = 휌푅 푑휃 = 푅 휀 푑휃 휙 (44) Maxwell equations was shown. 0 푒 푒 0 ℏ 0 It was then assumed that the charged electron is Assumingϕ, the electromagnetic potential, to be not a point like particle, but rather a finite symmetrical and constant ϕ , independent ofθ, 0 doughnut shaped particle of radius Re, carrying one obtains a ring current. 푚푐 2 푒 = 2휋 푅푒 휀0 휙0 (45) Its internal charge dynamics creates a vector ℏ potentialAμ. Under certain assumptions on the This expression creates a connection between proportionality of the vector potential to the the electron radius, its mass, and the two current density, the Dirac equation wave constants of nature c, and ℏ . functions are similar to the internal electromagnetic vector potential. Calculation of this radius gives R푒 = −34 4.29x10 ϕ0 푚, with some ϕ0 constant value. Based on this model, the electron radius is According to this model, the mass/radius ratio directly connected to the Planck's and speed of of the proton to electron (both having the same light universal constants, together with the charge magnitude and assuming the same permittivity electric constant and the electron potential) would be: mass. 2 R푃 m푃 The electron radius is then predicted in = ≈ 3.46 푥106 comparison to the known proton radius, and is R푒 m푒 estimated at 3.2x10−22 m. Thus, based on the known proton radius3 −15 (RP ≈ 1.11x10 m)the estimated electron REFERENCES radius will be [1] Dirac, P. A. M., "The Quantum Theory of the −22 Electron" , Proceedings of the Royal Society A: R푒 ≈ 3.2x10 m Mathematical, Physical and Engineering Electron Spin Sciences. 117 (778): 610. 1928. According to this ring shape model, the electron [2] M. Born, "Modified Field Equations with a will have a spin according to the direction of the Finite Radius of the Electron". Nature, ring current and the ring mass m. Vol 132, p.282, 1933. [3] RJ Hill and G Paz "Model independent The spin magnetic moment will depend on the extraction of the proton charge radius from charge ring current. electron scattering", Physical Review D, 2010 The spin magnetic moment is given by the ring [4] Hans Dehmelt, "A Single Atomic Particle area times the current I: Forever Floating at Rest in Free Space: New 휇 = 휋푅2퐼 (46) Value for Electron Radius", PhysicaScripta. 푠 푒 Vol. T22, 102-1 10, 1988. 휕휌 푚푐 2 휕휙 퐼 = = 휀 (47) 휕푡 0 ℏ 휕푡 [5] Andr´e Gsponer, "On the 'Equivalence' of the 2 푚푐 2 Maxwell and Dirac Equations" .International 휇 = − 휋푅2휀 훻 ∙ 퐴 (48) Journal of Theoretical Physics, Vol. 41, No. 4, 푠 0 ℏ 2002 . Therefore, the spin of the electron will depend [6] Richard P. Bocker, B. Roy Frieden, "A new on the direction of the current clockwise or matrix formulation of the Maxwell and Dirac anticlockwise. equations", Special Relativity, Electro This model cannot hold for the proton, as it is magnetism, Quantum Mechanics. Volume 4, composed of quarks. Issue 12, 2018 [7] Carlos A Lopez, "Extended model of the Experimentally we know that the spin direction electron in general relativity", Physical Review. is dictated by the direction of the externally D, Particles Fields; Vol. 30(2); p. 313-316, applied magnetic field. This means that the 1984.

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[8] D Kwiat, "The Schrödinger Equation and [9] D Kwiat, "Dirac Equation and Electron Internal Asymptotic Strings", International Journal of Structure", International Journal of Theoretical Theoretical and Mathematical Physics, pp. 71- and Mathematical Physics, pp. 51-54, Vol 9 (2), 77, Vol 8 (3), 2018. 2019.

Citation: Doron Kwiat, "Similarity between Dirac and Maxwell Equations", Open Access Journal of Physics, 3(3), 2019, pp. 15-20. Copyright: © 2019 Doron Kwiat. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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