aa r X i v : . [ phy s i c s . g e n - ph ] S e p Anomalous magnetic moment and vortex structure of the electron
S. C. TiwariDepartment of Physics, Institute of Science, Banaras Hindu University, Varanasi 221005, andInstitute of Natural PhilosophyVaranasi India
The electron magnetic moment decomposition calculated in QED is proposed to have origin ina multi-vortex internal structure of the electron. This proposition is founded on two importantcontributions of the present work. First, a critical review on Weyl model of electron, Einstein-Rosen bridge and Wheeler’s wormholes leads to a new idea of a topological defect in cylindricalspace-time geometry as a natural representation of the electron. A concrete realization of this ideain terms of the derivation of a new vortex-metric, and its physical interpretation to establish thegeometric origin of the electron magnetic moment decomposition constitute the second contribution.Remarkable occurrence of a vortex structure in the basis mode function used in BLFQ is also pointedout indicating wider ramification of the proposed electron model.
PACS numbers: 04.20.-q , 02.40.Ky, 11.15.Tk
I. INTRODUCTION
Recent discovery of the Higgs boson at LHC givesstrong credence to the Standard Model (SM) of particlephysics. The SM has withstood precision experimentaltests carried out over past decades. The successful devel-opment of the gauge theory of electro-weak sector, andthe advances in perturbative QCD methods combinedwith near absence of new/exotic events at LHC makethe quest for unified theories beyond the SM superfluousfrom the view-point of the empiricism/operationalism.However the known drawbacks on the conceptual foun-dations of the SM and the exclusion of gravity describedby general relativity in this unified scheme, i. e. the SM,motivate enormous efforts in superstring theory. Unfor-tunately superstrings continue to have tenuous, if any,link with the real particle world. Moreover superstringsseem to have entered into a state of stalemate. Thoughthe necessity of a new idea is felt by many physicists thedirection of the new pathways is uncertain and specu-lative: intensely explored ideas related with, for exam-ple, AdS/CFT correspondence and qubits/quantum en-tanglement. One of the important consequences of thesestudies is that gauge fields/forces as well as space-timemay appear as emergent phenomena. The contrastingviewpoints on the nature of space-time ’an arena or it isall’ [1], however remain unsettled and open. In this sce-nario I believe the profound thoughts of the great mindsenvisaged in the past could offer a guideline: geometryand topology of the physical space-time have to be fun-damental. We suggest that exploring the internal space-time structure of the electron could serve the basis for anew approach.Historically classical models of electron were concernedwith the extended charge distributions to explain theelectron rest mass, beginning with J J Thomson’s model.Lateron spin, Bohr magneton and anomalous magneticmoment also inspired geometric and field theoretic mod-els. Electron spin arises naturally in the Dirac equation, and the intrinsic magnetic moment vector µ has the z-component µ = ± e ¯ h mc = ± µ B (1)In terms of g-factor, expression (1) implies that g = 2 forthe Dirac electron. Free electron magnetic moment mea-sured using a one-electron quantum cyclotron [2], how-ever gives a different value g . g . α = e ¯ hc gave the altered expression µ ′ e = µ B [1 + α π ] (4)Interestingly for the assumed value of α = one gets g ≈ . αµ e = µ B [1 + α π − . α π ] (6)Does this structure of the QED expression of the anoma-lous magnetic moment have deep physical significance?Since the empirical number for µ e (2) gives a measuredquantity, from the operational point of view the ques-tion of the physical significance of individual terms in(4) or (6) would seem to be unimportant or meaningless.However this question has fascinated many physicists,and led to the speculations on the internal structure ofthe electron. Comprehensive review [7] shows that gen-erally it is believed that these attempts failed to offerviable alternative to QED and SM. Moreover, high en-ergy electron scattering experiments [8] severely limit thesize of the electron to < − cm. It could be arguedthat the failed attempts perhaps indicate that the rightconceptual issues were not raised, and new ideas weremissing [7]. For example, the quantization of charge inthe unit of electronic charge e intrigued Dirac to pro-pose magnetic monopole, however the most fundamentalquestion as to the nature and origin of charge has notbeen asked [7, 9]. In a radical departure from the clas-sical models and the QED paradigm we proposed a newspatio-temporal bounded field approach. Weyl geome-try, topology and deep physical meaning attributed tothe form of fine structure constant α , and the structureof the anomalous magnetic moment expression (4) mo-tivated this approach; see the monograph [7]. We have,in a sense, followed the concerns of prominent foundersof QED, notably Dirac and Pauli on the foundations ofQED to seek alternatives.The main ingredients that distinguish our approachfrom the past ones may be summarized as follows. Mass-less internal fields with nontrivial topology constitute theelectron, and the rest mass of the electron is not intrin-sic but a secondary physical property. Planck constantappears in α with other fundamental constants e and c in a ratio. Planck constant has the dimension of actionbut the dimension of angular momentum is also the sameas that of action. Dimensionless fine structure constantis interpreted as a ratio of angular momenta, i. e. e c isinterpreted as angular momentum associated with elec-tron. Remarkably, the anomalous magnetic moment (4)could be re-written as µ ′ e = emc [ ¯ h f f = e πc (8)Logically the bracketted term in (7) should correspondto the angular momentum. We are led to the hypothesisthat electron spin has a fractional part f , if expressedin units of Planck constant, and the electron charge hasa mechanical origin as some kind of rotation. For fur-ther elucidation of this hypothesis see [9, 10], and vortexmodel of the spatio-temporal bounded internal fields isdeveloped in [11, 12].Physical interpretation of one-particle Dirac equationcontinues to be of interest primarily at the conceptual level; curiously the significance of the classical electroncharge radius r e = e mc is least understood or obscure ascompared to the Compton wavelength λ c = ¯ hmc . Even inQED, quite often, the intuitive picture of quantum vac-uum effects is related to the size of the Compton wave-length. It is easily seen that in an alternative outlook[11, 12] both length scales arise in the fine structure con-stant α = r e λ c , and magnetic moment becomes µ ′ e = e λ c + r e π ] (9)There is a renewed interest in the recent literature [13–18] on the possible internal structure of electron. Thepresent article makes two significant contributions: i)it is suggested that a topological defect in a cylindricalspace-time geometry is the most natural representationof electron, and ii) a new metric, termed vortex-metric,is derived and physically interpreted to explain the geo-metric origin of the electron anomalous magnetic moment(7). It is important to note that the metric structure ofspace-time could be obtained either solving a field equa-tion (e. g. Einstein field equation) or by constructingmetric independently of any field equation. We adoptthe second approach in which black holes do not exist.Since Burinskii [13] underlines the role of black holes inelementary particle models a brief digression is made tore-interpret the standard Schwarzschild metric in our ap-proach for the sake of clarity.The paper is organized as follows. In the next sec-tion a critical appraisal on the Weyl model [19–21], theEinstein-Rosen bridge for a charged particle [22], andwormholes [1] is made to gain new physical insights onthe geometric structure of electron. It is concluded thata topological defect, cylindrical space-time geometry andfundamental role of angular momentum have to be themain ingredients for building electron model. A brief di-gression is made in Section III to revisit the Schwarzschildmetric in the present context. A new metric possessingmulti-vortex structure is derived in Section IV. Plausiblearguments are presented to relate this metric with theinternal space-time structure of the electron. In the lastsection the proposed new conceptual framework is putin the perspective of the Kerr-Newman geometric mod-els [13, 18] and recent light-front quantization methods[14–17, 23, 24]. In this discussion we are guided by thecritique on QED by Dirac [25], Pauli [26] and Feynman[14]. The occurrence of vortex-like structure in the basislight-front quantization (BLFQ) [16] is suggested basedon the recent analysis of the Landau problem [27]. Thesignificance of vortex as a topological defect and Kelvin’svortex atom model [28] show that vortices may serve thebasis for explaining elementary particle spectrum. II. GEOMETRIC MODEL OF ELECTRON:CHARGE AND MASS
Electric charge, e and mass, m were the only knownphysical attributes of the electron in the early decadesafter its discovery in 1897. Understandably the physi-cists during that period, notably Thomson, Poincare,Lorentz and Abraham sought electrodynamical modelsof the electron [7]. Geometrization of gravitation in gen-eral relativity of Einstein inspired early geometric modelsof electron. At this point we must emphasize that thegeometry in the modern gauge field theories and SM be-longs to the postulated internal spaces different than thephysical space-time geometry [29]. In my opinion [9] thefollowing fundamental question has not been addressedin the physics literature: What is the physical origin ofelectric charge? For example, in QED local U(1) gaugeinvariance leads to a conserved Noether current, and theNoether charge is a Hermitian operator and the quantumvacuum is also gauge invariant. This definition amountsto just a formal construct in the hypothetical internalspace.A fresh outlook on the classical models of electron[19, 22] is presented in this section. The first unified the-ory of gravitation and electromagnetism is due to Weylgeneralizing the metric Riemann space postulating non-compact homothetic gauge transformation in addition tothe general coordinate transformations of general rela-tivity. In the monograph [19] Weyl influenced by Mie’stheory discusses electron model in Section 32 based onEinstein-Maxwell theory, and in Section 36 based on hisunified theory. On the other hand, Einstein and Rosenin 1935 argue that field singularities are unacceptable,and present an alternative approach discussing electronas well [22].Weyl considers the Reissner-Nordstroem solution ofthe Einstein-Maxwell equations for a static chargedsphere ds = − f rn c dt + f − rn dr + r d Ω (10) f rn = 1 − r g r + e g r (11) d Ω = r ( dθ + sin θdφ ) (12)Here the gravitational radius of the mass m of the elec-tron is r g = Gmc (13)and the gravitational radius of the charge is e g = √ Gec (14)Physical interpretation demands that electron is a realfield singularity and the mass of the electron is continu-ously distributed in the Universe. However, there is noanalog of the electron charge radius, r e . Weyl also investigates cylindrical geometry and findsthat though the electrostatic potential differs from thatof the Maxwell theory, the difference is appreciable onlyin the internal space of the electron where the dimensionbecomes of the order of the gravitational radius of thecharge e g = 10 − cm.Einstein-Rosen [22] assert that the field singularitiesare unphysical, and must be avoided as a fundamentalprinciple. Towards this aim the authors are prepared tomodify the field equation. A coordinate transformationis used to obtain regular solutions. For the charged par-ticle the new metric is obtained such that f rn in (10) isreplaced by f er = 1 − r g r − e g r (15)Note the sign change in the last term of (15) as comparedto (11) that arises due to the modified field equation.Since mass and charge occur independently in the metrictensor, the authors argue that one could set m = 0. Thusone may envisage a massless electron model. Assuminga coordinate transformation replacing r by uu = r − e g a µ as a linear 1-form a µ dx µ gives a distance curvature f µν = ∂ µ a ν − ∂ ν a µ (17)Weyl assumes e in e.s.u. as a definite unit of electric-ity interpreting ea µ as the electromagnetic potential A µ .The electric current density J ∝ √− g e a (18)The physical interpretation of electron is speculative andintriguing. Electron is represented by a singularity canal.Charge and mass characterize singularities and remainconstant along the canal. However the world-direction ofthe canal is determined from the Einstein-Lorentz equa-tion, i. e. the geodesic equation generalizing the Newton-Lorentz equation. Weyl dwells on the philosophical ar-guments to understand diffused charge throughout theUniverse and the sign of electric charge possibly havingorigin in the arrow of time: past to future ordering.To put the import of these classical geometric mod-els in perspective it may be remarked that there existsenormous literature on the singularities specially oncethe idea of black hole physics became acceptable amongphysicists. On the Reissner-Nordstroem solution (10)a nice discussion given in [30] shows that the singular-ity at r = 0 cannot be removed, however other singu-larities are removable by appropriate coordinate trans-formations, and a maximally extended manifold is ob-tained; see Penrose diagram on page 158 of [30]. Re-garding the original Weyl theory [19] it is known thathe himself abandoned it, and entirely different version ofgauge symmetry occurs in the modern developments [31].Dirac in 1973 revived Weyl geometry for its ’simplicityand beauty’ essentially with the motivation of developinghis belief in the large number hypothesis [32]. My owninterest in the Weyl-Dirac theory has been to understandthe electron structure [7, 9, 10, 21].A fresh outlook on the speculations of Weyl [19] andEinstein-Rosen [22] leads to radically new ideas as wediscuss below. First let it be noted that the nature ofelectric charge as such remains obscure in these works.However new light is thrown on the electron mass. Tounderstand it we recall an important result in the conven-tional approach [33] treating electron as a charged sphereand assuming Reissner-Nordstroem exterior metric. Au-thors find that the application of the junction conditionson the boundary imply negative mass density of electron.The argument runs as follows. For large r the third termin the expression (11) is negligible compared to the sec-ond term. The value of ( e g r − r g r ) at the assumed bound-ary r = 10 − cm is ∼ × − . Thus f rn >
1, and thejunction condition shows that the mass density for somevalue of r in the interior has to be negative. This resultputs a question mark on the validity of the singularitytheorems where positivity of mass plays a crucial role[30]. Curiously in the Einstein-Rosen model [21] the pos-sibility of massless charge arises as an interesting idea. InWeyl geometry [19] null vectors or gauge-invariant zerolength have a unique place: the proposition of a masslesselectron seems quite natural [21]. Thus it may be arguedthat mass is not an intrinsic physical attribute of elec-tron. The origin of observed mass, in that case, needsa different explanation than what was attempted in thepast, namely the electrodynamical models.On the nature of singularities discussed in [19, 22] topo-logical perspective could be another line of thought. Ein-stein firmly believed that singularities in field theorieswere physically unacceptable see, [34]. He termed theEinstein field equation incomplete, and suggested provi-sional role to the energy-momentum tensor T µν on theright hand side of the equation G µν = 8 πGc T µν (19)Obviously Einstein-Rosen arguments [22] are in confor-mity with this belief. A critique [34] on the foundations ofthe Einstein field equation (19) shows that geometriza-tion without the field equation is logically admissible.Note that the metric tensor, and the geodesic equationare sufficient for this purpose. Moreover crucial experi- mental tests of general relativity are based mainly on theSchwarzschild metric [30].The question is: How to get a topological perspective?Nontrivial topology of spacetime finds beautiful expres-sion in the concept of wormholes [1]. Authors discussReissner-Nordstroem metric and Einstein-Rosen bridgeamong several topics in this paper. Wheeler speculateson a model of electron as a collective excitation of thespacetime foam in quantum geometrodynamics [35]. Itmay, however be asked whether quantum theory is neces-sary in a topological approach. Wheeler recognized thatthis theory [1, 35] was unable to explain ’the world ofparticle physics’. In spite of great advances in topologi-cal quantum field theories since then the progress on theissue raised by Wheeler, i. e. to make contact with ob-served elementary particles has remained unsatisfactory.Could it be due to the belief in the fundamental roleof quantum theory? In a new approach Post abandonsquantum theory altogether and puts forward the conceptof quantum cohomology [36]. My own conceptual pic-ture of space-time departs radically from the past ideas;it is elaborated in a monograph [37]. Here an attempt ismade to apply this picture to the problem of electron.We postulate electron to be a topological defect,namely a vortex, and accord fundamental significance tothe angular momentum e c than the electric charge. Thusa singularity canal in Weyl theory [19] is re-interpreted asa vortex. The second proposition is also supported fromsome basic considerations. The most remarkable fact,not commonly recognized, regarding the unit of electric-ity e is that it can be factored out from the Maxwellequations. As a consequence the physical description isindistinguishable for the Maxwell field F µν from that of F µν e . However in the equation of motion of a chargedparticle factorization results into an irremovable factor e in the Lorentz force expression as it does not cancel.Similarly the expression for the generalized momentumbecomes p µ − ec A µ → p µ − e c a µ (20)Note that in the geometric theory of Weyl a µ and f µν have the dimensions of ( length ) − and ( length ) − re-spectively. Following the conventional approach Weylmultiplies them by e to get the electromagnetic quan-tities. In contrast to Weyl, in the light of the precedingarguments we suggest multiplication by e c . In general, asuitable angular momentum unit should be used to maketransformation from geometry to physics. III. SCHWARZSCHILD METRIC: NEWINTERPRETATION
Let us revisit gravitation in this approach. The moststudied Schwarzschild metric is spherically symmetricstatic solution of the vacuum Einstein field equation. TheSchwarzschild metric is given by form (10) replacing f rn by f s f s = 1 − r g r (21)Important experimental tests of classical general relativ-ity are based on this metric, and mathematical theory ofblack holes also evolved from its study [30]. The sin-gularity at r = 0 cannot be removed but the secondone at r = 2 r g can be removed by appropriate coordi-nate transformation, for example, the advanced or theretarded null coordinates in the Eddington-Finkelsteinform of the metric. Two analytic manifolds are separatedby the surface at r = 2 r g . The Kruskal construction andthe Penrose diagram are nicely explained in the text [30].There exist several derivations of the Schwarzschildmetric in the literature, that given by Weyl in Section 31of [19] is an elegant one. The crucial part in all deriva-tions is common: the identification of the integration con-stant is based on the correspondence with the Newtoniantheory of gravitation giving expression (13). Note thatthe integration constant has a unit of length in the so-lution of the vacuum Einstein field equation. Eddingtonmakes a perceptive remark in a footnote on page 87 [38]regarding the objections for using unit of length for thegravitational mass. Moreover, the interpretation of massas a source in the metric also does not seem to have asolid foundation [34]. Logically there is no objection ifwe relate the integration constant with the unit of angu-lar momentum. Could we attribute physical significanceto the angular moentum? The suggested role of angularmomentum would seem almost heretical due to sphericalsyymetry and static nature of the metric. Neverthelesslet us proceed to explore hidden angular momentum inthe metric.One can construct a quantity having the angular mo-mentum dimension using purely dimensional argumentsto get L g = G M c (22)Defining L ( r ) = M cr , the expression for f s becomes f s = 1 − L g L ( r ) (23)In this setting the problem of singularity acquires an en-tirely different version. The angular momentum L ( r ) canbe assigned a limiting minimum value, let us assume it tobe equal to the Planck constant ¯ h . Thus the singularityat r = 0 is rendered superfluous. On the other hand, at L ( r ) = 2 L g (24)the singularity does exist. The limit on L ( r ) being ¯ h onegets mass of the order of Planck mass M = M P r S = 2 GMc (26)whereas the singularity defined by (25) occurs at a def-inite Planck scale ∼ GeV. In this re-interpretationthe problem of negative mass density [33] is rendered anonissue.In Cosmological models the incorporation of Machprinciple has attracted attention of some physicists;Brans and Dicke [39] discuss in detail different versionsof this principle, and relate it with the following GM Universe Rc ∼ M Universe is the finite mass of the visible Universeand R is its radius. Authors provide a simple argumentfor the derivation of (27). Newton’s theory gives acceler-ation due to gravity at a distance r from, let us say theSun, GM Sun r , and from dimensional considerations onecan construct acceleration to be M Sun Rc M Universe r ; equating thetwo the relation (27) follows immediately. In the angularmomentum approach, the relation (27) is interpreted asthe upper limit on L ( r ) leading to the maximum value L ( r ) max = GM Universe c (28)The ratio between the maximum and minimum values of L ( r ) turns out to be a large number ∼ assumingthe rough estimates of the mass and the radius of thevisible Universe. Further discussion on the large scalestructure of the Universe is deferred to a separate papersince the focus here is on the electron structure. IV. ELECTRON STRUCTURE: GEOMETRYAND TOPOLOGY
Let us briefly mention the salient features of quantumgeometrodynamics [35]. Wheeler’s qualitative picture ofthe gravitational field fluctuations is in analogy to QEDvacuum fluctuations. Feynman’s path integral approachis adopted in which the action integral in the phase expo-nent uses the Planck constant as a quantum of action fol-lowing the standard QFT practice. Virtual particle pairshaving charge ∼ e and mass of the order of Planckmass are expected to be created by vacuum fluctuations.However such particles cannot be identified with the elec-tron or any observed elementary particle in nature. How-ever drawing analogy with QED renormalization methodingenious arguments are put forward by Wheeler to re-interpret the virtual particles in the wormhole picture.Departing from Wheeler’s approach we propose to in-corporate topology of space-time at a fundamental level.For this purpose, quantum theory/QED paradigm isshifted to the old quantum theory. To appreciate thesignificance of the old quantum rule of Bohr on angularmomentum quantization we refer to a short historical ac-count given in [40], and for its relationship with topologyto the Post’s monograph [36]. Bohr-Wilson-Sommerfeld(BWS) quantization requires nontrivial topology; Posthighlights an important contribution of Einstein in thisconnection [36, 41]. Einstein replaces Bohr’s circular or-bits by torus, and the orbital manifold (orbifold) signifiestopological obstruction. Orbifolds have de Rham coho-mology ramifications [36, 41, 42]: the principal result ofthis work is that physical BWS quantum conditions onangular momentum have natural mathematical counter-part in the form of de Rham periods for topological de-fects. Application of this idea can be found in the mono-graph [36] and for topological photon in [43]. Thus thefundamental significance accorded to Planck constant asa quantum of angular momentum becomes logically jus-tified.Admittedly introducing ¯ h in the Schwarzschild met-ric is heuristic, however it can be pursued further qual-itatively for other metrics. Reissner-Nordstroem, andEinstein-Rosen metrics become f rn = 1 − L eg ¯ h [1 − α f er = 1 − L eg ¯ h [1 + α L eg = Gm c (31)Apart from the difference in the numerical factor, i. e. 4instead of 2 π formal resemblence of the bracketted termin (30) with the anomalous part in the magnetic momentexpression (4) seems curious. Does it have some deepsignificance? Let us explore this question. The numeri-cal value of L eg is a very small number; the ratio L eg ¯ h is ofthe order of 10 − . Note that the electron charge radiusand the Compton wavelength can be defined using angu-lar momenta e c = mcr e and ¯ h = mcλ c respectively. Alength analogous to them using L eg can be defined λ G = L eg mc (32)Since r e = αλ c we anticipate λ G = α N λ c (33)for some integer N; in numbers λ c ∼ − cm and λ G ∼ − cm. Following geometric picture presents itself: acore sphere of radius λ c surrounded by concentric thinspherical shells of width α n λ c ; n = 1 , , ....N .The concentric spherical shell model shows that mul-tiple geometric sub-structures have to be explored for the electron model: a possible propagating vortex is sug-gested for this purpose. The crucial question is: How dowe get it in the metric tensor for the geometric model?It is obvious that the cylindrical metric rather than thespherical Reissner-Nordstroem metric has to be consid-ered. Unfortunately the known Weyl metric [19] did notsucceed as electron model. Moreover, the problem of log-arithmic divergence at large radial distance for a cylin-drical system makes physical interpretation very difficult.For the envisaged vortex we adopt an alternative methodproposed in [34]. In this approach massless scalar field Φsatisfies wave equation in flat spacetime with the metric η µν ∂ µ ∂ µ Φ = 0 (34)and assumed to consist of two scalarsΦ = F Ψ (35)such that ∂ µ Ψ is a null vector ∂ µ Ψ ∂ µ Ψ = 0 (36)The choice for a massless scalar field as a fundamentalfield variable is primarily based on the guiding principlesof simplicity and symmetry. The proposition (35) givesa bi-scalar structure to the field Φ necessary to constructa nontrivial metric. Now the metric tensor is defined inthe Kerr-Schild form g µν = η µν + F ∂ µ Ψ ∂ ν Ψ (37)Note that ∂ µ Ψ is also a null vector with respect to g µν .Earlier we have derived some known metrics, for example,Vaidya metric, the Schwarzschild metric, and Brinkman-Robinson metric using this approach [34].New solutions in cylindrical system ( ρ, φ, z ) with theaim of realizing vortex are presented here. Assuming Ψto be a function of (t, z) only the null condition is satisfiedfor functions Ψ( z ± ct ). We assume simplest form for apropagating field Ψ = e i ( kz − ωt ) (38)Assumed time-independent F ( ρ, φ, z ) the wave equation(34) becomes ∂ F∂ρ + 1 ρ ∂F∂ρ + 1 ρ ∂ F∂φ + 2 ik ∂F∂z + ∂ F∂z = 0 (39)The usual azimuthal-dependence e ilφ reduces Eq.(39) to ∂ F∂ρ + 1 ρ ∂F∂ρ − l ρ F + 2 ik ∂F∂z + ∂ F∂z = 0 (40)To solve Eq.(40) we make a simplification that F is notz-dependent. In this case the solution is given by F = ρ l e ilφ (41)and the line element assumes the form ds = − c dt + dρ + ρ dφ + dz + F ( ρ, φ )( ∂ Ψ ∂z ) ( cdt − dz ) (42)The solution (41) does represent a vortex: phase singu-larity at ρ = 0. Unfortunately the divergence as ρ → ∞ isworse than the logarithmic divergence making this metricphysically unacceptable.Retaining z-dependence of F in Eq.(40), and in anal-ogy to the paraxial approximation in optics [44] neglect-ing ∂ F∂z compared to ∂F∂z Eq.(40) becomes ∂ F∂ρ + 1 ρ ∂F∂ρ − l ρ F + 2 ik ∂F∂z = 0 (43)This equation has the well-known Laguerre-Gaussian(LG) mode solution F ( ρ, φ, z ) = LG lp (44)Here azimuthal index l , and the index p of the associatedLaguerre polynomial characterize the vortex and the con-centric rings in the transverse profile of the LG mode.Two of the remarkable properties of LG modes notedin optics literature [44] are the z-dependent radius of thebeam, and the topological quantum number l . Phase sin-gularity has a topological charge π H ldφ . In the presentcase, the ansatz (37) provides a geometric significance forthe LG mode function in a new line element ds = − c dt + dρ + ρ dφ + dz + LG lp ( ∂ Ψ ∂z ) ( cdt − dz ) (45)Note that the metric is not complex; it is understood thatfollowing the convenient practice only complex represen-tation is used for the solution of the wave equation.Could we visualize internal structure of the electrongeometrically using this metric? Qualitative interest-ing correspondence follows immediately. The metricfor l = 1 , p = 0 shows vortex structure of the form ρL ( ρ ) e − ρ w e iφ . Tentative identification of w → λ c makes this vortex as a core vortex for the electron. Theconcentric ring for p = 1 may be identified with the sec-ond vortex having dimension equal to r e . This picturenicely fits the two-vortex Dirac electron model [12]. Themulti-vortex structure can be envisaged for the geome-try of the metric (45). To associate angular momentumwith the vortex we cannot adopt the optics method: or-bital angular momentum per photon l ¯ h is obtained usingPoynting vector for linear momentum density and cal-culating the orbital angular momentum from it [44]. Inthe geometric formulation we have to seek a new idea.The geometric quantities like the curvature tensors canbe calculated following lengthy but straightforward tech-nical method [38]. However these are of no interest forangular momentum. In view of the structure of the Kerr-Schild form (37) a new geometric quantity can be natu-rally defined v µ = F ∂ µ Ψ (46) L µν = ∂ µ v ν − ∂ ν v µ = ∂µF ∂ ν Ψ − ∂ ν F ∂ µ Ψ (47)The antisymmetric tensor L µν has formal similarity withthe electromagnetic field tensor for a specific choice of A µ having the form (46). To get its physical interpretationit is instructive to make explicit calculation of its com-ponents: L i and L ij ; i, j = 1 , , L L = − lkF Ψ (48)Here we take ∂ = ∂∂φ . We also calculate the nonvanish-ing components of v µ v = − iωc F Ψ (49) v = ikF Ψ (50)An interesting quantity is the following ratio | L /v | = | L /v | = l (51)Multiplication by appropriate angular momentum unit in(51) provides physical interpretation of this ratio as quna-tized angular momentum. Note that the ratio | L /v | = l . In our approach we have two fundamental units ¯ h and e c , therefore, the vorticity of of two different vortices p = 0 , l = 1 and p = 1 , l = 1 are suggested to be quan-tized in these units respectively. A qualitative geometricmodel of electron inspired by the QED calculated elec-tron magnetic moment written in the form (7) is thusestablished. V. DISCUSSION AND CONCLUSION
Geometric models of electron have attracted atten-tion of many physicists in the past hundred years: theirpresent status remains tentaive and speculative. In spiteof this there are at least two strong reasons to continueefforts in this direction. First, geometry and topologyas foundations of physics has tremendous philosophicalappeal. Unfortunately there is no breakthrough in su-perstrings and M-theory in this direction; there is a needfor radical alternative. Secondly the great advances inmodern quantum field theories have so far failed to ad-dress the conceptual issues raised by eminent founders ofQED, e. g. Dirac [25] and Pauli [26]. Candid remarksin the concluding part of Pauli’s Nobel Lecture [26] areworth reproducing in this connection: “At the end ofthis lecture I may express my critical opinion, that acorrect theory should neither lead to infinite zero-pointenergies nor to infinite zero charges, that it should notuse mathematical tricks to subtract infinities or singular-ities nor should it invent a ’hypothetical world’ which isonly a mathematical fiction before it is able to formulatethe correct interpretation of the actual world of physics.From the point of view of logic, my report on ’exclusionprinciple and quantum mechanics’ has no conclusion. Ibelieve that it wil only be possible to write the conclu-sion if a theory will be established which will determinethe value of the fine-structure constant and will thus ex-plain the atomistic structure of electricity, which is suchan essential quality of all atomic sources of electric fieldsactually occurring in Nature.”The criticisms of Dirac and Pauli, in my view, mo-tivate us to discover underlying elements of reality inQED rather than its rejection altogether. Alternativeapproach based on geometry and topology has a poten-tial to address fundamental questions. One example thatwe explore in this paper is the geometric interpretationof the electron magnetic moment: it is related with thequestion raised by Feynman at the 1961 Solvay Confer-ence highlighted recently [14, 15]. Feynman’s questionis: do individual terms in the perturbative QED calcu-lations of the magnetic moment have intuitive physicalinterpretation? A vortex in the multi-vortex structureis proposed to correspond to the individual terms in theelectron magnetic moment expression. The utility of thepresent ideas lies in offering a framework for a synthe-sis of geometry-based approach and QED. Burinskii ina series of papers, see references in [13], has attempteddeveloping the Kerr-Newman (KN) geometric model ofelectron [18]. A rotating bubble model as an oblate ellip-soid of dimension λ c and thickness r e is obtained. Theauthor suggests connection between black holes and el-ementary particles, and asserts that extended electronmodel must be taken seriously. Aspects of superstringsand SM are also viewed in the light of electron model.Though some interesting points emerge from Burinskii’scontributions, it is fair to state that his electron modelremains tentative. Further progress in his approach mayneed radically different outlook: incorporating vortex-metric and the angular momentum framework. In fact,the Kerr-Schild form of the KN metric, and emergenceof a vortex in his analysis are attractive features in thisrespect. Regarding black holes the angular momentumperspective proposed in Section III substantially alters itsphysics, and more closely relates it with particle physicsin view of topological quantization a la BWS quantiza-tion of orbifolds.Nonperturbative QFT may get viable framework basedon LFQ methods [23, 24]. In the present context, the cal-culation of electron magnetic moment using LFQ meth-ods hints at internal structure of the electron [14–16]. Anovel development is the basis LFQ (BLFQ) nonpertur-bative approach aimed at QCD; its application to QED[16] has led to a significant result calculating expression(4). This result signifies ’a nontrivial structure of theelectron in QED’ [16]. Authors represent a physical elec-tron by a truncated Fock-sector expansion | e phys > = a | e > + b | eγ > (52)Two-dimensional (2D) harmonic oscillator wavefunctionfor the transverse direction and a plane wave for the lon-gitudinal direction serve as the basis mode function for a Fock particle. HereΦ nm ( ρ, φ ) = Ce imφ ρ | m | e − ρ L | m | n ( ρ ) (53)The mode function (53) has interesting properties: theFourier transform of coordinate and momentum spacewavefunctions have the same structure, it has finite trans-verse size, and it has vortex structure [27]. The first twoproperties have been noted in the BLFQ literature [16],however the last one has not received attention. Its signif-icance has recently been pointed out in [27]. The choiceof Cartesian coordinate system results into 2D harmonicoscillator wavefunction as a product of Hermite-Gaussian(HG) functions: absence of vortex and ill-defined orzero angular momentum characterize this solution. Now,Zhao et al [16] explain that the mode function (53) assuch is not obtained from the field equation, however theplausibility arguments justify 2D oscillator wavefunction.Obviously the BLFQ calculation needs to be carried outfor HG modes and compared with that of (53). We sug-gest the role of vortex in BLFQ deserves investigation. Itmay be pointed out that the paraxial LG mode function(44) differs from (53) due to its z-dependence.The idea that electric charge has mechanical originin the second part of the expression (7) [7] is given asound basis in the present work. Its physical consequencein the context of Wheeler’s quantum geometrodynamics[35] is illuminating. Qualitative considerations on thequantum fluctuation of field and integrated flux throughwormholes leads to the electric charge of the order of √ ¯ hc ∼ e independent of the wormhole size. Sincethis value of charge looks unphysical Wheeler seeks in-terpretation invoking quantum vacuum effects of QED.In contrast, the present geometric interpretation leadsto a charge for the core vortex with spin ¯ h of value g = q e . One may calculate charge radius for g to be λ c . In our geometric picture of the internal structure ofthe electron both charges represent internal vortices. Re-garding electric charge quantization, specific topologicalquantum number l = 1 for the vortex and N as algebraicsum of N vortices gives N e .An important question is to explain half-integral elec-tron spin since l is an integer. In the KN geometric modelLopez [18] simply assumes empirically given value of ¯ h .It could be done here also, however in the vortex model itmust have origin in the nature of the topological obstruc-tion. There is no definite result, however for a clue wesuggest insights based on the topological approach to theproton spin puzzle [45]. Moreover, the vortex structurein the 2D simple harmonic oscillator recently pointed out[27] and the occurrence of half-integral representation ofthe rotation group in 2D oscillator [46] offer a possiblelink to resolve this question.Another issue not addressed in this paper is concernedwith the explanation of electron mass. We have arguedthat mass is a secondary physical attribute of electron,not an intrinsic one. A tentaive suggestion is that it maybe explained in terms of vortex-vortex interaction result-ing into some kind of spiraling vortex characterized bymass parameter. The problem is being further investi-gated.In conclusion, the peculiar combination of the funda-mental constants ( e, ¯ h, c ) in α and the structure of theQED electron magnetic moment decomposition motivateus to propose internal space-time vortex structure of theelectron. A new vortex-metric and delineation of vortexin BLFQ are some of the important contributions of thepresent work. The conceptual framework proposed heremay have significant implications on the recent develop- ments [13–17]. In particular we emphasize that adoptingBWS quantum conditions has two important advantages:counter-intuitive quantum weirdness does not exist, andit is more suitable for a topological approach [47]. Knot-ted structures of the fundamental electron vortex couldbe envisaged to account for higher spin elementary par-ticle spectrum reviving aspects of Wheeler’s wormholes[1, 35] and Kelvin vortex model [28] in the light of thepresent framework in a unifying picture. Acknowledgment
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