arXiv:2009.07077v1 [.gen-ph] 10 Sep 2020 l.Eeto pnaie aual nteDrcequation, Dirac the in naturally arises mod- spin theoretic field magnetic and els. anomalous geometric inspired and also the magneton model. moment Thomson’s Bohr explain J spin, J to with Lateron distributions beginning mass, charge rest electron extended the with a for basis the serve could approach. electron space- new the fun- internal of be the structure to exploring time geometry have that space-time suggest guideline: We physical a damental. the offer of minds could topology great past and the sce- the of this thoughts in In profound envisaged is the open. it believe and or I unsettled arena nario remain ’an contrasting however space-time [1], The of all’ nature space-time the phenomena. as on emergent well viewpoints as as fields/forces appear gauge these may of that consequences is important studies the exam- en- of qubits/quantum One for specu- and tanglement. with, and correspondence related AdS/CFT uncertain ideas ple, the is explored physicists pathways intensely many new by lative: Though felt the is stalemate. of idea of new direction state a of a superstrings necessity into Moreover the entered any, have world. to if particle seem real tenuous, the have with to link Unfor- continue SM, theory. the superstrings superstring e. tunately in i. efforts scheme, described unified enormous gravity this motivate of in exclusion relativity the foun- general and by conceptual SM the the of on empiricism/operationalism. dations drawbacks the known make of the superfluous However LHC SM view-point the at the beyond events theories from unified new/exotic combined for of quest methods the absence and QCD near sector, perturbative electro-weak with in of advances theory devel- gauge the successful the The experimental of decades. precision opment past over withstood out particle has carried of tests SM (SM) The Model Standard physics. the to credence strong itrclycasclmdl feeto eeconcerned were electron of models classical Historically gives LHC at boson Higgs the of discovery Recent eateto hsc,Isiueo cec,BnrsHnuUnive Hindu Banaras Science, of Institute Physics, of Department otiuin ftepeetwr.Frt rtclreview critical a pro First, This work. electron. present the the of of structure contributions internal multi-vortex a ASnmes 42.q,0.0K,11.15.Tk 02.40.Ky, , 04.20.-q m numbers: PACS electron proposed the of mod ramification phy basis wider the its indicating in and structure out vortex vortex-metric, a decomposi t of moment new occurrence magnetic a a Remarkable electron of of the of derivation idea elec origin the the geometric new of of a representation terms to natural in a leads as wormholes geometry Wheeler’s space-time and bridge Rosen nmlu antcmmn n otxsrcueo h elect the of structure vortex and moment magnetic Anomalous h lcrnmgei oetdcmoiincluae nQ in calculated decomposition moment magnetic electron The .INTRODUCTION I. nttt fNtrlPhilosophy Natural of Institute aaaiIndia Varanasi .C Tiwari C. S. antcmmn otescn re in order second the to moment magnetic 6] [2, SM (2). by experimental the the given in value with agreement taking required precise also extremely corrections and show QED, the in account (4) into to corrections order Higher neetnl o h sue au of value assumed the for Interestingly cwne’ E aclto 5 otefis re nthe in order the first constant in the structure fine to levels after, [5] Soon calculation [4]. QED of deuterium Schwinger’s and splitting hydrogen in in hyperfine effect appeared Zeeman the first (1) in magneton 1947 Bohr the from Deviation opr twt htdtrie ntelnmr exper- landmark the [3] in iment determined that with it Compare ue sn n-lcrnqatmccorn[] how- [2], value different cyclotron a quantum gives one-electron ever mea- a moment using magnetic electron sured Free electron. Dirac the ntrso -atr xrsin()ipisthat implies (1) expression g-factor, of terms In n h nrni antcmmn vector moment magnetic component intrinsic the and e scnie h E acltdepeso o the for expression calculated QED the consider us Let rn oceeraiaino hsidea this of realization concrete A tron. incnttt h eodcontribution. second the constitute tion ucinue nBF sas pointed also is BLFQ in used function e oiini one ntoimportant two on founded is position odel. nWy oe feeto,Einstein- electron, of model Weyl on ia nepeaint sals the establish to interpretation sical µ e Di rpsdt aeoii in origin have to proposed is ED = plgcldfc ncylindrical in defect opological g 2 µ 1 = B st,Vrns 205 and 221005, Varanasi rsity, g 2 µ 1+ [1 µ 1 = . = e ′ 01951032)(2) 00115965218073(28) g 2 α = ± . 2 ≈ 019520(3) 001159652200 α = π µ 2 e mc B 1 ¯ h − hc ¯ e . 1+ [1 2 01 (5) 00116 0 aeteatrdexpression altered the gave = . 328478444 ± 2 α π µ (4) ] B ron α = α α π 137 2 µ 1 (6) ] a h z- the has n gets one g for 2 = (1) 2

Does this structure of the QED expression of the anoma- level; curiously the significance of the classical electron 2 lous magnetic moment have deep physical significance? e charge radius re = mc2 is least understood or obscure as Since the empirical number for µe (2) gives a measured h¯ compared to the Compton wavelength λc = mc . Even in quantity, from the operational point of view the ques- QED, quite often, the intuitive picture of quantum vac- tion of the physical significance of individual terms in uum effects is related to the size of the Compton wave- (4) or (6) would seem to be unimportant or meaningless. length. It is easily seen that in an alternative outlook However this question has fascinated many physicists, [11, 12] both length scales arise in the fine structure con- and led to the speculations on the internal structure of stant α = re , and magnetic moment becomes the electron. Comprehensive review [7] shows that gen- λc erally it is believed that these attempts failed to offer viable alternative to QED and SM. Moreover, high en- ergy electron scattering experiments [8] severely limit the e re µ′ = [λ + ] (9) size of the electron to < 10−16cm. It could be argued e 2 c 2π that the failed attempts perhaps indicate that the right conceptual issues were not raised, and new ideas were missing [7]. For example, the quantization of charge in the unit of electronic charge e intrigued Dirac to pro- pose magnetic monopole, however the most fundamental There is a renewed interest in the recent literature [13– question as to the nature and origin of charge has not 18] on the possible internal structure of electron. The been asked [7, 9]. In a radical departure from the clas- present article makes two significant contributions: i) sical models and the QED paradigm we proposed a new it is suggested that a topological defect in a cylindrical spatio-temporal bounded field approach. Weyl geome- space-time geometry is the most natural representation try, topology and deep physical meaning attributed to of electron, and ii) a new metric, termed vortex-metric, the form of fine structure constant α, and the structure is derived and physically interpreted to explain the geo- of the anomalous magnetic moment expression (4) mo- metric origin of the electron anomalous magnetic moment tivated this approach; see the monograph [7]. We have, (7). It is important to note that the metric structure of in a sense, followed the concerns of prominent founders space-time could be obtained either solving a field equa- of QED, notably Dirac and Pauli on the foundations of tion (e. g. Einstein field equation) or by constructing QED to seek alternatives. metric independently of any field equation. We adopt The main ingredients that distinguish our approach the second approach in which black holes do not exist. from the past ones may be summarized as follows. Mass- Since Burinskii [13] underlines the role of black holes in less internal fields with nontrivial topology constitute the elementary particle models a brief digression is made to electron, and the rest mass of the electron is not intrin- re-interpret the standard Schwarzschild metric in our ap- sic but a secondary physical property. proach for the sake of clarity. appears in α with other fundamental constants e and c The paper is organized as follows. In the next sec- in a ratio. Planck constant has the dimension of action but the dimension of angular momentum is also the same tion a critical appraisal on the Weyl model [19–21], the Einstein-Rosen bridge for a charged particle [22], and as that of action. Dimensionless fine structure constant2 is interpreted as a ratio of angular momenta, i. e. e is wormholes [1] is made to gain new physical insights on c the geometric structure of electron. It is concluded that interpreted as angular momentum associated with elec- a topological defect, cylindrical space-time geometry and tron. Remarkably, the anomalous magnetic moment (4) could be re-written as fundamental role of angular momentum have to be the main ingredients for building electron model. A brief di- e ¯h f gression is made in Section III to revisit the Schwarzschild µ′ = [ + ] (7) e mc 2 2 metric in the present context. A new metric possessing multi-vortex structure is derived in Section IV. Plausible e2 arguments are presented to relate this metric with the f = (8) internal space-time structure of the electron. In the last 2πc section the proposed new conceptual framework is put Logically the bracketted term in (7) should correspond in the perspective of the Kerr-Newman geometric mod- to the angular momentum. We are led to the hypothesis els [13, 18] and recent light-front quantization methods f that electron spin has a fractional part 2 , if expressed [14–17, 23, 24]. In this discussion we are guided by the in units of Planck constant, and the electron charge has critique on QED by Dirac [25], Pauli [26] and Feynman a mechanical origin as some kind of rotation. For fur- [14]. The occurrence of vortex-like structure in the basis ther elucidation of this hypothesis see [9, 10], and vortex light-front quantization (BLFQ) [16] is suggested based model of the spatio-temporal bounded internal fields is on the recent analysis of the Landau problem [27]. The developed in [11, 12]. significance of vortex as a topological defect and Kelvin’s Physical interpretation of one-particle Dirac equation vortex atom model [28] show that vortices may serve the continues to be of interest primarily at the conceptual basis for explaining elementary particle spectrum. 3

II. GEOMETRIC MODEL OF ELECTRON: Weyl also investigates cylindrical geometry and finds CHARGE AND MASS that though the electrostatic potential differs from that of the Maxwell theory, the difference is appreciable only , e and mass, m were the only known in the internal space of the electron where the dimension physical attributes of the electron in the early decades becomes of the order of the gravitational radius of the −33 after its discovery in 1897. Understandably the physi- charge eg = 10 cm. cists during that period, notably Thomson, Poincare, Einstein-Rosen [22] assert that the field singularities Lorentz and Abraham sought electrodynamical models are unphysical, and must be avoided as a fundamental of the electron [7]. Geometrization of gravitation in gen- principle. Towards this aim the authors are prepared to eral relativity of Einstein inspired early geometric models modify the field equation. A coordinate transformation of electron. At this point we must emphasize that the is used to obtain regular solutions. For the charged par- geometry in the modern gauge field theories and SM be- ticle the new metric is obtained such that frn in (10) is longs to the postulated internal spaces different than the replaced by physical space-time geometry [29]. In my opinion [9] the 2 following fundamental question has not been addressed 2rg eg fer =1 (15) in the physics literature: What is the physical origin of − r − 2r2 electric charge? For example, in QED local U(1) gauge Note the sign change in the last term of (15) as compared invariance leads to a conserved Noether current, and the to (11) that arises due to the modified field equation. Noether charge is a Hermitian operator and the quantum Since mass and charge occur independently in the metric vacuum is also gauge invariant. This definition amounts tensor, the authors argue that one could set m = 0. Thus to just a formal construct in the hypothetical internal one may envisage a massless electron model. Assuming space. a coordinate transformation replacing r by u A fresh outlook on the classical models of electron [19, 22] is presented in this section. The first unified the- 2 eg ory of gravitation and electromagnetism is due to Weyl u2 = r2 (16) 2 generalizing the metric Riemann space postulating non- − compact homothetic gauge transformation in addition to the singularity-free space having two sheets results, and the general coordinate transformations of general rela- the electron is represented by a bridge between them. tivity. In the monograph [19] Weyl influenced by Mie’s Now returning to the Weyl unified theory we discuss theory discusses electron model in Section 32 based on its relatively lesser appreciated part: geometric model of Einstein-Maxwell theory, and in Section 36 based on his electron discussed in Section 36 of [19]. Two markedly unified theory. On the other hand, Einstein and Rosen distinct features characterize this model. The cosmologi- in 1935 argue that field singularities are unacceptable, cal constant dependent on the electromagnetic potentials and present an alternative approach discussing electron arises naturally in the field equation. And, the conserved as well [22]. electric current density also depends on the electromag- Weyl considers the Reissner-Nordstroem solution of netic potentials. In Weyl geometry, a gauge connection µ the Einstein-Maxwell equations for a static charged aµ as a linear 1-form aµdx gives a distance curvature sphere 2 2 2 −1 2 2 2 fµν = ∂µaν ∂ν aµ (17) ds = frnc dt + f dr + r dΩ (10) − − rn Weyl assumes e in e.s.u. as a definite unit of electric- 2 2rg eg ity interpreting eaµ as the electromagnetic potential Aµ. frn =1 + (11) − r r2 The electric current density

2 2 2 2 2 J √ g e a (18) dΩ = r (dθ + sin θdφ ) (12) ∝ − Here the gravitational radius of the mass m of the elec- The physical interpretation of electron is speculative and tron is intriguing. Electron is represented by a singularity canal. Charge and mass characterize singularities and remain Gm constant along the canal. However the world-direction of rg = 2 (13) c the canal is determined from the Einstein-Lorentz equa- and the gravitational radius of the charge is tion, i. e. the geodesic equation generalizing the Newton- Lorentz equation. Weyl dwells on the philosophical ar- √Ge guments to understand diffused charge throughout the eg = 2 (14) c Universe and the sign of electric charge possibly having Physical interpretation demands that electron is a real origin in the arrow of time: past to future ordering. field singularity and the mass of the electron is continu- To put the import of these classical geometric mod- ously distributed in the Universe. However, there is no els in perspective it may be remarked that there exists analog of the electron charge radius, re. enormous literature on the singularities specially once 4 the idea of black hole physics became acceptable among mental tests of general relativity are based mainly on the physicists. On the Reissner-Nordstroem solution (10) Schwarzschild metric [30]. a nice discussion given in [30] shows that the singular- The question is: How to get a topological perspective? ity at r = 0 cannot be removed, however other singu- Nontrivial topology of spacetime finds beautiful expres- larities are removable by appropriate coordinate trans- sion in the concept of wormholes [1]. Authors discuss formations, and a maximally extended manifold is ob- Reissner-Nordstroem metric and Einstein-Rosen bridge tained; see Penrose diagram on page 158 of [30]. Re- among several topics in this paper. Wheeler speculates garding the original Weyl theory [19] it is known that on a model of electron as a collective excitation of the he himself abandoned it, and entirely different version of spacetime foam in quantum geometrodynamics [35]. It gauge symmetry occurs in the modern developments [31]. may, however be asked whether quantum theory is neces- Dirac in 1973 revived Weyl geometry for its ’simplicity sary in a topological approach. Wheeler recognized that and beauty’ essentially with the motivation of developing this theory [1, 35] was unable to explain ’the world of his belief in the large number hypothesis [32]. My own particle physics’. In spite of great advances in topologi- interest in the Weyl-Dirac theory has been to understand cal quantum field theories since then the progress on the the electron structure [7, 9, 10, 21]. issue raised by Wheeler, i. e. to make contact with ob- A fresh outlook on the speculations of Weyl [19] and served elementary particles has remained unsatisfactory. Einstein-Rosen [22] leads to radically new ideas as we Could it be due to the belief in the fundamental role discuss below. First let it be noted that the nature of of quantum theory? In a new approach Post abandons electric charge as such remains obscure in these works. quantum theory altogether and puts forward the concept However new light is thrown on the electron mass. To of quantum cohomology [36]. My own conceptual pic- understand it we recall an important result in the conven- ture of space-time departs radically from the past ideas; tional approach [33] treating electron as a charged sphere it is elaborated in a monograph [37]. Here an attempt is and assuming Reissner-Nordstroem exterior metric. Au- made to apply this picture to the problem of electron. thors find that the application of the junction conditions We postulate electron to be a topological defect, on the boundary imply negative mass density of electron. namely a vortex, and accord fundamental significance to 2 The argument runs as follows. For large r the third term the angular momentum e than the electric charge. Thus in the expression (11) is negligible compared to the sec- c 2 a singularity canal in Weyl theory [19] is re-interpreted as e g 2rg a vortex. The second proposition is also supported from ond term. The value of ( r2 r ) at the assumed bound- −16 −−36 ary r = 10 cm is 2 10 . Thus frn > 1, and the some basic considerations. The most remarkable fact, junction condition shows∼ × that the mass density for some not commonly recognized, regarding the unit of electric- value of r in the interior has to be negative. This result ity e is that it can be factored out from the Maxwell puts a question mark on the validity of the singularity equations. As a consequence the physical description is theorems where positivity of mass plays a crucial role indistinguishable for the Maxwell field Fµν from that of Fµν [30]. Curiously in the Einstein-Rosen model [21] the pos- e . However in the equation of motion of a charged sibility of massless charge arises as an interesting idea. In particle factorization results into an irremovable factor Weyl geometry [19] null vectors or gauge-invariant zero e2 in the Lorentz force expression as it does not cancel. length have a unique place: the proposition of a massless Similarly the expression for the generalized momentum electron seems quite natural [21]. Thus it may be argued becomes that mass is not an intrinsic physical attribute of elec- 2 tron. The origin of observed mass, in that case, needs e e pµ Aµ pµ aµ (20) a different explanation than what was attempted in the − c → − c past, namely the electrodynamical models. Note that in the geometric theory of Weyl aµ and fµν On the nature of singularities discussed in [19, 22] topo- −1 −2 logical perspective could be another line of thought. Ein- have the dimensions of (length) and (length) re- stein firmly believed that singularities in field theories spectively. Following the conventional approach Weyl were physically unacceptable see, [34]. He termed the multiplies them by e to get the electromagnetic quan- tities. In contrast to Weyl, in the light of the preceding Einstein field equation incomplete, and suggested provi- 2 arguments we suggest multiplication by e . In general, a sional role to the energy-momentum tensor Tµν on the c right hand side of the equation suitable angular momentum unit should be used to make transformation from geometry to physics. 8πG G = T (19) µν c4 µν Obviously Einstein-Rosen arguments [22] are in confor- III. SCHWARZSCHILD METRIC: NEW INTERPRETATION mity with this belief. A critique [34] on the foundations of the Einstein field equation (19) shows that geometriza- tion without the field equation is logically admissible. Let us revisit gravitation in this approach. The most Note that the metric tensor, and the geodesic equation studied Schwarzschild metric is spherically symmetric are sufficient for this purpose. Moreover crucial experi- static solution of the vacuum Einstein field equation. The 5

Schwarzschild metric is given by form (10) replacing frn Recall that in the standard black hole physics instead of by fs the condition (24) one obtains event horizon for any mass defined by the Schwarzschild radius 2rg fs =1 (21) 2GM − r r = (26) S c2 Important experimental tests of classical general relativ- ity are based on this metric, and mathematical theory of whereas the singularity defined by (25) occurs at a def- black holes also evolved from its study [30]. The sin- inite Planck scale 1019GeV. In this re-interpretation gularity at r = 0 cannot be removed but the second the problem of negative∼ mass density [33] is rendered a one at r = 2rg can be removed by appropriate coordi- nonissue. nate transformation, for example, the advanced or the In Cosmological models the incorporation of Mach retarded null coordinates in the Eddington-Finkelstein principle has attracted attention of some physicists; form of the metric. Two analytic manifolds are separated Brans and Dicke [39] discuss in detail different versions by the surface at r =2rg. The Kruskal construction and of this principle, and relate it with the following the Penrose diagram are nicely explained in the text [30]. GM There exist several derivations of the Schwarzschild Universe 1 (27) metric in the literature, that given by Weyl in Section 31 Rc2 ∼ of [19] is an elegant one. The crucial part in all deriva- Here MUniverse is the finite mass of the visible Universe tions is common: the identification of the integration con- and R is its radius. Authors provide a simple argument stant is based on the correspondence with the Newtonian for the derivation of (27). Newton’s theory gives acceler- theory of gravitation giving expression (13). Note that ation due to gravity at a distance r from, let us say the the integration constant has a unit of length in the so- GMSun Sun, r2 , and from dimensional considerations one lution of the vacuum Einstein field equation. Eddington 2 MSunRc can construct acceleration to be 2 ; equating the makes a perceptive remark in a footnote on page 87 [38] MUniverser regarding the objections for using unit of length for the two the relation (27) follows immediately. In the angular gravitational mass. Moreover, the interpretation of mass momentum approach, the relation (27) is interpreted as as a source in the metric also does not seem to have a the upper limit on L(r) leading to the maximum value solid foundation [34]. Logically there is no objection if 2 GMUniverse we relate the integration constant with the unit of angu- L(r)max = (28) lar momentum. Could we attribute physical significance c to the angular moentum? The suggested role of angular The ratio between the maximum and minimum values of momentum would seem almost heretical due to spherical L(r) turns out to be a large number 10120 assuming syymetry and static nature of the metric. Nevertheless the rough estimates of the mass and∼ the radius of the let us proceed to explore hidden angular momentum in visible Universe. Further discussion on the large scale the metric. structure of the Universe is deferred to a separate paper One can construct a quantity having the angular mo- since the focus here is on the electron structure. mentum dimension using purely dimensional arguments to get IV. ELECTRON STRUCTURE: GEOMETRY G 2 AND TOPOLOGY L = M (22) g c

Defining L(r)= cr, the expression for fs becomes Let us briefly mention the salient features of quantum M geometrodynamics [35]. Wheeler’s qualitative picture of 2Lg the gravitational field fluctuations is in analogy to QED fs =1 (23) − L(r) vacuum fluctuations. Feynman’s path integral approach is adopted in which the action integral in the phase expo- In this setting the problem of singularity acquires an en- nent uses the Planck constant as a quantum of action fol- tirely different version. The angular momentum L(r) can lowing the standard QFT practice. Virtual particle pairs be assigned a limiting minimum value, let us assume it to having charge 12e and mass of the order of Planck be equal to the Planck constant ¯h. Thus the singularity mass are expected∼ to be created by vacuum fluctuations. at r = 0 is rendered superfluous. On the other hand, at However such particles cannot be identified with the elec- tron or any observed elementary particle in nature. How- L(r)=2Lg (24) ever drawing analogy with QED renormalization method the singularity does exist. The limit on L(r) being ¯h one ingenious arguments are put forward by Wheeler to re- gets mass of the order of Planck mass interpret the virtual particles in the wormhole picture. Departing from Wheeler’s approach we propose to in- M 2 corporate topology of space-time at a fundamental level. 2 = P (25) M 2 For this purpose, quantum theory/QED paradigm is 6 shifted to the old quantum theory. To appreciate the the electron model: a possible propagating vortex is sug- significance of the old quantum rule of Bohr on angular gested for this purpose. The crucial question is: How do momentum quantization we refer to a short historical ac- we get it in the metric tensor for the geometric model? count given in [40], and for its relationship with topology It is obvious that the cylindrical metric rather than the to the Post’s monograph [36]. Bohr-Wilson-Sommerfeld spherical Reissner-Nordstroem metric has to be consid- (BWS) quantization requires nontrivial topology; Post ered. Unfortunately the known Weyl metric [19] did not highlights an important contribution of Einstein in this succeed as electron model. Moreover, the problem of log- connection [36, 41]. Einstein replaces Bohr’s circular or- arithmic divergence at large radial distance for a cylin- bits by torus, and the orbital manifold (orbifold) signifies drical system makes physical interpretation very difficult. topological obstruction. Orbifolds have de Rham coho- For the envisaged vortex we adopt an alternative method mology ramifications [36, 41, 42]: the principal result of proposed in [34]. In this approach massless scalar field Φ this work is that physical BWS quantum conditions on satisfies wave equation in flat spacetime with the metric angular momentum have natural mathematical counter- ηµν part in the form of de Rham periods for topological de- µ fects. Application of this idea can be found in the mono- ∂µ∂ Φ=0 (34) graph [36] and for topological photon in [43]. Thus the fundamental significance accorded to Planck constant as and assumed to consist of two scalars a quantum of angular momentum becomes logically jus- tified. Φ= F Ψ (35) Admittedly introducing ¯h in the Schwarzschild met- such that ∂ Ψ is a null vector ric is heuristic, however it can be pursued further qual- µ itatively for other metrics. Reissner-Nordstroem, and ∂ Ψ∂µΨ=0 (36) Einstein-Rosen metrics become µ e The choice for a massless scalar field as a fundamental 2Lg α frn =1 [1 ] (29) field variable is primarily based on the guiding principles − ¯h − 2 of simplicity and symmetry. The proposition (35) gives a bi-scalar structure to the field Φ necessary to construct e 2Lg α a nontrivial metric. Now the metric tensor is defined in fer =1 [1 + ] (30) − ¯h 4 the Kerr-Schild form where we define gµν = ηµν + F ∂µΨ∂ν Ψ (37) Gm2 Le = (31) Note that ∂µΨ is also a null vector with respect to gµν . g c Earlier we have derived some known metrics, for example, Vaidya metric, the Schwarzschild metric, and Brinkman- Apart from the difference in the numerical factor, i. e. 4 Robinson metric using this approach [34]. instead of 2π formal resemblence of the bracketted term New solutions in cylindrical system (ρ,φ,z) with the in (30) with the anomalous part in the magnetic moment aim of realizing vortex are presented here. Assuming Ψ expression (4) seems curious. Does it have some deep to be a function of (t, z) only the null condition is satisfied significance? Let us explore this question. The numeri- Le for functions Ψ(z ct). We assume simplest form for a e g ± cal value of Lg is a very small number; the ratio h¯ is of propagating field the order of 10−44. Note that the electron charge radius and the Compton wavelength can be defined using angu- Ψ= ei(kz−ωt) (38) 2 e lar momenta c = mcre andh ¯ = mcλc respectively. A e Assumed time-independent F (ρ,φ,z) the wave equation length analogous to them using Lg can be defined (34) becomes e Lg λG = (32) ∂2F 1 ∂F 1 ∂2F ∂F ∂2F mc + + +2ik + = 0 (39) ∂ρ2 ρ ∂ρ ρ2 ∂φ2 ∂z ∂z2 Since re = αλc we anticipate The usual azimuthal-dependence eilφ reduces Eq.(39) to N λG = α λc (33) ∂2F 1 ∂F l2 ∂F ∂2F −11 + F +2ik + = 0 (40) for some integer N; in numbers λc 10 cm and λG ∂ρ2 ρ ∂ρ − ρ2 ∂z ∂z2 10−55cm. Following geometric picture∼ presents itself:∼ a core sphere of radius λc surrounded by concentric thin To solve Eq.(40) we make a simplification that F is not n spherical shells of width α λc; n =1, 2, ....N. z-dependent. In this case the solution is given by The concentric spherical shell model shows that mul- tiple geometric sub-structures have to be explored for F = ρl eilφ (41) 7 and the line element assumes the form Lµν = ∂µvν ∂ν vµ = ∂µF ∂ν Ψ ∂ν F ∂µΨ (47) − − 2 2 2 2 2 2 2 ∂Ψ 2 2 ds = c dt +dρ +ρ dφ +dz +F (ρ, φ)( ) (cdt dz) The antisymmetric tensor Lµν has formal similarity with − ∂z − the electromagnetic field tensor for a specific choice of A (42) µ having the form (46). To get its physical interpretation The solution (41) does represent a vortex: phase singu- it is instructive to make explicit calculation of its com- larity at ρ = 0. Unfortunately the divergence as ρ is ponents: L and L ; i, j = 1, 2, 3 are nonvashing. Of worse than the logarithmic divergence making this→ metric ∞ 0i ij particular interest is the component L physically unacceptable. 23 Retaining z-dependence of F in Eq.(40), and in anal- L23 = lkF Ψ (48) ogy to the paraxial approximation in optics [44] neglect- − 2 ∂ F ∂F ∂ ing ∂z2 compared to ∂z Eq.(40) becomes Here we take ∂2 = ∂φ . We also calculate the nonvanish- ing components of vµ ∂2F 1 ∂F l2 ∂F + F +2ik = 0 (43) ∂ρ2 ρ ∂ρ − ρ2 ∂z iω v = F Ψ (49) 0 c This equation has the well-known Laguerre-Gaussian − (LG) mode solution l v3 = ikF Ψ (50) F (ρ,φ,z)= LGp (44) An interesting quantity is the following ratio Here azimuthal index l, and the index p of the associated

Laguerre polynomial characterize the vortex and the con- L23/v0 = L23/v3 = l (51) centric rings in the transverse profile of the LG mode. | | | | Two of the remarkable properties of LG modes noted Multiplication by appropriate angular momentum unit in in optics literature [44] are the z-dependent radius of the (51) provides physical interpretation of this ratio as quna- beam, and the topological quantum number l. Phase sin- tized angular momentum. Note that the ratio L02/v0 = gularity has a topological charge 1 ldφ. In the present l. In our approach we have two fundamental units| ¯h and| 2π 2 case, the ansatz (37) provides a geometricH significance for e c , therefore, the vorticity of of two different vortices the LG mode function in a new line element p =0, l = 1 and p =1, l = 1 are suggested to be quan- ∂Ψ tized in these units respectively. A qualitative geometric ds2 = c2dt2 +dρ2 +ρ2dφ2 +dz2 +LGl ( )2(cdt dz)2 − p ∂z − model of electron inspired by the QED calculated elec- (45) tron magnetic moment written in the form (7) is thus Note that the metric is not complex; it is understood that established. following the convenient practice only complex represen- tation is used for the solution of the wave equation. Could we visualize internal structure of the electron V. DISCUSSION AND CONCLUSION geometrically using this metric? Qualitative interest- ing correspondence follows immediately. The metric Geometric models of electron have attracted atten- for l = 1,p = 0 shows vortex structure of the form tion of many physicists in the past hundred years: their 2 ρ 1 2 − 2 iφ present status remains tentaive and speculative. In spite ρL0(ρ ) e w e . Tentative identification of w λc makes this vortex as a core vortex for the electron.→ The of this there are at least two strong reasons to continue concentric ring for p = 1 may be identified with the sec- efforts in this direction. First, geometry and topology as foundations of physics has tremendous philosophical ond vortex having dimension equal to re. This picture nicely fits the two-vortex Dirac electron model [12]. The appeal. Unfortunately there is no breakthrough in su- multi-vortex structure can be envisaged for the geome- perstrings and M-theory in this direction; there is a need try of the metric (45). To associate angular momentum for radical alternative. Secondly the great advances in with the vortex we cannot adopt the optics method: or- modern quantum field theories have so far failed to ad- bital angular momentum per photon l¯h is obtained using dress the conceptual issues raised by eminent founders of Poynting vector for linear momentum density and cal- QED, e. g. Dirac [25] and Pauli [26]. Candid remarks culating the orbital angular momentum from it [44]. In in the concluding part of Pauli’s Nobel Lecture [26] are the geometric formulation we have to seek a new idea. worth reproducing in this connection: “At the end of The geometric quantities like the curvature tensors can this lecture I may express my critical opinion, that a be calculated following lengthy but straightforward tech- correct theory should neither lead to infinite zero-point nical method [38]. However these are of no interest for nor to infinite zero charges, that it should not angular momentum. In view of the structure of the Kerr- use mathematical tricks to subtract infinities or singular- Schild form (37) a new geometric quantity can be natu- ities nor should it invent a ’hypothetical world’ which is rally defined only a mathematical fiction before it is able to formulate the correct interpretation of the actual world of physics. vµ = F ∂µΨ (46) From the point of view of logic, my report on ’exclusion 8 principle and quantum mechanics’ has no conclusion. I Fock particle. Here believe that it wil only be possible to write the conclu- 2 ρ sion if a theory will be established which will determine Φ (ρ, φ)= Ceimφρ|m|e− 2 L|m|(ρ2) (53) the value of the fine-structure constant and will thus ex- nm n plain the atomistic structure of electricity, which is such The mode function (53) has interesting properties: the an essential quality of all atomic sources of electric fields Fourier transform of coordinate and momentum space actually occurring in Nature.” wavefunctions have the same structure, it has finite trans- The criticisms of Dirac and Pauli, in my view, mo- verse size, and it has vortex structure [27]. The first two tivate us to discover underlying elements of reality in properties have been noted in the BLFQ literature [16], QED rather than its rejection altogether. Alternative however the last one has not received attention. Its signif- approach based on geometry and topology has a poten- icance has recently been pointed out in [27]. The choice tial to address fundamental questions. One example that of Cartesian coordinate system results into 2D harmonic we explore in this paper is the geometric interpretation oscillator wavefunction as a product of Hermite-Gaussian of the electron magnetic moment: it is related with the (HG) functions: absence of vortex and ill-defined or question raised by Feynman at the 1961 Solvay Confer- zero angular momentum characterize this solution. Now, ence highlighted recently [14, 15]. Feynman’s question Zhao et al [16] explain that the mode function (53) as is: do individual terms in the perturbative QED calcu- such is not obtained from the field equation, however the lations of the magnetic moment have intuitive physical plausibility arguments justify 2D oscillator wavefunction. interpretation? A vortex in the multi-vortex structure Obviously the BLFQ calculation needs to be carried out is proposed to correspond to the individual terms in the for HG modes and compared with that of (53). We sug- electron magnetic moment expression. The utility of the gest the role of vortex in BLFQ deserves investigation. It present ideas lies in offering a framework for a synthe- may be pointed out that the paraxial LG mode function sis of geometry-based approach and QED. Burinskii in (44) differs from (53) due to its z-dependence. a series of papers, see references in [13], has attempted The idea that electric charge has mechanical origin developing the Kerr-Newman (KN) geometric model of in the second part of the expression (7) [7] is given a electron [18]. A rotating bubble model as an oblate ellip- sound basis in the present work. Its physical consequence soid of dimension λ and thickness r is obtained. The c e in the context of Wheeler’s quantum geometrodynamics author suggests connection between black holes and el- [35] is illuminating. Qualitative considerations on the ementary particles, and asserts that extended electron quantum fluctuation of field and integrated flux through model must be taken seriously. Aspects of superstrings wormholes leads to the electric charge of the order of and SM are also viewed in the light of electron model. √¯hc 12e independent of the wormhole size. Since Though some interesting points emerge from Burinskii’s this value∼ of charge looks unphysical Wheeler seeks in- contributions, it is fair to state that his electron model terpretation invoking quantum vacuum effects of QED. remains tentative. Further progress in his approach may In contrast, the present geometric interpretation leads need radically different outlook: incorporating vortex- to a charge for the core vortex with spin h¯ of value metric and the angular momentum framework. In fact, 2 137 the Kerr-Schild form of the KN metric, and emergence g = q 2 e. One may calculate charge radius for g to be of a vortex in his analysis are attractive features in this λc 2 . In our geometric picture of the internal structure of respect. Regarding black holes the angular momentum the electron both charges represent internal vortices. Re- perspective proposed in Section III substantially alters its garding electric charge quantization, specific topological physics, and more closely relates it with particle physics quantum number l = 1 for the vortex and N as algebraic in view of topological quantization a la BWS quantiza- sum of N vortices gives Ne. tion of orbifolds. An important question is to explain half-integral elec- Nonperturbative QFT may get viable framework based tron spin since l is an integer. In the KN geometric model on LFQ methods [23, 24]. In the present context, the cal- Lopez [18] simply assumes empirically given value of h¯ . culation of electron magnetic moment using LFQ meth- 2 It could be done here also, however in the vortex model it ods hints at internal structure of the electron [14–16]. A must have origin in the nature of the topological obstruc- novel development is the basis LFQ (BLFQ) nonpertur- tion. There is no definite result, however for a clue we bative approach aimed at QCD; its application to QED suggest insights based on the topological approach to the [16] has led to a significant result calculating expression proton spin puzzle [45]. Moreover, the vortex structure (4). This result signifies ’a nontrivial structure of the in the 2D simple harmonic oscillator recently pointed out electron in QED’ [16]. Authors represent a physical elec- [27] and the occurrence of half-integral representation of tron by a truncated Fock-sector expansion the rotation group in 2D oscillator [46] offer a possible link to resolve this question. ephys >= a e> +b eγ > (52) | | | Another issue not addressed in this paper is concerned Two-dimensional (2D) harmonic oscillator wavefunction with the explanation of electron mass. We have argued for the transverse direction and a plane wave for the lon- that mass is a secondary physical attribute of electron, gitudinal direction serve as the basis mode function for a not an intrinsic one. A tentaive suggestion is that it may 9 be explained in terms of vortex-vortex interaction result- ments [13–17]. In particular we emphasize that adopting ing into some kind of spiraling vortex characterized by BWS quantum conditions has two important advantages: mass parameter. The problem is being further investi- counter-intuitive quantum weirdness does not exist, and gated. it is more suitable for a topological approach [47]. Knot- In conclusion, the peculiar combination of the funda- ted structures of the fundamental electron vortex could mental constants (e, ¯h,c) in α and the structure of the be envisaged to account for higher spin elementary par- QED electron magnetic moment decomposition motivate ticle spectrum reviving aspects of Wheeler’s wormholes us to propose internal space-time vortex structure of the [1, 35] and Kelvin vortex model [28] in the light of the electron. A new vortex-metric and delineation of vortex present framework in a unifying picture. in BLFQ are some of the important contributions of the Acknowledgment present work. The conceptual framework proposed here I thank the referees for useful comments that helped may have significant implications on the recent develop- improve the presentation of the manuscript.

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