aa r X i v : . [ phy s i c s . g e n - ph ] F e b Electric Charge, Fractional Spin and Flux
S. C. TiwariInstitute of Natural Philosophy1 Kusum Kutir, MahamanapuriVaranasi 221005, India
Ratio of electron charge radius and Compton wavelength of electron is known to be equal to thedimensionless electromagnetic coupling constant e / ¯ hc . It is pointed out that the coupling constanthas two alternative interpretations: as a ratio of two angular momenta since Planck constant has thedimension of angular momentum, and two flux quanta e and hc/e . We argue that it has deep physicalsignificance such that the electronic charge becomes flus itself and at a fundamental level fractionalspin of quantized vortex. A unified perspective of the three interpretations of the coupling constantis presented invoking the new interpretation of the magnetic moment of the electron comprising ofthree terms. A critical discussion on the past attempts to give fundamental importance to magnetismand flux quantum is given and the implication on the unification quest of our ideas is outlined. PACS numbers:
1. INTRODUCTIONDirac’s relativistic equation of electron explains spinand magnetic moment of electron in a natural way [1],and re-interpretation of negative energy states as anti-particle led to the prediction of positron discovered sub-sequently. It was a great triumph of Dirac’s theory.However as noted by Dirac [2] it was in fact Weyl whofirst gave a definite statement on the mass of the anti-particle equal to the mass of the electron while Dirac hadoriginally identified that as proton. Spinning electronwas first conceived by M. Abraham and a comprehen-sive theory of electron having spin h/ π = ¯ h/
2, where h is Planck constant was given by Thomas [3]. Thomasnotes that Compton had suggested quantized spin in 1921and Uhlenbeck and Goudsmit explained anomalous Zee-man effect using that suggestion. In contrast let us ask:What is electric charge? Is it not strange that even aftermore than a century of the discovery of the electron thisfundamental question has not been raised? In a mono-graph devoted to electron [4] I have reviewed numerousattempts to model electron and various equations of mo-tion proposed in the literature to get rid off ambiguitiesand infinities. A question that has now become stan-dard had been first asked by Dirac [5], namely the originof charge quantization. It is instructive to quote Dirac:’This smallest charge is known to exist experimentallyand to have value e given approximately by hc/e .......The theory of this paper ....... is found when worked outto give a connection between the smallest electric chargeand the smallest magnetic pole.’ Magnetic monopole andcharge quantization have been of renewed interest in thelight of the standard model of particle physics and be-yond.It is well known that one-particle Dirac equation isnot satisfactory, and quantum electrodynamics (QED)also has foundational problems [2]. Here we mentiontwo aspects of Dirac equation. Though spin half emergesbeautifully, the magnetic moment has to be equal to theBohr magneton µ B whereas the experimental value de- viates from it by an anomalous term. The QED calcu-lated value in the power of the fine structure constant α = e / ¯ hc agrees extremely well with the empirical dataand neglecting higher order terms is given by µ e = µ B [1 + α π − . απ ] (1) µ B = e ¯ h mc (2)It is only for the electron that anomalous part is a smallcorrection, for proton that is also a spin half particle themagnetic moment predicted by Dirac equation would be µ B m/m p , where m p is the mass of the proton. However itis in gross error from its actual value. A nice expositionon the magnetic moment of elementary particles withextensive list of references can be found in [6]. ThusDirac equation cannot be accepted as a general equationfor any spin half charged particleThe second issue is that of nonlocality: though Diracequation represents point particle a length scale of theorder of Compton wavelength λ c = ¯ h/mc becomes neces-sary for physical interpretation. In some of the literaturethe Compton wavelength is taken to be h/mc . Dirac him-self [1] invoked high frequency oscillation (Schroedinger’szitterbewegung) in order to understand the motion ofa free electron with the velocity of light predicted bythe theory. It could be argued that localization of elec-tron wavepacket in a spatial region of less than λ c wouldnecessarily result into the interference between negativeand positive energy states and the picture of one-particlebreaks down. Curiously a length scale of the order ofelectron charge radius r e = e /mc much smaller thanthe Compton wavelength appears at classical level in theelectron theory; there is no explanation for this.Physicists have pondered over the meaning of the di-mensionless coupling constant α , and speculated on thecoincidence that the ratio of electron charge radius andCompton wavelength is equal to α relating it with struc-ture of the electron. Attempts to build models of ele-mentary particles using charge distribution over extendedstructures and also considering the role of magnetic mo-ment have not succeeded. The advent of gauge field the-ories and the Standard Model of particle physics withremarkable successes has almost put a break on suchefforts. Nevertheless there is no denying the fact thatfoundational issues on QED remain unresolved. In thisshort paper my aim is to put forward a new insight onthe meaning of electric charge (in the next section) andbriefly discuss its physical consequences which are likelyto offer a radically new pathway to fundamental physicsin the last section.2. ELECTRIC CHARGE IS FLUXA new ingredient in our model of electron [4] is theproposition that electric charge is a manifestation of frac-tional spin of the order of e /c . In Sections 9.2 and 9.3 ofthe monograph [4] the application of the tentative modelof the electron was discussed in condendnsed matter sys-tems, specially superconductivity and quantum Hall ef-fect. In an otherwise positive review of the book, Post [7]was critical of the fractional charge in Laughlin’s theoryof fractional quantum Hall effect apparently supported inmy work. It has to be clearly stated that in [4] electron isendowed with fractional spin not fractional charge, andthis spin is interpreted to be the origin of charge e .The argument behind this new interpretation camefirst from the observation that the fine structure constantbeing dimensionless immediately leads to the fact that e /c has the dimension of angular momentum since thePlanck constant has this dimension. Does it have deepphysical significance? Further support to this proposi-tion was discussed in [8] noting the remarkable fact thatcharge invariably occurs as e in charge-field interactionif fields are expressed in the units factoring out the chargeunit, and disappears for source free electromagnetic fieldequations. Visualizing pure spacetime fluid as fundamen-tal entity the physically intuitive possible operations arethose of spacetime translation and rotation, therefore itwould be a great advancement towards unification if allinteractions are ultimately caused by rotation and trans-lation. A significant clue to it is offered by Equation (1)rewritten in the form µ e = emc [ h π + e πc ] (3)Above expression could be interpreted to imply that elec-tron has spin angular momentum of e / πc besides theusual half spin. For notational convenience let us de-note e / πc by f . We have elucidated the significance offractional spin in [9], however there has not been furtherprogress in this since linking the fractional spin with thecharge has not been achieved in concrete form.The new insight that makes the model of the electroncomplete to a great extent could be stated in the form offollowing hypotheses.H1: Rest energy mc of electron is purely rotational. Half of it is attributed to spin ¯ h/ f / hc/e and e . The vor-tices have different core radii and the one with flux quan-tum e behaves as a point votrex (or point charge); thuselectric charge is nothing but a flux quantum.We now give plausible arguments to justify the abovehypotheses. It is interesting that in the theory of elec-tron, spin angular momentum is not associated with en-ergy, for example, Dirac on p.266 of [1] states that,’Thespin angular momentum does not give rise to any po-tential energy–’. It is only in the interaction that spineffect manifests, say in Zeeman splitting or the motion ofelectron in spherically symmetric potential. Similar situ-ation prevails in the case of photon where its energy hν isassumed purely kinetic, however we have recently arguedthat half of the energy of photon is due to spin angularmomentum ¯ h of photon [10]. It would be reasonable toassociate rotational energy to electron. Interesting resultfollows based on hypothesis H1. Since fractional spin f / L / I where L is an-gular momentum and I is rotational inertia of the disk M R / h / ma and that for fractional spin it is given by f / mb where a and b are the radii of the respective disks. Here mass foreach disk is assumed to be half of the mass m . Equatingeach of these two with half of the rest energy we get a = ¯ hmc = λ c (4) b = e πmc = r e π (5)Notice that the two lengths associated with the elec-tron arise here in relation to the rotational energy;though we have used a simple picture this result is sig-nificant.Next let us make further analysis of the fine structureconstant: the fine structure constant could be consideredas a ratio of e and flux quantum hc/e thus electric chargeitself has the dimension of a flux. Hypothesis H2 becomesquite natural if this guess has physical content. Let uscalculate the energy for two flux quanta recalling that amagnetic dipole placed in an external magnetic field hasthe energy µB . Since the hypothetical magnetic fieldis internal in the present case it is more appropriate totake half of this energy for electron magnetic moment.Multiplication by area of the disk renders the energy ex-pression in the form of µ Φ where Φ = B × area is flux.For the assumed flux quanta Φ and Φ for electron wehave µ B Φ mc πa µ B α Φ π = mc πb a and b calculated from the rotating diskmodel we arrive at Φ = hc/e (8)Φ = e (9)It is satisfying that the hypotheses H1 and H2 are con-sistent and that gives confidence in the present model.However it has to be pointed out that rotating disk modeland ’magnetic energy’ equal to µB/ µB then with the flux quanta of hc/e and e one can calculate the radii from Equations (6)and (7) respectively. These turn out to be somewhat oddlooking √ λ c and √ r e / π , however for self-consistencyit is possible to adjust the moment of inertia. Since thelength scales (4) and (5) arise naturally in a new light weprefer the previous calculation.The most precise experimental value of anomalouspart in the magnetic moment of electron [12] is equalto 1 . × − where (74) gives the 1-standard-deviation in the last digits. The expression (1)for µ e on the otherhand originates from the perturbativeQED calculations. Why should the individual terms inEq.(1) be given fundamental physical significance? Re-call that the leading term in the magnetic moment i. e. µ B , first predicted by Dirac equation is intimately re-lated with intrinsic spin of electron, and in QED it isthe lowest order term. Thus logically the magnetic mo-ments at one-loop and higher orders could be attributedindependent physical interpretation, and since the firstterm is related with spin we expect higher order termsto be in some way related with internal angular momen-tum or fractional spin of electron. We have already seenthat fractional spin f / λ c for a point charge executing circular zitterbe-wegung could explain the Bohr magneton for electron,however absence of the magnetic moment interaction inhigh energy electron-electron and electron-positron colli-sions indicates that electron has to be viewed as a pointparticle with dimension less than 10 − cm. The dilemmafaced between a point electron and extended structure isobvious from the remarks made in [13].A miniature rotating golf ball model for spinning elec-tron is certainly ruled out, however the present workbrings together the three interpretations that are hiddenin the expression of the fine structure constant: in termsof the ratios of two lengths (classical electron charge ra-dius and Compton wavelength), two angular momenta(fractional spin and spin half) and two flux quanta ( e and hc/e ). This must have a unifying picture. To achieve thislet us consider an equivalent intrinsic spin angular mo-mentum obtained from expression (1) for the magneticmoment comprising of three parts S v = ¯ h f − . f α (10)The magnetic moment expression is now translated toangular momentum, and is proposed to represent angu-lar momenta of three vortices: quantized vortices in therotating spacetime fluid or aether if one is not preju-diced against the usage of this term with the strengths(or circulation) Γ g , Γ e and Γ w for the vortices, let ussay central (C), orbiting (O) and tail (T) respectively.To fix the value of the strengths of the vortices one maydivide S v by Planck constant and obtain dimensionlessnumbers. The sign of the circulation determines the signof the charge: vortex-vortex repel and vortex-antivortexattract each other. In analogy to the electric charge e embodied in the fractional spin f / g and w respectively to the first and the last term of S v defined as follows g πc = ¯ h w πc = 0 . hα π (12)The strength of the vortex C is very large as comparedto O and that of O relative to T. Therefore neglectingthe vortex T for the timebeing, electron is envisaged asthe vortex O rotating around C. Note that O has thespinning motion in a core radius of b = r e / π besidesthe rotation around C. To the outside observer rotat-ing vortex O appears as a source of point charge e : theCoulomb field is a manifestation of the time varying fluxquantum e . Electron has two distinct internal configura-tions O − C + and O − C − where the suffix - or + denotevortex or antivortex. Other configurations O + C + and O + C − represent the positron.It is necessary to clarify at this point that the pre-ceding description deals with the internal structure ofelectron that lies in a 2+1 dimensional plane and thiswhole 3-vortex structure travels along the normal to theplane with light velocity. To put it in the context ofDirac’s theory [1] he obtains the time variation of thevelocity of free electron having a component equal to theconstant value c p/H , where p and H are momentumand kinetic energy Hamiltonian respectively for a rela-tivistic point particle, and an oscillatory part that hasinstanteneous value equal to the velocity of light. Diracargues that the observed motion corresponds to a mea-surement of the average velocity over a time interval verylarge compared to the time scale of the oscillatory mo-tion h/ mc . For a Dirac electron at rest there remainsonly oscillatory motion with light velocity. However inour model electron does not have a rest state in vacuumthough the planar structure could possibly be confinedin a three dimensional spatial region due to some sort of’external potential’ of a surrounding medium. Observedmotion with velocity v less than c corresponds to thecollision-limited average drift motion. The third lengthassociated with electron, namely de Broglie wavelengthbecomes meaningful only in that case, and obviously itis not an intrinsic property of electron. The surroundingmedium that we have invoked arises due to the presenceof the multitude of the structures on the spacetime fluidor the substratum aether. The ubiquitous aether is notunphysical. Note that the whole of the universe is be-lieved to be filled with some kind of fields in the moderntheories both classical and quantum. Since we do notintend to identify the surrounding medium with specificnotions such as quantum vacuum, microwave backgroundradiation or curvature of spacetime having definite theo-retical foundations, we prefer to call it aether.A noteworthy standard result in the radiation theoryof a charged particle is that for a uniform motion there isno radiation even if the charge moves with a relativisticvelocity, however the field pattern surrounding the chargeincreases in the direction perpendicular to the directionof motion as the velocity increases, and becomes confinedin the normal plane for the velocity approaching the ve-locity of light. For a charge at rest the field distributionis spherically symmetric. An accelerated charge radiateselectromagnetic radiation: for a co-linear velocity and ac-celeration there is no radiation emitted along the forwarddirection of motion even if the charge is accelerated to thevelocity near the light velocity. Planar vortex model ofelectron accounts well these features qualitatively in arather obvious manner. Static electron here correspondsto a kind of revolving disk with approximately the di-mension of λ c tracing out a spherical surface; it is dueto the confinement caused by the surrounding medium.The disturbance caused by the vortex O propagates inthe aether establishing isotropic Coulomb field when av-eraged out over the large time scales. Since free electron travels with velocity c , the accelerated motion actuallyrepresents the state when the density of the scatterershas been depleted along certain direction, and the aver-age drift velocity increases. The physical mechanism ofradiation would be the liberation of photons due to col-lision or intense disturbance caused by the vortex O inthe plane of the electron.In our model the circulation of central vortex C de-termines the spin state, up or down, of the electron andthe charge g associated with it is very strong at smalldistances. Spin polarized electrons O − C + and O − C − are markedly different from each other due the sign ofthe charge of C and the nature of vortex-vortex andvortex-antivortex interactions. Further we envisage a 2-vortex structure consisting of C and T only. The pos-sible four states are proposed to be neutrinos and sincethe vortex O is absent these are ‘’electrically neutral’.Thus the two electron neutrinos and two muon neu-trinos are identified with the for composite structures C − T + , C − T − , C + T + , C + T − .3. DISCUSSION AND CONCLUSIONReflecting on the past, the present ideas have to beseen as the continuation of the speculations on geometrybeing fundamental perceived by Riemann and Clifford,and Kelvin’s vortex atom. In Kelvin’s theory knottedvortex tubes of aether represented atoms accounting fortheir stability and variety [14]. Would vortex dynam-ics and construction of knotted structures based on ourmodel explain the existence of a large number of elemen-tary particles and unify strong, weak and electromagneticinteractions? To address this question satisfactorily wemust delineate what is distinctly new in our model, crit-ically evaluate the past failed attempts that gave funda-mental significance to magnetism or magnetic flux quan-tum, and ensure that the established physics is containedin the appropriate domain of the new theory.We take up the last question first. Static Coulomb fieldand radiation find re-interpretation, at the same time theproblem of infinity that plagues point field theory is elim-inated. The asymmetry between the sources for electricand magnetic fields is not of fundamental nature as elec-tric charge itself is a flux quantum: it is only due to thesmall value of this flux quantum e as compared to hc/e ,and the rotatory motion of the vortex O that for largedistances the observed field is electric field. Assumptionof the rotating flux e as a point charge discarding thevortex C and the associated spin and flux quantum hc/e we get the classical picture in which charges and cur-rents are the sources of the electromagnetic fields. Spinof the electron does not play any role either in the de-scription of the current flow or the Lorentz force law.Note that the magnetic moment arises as a secondary ef-fect. Obviously none of the experimental facts would beviolated in our approach. However rather than seekingmagnetic monopole, the elusive object not observed tilldate, here we have electron as a composite particle con-sisting of electric-charge like flux quantum and magnetic-monopole like flux quantum. Further the two electronswith opposite spin have distinct internal structure, there-fore the spin polarization of current carrying electronsshould show up in new electromagnetic phenomena. Re-cent advances and interest in spin-polarized transport ofelectrons and spintronics have led to new effects and in-terpretations of electromagnetism [15–17]. The presentelectron model offers the possibility for new insights inthis field.Though as early as 1917 the role of magnetic energyin the spinning electron model of Abraham was discussed[3, 18], in general magnetic field has been of lesser impor-tance. Barut [19] noted that,’It would have been strangeif Nature provided magnetic forces just to be tiny correc-tions to the building principle of atoms ....’. In Barut’smodel the basic constituents of matter are assumed to bethe stable particles proton, electron, neutrinos and pho-ton, and the only binding force is that of electromagneticorigin. It is shown that magnetic forces between the sta-ble particles become very strong at short distances; thestrong interaction between hadrons is interpreted as adynamical spin-spin and spin-orbit force. Lepton-hadrondistinction is not of significance, and muon is visualizedas a magnetic excitation of electron due to the interactionof the anomalous magnetic moment with its own field.Schwinger in 1969 speculated on a magnetic modelof matter [20]. A new ingredient in his model is themodification of Maxwell equations incorporating Dirac’smonopole, however postulating a new species of parti-cle: dyon. Dyon is a dual charged particle possessingelectric charge with coupling constant α and magneticcharge with coupling constant 4 /α . A tentative the-ory of hadrons is outlined noting that the force betweenmagnetic charges is superstrong in comparison with thestrong nuclear force. Leptons are not composite thoughit is suggested that neutrinos could belong to both leptonand hadron families.Jehle in a remarkable series of papers in 1970s not onlyhighlighted certain fundamental questions in the histor-ical perspective but also formulated a new approach toelectromagnetism and elementary particle physics [21].The standard electromagnetism is built on electric chargeand its dynamics. Jehle puts forward the hypothesis thatquantized flux hc/e is fundamental and the electricityand electric properties are derived from it. A closed fluxloop is an elementary object from which a manifold ofloopforms is constructed. Electron and muon are repre-sented by a single loop. Topology of linked and knottedflux loops is used to interpret quarks and classification ofelementary particles. It seems surprising that even afterthe advent of speculative superstring theory, Jehle’s workhas not received the attention that it deserves.A more radical though tentaive idea is that of quan-tum cohomology due to Post [11]. In his book Post makestwo main contributions: a strong criticism of the ortho-dox Copenhagen interpretation of quantum mechanics,and an alternative topological approach for fundamentalphysics. Unfortunately excessive and repetitive emphasison the first has obscured the novelty of the topological approach. Electromagnetism as metric-free theory, therecognition of flux quantum as de Rham period integral,and the significance of topological torsion in 4-dimensionscomprise Post’s quantum cohomology. To avoid likelyconfusion with the term quantum cohomology, it hasto be emphasized that Post’s idea is entirely differentthan quantum cohomology of superstring literature [22].Electric charge is fundamental in Post’s theory, and a3-dimensional period integral for spin angular momen-tum proposed by Kiehn [23] is a new addition to the wellknown 1-dimensional Aharonov-Bohm flux integral and2-dimensional Ampere-Gauss charge integral. Electronand muon are represented by a trefoil knot and a ’pre-liminary cohomological classification’ for electron, muon,neutrinos, pions and photon is presented.The question arises as to why these attempts did notsucceed. Schwinger in his paper highlights the specu-lative character of his ideas and at one place remarksthat,’However wide of the truth this hypothesis may be,it can serve to bring into better focus the nature of thequest for order and understanding that underlies the ac-tivity of the high-energy physicists’. It is possible thatthe ideas of Barut, Schwinger, Jehle and Post do notbelong to the realm of the laws of Nature or have phys-ical realization. This sort of conclusion would be ratherpremature since the mainstream physicists have not ex-plored these ideas as vigorously as the most successfulstandard theories have been. Nevertheless let us havea critical look if there are weaknesses in these endeav-ours. One drawback common to them except that of thePost’s work is that the new ideas were applied to the par-ticle physics retaining the standard paradigm: the clas-sification scheme based on so the called internal quan-tum numbers, conceptual framework of quantum fieldtheory and quark models. Post argues for an alterna-tive in which quark is not a legitimate object of physicalreality; this, of course, would require tremendous effortto recast enormous knowledge in high energy physics inthe alternative paradigm. Barut and Schwinger do notprobe further if magnetism and magnetic charge couldhave deeper meaning than that following the Maxwell-Lorentz theory of electromagnetism. Jehle does take astep forward replacing the electric charge as fundamen-tal to elementary flux loop as fundamental and also dis-pensing with magnetic monopole. I think there are twodrawbacks in Jehle’s approach that probably limited itsscope. The derivation of electric field from quantizedflux loop involves somewhat artificial introduction of afraction of the quantized flux, denoted by F, since theloopform is assumed to spin at an angular velocity of2 mc / ¯ h that corresponds to the Compton wavelength.Though quantized charge is explained due to the quan-tized flux without postulating magnetic monopole, thereis no explanation as to what charge is. Secondly Jehledrifts away to construct quark model assuming them tobe real physical objects.Regarding quantum cohomology of Post it remainscompletely unexplored and ignored too. In my work[10] a different idea than that of Post for topologicaltorsion has been proposed for a new model of photon.Post treats flux as more fundamental than the magneticmoment, however electric charge is assumed fundamen-tal elementary unit independent of spacetime providedby Nature. Has this rigid assumption on electric chargeblocked deeper insights from quantum cohomology?Now it becomes straightforward to state the new ele-ments in our approach: in contrast to a single flux quan-tum hc/e in Jehle’s model we have two flux quanta suchthat electric charge itself is a quantum of flux, the classi-cal concept of charge in conjunction with flux hc/e couldbe used to derive the notion of magnetic moment but itis not fundamental, and finally the magnetic flux itselfis a derived concept from the vorticity or the circulationof the spacetime fluid or aether vortex. The concept ofelectric charge proposed here is radically different thanthat of Post since spacetime rotation manifests as chargewhile according to Post charge is independent of space-time. Neglecting the vortex T electron is a composite oftwo vortices or two flux quanta - it is akin to Schwinger’sdyon. The concept of composite particles discussed in theliterature on fractional quantum Hall effect should not beconfused with our electron model. It has to be empha-sized that in these theories the flux quantum of a vortexis created by the application of the external magneticfield on a 2-dimensional electron system; in the words ofStormer [24] ’Electrons plus flux quanta can be viewedas new entities, which have come to be called compositeparticles, CPs’.To conclude the paper we briefly outline the physi-cal consequences of the new conceptual framework andsuggest a fresh outlook on the unification quest. Fluidor hydrodynamical approach to electromagnetism in thenineteenth century, and to quantum theory in the 1920sis well known. The present ideas in which electric chargeis given a mechanical interpretation could stimulate re-vival of the fluid dynamical paradigm for fundamentalphysics: in [8] the electromagnetic field tensor has beeninterpreted as angular momentum tensor of photon fluid,the representation of sources in terms of flux integralswould render this description to a completion. Instead ofa point charge what we have is a flux integral for electriccharge, therefore the divergence problem will not arise.Quantized vortices are best treated as topological objectsmaking geometric and topological rendition of the elec-tromagnetic phenomena quite natural.Dirac theory of electron does not explain anomalousmagnetic moment; theoretically it emerges as a QED ef-fect. One could postulte additional term in the Diracequation for electron under external electromagnetic fieldfollowing Pauli to account for the anomalous magneticmoment. However the simplicity gets lost and the QED with this term is problemetic: though Pauli’s term isLorentz covariant and gauge invariant the resulting the-ory is not renormalizable. Could there be a possible mod-ification to Dirac equation that incorporates anomalousmagnetic moment? Our model brings into sharp focustwo length scales associated with the two flux loops inthe electron structure. Unlike Compton wavelength thatemerges in the Dirac equation in various ways, the elec-tron charge radius is absent. We suggest the inclusion offractional spin and this length scale into the Dirac equa-tion such that the rest energy mc is treated as rotationalenergy could prove insightful. Since the sign of the elec-tric charge is related with the circulation of the vortex O,reformulation of Dirac theory in this perspective may ul-timately resolve the issue of negative energy states. In animportant work Gurtler and Hestenes [25] show that theconsistency of Dirac and Schroedinger theories leads tothe conclusion that the Schroedinger equation describes aparticle in a spin eigenstate not a spinless one. Differencebetween charge current and momentum current in Diracequation is also elucidated by the authors. The inconclu-sive ideas of Hestenes could be revisited in the light ofpresent work and also that related with Weyl space [26].If the present ideas have elements of physical realitythe most profound implication would be on the nature offundamental interactions. The expression for S v , Equa-tion(10) consists of three terms and has a sort of univer-sality: only fundamental constants h, c, e appear in thisexpression. Electric charge is a quantized flux of the vor-tex O. Postulating tiny quantized vortices C, O, and Tas fundamental constituents of the elementary particlesthe interpretation of the electric charge is extended to thenew charges g, w given by Equations (11) and (12) respec-tively for the vortices C and T . The coupling constants inanalogy to α are 2 π and 0 . α for these charges. Sincethe strong coupling constant in quantum chromodynam-ics and Fermi’s coupling constant for weak interactionsare energy dependent, it is not sytraightforward to iden-tify g / ¯ hc and w / ¯ hc with them, however the order ofthe strengths and the short range nature indicate thatall the three fundamental interactions are embodied in S v .Electron is a composite structure of all the three vor-tices, thus confined in small spatial regions it could be-come a strongly interacting particle. Neutrinos are com-posite of C and T, and hence possess strong and weak in-teractions. Entangled links and knotted structures builtfrom these basic building blocks represent the whole spec-trum of the elementary particles. Simplicity and intrinsicunity of the present ideas, though speculative, invite at-tention as an alternative to the unification paradigm.The library facility of Banaras Hindu University is ac-knowledged. [1] P. A. M. Dirac,The Principles of Quantum Mechanics(Oxford University Press, Fourth Edition,1958) [2] P. A. M. Dirac, Directions in Physics (Wiley Interscience,[1] P. A. M. Dirac,The Principles of Quantum Mechanics(Oxford University Press, Fourth Edition,1958) [2] P. A. M. Dirac, Directions in Physics (Wiley Interscience,