Weyl geometry, topology of space-time and reality of electromagnetic potentials, and new perspective on particle physics
aa r X i v : . [ phy s i c s . g e n - ph ] N ov Weyl geometry, topology of space-time and reality of electromagnetic potentials, andnew perspective on particle physics
S. C. TiwariDepartment of Physics, Institute of Science, Banaras Hindu University, and Institute of Natural Philosophy,Varanasi 221005, India
Consistency of Weyl natural gauge, Lorentz gauge and nonlinear gauge is studied in Weyl geom-etry. Field equations in generalized Weyl-Dirac theory show that spinless electron and photon aretopological defects. Statistical metric and fluctuating metric in 3D space with time as a measure ofspatial relations are discussed to propose a statistical interpretation of Maxwell field equations. It isargued that physical geometry is an approximation to mathematical geometry, and 4D relativisticspacetime is essentially 3D space with changing spatial relations. The present work is suggested tohave radical new outlook on elementary particle physics.
PACS numbers: 12.10.-g, 12.90.+b
I. INTRODUCTION
Riemann, way back in 1861, had speculated on thephysical significance of the electromagnetic (EM) poten-tials [1]. Historical evolution of the concept of EM po-tentials has witnessed landmark achievements [2, 3], yettheir physical reality remains unsettled. Aharonov andBohm in a comprehensive article [4] present the role ofgeometry, topology, and measurability in connection withthe physical reality of the potentials. Unfortunately themanner in which this question was introduced underlin-ing classical versus quantum has become a source of con-fusion in the literature. Aharonov-Bohm (AB) effect hasbeen demonstrated experimentally many times [5], how-ever multiple shades of philosophical/mathematical ar-guments tend to obscure the nature of the EM potentials[6].The issue has also acquired great significance in thecontext of the standard model (SM) of particle physics.A crucial element in SM and modern gauge theories is thepostulated concept of hypothetical internal spaces for thegauge symmetry. For example, in SU(3) gauge group theset of parameters θ a are Lorentz scalars and the index a = 1 , , ... U SU (3) = e iθ a ( x ) λ a / where λ a are the Gell-Mann matrices. In quantum electrody-namics (QED) there is just one parameter for the gaugetransformation U (1) = e iθ ( x ) . If the parameters do notdepend on space-time coordinate x then we have globalgauge transformation. Gauge invariance of the actionfunction or the Lagrangian density leads to conservedNoether currents and corresponding gauge charges, forexample, the electric charge is interpreted as a gaugecharge for U(1) gauge symmetry in QED. Note that QEDis a paradigm for SM and modern gauge theories. How-ever, originally the idea of gauge symmetry proposed byWeyl [7] represented scale transformation of the line ele-ment in the pseudo-Riemannian spacetime of general rel-ativity resulting into the change in the length of a vectorunder parallel transport. Thus gauge symmetry was aspacetime symmetry transformation with a wider group of transformations than the general coordinate transfor-mations of general relativity [7]. We may now have twoview-points. First, the empirical success of SM in preci-sion experiments, and the detection of the gauge bosonsof the electroweak unified theory could be taken to im-ply that the question of the reality of potentials has beenrendered vacuous: the dynamical gauge potentials in thistheory represent the gauge bosons i. e. photon and mas-sive weak gauge bosons W ± ; Z , and they have physicalreality. This point is further elaborated in the next sec-tion. The second view could be as follows. The funda-mental role of the internal spaces in modern theories, e.g. SM and superstrings, departs from the geometriza-tion of physics based on the intuitive reality of space andtime speculated by Riemann, Clifford, Einstein and Weyl[1, 7]. Could one seek alternative scenario in that tradi-tion?The present work is focused on this question. Obvi-ously the concepts of space and time would need radicalrevision in which nontrivial geometry and topology ac-quire central role [8]. In the new perspective it emergesthat classical Weyl geometry of space-time has not onlya nontrivial topological structure it also admits statisti-cal interpretation of geometric quantities. Appraisal of acomprehensive review on Weyl geometry [9] and a crit-ical reading of Dirac’s paper [10] show that most of thepresent work is a new contribution. The significance ofour work lies in the fact that rather than discarding thereality of space and time we articulate a framework inwhich space and time constitute a fundamental physicalreality.The paper is organized as follows. In the next sectionwe explain the difference between geometry and topol-ogy; and discuss physical geometry for physical phenom-ena and its description. The conception of physical real-ity depends in a fundamental way on the approximationand/or limitation of the mathematical reality. The roleof language in human affairs is primarily to express thethoughts and the feelings in a tangible form; this processhas intrinsic limitation in the exact reproducibility of thesubtle thoughts. The origin of thoughts, mind-brain du-ality, and the Platonic reality of mathematics invite at-tention to deep questions [8]. Note that the symboliclogic had origin in Boole’s work “The Laws of Thought”.Our proposition that physics is the language of mathe-matics echoes the commonplace understanding regardingthe expression of subtle thoughts in tangible form of thelanguage. The philosophical discussions could be foundin [8] and references cited therein. Here the considera-tions are primarily concerned with electromagnetism andWeyl geometry leading to the important result that Weylgeometry could be endowed with a nontrivial topologyand statistical nature.In section III we investigate various kinds of gauge con-ditions, i. e. Weyl’s natural gauge, Lorentz gauge andnonlinear gauge, and their consistency in Weyl [7] andWeyl-Dirac [10] theories. The main result of this sec-tion is that the principle of gauge invariance may havestatistical interpretation, and the elementary geometri-cal objects defined by pure gauge potentials may have atopology akin to that of AB effect [4]. The field equa-tions derived in a generalized Weyl-Dirac framework [11]are discussed in section IV. Three important results are:electron could be decoupled from EM field and poten-tials, single electron could be visualized as a propagatingtopological defect, and new insight is gained on the topo-logical model of photon [12].The physical significance of the nonlinear term inscalar curvature in Weyl geometry remains obscure [7,13]. In flat spacetime geometry Dirac in 1951 [14] inves-tigated the nonlinear gauge to develop a new theory ofelectron. On the other hand, Gubarev et al [15] discussedtopological structures for the minimum value of the vol-ume integral of the squared potentials. The role of non-linear gauge in the generalized Weyl-Dirac theory in con-nection with [14, 15] is discussed in section V. A logicallyconsistent approach is proposed for macroscopic systemdrawing analogy to fluid dynamics. In the last section abrief discussion on the implications of the present workon modern gauge field theories and the current ideas onemergent spacetime is presented followed by concludingremarks. II. GEOMETRY AND PHYSICS
Classical physics comprises of a classical system or ob-ject, observable/measurable physical quantities, and atheoretical description quite often having a mathemati-cal formalism. A simple example is that of a macroscopicbody that has well defined macroscopic variables deter-mined by local differential equations. One may also con-sider a microscopic system and try to develop a macro-scopic system from it using the methods of statisticalmechanics. The intricate question is that of the neces-sary criteria characterizing a system macroscopic or mi-croscopic.Majority of physicists follow Newton’s experimentalnatural philosophy: experimentally measured quantities are defined in terms of mathematical variables and con-cepts, and depending on the physical laws or hypothesesthe mathematical formalism is developed. In this methodabstract mathematics is a convenient tool. It seems tohave given rise to the belief that mathematics is a lan-guage of physics. In classical electrodynamics (CED) theexperimental force laws of Coulomb and Ampere, andFaraday’s law of induction serve the basis for Maxwellequations ∇ . E = ρ (1) ∇ × B = J + 1 c ∂ E ∂t (2) ∇ × E = − c ∂ B ∂t (3) ∇ . B = 0 (4)EM fields are mathematical variables defined from theforce laws using a limiting procedure. First Chapter inthe textbook [16] rightly notes that independent phys-ical reality of EM fields emerges when energy, momen-tum, and angular momentum associated with them aredefined. We may add that only after the advent of specialrelativity and the observed mechanical effects of the ra-diation field separated from the sources ρ and J the truephysical significance of the EM fields was recognized andaccepted.Mathematically, without recourse to CED, one mayproceed defining a second-rank antisymmetric tensor F µν in 4D space F µν = ∂ µ A ν − ∂ ν A µ (5)and define a scalar L = − F µν F µν (6)that gives the Euler-Lagrange equation from the actionprinciple ∂ µ F µν = 0 (7)Definition (5) gives ∂ µ F νσ + ∂ ν F σµ + ∂ σ F µν = 0 (8)The set of equations (5) to (8) constitutes a mathematicalsystem. Formally they are equivalent to the source-freeMaxwell equations if 4D space is identified with space-time: Eqs. (1) and (2) → Eq.(7); Eqs. (3) and (4) → Eq.(8). Physical interpretation of the tensor F µν and the4-vector A µ representing EM phenomena is essential toidentify Eqs. (7) and (8) as Maxwell equations. His-torically physics laws preceded Maxwell equations and itappeared as if the role of mathematics was just that of aconvenient description tool.The constructs (5) and (6) look artificial because wealready have the EM field tensor in mind, however Weyltheory [7] presents a new picture not recognized ade-quately. Spacetime metric geometry ds = g µν dx µ dx ν (9)is generalized postulating a linear groundform W µ dx µ such that under gauge transformation ds → ds ′ = λ ds (10) W µ → W ′ µ = W µ + φ ,µ (11)where φ = log λ and λ is an arbitrary function of co-ordinates. Notation and metric convention are those of[10, 13]. Note that in Weyl geometry the linear ground-form is in addition to the metric groundform (9), and itdepends on the gauge or calibration λ via Eq.(11). InWeyl geometry both length and direction of a vector un-dergo changes under parallel transport from spacetimepoint x µ to x µ + dx µ and the total change in the lengthof a vector round a small closed loop is I δl = lW µν δS µν (12)Here δS µν is the area enclosed by the loop and W µν is ageometric quantity termed distance curvature, see section15 in [7] W µν = W µ,ν − W ν,µ (13)Proof of Eq.(12) is given in Appendix-I. An invariant in-tegral R W µν W µν √− g d x , and intrinsic geometric prop-erty, i. e. Bianchi identity ∂ µ W νσ + ∂ ν W σµ + ∂ σ W µν = 0 (14)could also be given in Weyl geometry.Dirac notes that physicists rejected Weyl unified the-ory [10], Weyl himself abandoned it [3, 7], and Einsteinobjected to it on physical grounds [17]. Dirac was, how-ever fascinated with its simplicity and beauty. Edding-ton [13] makes a distinction between natural geometryand world geometry: exact natural geometry applies toreal physical world, and world geometry to conceptual ormathematical space. Eddington is critical of Weyl geom-etry on mathematical ground [13, 17]. In our view thefunctions g µν , W µ characterize the abstract Weyl space,and it cannot be rejected on either physical or mathemat-ical grounds. The reason is as follows. The Maxwell-likeequations for the distance curvature W µν on their own donot represent physics and the gauge-invariant zero lengthneed not be related to light propagation, in that case Ed-dington’s geometry [13] is a natural mathematical gen-eralization of Weyl geometry, and obviously it is not arefutation of Weyl geometry. The question now becomes whether Weyl space has a physical realization, that is,does there exist a physical interpretation of the geomet-rical quantities g µν , W µ ? In analogy to Einstein’s inter-pretation of the coefficiants g µν of the quadratic ground-form (9) representing the gravitational potentials, thecoefficients of the linear groundform W µ are interpretedas EM potentials by Weyl. It seems deep influence ofEinstein’s relativity on Weyl prevented him from pro-pounding the true mathematical significance of his gen-eralized Riemannian geometry. In fact, Weyl states thata truly realistic mathematics should be conceived in linewith physics. It may be of interest to reproduce hereintriguing statements due to Weizsaecker in this context[18]: “A mathematical formalism like, e. g. Hamilton’sprinciple with its mathematical consequences, Maxwell’sequations with their solutions, or Hilbert space and theSchroedinger equation, is not eo ipso a part of physics. Itbecomes physics by an interpretation of the mathemati-cal quantities used in it, an interpretation which one maycall physical semantics.” It is true that on its own ac-count Maxwell-like equations for W µν in Weyl geometrydo not represent EM phenomena. At the end of the book[18] an interesting discussion, mainly between Linney andWeizsaecker, mentions classical and quantum languages,and the problem of continuity, but unfortunately remainsconfusing and inconclusive. What is the significance ofphysical semantics? Does it imply that physics is a lan-guage of mathematics?Translating abstract mathematical reality to physicsnecessarily involves metaphysical elements, and what onedoes in practice is to use the probabilty rules/statisticalquantities to get physical quantities. In the Newtonianmechanics the absolute time is a metaphysical conceptwithout which the role of relative time defined by New-ton to explain the observed phenomena would make nophysical sense [8]. Mathematically a point, a circle, 2Dspherical surface and a punctured 2D Euclidean plane R − { } are well defined. A point is dimensionless, acircle with unit diameter has the circumference π , and2D spherical surface has the direction holonomy, i. e. thedirection of a vector parallel transported on the surfaceafter the completion of a cycle is changed. Removingthe origin in 2D plane results into a topological defectin R − { } . All the four objects find numerous physicsapplications, in fact, a point particle is ubiquitous: classi-cal mechanics, CED, QED and SM. Physical realizationsmust be approximate, a physical particle is not a pointand a hole in R − { } is not an exact physical void inphysical systems like singular vortices in fluid and op-tics/light beams. According to Poincare [8] the rule ofinduction is a mental act exact in mathematics but ap-proximate in physics. Present arguments suggest thatgeometry and topology too are approximate in physicalworld.The most radical change that we propound is on thenature of time [8, 19]. Absolute time T has a unidirec-tional flow in discrete steps and creates the sequence ofnatural numbers intuitively experienced by mind; the dis-crete step has an extension that presents itself as spacewith each successive step acquiring the property of whole-ness as a result of mergence. Since spacetime continuumis an assumption some of the modern quantum grav-ity theories discard this assumption and envisage fun-damental length and discretized spacetime. HistoricallyPoincare and Weyl did speculate on discrete time [8],however we postulate discrete nature of time to relateit with the natural numbers. Real numbers are sec-ondary logical and random constructs. The concept ofmergence is introduced to define addition in a wholeness:for example, mixing of volumes of water; in a technicalsense, superposition of the tails of two Gaussian curvesis mergence [8]. Spatial regions of different numericalmagnitudes have random flow in which coalescence anddecomposition create connected quasi-permanent struc-tures that we call matter. The flow has meaning withreference to the absolute time and a common or rela-tive time variable t can be introduced that measures thechanges in the flux of the space. Thus matter has no in-trinsic existence but a form of spatio-temporal boundedstructure. In physical geometry a point has a spread anddistance between two points has a fluctuating statisticalcharacter. Geometrization of physics becomes a unifiedpicture in which space and time are fundamental.The ideas of Riemann, Einstein and Weyl on ge-ometrization of physics are entirely different than ourideas. Riemann’s infinitesimal geometry is a great ad-vancement over the finite Euclidean geometry; Weylterms the idea to gain knowledge from infinitesimal asthe mainspring of the theory of knowledge, page 92 in[7]. Riemann visualizes metric space only due to thepresence of matter, page 98 in [7]. Einstein’s geometriza-tion of macroscopic world is based on the continuationof Riemann’s philosophy assuming infinitesimal pseudo-Riemannian spacetime geometry for gravitation [1, 7].Weyl follows this line of thinking in the unified theory ofgravitation and electromagnetism.Clifford’s speculation differs from that of Riemann,Einstein and Weyl: matter is a kind of crinkles of space,footnote 2, page 156 in [1]. Menger’s statistical met-ric space [20] endows statistical distribution function forthe distance between any pair of points. Poincare’s ob-servation, see [8], that A = B, B = C ⇒ A = C is mathematical while physical experiences show that A = B, B = C ⇒ A < C has a probability description inthe Menger space. Menger suggests that the conceptualproblems of microphysics could find resolution in this un-derlying geometry. The geometric ideas of Clifford andMenger are of interest in the proposed space and timereality [8, 19, 21], however they have to be substantiallygeneralized.Critique on the relativistic time [8, 19, 21] shows thattime in relativity is just a convenient parameter to la-bel/order changes in spatial lengths between two pointsin 3D space, and the role of vacuum velocity of light isthat of a unit conversion factor. Now a physical pointin our approach has a statistical spread and fluctuations, therefore, Menger’s statistical metric cannot be used; forthe distance between two points also we need a suitabledescription. Earlier assuming stochastic electron motion[21] it was suggested that the line element of special rel-ativity represents standard deviation. The propositionthat spatio-temporal objects in the connected whole con-stitutes physical reality at a fundamental level liberatesphysics from extraneous matter [8, 22]. In the stochastictheory [23, 24] one follows the legacy of the Newtonianpoint particle picture and introduces fluctuations invok-ing some kind of hidden thermostat a la de Broglie orNernst’s zero point field (ZPF). The present approachdispenses with them since spatial regions are themselvesin random flux, and spatio-temporal objects have fluctu-ating spread intrinsic to them. As a consequence pointparticle, instantaneous velocity and potential functionsare mathematical idealizations. We seek approximationsto define physical properties from the statistical averagesdepending on length and time scales.In special relativity the interpretation of the line ele-ment is nicely explained in section 7 of [13] ds = c dt − dx − dy − dz (15)( dsdt ) = c − v (16)Time-like interval, i. e. positive value of ds is physi-cal for material particles that cannot travel faster thanlight, i. e. v < c in (16); and ds = 0 defines null-coneor light geometry. Unfortunately existence of space-likeintervals have to be inferred in a contrived way if one ac-cepts Eddington’s suggestion, moreover physical time isnot the common sense time, please see sections 7 and 8 in[13]. Eddington assumes physical matter to have timelikestructure and suggests that spacelike particles have dif-ferent structure. Now spacelike particles could exist onlyin an instantaneous space, therefore, they are unphysicalor impossible structures. Statistical nature of physicalpoints would suggest that variance or standard deviationcould be the best characterization for the metric ds → σ = dl − ( dl ) (17)where dl is the average or mean of the distance betweentwo points k r − r k . A parameter t that represents timevariable with reference to the absolute time T can be in-troduced using c as a unit conversion factor dl = c dt .Since variance cannot be negative space-like interval hasno meaning, and light-cone has zero variance. In the flowof space a set of physical points having uniform relativechange could be selected, and the collection of all suchsets { I } may be interpreted as the inertial frames. As-suming one of the sets as a standard for calibration onecan define expression (17); time t and light velocity c areconvenient parameters. Since changes in spatial relationsmay occur in both directions the time variable t may haveinversion t → − t . There is, however no real time reversalsince t has meaning with reference to T . For more detailswe refer to [8, 19].We have argued that statistical theory has crucial rolein translating mathematics to physics; obviously a gen-eral and comprehensive treatment is needed for this pur-pose. Here a few examples are worth mentioning. Thewell known relativistic relation E − c p . p = m c (18)may be interpreted statistically in which momentum hasaverage value p and energy is c p . Invariant mass turnsout to be a measure of standard deviation opposed to itsintrinsic invariant value in relativity [13]. Note that ina different way already in nonrelativistic quantum me-chanics Ehrenfest theorem makes use of the expectationvalues of position, momentum and energy operators. In-finitesimal Riemannian geometry in physical phenomenahas logical interpretation in terms of the Menger metricspace [20] while pseudo-Riemannian space-time geometrymay be obtained based on the generalization of Eq.(17)such that g µν has statistical significance. Stationarity of R ds in relativity corresponds to the minimization of thevariance σ in Eq.(17). Extending the arguments to CED,the EM fields and potentials have statistical nature. Onemay envisage a true stochastic formulation without hid-den thermostat or ZPF [23, 24] for the Newton-Lorentzequation of motion in this approach.Preceding discussion leads to a challenging question:Do elementary objects like a single electron and a singlephoton make physical sense in the statistical approach?Note that advanced technology and experimental meth-ods have enabled numerous experiments at a single elec-tron and a single photon level. Electric charge and spin ofelectron and spin of photon appear to have unambiguousphysical reality. Electron mass in relativity also has in-variant and intrinsic characteristic, please see section 13in [13]. However, re-examining the relativistic relationwritten in a different form m c = M c − M v . v (19)in view of Eq.(17) statistical interpretation implies that σ = m c and p = M v . Electron mass m no longerrepresents intrinsic or internal physical attribute of theelectron. Thus charge and spin are the physical quanti-ties that we have to explain. Could their understandingcome from topology?Let us briefly explain our idea on the topological de-fect in 1D. A straight line segment could be continuouslydeformed to an arc in which shape and size are unim-portant. Removing any one of the interior points of thearc decomposes it inro two parts. A circle cannot be ob-tained from a straight segment using topological trans-formation since one needs joining the end points of thesegment. Let us consider a real line that is a geometricrepresentation of the set of real numbers ( R + , R − ): a realnumber a is positive if a >
0, and negative if a <
0. Inthe real analysis conventionally one divides the line such that specifying (not removing) a point O, i. e. the origin,on the line the right is positive and the left is negative;such a line is called a directed line. There is no continu-ous transformation that maps points on R + ↔ R − . Onemay introduce an imaginary axis and define a continuoustransformation e iθ on the complex plane to map pointson R + ↔ R − . A departure from this construction [28]envisages the origin as a point defect: real lines on eitherside have continuous transformations, however crossingthe origin is a discontinuous jump 0 + − − = δ , where δ is an infinitesimally small real number. This pointdefect is in 1D and distinct from that of the puncturedplane R − { } . Tifold in [12] may be related with thiskind of topological obstruction.Topology has two key characteristics [25, 26], namelycontinuity and global, and defects and discreteness. Theconcepts of connectedness, adjacent points, neighbor-hoods, and continuous mappings belong to the for-mer while counting of holes/defects to the later; thesecould be made technically precise [25]. However metric-independence of topological properties does not precludethe metric spaces from possessing nontrivial topology.We have argued that ideal mathematical geometry is re-alized statistically in physics; the statistical metric spacesmay also have nontrivial topology. Holes/defects in phys-ical space-time, for example, vortices and tifold [12, 22]may be such examples.Could topological models for single electron and singlephoton be related with the standard field theory? Let usconsider the photon model in which pure gauge poten-tial plays the role for the field description of orbifold andtifold [12]. Now, we face two questions. Firstly the realityof a discrete localized photon has been questioned in theliterature, e. g. Lamb’s anti-photon viewpoint [27] andin stochastic electrodynamics [24]. However, photodetec-tors do show direct evidence of an observed photon [28].Moreover, in the SM along with weak gauge bosons pho-ton is also a gauge boson and the observed data in par-ticle physics experiments [29] show the presence of pho-ton directly as compared to rather indirect observation ofweak gauge bosons inferred from their decay modes. Lowfrequence EM radiation and electron scattering treatedclassically explains the measured cross sections satisfac-torily using the Thomson formula. An important conceptof classical electron charge radius e /mc also followsfrom this. On the other hand, at high energy ν ≈ mc /h quantum theory of Compton scattering utilizing photonpicture becomes essential. For more details we refer to[16] and review on the photon concept in [30].The second question is an age-long one: potentials areconvenient calculational tools or they have independentreality. Aharonov and Bohm [4] state the problem veryclearly in the modern context that potentials seem tohave the role of auxiliary variables devoid of physicalsignificance either because they can be eliminated fromthe equations in classical theory (CED) or due to theprinciple of gauge invariance in quantum theory. Vastliterature on this contentious issue seems to ignore theimportant fact that the observed photon is a spin onegauge boson in SM [29] and spacetime symmetries asso-ciate spin one with a vector field not a tensor field. Doesit not prove the reality of EM potentials?Originally the Maxwell equations were obtained fromthe macroscopic EM phenomena. There is no compellingreason either experimental or theoretical to associate EMfields with a single photon; it is just a convention. Tradi-tional meaning of the principle of gauge invariance has tobe modified. A logically consistent approach could be todistinguish a single photon and a system of large numberof photons described by statistically averaged physicalquantities. Postulating E = B = 0 for a single photonhas profound implication on the physical nature of pho-ton and the significance of EM potentials [12]. Denotingthe EM potentials for a single photon by A µ we have thefollowing equations satisfied by them ∇ . A + 1 c ∂ A ∂t = 0 (20) ∇A + 1 c ∂ A ∂t = 0 (21) ∇ × A = 0 (22)First-order partial differential Eqs. (20)-(22) may becombined to derive the second-order wave equations ∇ A − c ∂ A ∂t = 0 (23) ∇ A − c ∂ A ∂t = 0 (24)Lorentz gauge (20) has the nice property that it is man-ifestly Lorentz covariant, moreover it has formal resem-blence with the continuity equation for fluid flow. Fora single photon 4-vector potential itself is interpreted asenergy-momentum 4-vector. The electric charge unit e has implicit presence in the definition of EM potentialsexpressed in terms of charge and current density. Nowfactoring out electric charge unit e from the EM fieldsand potentials one gets the geometrical unit for them; inWeyl geometry a geometric unit arises naturally. Mul-tiplication with ¯ h makes the vector potential A to havethe dimension of momentum. We have proposed a screwdisclination and tifold model for photon [12] in whichspin is a topological invariant. Particle nature of photonhas been suggested to be due to the topological defectassociated with the spin.In an abstract mathematical formalism Salingaros [31]obtains holomorphic field equations in 4D spacetime fortwo types of tensors that formally resemble Maxwellequations. For a vector field a µ the holomorphy condi-tion is the set of equations (18) in [31]. These equationsare nothing but Eqs. (20)-(22) as above. The author em-phasizes the spacetime interpretation of EM phenomena; this interpretation is in the spirit of Weyl’s interpretationin which EM field tensor is a spacetime distance curva-ture. Unfortunately, the equations satisfied by a µ havebeen unimaginatively treated as just the Lorentz gaugecondition and the identity that E = B = 0. Note that theequations for a µ in [31] are independent of Maxwell-likeequations for a tensor of type 2, and could be viewed asgeneralized Cauchy-Riemann equations. For this reason,mathematically the equations for a µ are as significant in4D as Cauchy-Riemann equations are in 2D. The pho-ton model based on Eqs. (20)-(22) in [12] acquires addedsignificance.The present perspective on mathematics and physics,specifically in the context of electromagnetism, leads totwo new directions: 1) foundations of CED have to bere-examined interpreting EM fields as statistically aver-aged macroscopic quantities for photon fluid, and 2) thefundamental role of pure gauge potentials in the topologynecessitates a thorough analysis of the gauge conditionsin Weyl [7, 13] and Weyl-Dirac [10] theories. III. GAUGE INVARIANCE AND WEYLGEOMETRY: NEW INSIGHTS
Relativity and 4D spacetime geometry had profoundinfluence on Weyl, and his aim of a unified theory wasgreatly inspired by Mie’s theory [7]. Curiously Weyl per-ceptively quotes Clifford’s speculation of 1875:“the the-ory of space curvature hints at a possibility of describingmatter and motion in terms of extension only”. Weylasserts that “physics does not extend beyond geometry”,but reverts to the conventional setup of matter immersedin space in his unified theory. The central problem in hiswork is that of the electron model; according to him theMaxwell-Lorentz theory is invalid in the interior of elec-tron. Unfortunately the unified theory of Weyl couldnot make much progress in this objective [7]. Relativ-ity of magnitude together with the relativity of motionresulted into a true infinitesimal geometry, namely Weylgeometry: a point in spacetime needs specification of co-ordinate system and a gauge or units of measure. Besides[7, 13] a self-contained necessary tensor analysis in Weylgeometry could be found in Dirac’s paper [10].Following the metric and gauge conventions in [10, 13]in-invariants and in-tensors are gauge invariant. Metrictensor g µν is a co-tensor of power 2, and √− g has power4. The generalized affine connection or Christoffel symbolinvariant under gauge transformation denoted by ∗ Γ αµν is ∗ Γ αµν = Γ αµν − g αµ A ν − g αν A µ + g µν A α (25)In Weyl geometry the transformation of the tensor quan-tities has to be considered for both coordinate and gaugetransformations, therefore, one has to assign Weyl pow-ers as well to them. The meaning of tensor densities andthe role of √− g could be found in sections (16) and (17)in [7]. Following Eddington [13] the scalar curvature ∗ R is termed a co-invariant of power − ∗ R = R − W µ : µ + 6 W µ W µ (26)Assuming that variational principle for an appropriateaction integral yields the field equations for the unifiedtheory the problem becomes that of constructing W inthe action S = Z W √− g d x (27)Gauge invariance of the action S demands that W mustbe a co-scalar of power − √− g has power 4. Weylopts for the squared scalar curvature ∗ R , however headmits that it may not be realized in nature. Dirac [10]introduces an arbitrary scalar function β of power − W linear in ∗ R . Their respective functions are W W = ∗ R − α W µν W µν (28) W D = β ∗ R − F µν F µν (29)Distinct symbols in (28) and (29) for the second rank an-tisymmetric tensor signify the fact that in Weyl action itis a geometric quantity, namely the distance curvature,while Dirac proceeds with the assumption that it repre-sents the EM field tensor.Here our main interest is regarding the issue of gaugeinvariance rather than the field equations. In Weyl ge-ometry the fundamental postulate is that the compari-son of lengths at a distance is not possible and a uniqueand uniform calibration does not exist. To reconcile thiswith the physical experiences Weyl puts forward two ar-guments that have been critically elaborated by Edding-ton. The first argument is that we may define a unit oflength assuming ∗ R = 4 λ (30)where λ is a constant. This gauging equation is sup-posed to represent a natural gauge. The existence of thenatural gauge could be justified in analogy to the exis-tence of a unique Galilean system of coordinates in thepseudo-Riemannian spacetime geometry of general rela-tivity. However, there arises a serious difficulty in fixinga gauge since the law of parallel displacement of a vectorcontradicts it. To resolve this problem the second ar-gument is developed based on the philosophy that thereare two ways of determining the quantities: by persis-tence and by adjustment. A priori one cannot concludethat pure transference following the tendency of persis-tence is integrable. Size of an electron is determined byadjustment in view of the presence of space curvature,and not by the persistence of its time history. Anotherexample is that of electric charge: conservation of chargecannot explain why electron has the same charge aftera lapse of long time, and why this charge is the same for all electrons. Thus charge is determined by adjust-ment, not by persistence, i. e. at every instant of timethe state of equilibrium of negative electricity adjusts tothis value. In Eddington’s view this interpretation of thegauge principle amounts to a graphical representation ofphysical facts; a measuring rod taken from one point toanother for calibration could be taken to satisfy the nat-ural gauge condition (30), and this process should not beviewed as a parallel displacement of a vector.The question is: What is the significance of the gaugecondition (30) in the unified field equations? First animportant point has to be noted that even if ∗ R is notconstant one could change the measure to transform itto a constant in a new gauge. Though unified field equa-tions correspond to Einstein-Maxwell like field equations;let us drop “like” and accept Weyl’s interpretation thatthey represent electromagnetism and gravitation. In thatcase the Maxwell equations for W µν depart radically fromstandard CED equations since the gauge potential W µ it-self acquires the role of source current density J µ ∝ W µ (31)Weyl postulates a unique calibration setting ∗ R = 1 andintroduces the radius of curvature of the universe for therecalibration. This leads to the gauge condition (30) and λ acquires the significance as a cosmological constant inthe Einstein equation.A remarkably unusual result (31) that charge currentdensity is equivalent to the potentials has intriguing as-pects. First it may be noted that the current continuityequation follows from the Maxwell equation J µ : µ = 0 (32)In view of Eq.(31) formally we have W µ : µ = 0 (33)Lorentz gauge condition of the standard CED assumesa radically new role in Eq.(33): it is inseparably re-lated with the charge conservation law and accordingto Weyl, Eq.(31) shows that the electric charge is “dif-fused throughout the world” since the gauge potential ispresent in all space.Alternative derivation of Eq.(32) is based on the con-traction of the Einstein equation and the use of the gaugecondition (30). Two routes to get current continuity posea serious problem since the Lorentz gauge condition is in-consistent with the gauge condition (30). Weyl [7] andEddington [13] seek a resolution of this issue exploringthe interplay between the energy tensor of matter andthat of EM field. The arguments run as follows. Totalenergy-momentum tensor in the Einstein field equationwith a cosmological constant could be assumed compris-ing of the EM part, E µν , and the matter one M µν . Thecontraction gives the trace equation interpreted in termsof the proper-density of matter ρ M ∝ R − λ (34)Using the gauge condition (30) ρ M becomes ρ M ∝ W µ : µ − W µ W µ (35)Empty space is defined to be the absence of electrons,though EM field is nonzero, then ρ M = 0, and Eq.(35)implies W µ W µ = W µ : µ (36)It is clear that the problem has become more intricate asthere are now three incompatible gauge conditions: nat-ural gauge (30), Lorentz gauge (33), and nonlinear gauge(36). Weyl asserts that the relation (31) has fundamen-tal importance, in that case, there is no empty space andthe Lorentz gauge would have general validity. Howeversubstituting (33) in Eq.(35) one has ρ M ∝ W µ W µ ⇒ ρ M ∝ − J µ J µ (37)The positivity of mass density implies that J µ is a space-like vector, and it cannot be true for the electric charge.A speculative idea is to consider interior of the electronwhere Eq.(37) holds, and the outside region separatedby the boundary of the electron. Eddington suggeststhat the internal structure of the electron may have theconstituents something like magnetic charges. In Weyltheory the Lorentz gauge condition occupies special po-sition, and is assumed to hold in general. On the otherhand, the nonlinear gauge (36) is interpreted to representthe vanishing of the current density that in view of (31)becomes consistent with the Lorentz gauge. Outer regionof the electron, however has small charge and current ex-tending to infinity.Preceding account shows that the issue of gauge in-variance in the context of the Weyl action function forthe unified theory could not be settled satisfactorily. Ofcourse, Weyl himself states that, “I do not insist thatit is realized in nature” [7], and Eddington [13] remarksthat Weyl action “has no deep significance”. We em-phasize that questioning the specific action function, e.g. Eq.(28), does not mean a rejection of Weyl geometry.For this reason, Dirac’s view on Weyl geometry is veryimportant: wider group of spacetime transformations forphysical laws noted in the conclusion of his paper [10].It is important to note that Eddington’s criticism is ap-plicable to Dirac’s action function (29): combining twoin-invariants of different forms, namely the Maxwell ac-tion F µν F µν and β ∗ R .Dirac revived Weyl geometry [10] primarily with themotivation of his large number hyp[othesis in which grav-itational constant varies with time. He assumes a con-straint equation ∗ R = 0 (38)and using β as a Lagrange multiplier constructs a gaugeinvariant action integral (29). Dirac lists three choicesof the gauge: 1) the natural gauge; if W µν = 0 one has W µ = 0, 2) the Einstein gauge β = 1 in which Ein-stein field equations are obtained for the vanishing of the EM quantities, and 3) the atomic gauge. SurprisinglyDirac makes no attempt to explain (38) in relation toWeyl’s natural gauge (30). The issue of gauge invariancein empty space with the contradicting gauge conditions(33) and (36) that troubled Weyl and Eddington in con-nection with CED and electron structure also did notreceive Dirac’s attention. Though symmetry breakdownin Weyl space for charge conjugation and time reversal isseparately pointed out by him, Weyl’s speculation thatinequality of positive and negative electricity may be re-lated with the past → future asymmetry, see pp 310-311in [7], is also not discussed.Weyl in the Preface to the American edition of [7]states that his attempt of a unified theory had failed. Itis well known that Weyl’s original idea of gauge symme-try metamorphosed into the modern gauge field theories[2, 3]. Revival of the original Weyl geometry by Dirac[10] and current interest in this geometry [9] are mainlyin connection with the problems in cosmology. The lit-erature on Weyl theory has not unraveled the deeper is-sues related with gauge invariance and gauge conditions.Could there be a new strand for Weyl geometry appli-cable to particle physics? This question is made precisein the light of the new perspective emerging from ouranalysis. [I] Weyl and later Eddington exclusively focused onelectron and diffused charge interpretation. They did notexplore possible implications on the nature of photon andEM fields. Note that Einstein’s light quantum hypothe-sis was well known at that time though the word photonwas coined later in 1926. It is also not clear as to whyDirac in spite of his sustained engagement with new the-ories of the electron did not investigate Weyl geometryfrom this angle. May be it was due to his preoccupationwith the large number hypothesis in [10]. Setting an arbi-trary constant equal to 6 appearing in his action integralhe obtained vacuum field equations. Unfortunately thatmissed uncovering geometric origin of current density inthe Maxwell equation. This issue is of fundamental im-portance for physical interpretation of the EM potentialsand current density. To see this let us compare the stan-dard Maxwell-Lorentz theory and Weyl theory. Eq.(7)shows that vanishing of EM field tensor does not neces-sarily imply vanishing of the EM potentials since one mayhave A µ = ∂ µ χ and F µν = 0. In contrast, in Weyl theorythe current density (31) severely restricts the potentialssince W µν = 0 ⇒ W µ = 0. As a consequence thefreedom to envisage multi-valued pure gauge potential inWeyl theory is lost. [II] Relaxing Dirac’s assumption of setting the con-stant equal to 6 Papini and his collaborators have ex-tensively studied Weyl-Dirac theory [32]. However theirwork does not throw new light on the gauging condi-tions, and merely repeat Dirac’s argument that to recon-cile Weyl gauge principle with the atomic standrads oflength one could postulate two metrics. Papini’s idea onmulti-valued scalar field is however quite interesting. [III]
Weyl’s motivation for his geometry was unifiedtheory for gravitation and electromagnetism [7]. Thesubject as such has three aspects: mathematical com-prising of geometry, physical interpretation of geomet-ric quantities and field equations, and physics motivatedaction principle. Most of the criticism pertain to Weylaction (28) and Dirac too modifies the action to (29) in[10] introducing an additional field variable as a Lagrangemultiplier in the assumed constraint equation (38). Forphysical interpretation, analogy is drawn to Einstein’stheory of gravitation. Einstein field equation is foundedon the identification of the metric tensor as gravitationalpotential and the equivalence between the Einstein ten-sor and energy-momentum tensor. Now, Einstein fieldequation could be derived from the Hilbert-Einstein ac-tion. But, in general, a symmetry of the action may notnecessarily be a symmetry of the field equation. For alucid exposition on the calibration/gauge symmetry werefer to Bock [33].Axiomatically the metric tensor g µν for the quadraticform (line element) and W µ for the linear form are thefundamental geometric quantities in Weyl space. It iswell known, see e. g. [13] that in Riemannian geometryone can construct Galilean coordinate system for con-stant g µν , and the necessary and sufficient condition forflat geometry is that Riemann curvature tensor is zero.Note that the Christoffel symbol vanishes for constantmetric tensor, however it is not a tensor, therefore, itmay assume a nonvanishing value in a new coordinatesystem. In Weyl geometry the definition of the distancecurvature W µν ensures that it vanishes if W µ = 0, how-ever, for W µν = 0 one could have a nonvanishing puregradient for W µ . It may be argued that by a gaugetransformation W µ could be made equal to zero. In anontrivial topology of Weyl space an interesting possi-bility exists: a nonzero W µν exists in a small localizedregion and it is zero everywhere outside that region, but W µ is nonzero there. This case is similar to the topol-ogy due to a confined magnetic flux in a solenoid for ABeffect [4].Concluding this section, a new physical manifestationof Weyl space is indicated from our analysis where onemay consider two ideas: a single electron and a singlephoton could be visualized as topological defects in Weylspace, and EM fields and current density are treated asmacroscopic quantities having statistically averaged sig-nificance. In the light of the proposition of statisticalmetric tensor discussed in section II it seems natural toassociate statistical meaning to the gauge transforma-tion. IV. NEW FIELD EQUATIONS IN WEYLSPACE: NONTRIVIAL TOPOLOGY
Rather than Dirac’s emphasis on the large scale struc-ture of the universe in connection with the Weyl geom-etry, a new action function was setup to explore the na-ture of electron charge in [11]. Let us call it a generalized Weyl-Dirac theory. A beautiful feature of Weyl geome-try is the gauge invariance of zero length: massless fieldsbecome natural objects in this geometry. If one specu-lates that at a fundamental level electron comprises ofmassless fields then photon and electron assume specialplace in Weyl geometry. Could there be a fundamentalrole of nontrivial topology and the gauge potentials inthis scenario? To address this question we discuss thegeneralized Weyl-Dirac theory.Introducing an in-invariant field Ψ representing elec-tron in the proposed action integral [11] we have S = Z ( ∗ Rξ + pF µν ∗ R µν ) √− g d x (39) ξ = Ψ : µ Ψ ,µ (40)Field ξ is a co-scalar of power -2 and it seems akin toDirac’s β , however unlike cosmological interpretationof β , here the field ξ is proposed to be related withthe electron and photon. Variational principle treating g µν , A µ , Ψ as independent dynamical variables leads toa set of field equations comprising of the gravitationalfield equation, modified Maxwell field equation, and afield equation for Ψ, i. e. Equations (25)-(27) respec-tively in [11]. The expression for ∗ R µν contains anti-symmetric tensor F µν following [13] in contrast to thesymmetrized ∗ R µν used by Dirac [10]. We have returnedto the standard notation A µ , F µν for the EM quantitiesbut expressed in geometrical units. The source currentdensity in the modified Maxwell equation, setting the ar-bitrary constant p = 1 /
4, shows dependence on the gaugepotentials F µν ; ν = − ξ : µ + 2 ξA µ ) (41)The derivation of this equation is presented in Appendix-II. The conservation law for the current density J µ = − ξ : µ + 2 ξA µ ) (42)follows from Eq.(41); it could also be obtained as an iden-tity from the contraction of the Einstein field equation(25) in [11]. Thus we have ξ : µ : µ + 2 ξ ,µ A µ + 2 ξA µ : µ = 0 (43)The modified Maxwell field equation (41) differs fromthat of Weyl [7] and admits nonvanishing EM potantialfor F µν = 0 given by ξ : µ + 2 ξA µ = 0 (44)Equation (44) needs a careful analysis since ξ is also re-lated with the postulated electron field Ψ in Eq.(40). Letus consider the full set of the field equations and as-sume that the EM fields are zero then the energy tensor E µν = 0. Making use of Eq.(44) to eliminate derivativesof ξ the Einstein field equation becomes G µν = − ∗ Rξ Ψ ; µ Ψ ; ν + A α A α g µν +2 A µ A ν +2 A µ ; ν − A α : α g µν (45)0where the Einstein tensor is G µν = R µν − g µν R (46)Note that the trace of the Einstein field equation (45)gives just the identity (26). Imposing Weyl natural gaugecondition (30) the Ψ field satisfies the massless waveequation Ψ ; µ : µ = 0 (47)Introducing the energy tensor for Ψ and A µ T µν (Ψ) = Ψ ; µ P si ; ν − g µν Ψ ; α Ψ ,α (48) T µν ( A α ) = − A α A α g µν − A µ A ν − A µ ; ν + 2 A α : α g µν (49)Eq.(45) is re-written as G µν + 2 λg µν = − λξ T µν (Ψ) − T µν ( A α ) (50)A simple calculation gives the trace expressions T (Ψ) = − ξ (51) T ( A α ) = − A α A α + 6 A α : α (52)One of the important consequences of the set of the fieldequations is that both the natural gauge condition (30)and the Lorentz gauge condition (33) or (20) could be si-multaneously assumed without any inconsistency. Mak-ing this assumption we have T µν ( A α ) = − A α A α g µν − A µ A ν − A µ ; ν (53) T ( A α ) = − A α A α (54) T ( A α ) = R − λ (55)The expression for the energy tensor (53) contains onlythe vector gauge field and logically one would expect itsinterpretation as that of a massless vector field. In rel-ativistic field theory it is known that for a massive vec-tor field the representation splits into a 3-vector and ascalar under rotation, and the Lorentz gauge condition isneeded to resolve the question of the positivity of energy[34]. In our case, the interpretation of (53) as energy ten-sor of A α is inferred from the Einstein field equation (50).It is important to note that the trace expressions (52)and (54) remain unaltered even if EM fields are nonzerosince E µν is traceless. On the other hand, the energyexpression (90.6) in Weyl theory [13] containing poten-tials is attributed to the matter energy tensor. Recallthat in the standard CED Eqs. (20)-(24) show that forEM field free case, and the assumption of the Lorentz gauge, each component of A µ satisfies the massless waveequation. Field theoretic arguments [34] and topologicalmodel of photon [12] suggest that pure gauge field A µ may be identified as photon field.The presence of ξ in Eq.(44) and Eq.(50) indicates sub-tle role of the coupling between electron field Ψ and pho-ton field A µ . Is it possible to decouple electron from theinfluence of the EM field as well as pure gauge poten-tial? Towards this aim let us examine the consequencesof setting ξ = 0 . Eq.(40) shows thatΨ ; µ Ψ ,µ = 0 (56)Eq.(50) reduces to 4 λ Ψ ; µ Ψ ; ν = 0 (57)Mathematically there is no contradiction if the expressionmultiplied by by ξ in Eq.(50) is also equal to zero. In thatcase, one gets the Einstein equation G µν = − T µν ( A α ) (58)Physical arguments show that one should also take λ = 0as a solution of Eq.(57) andΨ ; µ Ψ ; ν = 0 (59)Trace of Eq.(58) shows that ∗ R = 0 in agreement with λ = 0 . Eq.(55) reduces to R = − A α A α (60)Decoupled electron field Ψ satisfies the massless waveequation (47) as well as Eq.(56). Multiplying Eq.(47)by Ψ and integrating over a compact manifold withoutboundary gives an uninteresting result when use is madeof Eq.(56). Evidently Ψ field cannot be a continuous fieldof spacetime variables. Let us examine the possibility ofa discontinuous solution. In the following for most of thediscussions on topological aspects we consider the illus-trative examples in flat spacetime geometry, and conjec-ture that the topological characteristics would be carriedover to the Weyl space. The conjecture is motivated bymetric-independence of topological properties.Define a boundary surface at which the field is discon-tinuous F ( x µ ) = L ( x, y, z ) − ct = 0 (61)Instead of a plane wave solution of Eq.(47) the field onthe discontinuity surface could be assumed asΨ = e iL − iωt (62)Substituting expression (62) in the wave equation, andmaking the approximation that ∇ L is small comparedto other terms we get ∇ L. ∇ L = ω c (63)1Using (62) in Eq.(56) we get once again Eq.(63). Follow-ing a nice discussion in the Appendix VI of [35] we definea unit normal vector to the surface (61)ˆ n = ∇ F |∇ F | (64)and the speed of the moving discontinuity by v f = − |∇ F | ∂F∂t (65)In the present case the discontinuity moves at the velocityof light. In analogy to geometrical optics limit of the lightwave propagation [35], Eq.(56) or Eq.(63) is the eikonalequation. Thus, the electron field is a defect propagatingat the speed of light.In the Einstein field equation (45) the energy contri-bution of Ψ field disappears, and the decoupled electronhas no self-field of the EM origin. The significance of thisfield is two-fold: its existence signals a nontrivial topol-ogy of Weyl geometry, and the possibility of nonzero ξ makes it observable through EM interaction via Eq.(42).The drawback of scalar field representation is that thespin property is not explained, in fact, it corresponds tospinless electron in CED; for a recent attempt to explorespin of electron, see [22].Decoupled photon field satisfies Eq.(58) and Eq.(60).Assuming traceless energy-momentum tensor for A µ Eq.(58) leads to R = 0. In this case Eq.(60) reducesto the nonlinear gauge A α A α = 0 (66)Mathematically Eq.(66) shows that A α is self-perpendicular. The most important result is thatthe gauge conditions are unambiguously consistent andcompatible: λ = 0 ⇒ ∗ R = 0; each term in ∗ R vanishes,therefore R = 0, Lorentz gauge (33) and nonlinear gauge(66) are also satisfied. Note that for a pure gauge field, F µν = 0 and using the Lorentz condition one findsthat A µ satisfies the massless vector wave equation. Tounravel the topology of Weyl geometry we examine thenature of A µ in three ways. I Simplest and conventional solution is given by A µ = η ,µ (67)Substituting (67) in the Lorentz condition we have η ; µ : µ = 0 (68)and the nonlinear gauge (66) becomes η ; µ η ,µ = 0 (69)Eqs.(68) and (69) for η are exactly the same as the onesfor Ψ, therefore, for a nonvanishing η the propagating dis-continuity interpretation would be essential. The photonfield becomes a propagating topological defect. It is, however not necessary to proceed with the solu-tion (67); the field A µ itself shows the topological prop-erty if we consider the discontinuous surface (61) andcalculate A µ on it following [35]. The vector potential A and the scalar potential Φ may be defined on eithersides of the surface (61) and using the unit step or Heav-iside function we may write, for example, for the vectorpotential A = H ( − F ) A + H ( F ) A (70)The derivative of the step function is the Dirac deltafunction, therefore space and time derivatives of the po-tentials can be calculated using the standard calculus.For example, ∇ × A = H ( − F ) ∇ × A + H ( F ) ∇ × A + δ ( F ) ∇ F × ∆ A (71) ∂ A ∂t = H ( − F ) ∂ A ∂t + H ( F ) ∂ A ∂t + δ ( F ) ∂F∂t ∆ A (72)∆ A = A − A (73)Expression(73) denotes the discontinuous change. TheLorentz condition using (64) and (65) finally leads toˆ n . ∆ A − ∆Φ = 0 (74)Note that E = 0 and B = 0 also give Eq.(74). On themoving surface the potentials are calculated using thefact that A , Φ are zero, therefore, Eq.(74) becomesˆ n . A − Φ = 0 (75)It is easy to verify that in view of Eq.(75) the nonlineargauge (66) is satisfied. Thus the photon is a discontinuityof the potentials propagating at the speed of light. II The field theoretic description of topological pho-ton [12] is given based on F µν = 0 and the Lorentz condi-tion. Here in addition we have the constraint (66). Screwdisclination for the vector field A has been interpreted interms of 2D defect termed orbifold and 1D defect as tifoldin [12]. The 4-vector field A µ is split into transverse A t and longitudinal components ( A z , Φ) choosing a propa-gation direction along z-axis. Following [36] the assumedsolution in complex representation is A x = kre iχ (76) A y = ikre iχ (77)where χ = θ + kz − ωt . It is easy to verify that ∂A x ∂x + ∂A y ∂y = 0 (78)and the Lorentz condition reduces to ∂A z ∂z + 1 c ∂ Φ ∂t = 0 (79)2Could one find A z , Φ such that for the solutions (76) and(77) the nonlinear gauge (66) is also satisfied? Since thesum of the squares of the real parts of (76) and (77) is k r a simple solution A z = kz and Φ = − ωt leads to acurious result: Eq.(66) becomes the null-cone equation x + y + z − c t = 0 (80)Note that Nye [36] considers the electric field vector tostudy disclination for polarization effects of EM wave; inthat case ∇ . E = 0. Here we consider A µ that satisfiesthe Lorentz condition, and hence follows Eq.(79). III
Definition of F µν , expression (5) . in analogyto curl in 3D could be treated as curl of A µ in 4D , seesection 3.2 in [13]. The 4-divergence of A µ is ∂ µ A µ . Nowthe curl-free and divergence-free vector in 3D may havepeculiar characteristics made transparent in the abstractlanguage of differential forms and de Rham periods. Anintuitive and physics-oriented account on de Rham peri-ods on the manifolds can be found in [37]; also summa-rized in [12]. In a Euclidean domain formally a 1-formis decomposed into three parts : an exact form (nonzerodivergence, zero curl) , a closed form (zero divergence,nonzero curl), and a harmonic form (both divergence andcurl are zero). According to de Rham theorem the closedloop integral of the harmonic form over a non-boundingcycle counts the number of holes/topological defects. Itmay be asked if de Rham theorem could be extended to4D spacetime.Formally photon field, i. e. A µ is curl-free F µν = 0and divergenceless ∂ µ A µ = 0; it implies that A µ dx µ is aharmonic form with the loop integral I A µ dx µ = 2 πN (81)Note that in Weyl geometry A µ has the dimension of length − . Here N is an integer counting the number oftopological defects, and for a single photon N = 1. Weylgeometry for harmonic 1-form (81) acquires a nontrivialtopology. However, the non-Euclidean spacetime geom-etry limits the straightforward application of de Rhamtheorem. We may get some physical insight consideringthe vector potential A such that B = 0, and take themanifold R − { } . An example of a harmonic form isgiven in Appendix C of [37] A = yr ˆ i − xr ˆ j (82)Though ∇ . A = 0 and ∇ × A = 0, the loop integralenclosing the origin is nonzero I A . dl = 2 π (83)The harmonic form (82) may also be given a physicalrealization in the form of a singular vortex: the fluid flowin concentric circles around the origin at which one hasa singularity of the velocity field. Note that multiplying A µ by ¯ h its dimension is that of momentum, and Eq.(83)may be related with the spin of the photon [30].In this section possible nontrivial topology of the Weylspace has been investigated based on the generalizedWeyl-Dirac theory [11]. The most important new resultis that Ψ field for spinless electron and A µ for photonrepresent topological defects in space-time of Weyl ge-ometry. V. NONLINEAR GAUGE, PARTICLES ANDFIELDS
In the preceding section Eq.(56) and Eq.(68) are look-a-like relativistic Hamilton-Jacobi (HJ) equations for amassless particle. In classical mechanics point particledynamics may be described using the Hamilton princi-pal function S ( q, p, t ) that assumes the following form ifHamiltonian is a constant of motion S ( q, p, t ) = W ( q, p ) − Et (84)Here W ( q, p ) is the characteristic function, and ( q, p )canonical variables. In analogy to optics , W in HJequation has the same role as the eikonal L . Textbook[38] section (10-8) makes quite insightful remarks on thegeometrical optics limit of the light wave and the roleof HJ equation in understanding particle trajectory inSchroedinger wave mechanics. The main problem thatremains unsatisfactorily resolved till date is the meaningof a localized point particle in the wave description orthe continuous field theoretic description.In the preceding section we have suggested that par-ticle aspect is embodied in a topological defect. Unfor-tunately the continuous fields of space and time obeyingspecific partial differential equations seem to have intrin-sic limitations for incorporating the topological objectsas we have seen for the Ψ field or nonlinear gauge for A µ that makes the necessity of the discontinuous fieldnatural, however the particle interpretation and the de-scription of the observed physical phenomena, e. g. elec-tron and EM field interaction and EM waves, in termsof these topological objects are not obvious. Reflectingon his life-long efforts to understand waves and particlesde Broglie made profound remarks on them in his essay[39]. Here we note two of them that seem to be useful forthe present considerations. First one concerns the defi-nition of a particle as a localized object of high energyconcentration that to a first approximation is a shiftingsingularity. The second point is that the principle of leastaction is a particular case of the second law of thermo-dynamics; entropy of a particle is defined by the relationentropy/Boltzmann constant is equal to action/Planckconstant. Now thermodynamical concept of a particle isvery important: it brings the role of statistical mechanicsto the prominence even for an isolated free particle; in deBroglie theory it is caused due to a hidden thermostat.The present work marks a more radical departure than3that of de Broglie: space and time are endowed with non-trivial topology, and the geometric objects and fields areproposed to have the statistical nature.We elucidate these ideas considering the significanceof the nonlinear gauge in the context of [14, 15]. Diracattempts a modification of CED with the aim to cure theproblem of infinities in QED [14] proposing the nonlineargauge A µ A µ = m c e (85)The Maxwell action is modified incorporating the con-straint (85) using a Lagrange multiplier Λ. The chargecurrent density is identified to be J µ = − Λ A µ (86)Dirac argues that if the source-free field has the poten-tials A ∗ µ then the gauge transformed potentials A ∗ µ + Φ ,µ corresponds to the charges. Eq.(85) then becomes( p µ + eA ∗ µ )( p µ + eA µ ∗ ) = m c (87)where e Φ , µ is interpreted as the energy-momentum 4-vector p µ in the relativistic HJ equation (87) for the elec-tron. Note that Dirac develops his theory in flat space-time. The formal similarity of the charge current density(86) with that of Weyl theory, and for constant ξ with thegeneralized Weyl-Dirac theory [11] is noteworthy. How-ever in the generalized Weyl-Dirac theory ξ is a co-scalarfield, therefore, it cannot be a constant. Moreover theanalogue to HJ equation for electron is Eq.(40) that sug-gests that mass is a space-time dependent field variable,and Ψ field does not appear directly in the current density(42). In contrast to particle aspect introduced through p µ in Dirac theory we have the propagating Ψ field dis-continuity representing the particle.Gubarev et al [15] introduce a novel idea seeking thephysical significance of the gauge noninvariant quantity A µ A µ . The volume integral of this quantity in Euclideanspace is shown to have minimum value under the gaugetransformation for a specific gauge condition. For ex-ample, B = ∇ × A is gauge invariant, and the volumeintegral R A . A d x under the gauge transformation is sta-tionary for the Coulomb gauge ∇ . A = 0. Authors em-phasize the point that their objective is to find the mini-mum value of this quantity itself that may throw light onthe nontrivial topology. This motivation comes from theconsiderations on the vacuum condensates, for example,quark and gluon condensates in QCD.It would, of course, be logically more satisfying to treat A µ as a statistical field variable, and interpret the mini-mum value in terms of the averages σ ( A µ ) = A µ A µ − A µ A µ (88)The authors [15] do consider the expectation values ofthe field operators in QFT, however here we associateprobability distribution function with the potentials. Let us recall that the pure gauge potential A µ satisfiesthe Lorentz gauge condition (20) and the minimum valueof the integral of A µ A µ is zero in view of Eq.(66). Thepotential A µ is harmonic (81) for the nontrivial topol-ogy representing a single photon. For a monochromaticphoton beam of uniform photon number density it wouldbe natural to assume Eqs. (20) and (66) for the photonbeam. In a fluid flow with velocity v and probabilitydensity ρ f the local conservation of particles satisfies thecontinuity equation ∂ρ f ∂t + ∇ .ρ f v = 0 (89)Statistical distribution function for photon fluid in anal-ogy to this may be introduced interpreting A = f p ( x µ ) A (90)as the momentum density and the Lorentz condition asthe energy-momentum conservation equation; note thatthe geometric quantity A µ in Weyl geometry has beenmultiplied by Planck constant. In [15] the minimumvalue of the vacuum expectation value of < | A µ A µ | > has been discussed and suggested to be nonzero. In thepresent paper we have geometry/topology plus statisti-cal/thermodynamical approach: at zero temperature theminimum value is proposed to correspond to the frozenphase of photons with zero momenta and only spinningtopological objects possessing energy hν per photon [12].It may be contrasted with the assumed ZPF in stochasticelectrodynamics, see section 2.2 in [24].The main inference that could be drawn from theseconsiderations is that the EM fields in CED have to beinterpreted as averaged statistical quantities represent-ing the photon fluid, i. e. rotating fluid of microscopicspinning topological objects, namely the photons. Inter-preting A µ as averaged energy-momentum vector wouldimply to interpret F µν in analogy to the antisymmetricsecond rank angular momentum tensor L µν : B as aver-aged angular momentum (orbital plus spin) and E as theaveraged value of the energy and the momentum of thefluid as a whole about center of energy; this interpreta-tion is based on the classical relativistic mechanics, seesection 14 in [40]. VI. DISCUSSION AND CONCLUSION
Radically new approach to geometrize physics is articu-lated in the present paper: (1) spacetime of relativity hasstatistical nature generalizing Menger’s statistical metricspace to Eq.(17), (2) EM potential A µ has fundamentalreality representing a single photon as a topological de-fect in space-time, and (3) gauge conditions in originalWeyl geometry, i. e. Weyl natural gauge, Lorentz gaugeand nonlinear gauge are proved to be consistent in gener-alized Weyl-Dirac theory [11] necessitating the existenceof topological defects. It is argued that Maxwell field4tensor and metric tensor represent statistically averagedquantities.Space-time itself is visualized as a fluid comprising ofmicroscopic particles defined by topological defects, forexample, spinless electron as a moving surface discontinu-ity and photon as a harmonic 1-form. EM waves wouldbe like sound waves in a fluid whereas photons are lo-calized space-time topological defects, Eq.(81). Interac-tion of electron with EM field in the present frameworkis viewed as a random scattering of electron propagat-ing in the photon fluid. Momentum exchange with mostprobable fraction of photon momentum given by the finestructure constant is envisaged from p − ec A → p ¯ h − e ¯ hc A g (91)where A g is in geometrical units a la Weyl geometry. Themost important implication of the present work is thatit paves the path for developing an alternative theory ofelementary particles and their interactions solely on thegeometry and topology of physical space and time [22].We emphasize that the present ideas are radically new ascompared to those of Riemann-Einstein-Weyl on space-time, however they adhere to the conventional wisdom ofspace and time reality in contrast to the current radicalideas articulating emergent spacetime and/or discardingspace and time reality.To put the speculations in perspective we present abrief discussion on Einstein’s belief [41] and alternativeideas [23, 24, 39]. Einstein in reply to the criticisms [41]states his belief in field theory as a program for physics.Fields are continuous functions in 4D spacetime contin-uum. Illustrating his idea taking the example of generalrelativity he puts forward three points. E1: Physicalthings are described by continuous functions of space-time, E2: the fields are tensors, e. g. g µν for gravity,and E3: physical measurability of the invariant line ele-ment. According to him the construction of a mathemat-ical theory rests exclusively on E1 and E2. If a completephysics theory exists E3 is not required. Einstein ex-presses serious reservations on radical efforts of Mengerin this perspective. What is the meaning of a mathe-matical theory of physics? As pointed out in section IImathematical objects like a geometrical point or a circleare mental objects and mathematical logic is sufficient todevelop a mathematical realm, for example, Riemanniangeometry or Weyl geometry. However physical objectsneed physicalization or approximation to the mathemat-ical objects, e. g. the fields. Therefore, the geometriza-tion of physics as a program has to alter the Einsteinianideas, and adopt in some way Clifford and Menger spec-ulations. The present approach has sought this objectivebased on the Weyl geometry.The role of probability in fundamental physics distinctthan the orthodox quantum theory has also been of in-terest among some physicists; unfortunately such workshave remained sidelined in the mainstream physics lit-erature. A common lament by de Broglie [39] , Luis de la Pena [23] and Boyer [24] is that full potential of theirideas remains unexplored. Could the present modest con-tribution provide impetus to such efforts? It seems theinclusion of topological model of photon and associatedinterpretation of zero point energy would be essential todevelop a complete stochastic theory for Maxwell equa-tions and Newton-Lorentz equation of motion.In conclusion, consistency of various gauge conditionsin Weyl geometry is thoroughly studied and the physicalreality of EM potentials is established. Harmonic 1-formin 4D spacetime of Weyl needs further investigation inconnection with the physical mechanism of the AB ef-fect [6]. Novel aspects on the topology and statisticalnature of space and time geometry proposed here wouldstrengthen the outlook on particle physics based on thereal wave equations in [22]. APPENDIX-I
Levi-Civita and Weyl define parallel displacement ofa vector V µ from spacetime point x µ to x µ + δx µ ; thechange of the vector round a small loop is given by δV µ = 12 ( V µ : ν : σ − V µ : σ : ν ) dS νσ (92)Here V µ : ν : σ is second covariant derivative of V µ . In theRiemannian space, using the definition of the Riemanncurvature tensor we have V µ δV µ = 12 R µνσλ V µ V λ dS νσ = 0 (93)since R µνσλ is antisymmetrical in µ and λ . Thus thelength of the vector l = V µ V µ does not change under par-allel transport round the loop in the Riemannian space.In Weyl space, the generalized Christoffel symbol (25)shows that the curvature tensor ∗ R µνσλ has a part sym-metrical in µ and λ [7, 13]. This part gives the noninte-grability of the length I δl = 2 V µ δV µ = W νσ g µλ V µ V λ dS νσ (94) APPENDIX-II
Here we derive Eq.(41) using the variational principle δS = 0 from Eq.(39) considering the variations in A µ treated as independent field variable. We get the follow-ing variations pδ ( F µν F µν √− g ) = − pF µν : ν √− g δA µ (95) ξδ ( A α : α √− g ) = − ξ : µ √− g δA µ (96) ξδ ( A α A α √− g ) = 2 ξA µ √− g δA µ (97)Substituting expression (16) in Eq.(39), and using (95)-(97) we finally arrive at the modified Maxwell field equa-tion (41).Apparently current density (31) in Weyl theory; (41)in generalized Weyl-Dirac theory; and (86) in new elec-tron theory of Dirac seem to be in conflict with the stan-dard CED. On closer examination it is found that there5are physical arguments showing that there is no contra-diction. Recall that in the Maxwell-Lorentz theory thecharge current density four-vector originates from themacroscopic system of charges and currents, however fora superconductor London assumed J proportional to A .In quantum theory the probability current density for aSchroedinger charged particle interacting with the mag-netic field acquires a component proportional to the vec-tor potential A . For a Dirac electron in the presence of the EM field the Dirac current ¯Ψ γ µ Ψ is unaltered, how-ever performing Gordon decomposition it is found thatthe Gordon current is modified containing a term pro-portional to the EM potential A µ . For further details werefer to [42]. ACKNOWLEDGMENT
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