Impedance Representation of the S-matrix:Proton Structure and Spin from an Electron Model

Peter Cameron∗

Strongarm StudiosMattituck, NY USA 11952

(Dated: June 17, 2016)

The possibility of electron geometric structure is studied using a model based upon quantizedelectromagnetic impedances, written in the language of geometric Clifford algebra. The electron isexpanded beyond the point, to include the simplest possible objects in one, two, and three dimen-sions. These point, line, plane, and volume elements, quantized at the scale of the electron Comptonwavelength and given the attributes of electric and magnetic fields, comprise a minimally completePauli algebra of flat 3D space. One can calculate quantized impedances associated with elementaryparticle spectrum observables (the S-matrix) from interactions between the eight geometric objectsof this algebra - one scalar, three vectors, three bivector pseudovectors, and one trivector pseu-doscalar. The resulting matrix comprises a Dirac algebra of 4D spacetime. Proton structure andspin are extracted via the dual character of scalar electric and pseudoscalar magnetic charges.

INTRODUCTION

This paper focuses upon the primary problem of thehigh energy spin physics community, the ongoing failureof QCD point-particle quark models to provide a coherentpicture of nucleon spin[1–4]. It is organized as follows:

• Introduction - outlines the structure of the paper,introduces the impedance representation, and givesa guided tour of the figures.

• Geometric Clifford Algebra and the ImpedanceRepresentation - presents a brief historical accountof the remarkable absence from mainstream QFTof Clifford’s original geometric interpreta-tion, identifies the fundamental geometric objects(FGOs) of the 3D Pauli and 4D Dirac subalgebrasof geometric algebra (GA) with the FGOs of theimpedance representation, and discusses topologi-cal symmetry breaking inherent in the algebras.

• S-matrix and the Impedance Representation -presents a brief historical account of the remark-able absence from mainstream QFT of exactimpedance quantization, and the equivalence ofthe S-matrix and impedance representions of QFT.

• Dark Modes and Symmetry Breaking - mode struc-tures of all the elementary particles are present inthe impedance representation. Particles with darkFGOs (magnetic charge, electric dipole andflux quantum) decay/decohere due to the differ-ing vacuum impedances they excite, and the re-sulting differential phase shifts. The stable protoncontains no dark FGOs, permitting us to pick outits mode structure.

• Proton Structure - mode structure is discussed;transition modes and topological mass generation,

then the stable eigenmodes and their representa-tion of point particle quark models.

• Proton Spin - the simple and exact spin 1/2 quan-tum that emerges from the algebra is identified.

Guided Tour of the Figures:At the outset we proceed beyond point particles by

examining commonality between fundamental geometricobjects (FGOs) of geometric Clifford algebra[5–10] andthe impedance model of the electron[11] (figure 1).

FIG. 1. Shared FGOs of the 3D Pauli subalgebra of geometricClifford algebra [8] and those of the impedance approach togeometric structure of the electron [11]. Bivector and trivec-tor are pseudovector and pseudoscalar of the Pauli algebra.

As the figure shows, electromagnetic duality[12] resultsin magnetic inversion of geometric grade/dimension:

- scalar electric charge - grade 0 point- vector electric moment - grade 1 line- pseudovector magnetic moment - grade 2 area- pseudoscalar magnetic charge - grade 3 volume

All are orientable.These objects are identified with the eight FGOs of a

minimally complete Pauli algebra of space (top and leftof figure 2), and via geometric products (figure 3) gen-erate an impedance representation of their interactionsin the Dirac algebra of flat Minkowski spacetime[8].

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FIG. 2. Impedance representation of the S-matrix. At top and left are the eight FGOs of both the impedance model[11] and aminimally complete Pauli algebra of 3D space - 1 scalar, 3 vectors, 3 bivectors/pseudovectors, and 1 trivector/pseudoscalar [8].The matrix of background independent two-body interactions[13] is generated by geometric products of these FGOs. Matrixelements comprise a 4D Dirac algebra of flat Minkowski spacetime, arranged in even (blue) and odd (yellow) by geometricgrade of the emerging FGOs (the observables). ‘Pauli FGOs’ enter an interaction. ‘Dirac FGOs’ emerge, are the Pauli FGOsentering the next interaction. Impedances of modes indicated by colored symbols are plotted as a function of energy/lengthscale in figure 4. Scale invariant mode impedances (quantum Hall, centrifugal, chiral, Coriolois, three body,...) are associatedwith inverse square potentials. They can do no work, but shift quantum phase, can act as mode couplers.

As explained in what follows, in the model there aretwo each electric dipole/vector and electric flux quan-tum/pseudovector. If we take the resulting eight FGOsat the top of figure 2 to comprise the electron, then inthe manner of the Dirac equation those on the left arethe positron.

We then come back to point particles to set an anchorpoint in the common language of the theorist, namely theS-matrix representation of quantum field theory[14–20],and explain its equivalence with what we are calling theimpedance representation. When you see ‘impedance’,think S-matrix [21]. This permits one to look betweenasymptotically free states of initial and final wavefunc-tions, to look deep inside the black box of Wheeler andHeisenberg’s ‘observables only’ S-matrix through the eyesof both experimentalists and theoreticians.

A portion of the network that results from calculatinginteraction impedances of ‘Pauli FGOs’ when endowedwith electric and magnetic fields is shown in figure 4.The relationship between this representation and the un-stable particle spectrum is established via correlations ofparticle lifetimes (their coherence lengths on the causalboundary of the light cone) with network nodes, whereimpedances are matched and energy flows without re-flection (as required by the decay process) [22–24].

The stable proton is absent from figure 4, which in-cludes only photon, electron, and all the unstable parti-cles. Which is not to say it is absent from figure 2. Ifthat figure is indeed a reasonable first approximation ofnature’s S-matrix, then proton mode structure must bethere. The question is how to identify those modes in themaze of possibilities present in the impedance matrix.

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As shown in figure 1, we see electric charge and mag-netic dipole and flux quantum, but not their duals [12].Magnetic charge [25, 26] and electric dipole and fluxquantum are absent, not visible, ‘dark’. Dark FGOs cou-ple only indirectly to the photon not because they aretoo weak, but rather too strong (figure 5) [11].

The speed of light (or impedance of free space) canbe calculated from excitation of virtual electron-positronpairs (represented in part as the impedance network offigure 4) by the photon[27]. Dark FGOs couple morestrongly (see a different impedance). Modes containingone or more dark FGOs decohere from differential phaseshifts. To identify the mode structure of the proton weneed only consider modes comprised exclusively of visibleFGOs (figure 6), a tremendous simplification.

GEOMETRIC CLIFFORD ALGEBRA ANDTHE IMPEDANCE REPRESENTATION

The impedance approach (IA) to the S-matrix isgrounded in geometric Clifford algebra (GA), the alge-bra of interactions between geometric objects as origi-nally conceived by Grassman and Clifford[5–10]. Withthe early death of Clifford in the late 1800s and ascen-dance of the more simple vector algebra of Gibbs, thepower of geometric interpretation has for the most partbeen lost in modern theoretical physics. While both Pauliand Dirac algebras are subalgebras of GA, their geomet-ric origin went unrecognized by their creators. It wasonly in the 1960s with the work of David Hestenes thatthe power of geometric interpretation was rediscoveredand introduced to physics, as recognized by the Ameri-can Physical Society in awarding him the 2002 OerstedMedal for “Reformulating the Mathematical Language ofPhysics”[9]. Yet even with this endorsement GA remainsobscure, acceptance confoundingly slow.

As shown in figures 1 and 2 and discussedelsewhere[28], IA and GA share the same Pauli FGOs.Six geometric objects, three magnetic and three elec-tric, follow from the electron model. However, the modelyields not one but two electric flux quanta[11, 29].

The first is associated with the magnetic flux quantum(a fundamental constant) and quantization of magneticflux in the photon, which by Maxwell’s equations requiresquantization of electric flux as well. The second followsfrom applying Gauss’s law to the electron charge, andis a factor of 2α smaller, where α is the electromagneticfine structure constant. Similarly, there are not one buttwo electric dipole moments in the model.

Like the wave function, whose ‘reality’ is of interest inquantum interpretations [30], FGOs of the 3D Pauli alge-bra are not observable. Of interest here are impedancesof observables, taken to be impedances of interactionsbetween Pauli FGOs - the mode impedances of the 4DDirac algebra. Or if you will, the S-matrix elements de-

rived from the impedance representation.The matrix elements of figure 2, grouped by geomet-

ric grade, comprise a 4D Dirac algebra of flat Minkowskispacetime, generated by taking geometric products ofthe Pauli FGOs. From these interactions time (relativephase) emerges, and topological symmetry is broken.

Topological Symmetry Breaking in GA

Given two vectors a and b, the geometric product abmixes products of different dimension, or grade (figure3). In the product ab = a · b+ a∧ b, two 1D vectors havebeen transformed into a point scalar and a 2D bivector.

“The problem is that even though we can transformthe line continuously into a point, we cannot undothis transformation and have a function from the pointback onto the line...” [31]. This breaks both topologi-cal and time symmetry, and is presumably true for allgrade/dimension increasing operations. The presence ofthe singularity is implicit, becomes explict when we in-troduce the singularities of the impedance model.

FIG. 3. Geometric algebra components in the 3D Pauli al-gebra of flat space. In GA the term grade is preferred to di-mension, whose meaning in physics is sometimes ambiguousand confused with degrees of freedom. The two products (dotand wedge or inner and outer) that comprise the geometricproduct lower and raise the grade [32]

.

In the above example of the geometric product of twovectors, the number of singularities is not conserved. Inthe impedance model (figure 1) each vector is comprisedof two singularities (those of the magnetic flux quantumof figure 2 are at opposite infinities), for a total of foursingularities entering the geometric product. Emergingfrom the product is a scalar electric charge (one singu-larity) and a pseudovector (none). In the process threesingularities disappear.

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In the impedance model scalars and vectors containsingularities, and their dual pseudovectors and pseu-doscalars do not, a topological distinction between par-ticle and pseudoparticle. It would seem that there aretwo types of topological symmetry breaking in this ex-

ample. One follows directly from dimensional transfor-mations of the geometric product and the other from ap-pearance and disappearance of singularities introducedby the impedance model. They are distinguished by thepresence of an event horizon in the physical model.

FIG. 4. In QFT one is permitted to define but one fundamental length (customarily taken to be the short wavelength cutoff).The impedance approach is finite, divergences being cut off by impedance mismatches as one moves away from the fundamentallength of the model, the electron Compton wavelength. With FGOs of the model confined to that scale by the mismatches,interaction impedances can be calculated as a function of their separation, the ‘impact parameter’. Strong correlation ofthe resulting network nodes with unstable particle coherence lengths[33–37] follows from the requirement that impedances bematched for energy flow between modes as required by the decay process, permitting for instance precise calculation of π0, η,and η′ branching ratios and resolution of the chiral anomaly[23].

S-MATRIX AND THEIMPEDANCE REPRESENTATION

Chapter 11 of Hatfield’s textbook [20] on the quantumfield theory of point particles and strings opens with thisstatement of S-matrix universality:

“One of our goals in solving interacting quantum fieldtheories is to calculate cross sections for scattering pro-cesses that can be compared with experiment. To com-pute a cross section, we need to know the S-matrix el-

ement corresponding to the scattering process. So, nomatter which representation of field theory we work with,in the end we want to know the S-matrix elements. Howthe S-matrix is calculated will vary from representationto representation.”

Barut, in opening his comprehensive introduction[19],asks “What is the meaning of the S-matrix elements?”and answers “It is the transition probability amplitudefrom the initial state i to the final state f. It is in the useof probability amplitudes rather than probabilities thatthe quantum principle enters into the theory.”

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In the process of decohering/collapsing the wave func-tion, the amplitude is extracted and the phase is lost[38]. The use of complex transition probability ampli-tudes permits taking the product of the wave functionwith its conjugate, canceling the phase - the mathemat-ical equivalent of decohering the physical wave function.Normalized this delivers the probability. In GA the Diracalgebra is a real algebra, and phase information is con-tained in the pseudoscalar I .

Impedance may be defined as the amplitude and phaseof opposition to the flow of energy. Whereas the S-matrix is comprised of complex probability amplitudesand phases, the impedance matrix is comprised of thatwhich governs those amplitudes and phase shifts. Theessential point, missing from QFT and crucially relevantin models and theories of quantum interactions, is this:Impedances are quantized. Yet how, if impedancequantization is both fact of nature and powerful theoret-ical tool, is it not already present in the Standard Model?

This absence is most remarkable. Impedance is a fun-damental concept, universally valid. Impedance match-ing governs the flow of energy. The oversight can beattributed primarily to three causes. The first is histori-cal [24], the second follows from the penchant of particlephysicists to set fundamental constants to dimensionlessunity, and the third from topological and electromagneticparadoxes in our systems of units [11, 29, 39].

The first is a simple historical accident, a consequenceof the order in which experimentalists revealed relevantphenomena. The scaffolding of QFT was erected on ex-perimental discoveries of the first half of the twentiethcentury, on the foundation of QED, which was set longbefore the Nobel prize discovery of the scale invariantquantum Hall impedance in 1980 [40]. Prior to thatimpedance quantization was more implied than explicitin the literature [41–47]. The concept of exact impedancequantization did not exist.

A more prosaic second cause is the habit of parti-cle physicists to set fundamental constants to dimension-less unity. Setting free space impedance to dimensionlessunity made impedance quantization just a little too easyto overlook. And to no useful purpose. What mattersare not absolute values of impedances, but rather theirrelative values, whether they are matched.

The third confusion is seen in an approach [43] sum-marized [44] as “...an analogy between Feynman dia-grams and electrical circuits, with Feynman parametersplaying the role of resistance, external momenta as cur-rent sources, and coordinate differences as voltage drops.Some of that found its way into section 18.4 of...” thecanonical text [45]. As presented there, the units of theFeynman parameter are [sec/kg], the units not of resis-tance, but rather mechanical conductance [48].

It is not difficult to understand what led us astray[13, 43, 49–51]. The units of mechanical impedance are[kg/sec]. One would think that more [kg/sec] would mean

more mass flow. However, the physical reality is more[kg/sec] means more impedance and less mass flow. Thisis one of many interwoven mechanical, electromagnetic,and topological paradoxes [39] to be found in the SI sys-tem of units, which ironically were developed with theintent that they “...would facilitate relating the standardunits of mechanics to electromagnetism.” [52].

With the confusion that resulted from misinterpret-ing conductance as resistance and lacking the concept ofquantized impedance, the anticipated intuitive advantage[45] of the circuit analogy was lost. The possibility of thejump from a well-considered analogy to a photon-electronimpedance model was not realized at that time.

Had impedance quantization been discovered in 1950rather than 1980, one wonders whether it might havefound its way into the foundation of QED at that time,before it was set in the bedrock. As it now stands the in-evitable reconciliation of practical and theoretical, the in-corporation of impedances into the foundations of quan-tum theory, opens new and exciting possibilities.

Transformation between impedance and scattering ma-trices is standard fare in electrical engineering[21, 53, 54].There is nothing particularly difficult or mysteriousabout this. As we endeavor to make clear in this pa-per, when seeking to understand details of the elemen-tary particle spectrum significant advantages accrue forthe physicist working in the impedance representation.

FIG. 5. Inversion of the fundamental lengths of figure 4 bymagnetic charge [55], with the magnetic singularity removedto infinity by the Dirac string [25].

DARK MODES AND SYMMETRY BREAKING

Much of the structure we observe in the physical worldis organized around four fundamental interaction scales,ordered in powers of α - inverse Rydberg, Bohr ra-dius, Compton wavelength, and classical electron radius(figure 4). The Compton wavelength λe = h/mec con-tains no charge, is the same for both magnetic and elec-tric charge (figure 5). However, substituting magnetic

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charge for electric via the Dirac relation 2eg = h invertsthe scaling of the remaining fundamental lengths [55].

With electric charge the lengths correspond to specificphysical mechanisms of photon emission or absorption,matched in quantized impedance and energy. Inversionresults in mismatches in both energy and impedance.Magnetic charge cannot couple directly to the photon

- not despite its great strength, but rather because of it.Consequently the Dirac monopole is dark, cannot cou-

ple to the photon. The Bohr radius cannot be inside theCompton wavelength, Rydberg inside Bohr,... Specificphysical mechanisms of photon emission and absorptionno longer work. Related arguments can be advanced forthe electric flux quantum and moment of figure 1.

FIG. 6. Modes lacking dark Pauli FGOs are highlighted, correspond to the transition (yellow) and eigenmodes (blue) of thestable proton. Unstable particles contain at least one dark FGO, the proton none. The differing interaction of dark and visibleFGOs with the vacuum (essentially the virtual electron impedance network) determines the differing impedances they see [27].This generates differential phase shifts, resulting in decoherence of unstable particles at impedance nodes (figure 4). Thematrix is arranged in even (blue) and odd (yellow) by geometric grade of the emerging Dirac FGOs (the observables).

The electron model presented here starts with maximalelectric-magnetic symmetry[12] in the 3D Pauli algebra ofphysical space. Electric and magnetic FGOs are taken tobe duals. Scalar and pseudoscalar are duals, as are vectorand pseudovector. The inversion of fundamental interac-tion scales of figures 4 and 5 suggests that the dualityis both electromagnetic and topological. Given that wedefine magnetic charge via the Dirac relation, which it-

self breaks topological symmetry, it is not surprising tofind other manifestations of this symmetry breaking.

For example, the magnetic flux quantum φ = h2e and

magnetic charge as defined by the Dirac relation g = h2eare numerically equal, but topologically distinct [29].

Topological character is also suggested by the inversionof units of mechanical impedance - more [kg/sec] meansmore impedance and less mass flow.

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As mentioned earlier, there are additional electromag-netic symmetry breakings. There is only one magneticflux quantum, but two electric flux quanta. One magneticmoment, but two electric moments. How these might berelated to topology is not yet clear.

In what follows the distinction between dark modes(whose presence dominates the impedance matrix of fig-ure 6) and visible modes is utilized to identify the modestructure of the proton. With that and the symmetrybreakings in hand, we seek to provide mechanisms fortopological mass generation and possibilities for investi-gating proton structure and spin [1],...

PROTON MODE STRUCTURE

The electron is not a point particle. It gives that ap-pearance if one doesn’t appreciate the possibility thatelectron geometric structure, when endowed with elec-tric and magnetic fields and excited by the photon, mightgenerate the remainder of the massive particle spectrum.By far the lightest of all charged elementary particles,the electron impedance network is the natural candidatefor this role [56], in some sense might be considered thestructure of the vacuum [27]. We seek to understand de-tails of how the stable proton emerges from excitationof that network (elsewhere we explore how a related ap-proach sheds light upon the early Big Bang [57]).

To sort out the dynamics of the full impedance/S-matrix is a formidable computational task. Theimpedance network of figure 4 is non-linear (plot is log-log) and presents only a small subset of the modes of fig-ure 6. Scale-dependent impedances open the possibilityof noiseless parametric mixing and amplification (whoseconnection to topological mass generation remains to beexplored) [58, 59]. Topological effects in general are notclearly understood. Within these complexities one mustiterate mode compositions, orientations, couplings andphases. The problem is far beyond resources available(particularly to independent researchers) for the presentpurpose.

However, restricting attention to modes containingonly the ‘visible’ FGOs of figure 1 gives us both tran-sition modes and eigenmodes of the proton (the onlyknown particle absent from figure 4 - more on neutri-nos later), resulting in tremendous simplification. Modescontaining visible FGOs only are highlighted in green infigure 6.

The FGOs entering the geometric products to generatethe transition modes are shown in figure 7, as well asgrades of the FGOs emerging from the products and theircorresponding identities in the impedance representation.

Transition Modes and Topological Mass

In the impedance approach there are two ways tocalculate electron mass - from electromagnetic field en-ergy of modes of the electron model [29], and from theimpedance mismatch to the event horizon at the Plancklength [60, 61]. Both methods are correct at the part-per-billion limit of experimental accuracy. Both requireprior knowledge of the electron Compton wavelength, theinput-by-hand fundamental length of the model.

Similarly, one can use either or both methods to cal-culate proton mass. And both require knowledge of theproton Compton wavelength, not a given in the model.The problem is how one makes the jump from electronCompton wavelength to that of the proton. This is wheretopological mass generation enters.

The muon mass calculation of the impedance approachagrees with experiment at one part per thousand, thepion at two parts per ten thousand, and the nucleonat seven parts per hundred thousand [29]. The muonand pion masses are calculated from field energies of fluxquanta confined to the electron Compton wavelength.The nucleon calculation exploits the topological differ-ence between Bohr magneton and flux quantum.

“It has been suggested that the origin of mass is some-how related to spin [13]. After the neutron, the next moststable particle is the muon. If we take the muon as a plat-form state [35] for the nucleon, in terms of spin-relatedphenomena we return here to the notion that the fluxquantum is similar to a magnetic moment with no returnflux, and consider the ratio of the magnetic flux quantumto the muon Bohr magneton

ratioµ =φB

µµBohr

The nucleon mass can then be calculated as

mnucleonCalc =

√2

2· e2 · ratioµ

where the√22 term might be regarded as a projection op-

erator. Taking the measured nucleon mass to be the av-erage of the proton and the neutron, we then have thecalculated nucleon mass accurate to seven parts in onehundred thousand.”[29]

Topological mass generation is a phenomenon in 2+1dimensions in which Yang-Mills fields acquire mass uponthe inclusion of a Chern-Simons term in the action[62], the essential point being that this happens with-out breaking gauge invariance, without losing quantumphase coherence. The phase shift of the added mass iscompensated by that of the Chern-Simons term (whosemode impedance is scale invariant, and therefore shiftsphase without emitting or absorbing energy).

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In figure 7 the Chern-Simons term φBe is the quan-tum Hall impedance of the charge ‘orbiting’ in the fieldof the flux quantum, the charge being driven by the elec-tromagnetic fields of the impinging photon. The two spinzero (vectors have no spin) flux quanta φB are indistin-guishable bosons, can be taken to couple the bivector(GA equivalent of a Yang-Mills axial vector) Bohr mag-neton µB to the charge scalar e.

FIG. 7. Transition modes of figure 6 having only ‘visible’FGOs entering the geometric products, and showing grades ofemerging FGOs and the corresponding electromagnetic FGOsof the impedance model.

FGOs entering geometric products of the transitionmodes (left column of figure 7) number one scalar, twovectors, and one bivector. These comprise a minimally

complete geometric algebra in two spatial dimensions.Their geometric products yield two vector flux quantaφB and the pseudoscalar magnetic charge g. With thepseudoscalar we’ve gained a dimension. Via the inter-actions we have the 2+1 dimensions of topological massgeneration [62]. This suggests the pseudoscalar can beidentified with time, perhaps defines relative phases.

At the scale of the .511 MeV electron Compton wave-length there exist modes of the electron impedance modelthat are shifted in energy by powers of α, a consequenceof nodes of the impedance network being arranged insuch powers. Scalar Lorentz coupling of emergent mag-netic charge g to flux quantum φB (rightmost column offigure 7) yields a route to the 70 MeV mass quantum,and a few pages later the muon mass [29].

In accord with that calculation, if one takes µB en-tering the interaction (leftmost column of figure 7) tobe not the electron Bohr magneton but rather that ofthe muon and φB to be similarly confined to the muonCompton wavelength, then the energy of the bivectormagneton in the field of the vector flux quantum, theenergy of the φBµB transition mode, is the muon mass.

FIG. 8. Impedance network of muon-proton topological mass generation. Horizontal scale is photon wavelength/energy,logarithmic in powers of the fine structure constant α. Energy step from muon to proton is ∼

√2α.

One might suppose this a recipe for muon making.Muon lifetime/decoherence then derives from the dif-fering impedances/phase shifts seen by the numericallyequal but topologically distinct pseudoscalar charge gand vector flux quanta φB, the subtle topological dis-tinction perhaps accounting for the exceptionally longmuon lifetime.

According to this recipe, if one turns up the flameand continues cooking, given sufficient heat the protonemerges. How energy is transferred is shown in fig-ure 8. The numerical identity between topologicallydistinct flux quantum φB and charge g is pivotal here.The 1027 ohm green line corresponds to the scale invari-ant Chern-Simons impedance of the three-body φBgφB

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mode. It intersects the impedance node at the (log-arithmic) midpoint between muon and proton. Alsoimpedance matched at the node are the near field 105MeV muon electric flux quantum, and the Coulomb andmagnetic moment impedances of the proton. Coupling ofenergy from muon to proton is via the impedance matchbetween the near field impedances of the muon electricflux quantum and proton magnetic moment bivectors andCoulomb scalars.

The point here is that the proton magnetic momentimpedance plotted in the figure corresponds to the exper-imentally measured proton gyromagnetic ratio. Withoutthe anomalous portion of the proton magnetic moment,topological mass generation doesn’t work. As shown inthe figure, the impedance corresponding to the anomaly-free nuclear Bohr magneton is that which matches thenear field electrical impedance of the 938 MeV protonelectric flux quantum (which is yet a few zeptosecondsin the future of topological mass generation), not that ofthe muon. The anomaly is essential.

However, the 938 MeV proton-mass µBµB mode(figure 9) is not that of the measured moment, butrather the anomaly-free theoretical nuclear magneton!This suggests that the anomaly originates not in the pro-ton, but rather in the transition excitation/measurement.

Proton Eigenmodes

The eigenmode Dirac FGOs emerging from the geo-metric products (figure 9) number three scalars, twobivectors, and one pseudoscalar - an even subalgebra ofthe Dirac algebra, itself again a Pauli algebra.

FIG. 9. Eigenmodes of figure 6 having only visible PauliFGOs entering the geometric products, and showing grades ofemerging Dirac FGOs and the corresponding electromagneticFGOs of the impedance model.

The connection of the emergent three scalars withquarks seems obvious. The only scalar in our model iselectric charge. Given that the top and left Pauli al-gebras of figure 6 correspond to electron and positron‘wave functions’, then all three scalars follow from threeparticle-antiparticle geometric products (ee, φBφB , andµBµB), one for each of the three grades entering theproducts. All are found on the diagonal of the matrixof figure 6.

Also prominent on both the diagonal and theimpedance network of figure 4 is the Coulomb modegg of magnetic charge, part of the mode structure of thesuperheavies (top, Higgs, Z, W,...).

The first ‘quark’, the scalar e emerging from the chargepair ee, is unaccompanied. One wonders if it is in anyobservable way different from the second, emerging fromthe φBφB interaction in the company of the pseudovec-tor µB, or whether they are distinguishable from thethird, emerging from µBµB along with pseudoscalar I .

The two bivector pseudovectors µB emergingfrom the geometric products φBφB and µBe might beidentified with axial vectors of Yang-Mills theory. Asmentioned in the previous sub-section, the 938 MeV pro-ton rest mass of the emergent coupled µBµB mode cor-responds to the interaction energy not of the measuredmagnetic moment, but rather the g=2 gyromagnetic ra-tio of the nuclear magneton.

The grade-4 pseudoscalar I = γ0γ1γ2γ3 definesspacetime orientation as manifested in the phases, withγ0 the sign of time orientation. The γµ are orthogonalbasis vectors in the Dirac algebra of flat 4D Minkowskispacetime, not matrices in ‘isospace’. [10]

There are no gluons or weak vector bosons to bind theconstituents. The modes are confined by the impedancemismatches, by reflections as one moves away from thequantization scales as defined by the impedance nodes.Mismatches also remove infinities associated with singu-larities. The impedance approach is finite and confined.

Proton Spin

Neither scalar (one singularity) nor vector (two) hasintrinsic spin, but rather only the bivector (and possiblyhigher grade geometric objects), taken in the literatureto be a magnetic flux quantum and given the attributeof a spin 1/2 fermion [63]. However magnetic geometricgrades are inverted relative to electric by the topologi-cal duality. It is not magnetic flux quantum, but rathermagnetic moment, that is to be identified with the bivec-tor spin 1/2 fermion, an assignment in agreement withJackson as well [64] (who persisted in calling it a dipoledespite the absence of poles/singularities).

If one takes that moment to be in some sense not avector dipole but rather a pseudovector dipole comprisedof two pseudoscalar magnetic charge volume elements,then the proton angular momentum controversy arisingfrom trying to ‘locate’ the intrinsic exact half-integer spin[4] need no longer be portioned out to various inexactorigins, but rather might find resolution in the diffusesingularity-free character of such a magnetic moment.

To understand dynamics of proton spin more deeplywill likely require further application of GA, and partic-ularly the rotor, to the impedance model.

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SUMMARY AND CONCLUSION

The serendipitous commonality of fundamental geo-metric objects between the impedance model and geo-metric Clifford algebra lends a formal structure to theimpedance approach that maximizes the utility of both,providing simple yet powerful mathematical tools to thephysicist and physical intuition to the mathematician.

Thus far applications of generalized quantumimpedances have been primarily conceptual, limitedto theoretical particle physics, quantum gravity, andquantum information theory. Sage advice [65] suggeststhat the most fertile field for impedances will be incondensed matter - in atomic, molecular, and opticalphysics, and particularly in superconductivity. If there ispractical value in this, AMO is the place where it will befound. Though harking back to Wheeler [14], impedancematching might prove equally useful in understandingboth fission and fusion.

ACKNOWLEDGEMENTS

The author thanks

• Michaele Suisse for many helpful discussions andliterature searches/compilations/networking/...

• John Nees of the University of Michigan ultrafasthigh energy laser lab for helpful discussions andreferences,

• David Hestenes for the amazing gift of geometricClifford algebra, and

• Yannis Semertzidis for ideas and encouragementduring early phases of this work.

We are but protons, neutrons, and electrons. How thisis possible will ever be the mystery of infinite gratitude.

∗ electronGaugeGroup@gmail.com

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[3] C. Aidala et.al., “The Spin Structure of the Nucleon”,RMP 85 655-691 (2013)http://arxiv.org/abs/1209.2803

[4] E. Leader, “On the controversy concerningthe definition of quark and gluon angular mo-mentum”, Phys. Rev. D 83 096012 (2011)

http://arxiv.org/pdf/1101.5956v2.pdf

[5] H. Grassmann, Lineale Ausdehnungslehre (1844)

[6] H. Grassmann, Die Ausdehnungslehre, Berlin (1862)

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[9] D. Hestenes, “Oersted Medal Lecture 2002: Reformingthe mathematical language of physics”, Am. J. Phys.71, 104 (2003) http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

[10] C. Doran and A. Lasenby, Geometric Algebra forPhysicists, Cambridge University Press (2003)

[11] P. Cameron, “Electron Impedances”, Apeiron 18 2 222-253 (2011 ) http://redshift.vif.com/JournalFiles/V18NO2PDF/V18N2CAM.pdf

[12] E. Witten, “Duality, Spacetime, and Quantum Mechan-ics”, Physics Today, p.28-33 (May 1997)

[13] P. Cameron “The Two Body Problem and Mach’sPrinciple”, submitted to Am. J. Phys. (1975), in re-vision. The original was published as an appendix to [11].

[14] J. Wheeler, “On the Mathematical Description of LightNuclei by the Method of Resonating Group Structure”,Phys. Rev. 52 1107-1122 (1937)

[15] W. Heisenberg, “Die beobachtbaren Grossen in derTheorie der Elementarteilchen. I”, Z. Phys. 120 (710):513-538 (1943)

[16] W. Heisenberg, “Die beobachtbaren Grossen in derTheorie der Elementarteilchen. II”. Z. Phys. 120 (1112):673-702 (1943)

[17] W. Heisenberg, “Die beobachtbaren Grossen in derTheorie der Elementarteilchen. III”. Z. Phys. 123 93-112(1944)

[18] G. Chew, S-matrix Theory of Strong Interactions, (1961)

[19] A.O. Barut, The Theory of the Scattering Matrix forInteractions of Fundamental Particles, McMillan (1967)

[20] D. Hatfield, Quantum Field Theory of Point Particlesand Strings, Addison-Wesley (1992)

[21] for those unfamiliar with the scattering matrix, asimple introduction to a two port network showing bothimpedance and scattering representations is availablehttp://www.antennamagus.com/database/

utilities/tools_page.php?id=24&page=

two-port-network-conversion-tool

[22] P. Cameron, “Generalized Quantum Impedances: ABackground Independent Model for the Unstable Parti-

mailto:electronGaugeGroup@gmail.comhttp://arxiv.org/abs/hep-ph/0411005 http://arxiv.org/abs/1209.2803http://arxiv.org/pdf/1101.5956v2.pdfhttp://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdfhttp://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdfhttp://redshift.vif.com/JournalFiles/V18NO2PDF/V18N2CAM.pdfhttp://redshift.vif.com/JournalFiles/V18NO2PDF/V18N2CAM.pdfhttp://www.antennamagus.com/database/utilities/tools_page.php?id=24&page=two-port-network-conversion-toolhttp://www.antennamagus.com/database/utilities/tools_page.php?id=24&page=two-port-network-conversion-toolhttp://www.antennamagus.com/database/utilities/tools_page.php?id=24&page=two-port-network-conversion-tool

11

cles” (2012) http://arxiv.org/abs/1108.3603

[23] P. Cameron, “An Impedance Approach to the ChiralAnomaly” (2014) http://vixra.org/abs/1402.0064

[24] P. Cameron, “Historical Perspective on the ImpedanceApproach to Quantum Field Theory” (Aug 2014)http://vixra.org/abs/1408.0109

[25] P. Dirac, “Quantized Singularities in the Electromag-netic Field”, Proc. Roy. Soc. A 133, 60 (1931)

[26] P. Dirac, “The Theory of Magnetic Poles”, Phys. Rev.74, 817 (1948)

[27] M. Urban et.al, “The quantum vacuum as the origin ofthe speed of light”, Eur. Phys. J. D 67:58 (2013)

[28] P. Cameron, “E8: A Gauge Group for the Electron?”(Oct 2015) http://vixra.org/abs/1510.0084

[29] P. Cameron, “Magnetic and Electric Flux Quanta: thePion Mass”, Apeiron 18 1 p.29-42 (2011)http://redshift.vif.com/JournalFiles/V18NO1PDF/

V18N1CAM.pdf

[30] M. Suisse and P. Cameron, “Quantum Interpretationof the Impedance Model”, accepted for presentation atthe 2014 Berlin Conf. on Quantum Information andMeasurement. http://vixra.org/abs/1311.0143

[31] R. Conover, A First Course in Topology: An Introduc-tion to Mathematical Thinking, p.52, Dover (2014)

[32] A useful visual introduction to geometric algebra can befound here https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/

[33] M. H. MacGregor, “The Fine-Structure Constant asa Universal Scaling Factor”, Lett. Nuovo Cimento 1,759-764 (1971)

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[35] M. H. MacGregor, The Power of Alpha, World Scientific(2007) see also http://137alpha.org/

[36] C. Capps, “Near Field or Far Field?”, Elec-tronic Design News, p.95 (16 Aug 2001) http://edn.com/design/communications-networking/

4340588/Near-field-or-far-field-

[37] The mathcad file that generates the impedance plots isavailable from the author.

[38] P. Cameron, “Quantum Impedances, Entanglement, andState Reduction” (2013)http://vixra.org/abs/1303.0039

[39] P. Cameron, “The ‘One Slide’ Introduction to General-ized Quantum Impedances”, p.42-43http://vixra.org/abs/1406.0146

[40] K. von Klitzing et.al, “New method for high-accuracydetermination of the fine-structure constant based onquantized Hall resistance”, PRL 45 6 494-497 (1980)

[41] J.L. Jackson and M. Yovits, “Properties ofthe Quantum Statistical Impedance”, Phys.Rev. 96 15 (1954) http://www.deepdyve.com/lp/american-physical-society-aps/

properties-of-the-quantum-statistical-impedance-8qnQk0mK33

[42] R. Landauer, “Spatial Variation of Currents andFields Due to Localized Scatterers in Metal-lic Conduction”, IBM J. Res. Dev. 1 223 (1957)http://citeseerx.ist.psu.edu/viewdoc/download?

doi=10.1.1.91.9544&rep=rep1&type=pdf

[43] J. Bjorken, “Experimental tests of quantum electrody-namics and spectral representations of Green’s functionsin perturbation theory”, Thesis, Dept. of Physics, Stan-ford University (1959) http://searchworks.stanford.edu/view/2001021

[44] J. Bjorken, private communication (2014)

[45] J. Bjorken, and S. Drell, Relativistic Quantum Fields,McGraw-Hill, section 18.4 (1965)

[46] R. Feynman and F. Vernon, “The Theory of a GeneralQuantum System Interacting with a Linear Dissipa-tive System”, Annals of Physics 24 118-173 (1963)http://isis.roma1.infn.it/~presilla/teaching/

mqm/feynman.vernon.1963.pdf

[47] R. Landauer, “Electrical Resistance of DisorderedOne-dimensional Lattices”, Philos. Mag. 21 86 (1970)

[48] N. Flertcher and T. Rossing, The Physics of MusicalInstruments, 2nd ed., Springer (1998)

[49] C. Lam, “Navigating around the algebraic jungle ofQCD: efficient evaluation of loop helicity amplitudes”,Nuc. Phys. B 397, (12) 143172 (1993)

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[51] D. Huang, “Consistency and Advantage of Loop Regu-larization Method Merging with Bjorken-Drells AnalogyBetween Feynman Diagrams and Electrical Circuits”,EJP C, (2012) http://arxiv.org/abs/1108.3603

[52] M. Tobar, “Global Representation of the Fine StructureConstant and its Variation”, Meterologia 42 p.129-133(2005)

[53] https://en.wikipedia.org/wiki/Scattering_parameters

[54] https://en.wikipedia.org/wiki/Impedance_parameters

[55] T. Datta, “The Fine Structure Constant, MagneticMonopoles, and the Dirac Quantization Condition”,Lettere Al Nuovo Cimento 37 2 p.51-54 (May 1983)

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12

[56] P. Cameron, “Quantizing Gauge Theory Gravity”,Barcelona conference on applications of geometricClifford algebra (2015) http://www-ma2.upc.edu/agacse2015/3641572286-EP/

also available at http://vixra.org/abs/1506.0215

[57] P. Cameron, “The First Zeptoseconds: An ImpedanceTemplate for the Big Bang (2015)http://vixra.org/abs/1501.0208

[58] W. Louisell, “Coupled Modes and Parametric Electron-ics”, Wiley (1960)

[59] B. Zeldovich, “Impedance and parametric excitation ofoscillators”, Physics-Uspekhi 51 (5) 465-484 (2008)

[60] P. Cameron, “Background Independent Relations be-tween Gravity and Electromagnetism”http://vixra.org/abs/1211.0052

[61] P. Cameron, “A Possible Resolution of the Black HoleInformation Paradox”, Rochester Conference on Quan-tum Optics, Information, and Measurement (2013)http://www.opticsinfobase.org/abstract.cfm?URI=

QIM-2013-W6.01

also available at http:vixra.org/abs/1304.0036

[62] S. Deser, R. Jackiw, and S. Templeton, “Three-Dimensional Massive Gauge Theories”, PRL 48 975(1982)

[63] F. Wilczek, “Fractional Statistics and Anyon Supercon-ductivity”, p.7 World Scientific (1990).

[64] J.D. Jackson, “The Nature of Intrinsic Magnetic DipoleMoments”, CERN 77-17 (1977).

[65] G. Kane, private communication (2014)

http://www-ma2.upc.edu/agacse2015/3641572286-EP/http://www-ma2.upc.edu/agacse2015/3641572286-EP/http://vixra.org/abs/1506.0215http://vixra.org/abs/1501.0208http://vixra.org/abs/1211.0052http://www.opticsinfobase.org/abstract.cfm?URI=QIM-2013-W6.01http://www.opticsinfobase.org/abstract.cfm?URI=QIM-2013-W6.01http:vixra.org/abs/1304.0036

Impedance Representation of the S-matrix: Proton Structure and Spin from an Electron ModelAbstractintroductiongeometric clifford algebra and the impedance representationTopological Symmetry Breaking in GA

S-matrix and the impedance representationdark modes and symmetry breakingProton Mode Structure Transition Modes and Topological MassProton EigenmodesProton Spin

Summary and ConclusionAcknowledgementsReferences