The volume of about 10^80 Muon orTaun Compton Radius Vortices is roughly the volume of ourUniverse at the time of the ElectroWeakPhase Transition.
D. Lynden-Bell,in A Magic Electromagnetic Field, astro-ph/0207064, says: "... Anelectromagnetic field of simple algebraic structure is simplyderived. It turns out to be the G = 0 limit of the charged rotatingKerr-Newman metrics. Theseall have gyromagnetic ratio 2, the same as the Dirac electron. Thecharge and current distributions giving this high gyromagnetic ratiohave charges of both signs rotating at close to the velocity oflight. It is conjectured that something similar may occur in thequantum electrodynamic charge distribution surrounding the pointelectron. ... Classical models of the electron had a problem over thegyromagnetic ratio. Even if all the charge were confined to a ringrotating at close to the velocity of light the magnetic momentgenerated gives a gyromagnetic ratio of one rather than theelectron's value of 2.0023193044. It is of some interest to gain anunderstanding as to how the Kerr-Newmanmetric does it. The answer is that the charge distribution is notall of one sign. In fact a circular current dipole of two rings ofopposite charge rotating uniformly about their common axis gives anet magnetic moment but no net charge. The way our electromagneticfield gets its large magnetic dipole moment per unit net charge isthat its much larger internal charges are of opposite signs butrotate together giving a magnetic dipole with relatively little netcharge. We show elsewhere that this is a characteristic ofrelativistically rotating conductors! ...".
H. I. Arcos andJ. G. Pereira, in Kerr-Newman solution as a Dirac particle,hep-th/0210103, say: "... For m^2 < a^2 + q^2 , with m, a, andq respectively the source mass, angular momentum per unit mass, andelectric charge, the Kerr-Newman (KN) solution of Einstein's equationreduces to a naked singularity of circular shape, enclosing a diskacross which the metric components fail to be smooth. ... similarlyto the electron-positron system, this solution presents fourinequivalent classical states. ... due to its topological structure,the extended KN spacetime does admit states with half-integralangular momentum. ... under a rotation of the space coordinates,those inequivalent states transform into themselves only after a 4 pirotation. ... The state vector representing the whole KN solution isthen constructed, and its evolution is shown to be governed by theDirac equation. The KN solution can thus be consistently interpretedas a model for the electron-positron system, in which the concepts ofmass, charge and spin become connected with the spacetime geometry.... for symmetry reasons, the electric dipole moment of the KNsolution vanishes identically, a result that is within the limits ofexperimental data ... a and m are thought of as parameters of the KNsolution, which only asymptotically correspond respectively toangular momentum per unit mass and mass. Near the singularity, arepresents the radius of the singular ring, which according to Carteris unobservable. The above "renormalization" of the KN parameters...[is]... necessary to maintain the internal angularmomentum constant. As a consequence, to a higher velocity, theremight correspond a smaller radius of the singular ring. With thisrenormalization, it is a simple task to verify that, for the usualscattering energies, the resulting radius is below the experimentallimit for the extendedness of the electron. According to thesearguments, therefore, the electron extendedness will not show up inhigh-energy scattering experiments. This extendedness will show uponly in low energy experiments, where the electrons move at lowvelocities. ... Charge ... is interpreted as arising from themulti-connectedness of the spatial section of the KN solution. ...spin can be consistently interpreted as an internal rotational motionof the singular ring. ... this model can provide explanations...[of]... not well-understood phenomena of solid statephysics, as for example the fractional quantum Hall effect. ...".
D. Ranganathan, inA self consistent solution to the Einstein Maxwell Dirac Equations,gr-qc/0306090, says: "... A self consistent solution to Diracequation in a Kerr Newman space-time with M^2 > a^2 + Q^2 ispresented for the case when the Dirac particle is the source of thecurvature and the electromagnetic field. The solution is localised,continuous everywhere and valid only for a special choice of ... theparameters (me, Q, a) appearing in the Dirac equation. ... Such asolution corresponds to a generalisation of the free particle Diracequation in Minkowski space to include the effects of the curvatureproduced by the particle. The ordinary free particle solutions of theDirac equation are completely delocalised; the curvature however, nowcauses the Dirac wave functions to be localised over a regioncomparable in dimension to the Compton wavelength of the particle.Note that the wave function still has an enormous spread incomparison to the dimensions of the event or Cauchy horizons of theparticle. As is usually the case in single particle Dirac theory weignore the self interaction between the particle and its ownelectromagnetic field. Further we assume that we need only to lookfor solutions with zero azimuthal angular momentum about the rotationaxis. ... if the fields are centered at r = 0, it is not obvious whatit means to have a finite probability of finding the source at r =/=0. I suggest that this can be explained in a manner similar to theexplanation for the Klein paradox of Dirac theory. Namely, in theregion of the order of the Compton wavelength within which the wavefunction peaks, the field strengths are so large that virtual paircreation cannot be neglected. The antiparticle thus produced canrecombine with the "original" particle at r = 0 leaving us to detectthe particle now at r =/= 0. ... The solution ... is normalisable andthus represents a proper Dirac spinor wave function. There is ofcourse a second possibility due to the bifurcations at the horizon,which is to take the complex conjugate form inside the ergosphere.This however will give rise to the same Dirac currents etc., outsideand is thus indistinguishable form the form presented here. Nextconsider the question of the only allowed energy eigenvalue, zero. Inany bound state problem, we expect the energy of the state to be lessthan or equal to the asymptotic value of the binding potential. Inthis case if we consider the curvature to be producing the attractiveforce localising the particle close to r = 0; then the zero energyeigenvalue should evoke no surprise. ...".
Consider this statement by Bert Schroer in hep-th/9908021:
As Streater and Wightman say in PCT, Spin, Statistics, and AllThat (W. A. Benjamin 1964): "... From Theorem 4-15 [theJost-Schroer theorem] we get Theorem 4-16 (Haag's Theorem) ...Haag's Theorem is very inconvenient; it means that the interactionpicture exists only if there is no interaction. ...".
Schroer goeson to say: "... It is the constructive use of such nonlocal objectswhich is responsible for the disappearance of the ultravioletdivergency problem and together with it the short-distance aspects ofthe renormalization problem. ... [Polarization FreeGenerators] ... generalize the free field structure into acontrollable nonlocal direction with interactions, i.e. they areauxiliary quantities in the construction of local theories whichexist all the time in the original Hilbert space which is also theliving space of the more local operator algebras which they generate.With other words, although they are nonlocal ... and in some sensecontain a cut-off aspect, these properties are not ad hoc, and as aconsequence no limiting process for cutoff removal is required. Theirexistence is crucial for the linkage of the particle physics crossingsymmetry with the thermal and entropical aspects of QFT which werefirst noticed in the Bekenstein-Hawking-Unruh properties of Killinghorizons in black hole physics. These "classical" thermal propertiesin CST have a quantum counterpart in which bifurcated Killinghorizons are substituted by surfaces of Minkowski space localizationregions e.g. the light cone surface of a double cone. In fact thegeometric Killing symmetry in the quantum setting passes to the(geometrically) hidden quantum symmetry defined by the modular groupcorresponding to the concrete situation. In Unruh's case of a wedgeregion or in the analogous case of conformal matter enclosed in adouble cone, the hidden quantum symmetry passes to the one describedby a Killing vector associated with the Lorentz or conformal- group....".
Further, note that Diracmay have anticipated key aspects of the Compton Radius Vortex modelin 1938, as shown in this quote from pages 194-195 of Dirac: AScientific Biography, by Helge Kragh (Cambridge 1990): "... "... Itwould appear here that we have a contradiction with elementary ideasof causality. The electron seems to know about he pulse before itarrives and to get up an acceleration (as the equations of motionallow it to do), just sufficient to balance the effect of the pulsewhen it does arrive." Dirac seemed to accept this pre-acceleration asa matter of fact, necessitated by the equations, and did not discussit further. However, Dirac explained that the strange behaviorof electrons in this theory could be understood if the electron wasthought of as an extended particle with a nonlocal interior. Hesuggested that the point electron, embedded in its ownradiation field, be interpreted as a sphere of radius a, wherea is the distance within which an incoming pulse must arrive beforethe electron accelerates appreciably. With this interpretation heshowed that it was possible for a signal to be propagated faster thanlight through the interior of the electron. He wrote: "The finitesize of the electron now reappears in a new sense, the interior ofthe electron being a region of failure, not of the field equations ofelectromagnetic theory, but of some of the elementary properties ofspace-time." In spite of the appearance of superluminalvelocities, Dirac's theory was Lorentz-invariant. ...".
Gaja/Ganesha PhysicalCompton Vortex Elementary Particles,
GravitoEM Static RegionGaja/Ganesha ComptonVortex Phenomena, and
GravitoEM Induction RegionGaja/Ganesha ComptonVortex Phenomena.
Compton Radius Vortices look like DolphinBubble-Structures:
there are superluminal velocities, negative energy solutions(antiparticles), and nonHermitian operators.
In the D4-D5-E6 physics model theelectron is a spin 1/2 fundamental particle carrying electromagneticcharge, and the
According to B. G. Sidharth in quant-ph/9808020,quant-ph/9808031,and quant-ph/9805013,
In the usual theory of the Dirac equation the eigenvalues of thevelocity operator are +/- the velocity of light while the positionoperator is non-Hermitian:
The position operator consists of a real part which is the usualposition and a rapidly oscillating (or zitterbewegung) imaginarypart. This is due to the fact that our measurements are reallyaveraged over time intervals of the order h=mc 2 and correspondinglyover the space intervals of the order of h=mc, the Comptonwavelength, in which case the imaginary zitterbewegung partdisappears. Hermiticity and Conventional Physics begins after the anaveraging necessitated by our gross measurements.
The imaginary zitterbewegung position coordinates exist onlywithin the Compton Radius Vortex.
From a classical point of view, within the Compton Radius Vortex v> c is allowed in Lorentz transformations, so that positioncoordinates within the Compton Radius Vortex become imaginary butoutside the Compton Radius Vortex the position coordinates remainreal.
Outside the Compton Radius Vortex we have Conventional Physics,but as we approach the Compton Radius Vortex we encounter a regionwhere each of the space time axes becomes a complex plane.
Within the Compton Radius Vortex there are ZPFfluctuations of relativistic virtual ghostparticles. Only on averaging over the Compton Radius Vortex do we seeonly the usual electron moving with subluminal velocities. The DeBroglie-Bohm picture of a particle is that of an average over anensemble; while the Sidharthpicture is an averaging over a region that is inaccessible tomeasurement by Conventional Physics apparatus.
To see how this works, following section 3-7 of J. J. Sakurai'sbook Advanced Quantum Mechanics (Addison-Wesley 1967), consider thatan Electron of mass m is represented outside the Compton RadiusVortex by an oscillatory positive-energy solution in OscillatoryRegion I
while within the Compton Radius Vortex the negative-energyElectron solution in Damped Region II is exponentially damped.
The Hydrodynamical Formulation of BohmQuantum Theory corresponds to the ZPFFluctuations within the Vortex that effectively produce Short-(ComptonRadius)-Range Salam Strong Gravity.
According to Sidharth,a Compton Radius Vortex can be described by the mathematicalstructure of a Kerr-Newman Black Hole withSpin and Electric Charge, but with a different physicalinterpretation:
The Compton Radius Vortex interpretation is curvature ofSpaceTime by Electromagnetic Charge, ZPFfluctuations, and Spin so great that a region of4-Real-dimensional SpaceTime becomes
a 4-Complex-dimensional region in which
imaginary zitterbewegung ZPFfluctuation phenomena occur.
is consistent with the GyromagneticRatio of a Kerr-Newman Black Hole.
the ZPF fluctuations ofrelativistic virtual ghostparticles within the Compton Radius Vortex. According to Sidharth,the inertial mass of an elementary particle is the energy ofbinding of nonlocal amplitudes in the zitterbewegung within theCompton Radius Vortex, which for the Electron has Compton Radiusabout 10^(-11) cm.
According to Sidharth,the physical origin of rest mass can be seen by analogy with thetwo-state hydrogen molecular ion. There, the amplitude for the singleelectron to be with one hydrogen atom or the other shows up as abinding energy.
The energy of the Electron tied up within the Compton RadiusVortex is the inertial mass energy mc^2 . The amplitude for theelectron at x to be found at a neighboring point x+b can be taken tothe limit in which b approaches the boundary of the Compton RadiusVortex (and not the center 0). The effective mass is then the massitself!
Therefore, it is the force of binding of nonlocal positions withinthe Compton Radius Vortex, rather like the Hydrogen molecular ionbinding, that manifests itself as inertial mass.
looks like a Kerr-Newman Black Hole as depicted in Black Holes - ATraveller's Guide, by CliffordPickover (Wiley 1996)
except that
The Ring Singularity (purple) is therefore exposed, and
the hydrodynamic flow of the Bohm Quantum Potential, or,equivalently,
the flow of the ZPFfluctuations of relativistic virtual ghostparticles within the Compton Radius Vortex,
is a left-handed vortex flow around the Ring Singularity:
The Compton Radius Vortex illustrations are from clairvoyantviewing by Annie Besant and Charles Leadbeater in 1895, as describedin theirbook Occult Chemistry (ISBN 1-56459-6768-8), and as reprinted inBeyond the Big Bang, by Paul A. Laviolette (Park Street Press 1995).It is very similar to Babbitt'sAtom described by Edwin D. Babbitt in his book Principles ofLight and Color (1878) (ISBN 0-8065-0748-9). According to aweb page of J. Michael McBride "... in the 1860s Lord Kelvin hadbeen suggesting that an atom could be understood as a "vortex" in the"ether" ... Babbitt's "spirillae" are reminiscent of the windings ofelectromagnetic coils, and his "torrents" look like Faraday's linesof force.... Besant and Leadbeater ...[said]... that "Afairly accurate drawing is given in Babbitt's Principles of Light andColour, p. 102" ... but they went on to claim: "The illustrationsthere given of atomic combination are entirely wrong and misleading,but if the stovepipe run through the centre of the single atom beremoved, the picture may be taken as correct, and will give some ideaof the complexity of this fundamental unit of the physical universe."...".
The geometry is similar to that of the (3,10)Torus Knot geometry of Stan Tenen.
The Ring Singularity connects OrdinarySpaceTime with Exotic SpaceTime:
To see how a massless (at tree level) Neutrino Compton RadiusVortex compares with a massive Electron Comton Radius Vortex,compare an Electron of mass m as described in section 3-7 ofJ. J. Sakurai's book Advanced Quantum Mechanics (Addison-Wesley 1967)as being represented outside the Compton Radius Vortex by anoscillatory positive-energy solution in Oscillatory Region I andwithin the Compton Radius Vortex by an exponentially dampednegative-energy Electron solution in Damped Region II
with the m = 0 Neutrino picture
in which the Neutrino is represented outside the Compton RadiusVortex by an oscillatory positive-energy solution in OscillatoryRegion I and within the Compton Radius Vortex by an oscillatorynegative-energy Neutrino solution in Oscillatory Region III.
the Electron can be "... described by ... the positive energy twospinor and the negative energy two spinor ... under reflection ...[the positive energy two spinor transforms into itself andbehaves like a spinor, while the negative energy two spinortransforms into its negative and] ... behaves like apseudo-spinor; and
the Neutrino "... can be treated as an electron with vanishingmass so that the [Compton Radius Vortex] becomes arbitrarilylarge. ... [our physical universe is] ... in effect theregion within the [Compton Radius Vortex] ... [where]the negative energy [two spinor solution] ... dominates. ...under reflection ... [the negative energy two spinor] ...behaves like a pseudo-spinor [and transforms into itsnegative] ... [thus giving] ... a rationale for the lefthandedness of the neutrino. ..."
Since the Compton Radius Vortex is arbitrarily large, OscillatoryRegion I does not exist in our physical universe, and theeffective picture for the Neutrino is
In the D4-D5-E6 physicsmodel, Neutrinos have no Electric (or any other) Chargeand have zero tree level Mass. Therefore only the Spin 1/2 of theNeutrino can give it a non-zero Kerr-NewmanRadius.
For any finite non-zero Spin J, Mass M = 0 implies that a = J / Mis infinite, so that:
the Outer Event Horizon at r = r+ = + sqrt( - a^2 ) isinfinite and Complex;
the Inner Event Horizon at r = r- = - sqrt( - a^2 ) is alsoinfinite and Complex;
the Ergosphere at r = + sqrt( - a^2 cos^2(T) ) is infiniteand Complex, but with a hole at the poles of the z-axis.
From the point of view of a Neutrino, our entire physical universeis the interior of a Kerr-NewmanBlack Hole Compton Radius Vortex, so that
Since Neutrinos are Chargeless they do not interact with theCharged ZPF fluctuations ofrelativistic virtual ghostparticles within the Compton Radius Vortex.
Since Neutrinos are tree-level Massless they do not have thenon-local Dirac Operator cA.p + Bmc^2 interactions within the ComptonRadius Vortex, that is to say, a Neutrino just moves at the speed oflight and does not have the massive characteristic of a nonzeroamplitude to be at another spatial location.
As J. J. Sakurai says in Advanced Quantum Mechanics(Benjamin/Cummings 1967) (at pages 167-170), the massless characterof the Neutrino means that it is not a Dirac particle, but a Weylparticle, so that all Neutrinos are left-handed and all AntiNeutrinosare right-handed.
If a given Neutrino is regarded as a small ball linked by stringsto the boundary of its Compton Radius Vortex, then itsSpin 1/2 Spinor nature can be directly visualized.
Since Neutrinos see our entire universe as being within a Ring Singularity, it is interesting to speculate that they, interacting among themselves, might be able to travel in macroscopic Closed Timelike Loops.
According to David Kestenbaum's article in Science (281 11 Sep1998 pp. 1594 - 1595), "... Super-Kamiokande's measurement pins down... the difference in mass between [electron-type e-neutrinos andmuon-type mu-neutrinos] and so indicates only that one of themmust have a mass of at least 0.07 electron volts ... earlierexperiments had set an upper limit, showing that the electronneutrino, for instance, must have a mass less than 1/30,000 of theelectron's. ...". Since the electron mass is about 0.511 MeV = 511keV = 511,000 eV, the upper limit of the e-neutrino mass is 511,000 /30,000 = 17 eV.
The Super-Kamiokande results are consistent with the e-neutrinobeing massless and with the mu-neutrino having a mass of 0.07 eV.
Jack Sarfatti has noted that a mu-neutrino rest mass of 0.07ev is about 10^-7 the electron mass, so
"... This gives a neutrino Compton wavelength of a micron!Again the size of a living cell! ..."
Therefore, maybe:
Sidharthhas "... shown that at the [boundary surface of the ComptonRadius Vortex] ... we can deduce [the following equation]...
..."As Sidharth notes, referring to Narlikar (in Introduction toCosmology (2nd ed, Cambridge 1993) at page 57), the Stress-EnergyTensor Tij for particles moving on light-cones (as would the virtualparticles of ZPF fluctuations) at unit energy density can bediagonalized to be
1 0 0 0 0 1/3 0 0 Tij = 0 0 1/3 0 0 0 0 1/3
The contribution to electric charge of the T00 component is
1 0 0 0 0 0 0 0 T00 = 0 0 0 0 0 0 0 0
T00 corresponds to the electric charge at distances beyond theboundary of the Compton Radius Vortex, and is only representativeof the Electron, which is unconfined (unlike the confined Quarks) andis electrically charged (unlike the electrically neutral Neutrino).T00 corresponds to electric charge of unit magnitude.
At the boundary surface of the Compton Radius Vortex,electric charge corresponds to the terms T11, T22, and T33. Thecombinations of T11, T22, and T33 are:
0 0 0 0 0 1/3 0 0 T112233 = 0 0 1/3 0 0 0 0 1/3
T112233 corresponds to the Electron whose electric charge hasmagnitude 1/3 + 1/3 + 1/3 = 1.
0 0 0 0 0 1/3 0 0 T1122-- = 0 0 1/3 0 0 0 0 0
T1122-- corresponds to the Red Up Quark whose electric charge hasmagnitude 1/3 + 1/3 = 2/3.
0 0 0 0 0 1/3 0 0 T11--33 = 0 0 0 0 0 0 0 1/3
T11--33 corresponds to the Green Up Quark whose electric chargehas magnitude 1/3 + 1/3= 2/3.
0 0 0 0 0 0 0 0 T--2233 = 0 0 1/3 0 0 0 0 1/3
T--2233 corresponds to the Blue Up Quark whose electric charge hasmagnitude 1/3 + 1/3 = 2/3.
0 0 0 0 0 0 0 0 T----33 = 0 0 0 0 0 0 0 1/3
T----33 corresponds to the Red Down Quark whose electric chargehas magnitude 1/3.
0 0 0 0 0 0 0 0 T--22-- = 0 0 1/3 0 0 0 0 0
T--22-- corresponds to the Green Down Quark whose electric chargehas magnitude 1/3.
0 0 0 0 0 1/3 0 0 T11---- = 0 0 0 0 0 0 0 0
T11---- corresponds to the Blue Down Quark whose electric chargehas magnitude 1/3.
0 0 0 0 0 0 0 0 T------ = 0 0 0 0 0 0 0 0
T------ corresponds to the Neutrino whose electric charge hasmagnitude 0.
Octonion Fermion Triple of Basis Element Particle SU(2)=Spin(3) Spinors 1 e-neutrino 0 x 0 x 0 i red up quark 1 x 1 x 0 j green up quark 1 x 0 x 1 k blue up quark 1 x 0 x 1 I red down quark 0 x 0 x 1 J green down quark 0 x 1 x 0 K blue down quark 1 x 0 x 0 E electron 1 x 1 x 1
The effective (1+1)-dimensionality of the Pion's constituent Quarkand AntiQuark is due to the fact that the Quark and AntiQuark areconfined to the Pion and therefore interact at the surface boundariesof their Compton Radius Vortices, so that their Ring Singularitiesare effectively Naked Singularities with only 1 spatial dimension,that of the Ring Singularity at z =0 and x^2 + y^2 = a^2.
The Quark and AntiQuark interact as a Breathersolution of the (1+1)-dimensional Sine-Gordon equation to form aPion.
As T. D. Lee says on page 132 of his book Particle Physics andIntroduction to Field Theory (Harwood 1981), "... In the special caseof the sine-Gordon equation ..., because of the presence of aninfinite number of conservation laws, the shape and velocity of eachsoliton or anti-soliton remain unchanged even after ... a head-oncollision. We refer to this special class as indestructible solitons.Indestructible solitons exist only in one space-dimension.
If one requires relativistic invariance, then it exists only forthe sine-Gordon equation. ..."
Since the Pion Quark-AntiQuarkpair can be thought of as moving on light-cones and being repeatedlyannihilated and re-created, the Pion Quark-AntiQuark pair isrelativistic, so that the Pion is described as a Sine-Gordon BreatherQuark-AntiQuark Pair.
The mass of the Pion, about 140 MeV, is of the same order ofmagnitude as the characteristicenergy, about 245 Mev, of the SU(3) QCD Color Force, and
The Ring Singularities ofthe Compton Radius Vortices are interconnected like 3 (red, blue,and green, in the image below ) of the4 Clifford-Hopf circles ofthe 24-cell:
For example, a Proton might have Quark content:
The effective (1+1)-dimensionality of the Proton's constituentQuarks is due to the fact that the Quarks are confined to the Protonand therefore interact at the surface boundaries of their ComptonRadius Vortices, so that their Ring Singularities are effectivelyNaked Singularities with only 1 spatial dimension, that of theRing Singularity at z = 0 and x^2 +y^2 = a^2.
The 3 Quarks interact as a 3-Solitonsolution of the (1+1)-dimensional O(3) model to form aProton.
As T. D. Lee says on page 132 of his book Particle Physics andIntroduction to Field Theory (Harwood 1981), "... because of thepresence of an infinite number of conservation laws, the shape andvelocity of each soliton or anti-soliton remain unchanged even after... a head-on collision. We refer to this special class asindestructible solitons. Indestructible solitons exist only in onespace-dimension. ..."
As R. Rajaraman says on page 58 of his book Solitons andInstantons (North-Holland 1987), "... The O(3) model is ...interesting in (1+1) dimensions. It has been shown (Pohlmeyer, Comm.Math. Phys. 46 (1976) 207; Luscher and Pohlmeyer, Nucl. Phys. B137(1978) 46) that, like the sine-Gordon System, theO(3) model in (1+1) dimensions is also characterised by an infinitenumber of conserved quantities and by Backlund transformations forgenerating solutions. [In the quantized version of the theory, ithas been shown that it is asymptotically free and that the conservedquantities exist free of anomalies (Luscher , Nucl. Phys. B135 (1978)1; Polyakov, Phys. Lett. 72B (1977) 224).] ... , an exactfactorised S-matrix has been constructed using the existence of theseinfinite conserved quantities (Zamolodchikov and Zamolodchikov, Ann.Phys. (N.Y.) 120 (1979) 253). ... The static solutions ... in (2+1)dimensions ... serve as instantons [in (1+1) dimensions] .... ... "
T. D. Lee also says on page 132 of his book Particle Physics andIntroduction to Field Theory (Harwood 1981), "... If one requiresrelativistic invariance, then it exists only for the sine-Gordonequation. ..."
Since the Proton has 3 Valence Quarks, and no Valence AntiQuarks,there is no repeated annihilation and re-creation of the ValenceQuarks, which are at rest within the confining volume of the Protonbecause Bohm's "... the PSI-fieldis able to bring the [3 Valence Quarks] to rest and totransform [their] entire kinetic energy into potential energyof interaction with the PSI-field. ...".
Therefore, the Proton 3-Quark triple is NonRelativistic, so thatthe relativistic sine-Gordon model is not correct for the Proton,and
At pages 113, 115, and 49, Rajaraman says "... What would happenif we ... directly generalise the O(3) model to N real scalar fields... [ PHI_n with the sum over n of PHI_n^2 = 1 ? ] ... theallowed values of the field, subject to [ the sum over n ofPHI_n^2 = 1 ] ... , now fall on the surface of a hypersphereS(N-1) [in Internal SymmetrySpace] ... imbedded in N dimensions. Consequently theholonomy group of localised solutions would now be PI_2(S(N-1)). ...this group is trivial except when N = 3. Consequently non-trivialinstanton sectors for the ... O(N) model will exist only when N = 3.... Consequently, the O(3) model's results cannot be generalized bygoing to an O(N) model with N greater than 3. (see Din and Zakrewski,Nucl. Phys. B168 (1980) 173) ... the O(3) model in (1+1) dimensionshas several interesting properties, many of them similar to theYang-Mills theory in (3+1) dimensions. Both systems yield instantonscharacterised by integer-valued toplogical indices. ... both models[are] scale invariant and [yield] instantons ofarbitrary size. (At the quantum level, the similarities persist. ...both the O(3) and Yang-Mills theories are renormalisable andasymtotically free.) At the same time, the O(3) model iscomparatively simple. It consists of only three scalar fields in itstwo dimensions with a simple Lagrangian [L given by
where ] ... PHI can be considered as a vector in[Internal Symmetry Space] ...[which is] distinguished from vectors in coordinatespace, which are labelled by Lorentz indices, such as [ m inthe above equation for L ] ... both the Lagrangian ... and theconstraint [ sum over n of PHI_n^2 = 1 ] are invariant underglobal O(3) rotations in Internal SymmetrySpace]. ..."
The mass of the Proton, about 938 MeV, is of the same order ofmagnitude as the characteristicenergy, about 245 Mev, of the SU(3) QCD Color Force, and
Since Spin(3) = SU(2), the 3-real-dimensional InternalSymmetry Space on which O(3) acts in the O(3) Model can betransformed into a 2-complex-dimensional InternalSymmetry Space that is acted upon by SU(2). Since SU(2) actsglobally on the symmetric space SU(2) / U(1) = S3 / S1 = S2 = CP1,where CP1 is Complex Projective 1-space. As Rajaraman says on page116, "... the CP1 model is essentially the same as the O(3) model...."
Note about Borromean Rings:
The Borromean Rings image
is from thisweb page at the Centerfor the Popularizaton of Mathematics at the Universityof Wales, Bangor, which page notes "... If you try and make theBorromean Rings out of wire, you will find that you cannot make realflat circles into the figure. There must always be kinks. ... Thetheorem stating Borromean rings to be impossible with flat circles isproved rigorously in the article "Borromean circles are impossible,"Amer. Math. Monthly, 98 (1991) 340-341, by B. Lindstrom and H.-O.Zetterstrom. ... There is anotherproof ... that flat Borromean circles do not exist, usinghyperbolic geometry in four dimensions. ... It is however possible toproduce it with ellipses [or rectangles, for which two differentaxes have different lengths]. ..."
A more elliptical/rectangular set of 3 Borromean Rings can be putin 3 mutually orthogonal planes in 3-dimensional space, as in thisimage
from thisweb page. The flatter image may not be as accurate, because forrings of finite thickness, Borromean Rings cannot be exactly circularand perfectly flat.
......