Gravitational Strings Beyond Quantum Theory- Electron as a Closed String 1109.3872v3

7/28/2019 Gravitational Strings Beyond Quantum Theory- Electron as a Closed String 1109.3872v3

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arXiv:1109

.3872v3

[hep-th]

12Oct2011

Gravitational strings beyond quantum theory: Electron as a closed string.

Alexander BurinskiiLaboratory of Theor. Phys. , NSI, Russian Academy of Sciences, B. Tulskaya 52 Moscow 115191 Russia,

The observable parameters of the electron indicate unambiguously that its gravitational back-ground should be the Kerr-Newman solution without horizons. This background is not flat andhas a non-trivial topology created by the Kerr singular ring. This ring was identified with a closedgravitational string. We discuss the relation of this string to the low energy string theory and show

that the traveling waves along the Kerr closed string may lie beyond the Dirac theory of electron.Gravitational strings form a bridge between gravity and quantum theory, indicating a new way toconsistent Quantum Gravity. We argue that the predicted closed string in the core of electron maybe probed by the novel experimental method of the nonforward Compton scattering.

PACS numbers: 11.27.+d, 04.20.Jb, 04.60.- m, 04.70.Bw

I. INTRODUCTION

Modern physics is based on Quantum theory and Grav-ity. The both theories are confirmed experimentally withgreat precision. Nevertheless, they are conflicting and

cannot be unified in a whole theory. One of the principalcontradictions between Quantum theory and Gravity isthe question on the shape and size of an electron. Quan-tum theory states that electron is pointlike and struc-tureless [1, 2], and it is experimentally supported by thehigh energy scattering. In the same time, gravity basedon the Kerr-Newman (KN) solution indicates unambigu-ously that electron should forms a closed string of theCompton size. Contrary to the Schwarzschild solution,which displays that t size of a gravitating body (radiusof the horizon rg = 2m) is proportional to the mass, [40]the KN geometry of the rotating bodies presents a newdimensional parameter a = J/m, which grows with angu-lar momentum J and has the reverse mass-dependence.As a result, the zone of gravitational interaction is de-termined by parameter a and increases for large angularmomentum and small masses, turning to be essential forelementary particles. Reason of that, it a specific struc-ture of the KN gravitational field, which is concentratednear the Kerr singular ring, forming a closed gravitationalwaveguide a type of the closed gravitational string [3].

In 1968 Carter obtained that the KN solution for thecharged and rotating black holes has g = 2 as that ofthe Dirac electron, [4, 5], which allowed one to considerKN solution a consistent with gravity electron model, [318]. Mass of the electron in the considered units is m 10

22

, while a = J/m 1022

. Therefore, a >> m, andthe black hole horizons disappear, opening a nontrivialtopological defect and twosheeted spacetime generatedby the naked Kerr singular ring. Gravitational field ofthe KN solution concentrates near the Kerr singular ringand forms a closed gravitational waveguide an analogof the closed gravitational strings [3]. Kerrs closed string

email: [email protected]

takes the Compton radius, corresponding to the size of adressed electron in QED and to the limit of localizationof the electron in the Dirac theory.

There appears first question: how does the KN gravityknow about one of the principal parameters of Quantumtheory? The second question is: while does Quantum

theory works successfully on the flat spacetime ignoringthe stringlike peculiarity of the KN gravitational field.It can be understand that a small and slowly varyinggravitational field could be ignored. However, the string-like KN singularity forms a branch line of the spacetimeand creates the twosheeted topology, ignorance of whichcannot be justified. A simple answer to these questionsis to assume that there is a general underlying theoryproviding the consistency of quantum theory and grav-ity. In this paper we suggest a resolution of these prob-lems on the base of the rather unexpected point of view,that the underlying theory is the Einstein-Maxwell grav-ity as a fundamental part of the theory of superstrings.

In this case, quantum theory should follow from the the-ory of superstring, and we make a first step in this di-rection, showing emergence of the Dirac equation fromthe model of traveling waves, propagating along the KNcircular string.

So far as gravity predicts the existence of the closedKerr string on the boundary of the Compton area, sucha string, is really exists, should be experimentally ob-servable, and there appears the question while it was notobtained earlier by the high energy scattering. We tryto give an explanation to this fact and argue that theKN string may apparently be detected by the novel ex-perimental method based on the theory Generalized Par-ton Distributions (GPD) [19, 20] which represents a newregime for probing the transverse shape of the particlesby the nonforward Compton scattering [21].

II. THE KERR-NEWMAN BACKGROUND

The Kerr-Newman solution in the Kerr-Schild form hasthe metric

g = + 2Hkk, (1)
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where

H =mr e2/2

r2 + a2 cos2 , (2)

and is metric of auxiliary Minkowski space in theCartesian coordinates (t ,x,y,z), and the Kerr coordi-nates r and are the spheroidal oblate coordinates. Thefunction H is singular at r = 0, cos = 0, correspond-

ing to the Kerr singular ring. The KN electromagneticpotential

AKN = Ree

r + ia cos k, (3)

is aligned with null direction k. It is also singular at theKerr ring, which forms a branch line of the Kerr space-time in two sheets, corresponding to r > 0 and r < 0 inthe Kerr oblate coordinate system. Vector field k formsPrincipal Null Congruence (PNC) of KN space [41]. TheKerr PNC is smoothly propagated via the Kerr disk r = 0from the negative sheet (r < 0) of spacetime to the pos-itive one (r > 0) (see Fig.1), and therefore, it covers the

KN space twice: k(+)

for r > 0 and k()

for r < 0, lead-ing to different metrics and different electromagnetic fieldon the positive and negative sheets [24]. Therefore, theKerr ring creates a twosheeted background topology.

10

5

0

5

10

10

5

0

5

10

10

5

0

5

10

Z

FIG. 1: Vortex of the Kerr congruence. Twistor null lines arefocused on the Kerr singular ring, forming a circular gravita-tional waveguide, or string with lightlike excitations.

This twosheetedness forms a principal puzzle of theKerr geometry over a period of four decades. In 1968Israel truncated negative KN sheet, r < 0,, replacing itby the rotating disklike source at r = 0, spanned by theKerr singular ring of the Compton radius a = /2m,[9]. Then Hamity assumed in [22] that the disk is to berigidly rotating, which led to a reasonable interpretationof the matter of the source as an exotic stuff with thezero energy density and negative pressure. The matterdistribution was singular at the disk boundary, formingan additional closed string source, and Lopez suggested in[12] to regularize this source, covering the Kerr singularring by a disklike ellipsoidal surface. As a result theKN source was turned into a rotating and charged oblatebubble with a flat interior, and further, the bubble source

was realized as a regular soliton-like model [6], formedby a domain wall interpolating between the external KNsolution and a flat pseudovacuum state inside the bubble.

Alternatively, from the very beginning, there were con-sidered the stringlike models of the KN source, whichretained Kerrs twosheeted topology, forming a closedAlice string [3, 8, 23, 26]. The Kerr singular ring wasconsidered as a waveguide for electromagnetic traveling

waves generating the spin and mass of the KN solutionin accordance with the old Wheelers geon model ofmass without mass [10, 11, 27, 28].

III. THE KERR SINGULAR RING AS A

CLOSED STRING

Exact solutions for electromagnetic excitations on theKerr-Schild background, [2325], showed that there areno smooth harmonic solutions on the KN background.The electromagnetic wave excitations are generated bythe complex potential

A = (Y, ){ 1r + ia cos

kdx + (Y)dY}, (4)

where (Y, ) is a holomorphic functions of a complexprojective angular coordinate Y on celestial sphere, Y =ei tan 2 , and is a complex retarded time.[42] In thenontrivial Kerr-Schild solutions, the function containsat least one pole in Y, which creates a singular beamin the corresponding angular direction Yi = e

ii tan i2 ,

Due to the factor 1r+ia cos , any electromagnetic excita-tion of the KN geometry generates the traveling wavesalong the Kerr singular ring (at r = cos = 0), and si-multaneously, the pole in Y creates an axial singular

beam, which is topologycally coupled with the Kerr ring,see Fig.2. Therefore, any excitation of the KN back-ground has a paired character, creating simultaneouslythe circular traveling waves coupled with the propagat-ing outward axial traveling waves.[43] The vector fieldk is constant along the axial beams, and asymptot-ically (by r ) the beams tend to the well knownpp-wave (plane fronted) solutions, for which k is a co-variantly constant Killing direction.

The pp-waves take important role in superstring the-ory, forming the singular classical solutions to the low-energy string theory [29]. The string solutions are com-pactified to four dimensions, and the singular pp-wavesare regarded as the massless fields around a lightlike fun-damental string. It is suspected that the singular sourceof the string will be smoothed out in the full string theorywhich should take into account all orders in . Mean-while, it was shown that the pp-waves have remarkableproperty that all the - corrections in the string equationof motion are automatically zero [29]. In the nonpertur-bative approach based on analogues between the stringsand solitons, the pp-wave solutions are considered as fun-damental strings [30]. The pp-waves may carry travelingelectromagnetic and gravitational waves which represents


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2

1

0

1

2

3

4

5

4

2

0

2

42

1.5

1

0.5

0

0.5

1

1.5

2

J

kout

kout

FIG. 2: Skeleton of the Kerr geometry [23] formed by thetopologically coupled circular and axial strings.

propagating modes of the fundamental string [31]. Inparticular, the generalized pp-waves represent the singu-lar strings with traveling electromagnetic waves [23, 32].It has been noticed that the field structure of the Kerr

singular ring is also similar to a closed pp-wave string[3, 11]. This similarity is not incidental, since many so-lutions to the Einstein-Maxwell theory turn out to beparticular solutions to the low energy string theory witha zero (or constant) axion and dilaton fields. Indeed, af-ter compactification the bosonic part of the action forthe four-dimensional low-energy string theory takes theform, [33],

S =

d4x

g(R2()2e2F212

e4(a)2aFF),(5)

and contains the usual Einstein term

Sg =

d4xgR (6)

completed by the kinetic term for dilaton field 2()2and by the scaled by e2 electromagnetic part e2F2.The last two terms are related with axion field a andrepresent its nonlinear coupling with dilaton field

12

e4(a)2 (7)

and interaction of the axion with the dual electromag-netic field

F

=

F

. (8)

It follows immediately that any solution of the Einsteingravity, and in particular the Kerr solution, is to be ex-act solution of the effective low energy string theorywitha zero (or constant) axion and dilaton fields. Situationturns out to be more intricate for the Einstein-Maxwellsolutions since the electromagnetic invariant F2 plays therole of the source of dilaton field. Similarly, the term FFturns out to be the source of the axion field. The re-lationship between the classical cosmic superconducting

strings and the heterotic superstrings was first mentionedby Witten in [34]. The KN string is closed, charged andhas the lightlike current. Its excitations form the light-like circular modes traveling in only one direction, andall that was indicated in [34] as the characteristic pe-culiarities of the O(32) or E8 E8 heterotic strings.[44]The stringy analog to the Kerr-Newman solution withnontrivial axion and dilaton fields was obtained by Sen

[35], and it was shown in [14], that the field around thesingular string in the axidilatonic Kerr-Sen solution isvery similar to the field around a heterotic string. Thisproximity of the pure gravitational strings [3] to the low-energy string theory allows us to consider the Kerr singu-lar ring as a closed heterotic string and the correspondingtraveling waves as its lightlike propagating modes. Theaveraged dilaton field < e > corresponds to the stringtension, and therefore, obtaining of the exact solutions tothe low energy string theory with a nontrival dilaton fieldrepresents a very important problem, which may allow usto estimate the mass-energy of the excited string states.The structure of the Lagrangian (5) shows that the axionfield involves the dual magnetic field, and therefore, thecomplex axidilaton combination may generate the dual-ity rotation and create an additional twist of the electro-magnetic traveling waves. However, the exact solutionsof this type are unknown so far.

The low energy string theory is classical one, and as-suming that the lightlike string forms a core of the elec-tron structure, we have to obtain a bridge to quantumtheory of electron. The lightlike traveling waves alongthe KN closed string generate the spin and mass of theKN particle. Physically, it is resembling the originalWheelers idea of a geon, [27], in which mass withoutmass emerges from the system of circulating masslessparticles. Because of that, the corresponding quantum

theory of the massless fields could be adequate.

IV. WHEELERS MODEL OF MICROGEON,

TRAVELING WAVES AND EMERGENCE OF

THE DIRAC EQUATION

The Wheelers model of geon represents a cloud ofthe lightlike particles, which are held by own gravita-tional field. A microgeon with the Kerr geometry [11]represents a degenerate case of a sole photon (or otherlightlike particle) circulating on the lightlike orbit of theCompton size. It may be considered as a corpuscular

analogue to traveling waves along the KN gravitationalstring.

The observable parameters of the electron indicateunambiguously the Kerr-Newman background spacetimeand the Compton radius of the corresponding Kerr string.On the other hand, the Compton radius plays also pe-culiar role in the Dirac theory, as a limit of localizationof the wave packet. Localization beyond the Comptonzone creates a zitterbewegung affecting ...such para-doxes and dilemmas, which cannot be resolved in frame


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of the Dirac electron theory... (Bjorken and Drell, [36]).Dirac wrote in his Nobel Prize Lecture : The variables (velocity operators, AB) also give rise to some ratherunexpected phenomena concerning the motion of the elec-tron. .. It is found that an electron which seems to usto be moving slowly, must actually have a very high fre-quency oscillatory motion of small amplitude superposedon the regular motion which appears to us. As a result

of this oscillatory motion, the velocity of the electron atany time equals the velocity of light.These words by Dirac may be considered as a direct

hint regarding the inner structure of the electron, thecore of which should therefore be represented by a circu-lating lightlike particle, and consequently, its wave func-tion should satisfy the masslesswave equation, while themass should appear as an averaged energy of the circularmotion. It corresponds also to generation of the mass instringy models from excitations of the traveling modes.Therefore, the local 4-momentum of a massless particlecirculating around z-axis has to be lightlike,

p2x + p2y + p

2z = E

2, (9)

while the effective mass-energy has to be related with anaveraged orbital motion,

< p2x > + < p2y >= m

2. (10)

Averaging (9) under the condition (10) yields the relation

< p2x + p2y + p

2z >= m

2 + p2z = E2. (11)

The simplest quantum analog of this model correspondsto the wave equations obtained from the operator version

of these relations: p p = i, E = it.We start from the solutions of the scalar massless equa-

tion

= 0, (12)

which should satisfy the wave analog of the constraint(10),

(2x + 2y) = m

2. (13)

For the considered orientation of spin, the system of theequations (12) and (13) is equivalent to the system

(2x + 2y) = m

2 = (2t 2z ), (14)

which shows explicitly that the variables may be sepa-rated by the ansatz

= M(x, y)0(z, t). (15)The r.h.s. of (14) yields the equation for the wave func-tion of a massive particle, (2t 2z )0 = m20, and thecorresponding plane wave solutions

0(z, t) = expi

(zpz Et), (16)

have the usual de Broglie periodicity. In the same time,the l.h.s. of (14) yields the equation

(2x + 2y)M(x, y) = m2M(x, y), (17)

which determines a factor of internal structureM(x, y). The corresponding periodic cylindrical solu-tions are determined in polar coordinates , [45] by theHankel functions

H(

m

) of index ,

M = H( m

)exp{i}. (18)

Functions M are eigenfunctions of operator Jz = i with eigenvalues Jz = . For electron we have Jz =/2, = 1/2, and the factor

M1/2 = 1/2 exp{i( m

12

)} (19)

creates a singular ray along z-axis, which forms a branchline, and we note two peculiarities:

i/ the wave function turns out to be twovalued,

ii/ the wave function propagates along a singular stringformed by the singular structure factor M, while theusual plane wave solutions 0(z, t) of the Klein-Gordonequation [46] = m2 represent only a modulation ofthis string.

The resulting wave equations and solutions may begeneralized in diverse directions. First of all, there maybe obtained the corresponding wave functions based onthe eigenfunctions of the operator of the total angularmomentum and simultaneously of the spin projectionoperator. Next, the treatment may be considered in aLorentz covariant form for arbitrarily positioned and ori-ented wave functions. And finally, the corresponding so-

lutions of the massless spinor equation = 0 (20)

may be constructed from the scalar wave functions and0, (15). In particular, the known two basic plane-wavesolutions of the Dirac equation

= m (21)

corresponding to positive energy E > 0,[47]

r(x) = wr(p)exp{ i

px}, (22)

where p = (E, 0, 0, pz),

w1 =

10pzE+m

0

; w2 =

010

pzE+m

, (23)

being modified by singular function M(x, y), yield thewave functions

r(x) = Mr(x, y)wr(p)exp{ i

px}, (24)


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satisfying the massless Dirac equation (20). One sees thatlike the massless scalar solutions, the wave functions ofthe massless Dirac equation form the singular strings,modulated by the usual Dirac plane wave solutions.

It should be emphasized that separation of the vari-ables allows us to obtain the ordinary Dirac theory withthe massive Dirac equation and the usual Dirac planewaves from the solutions of the massless underlying the-

ory which regards zitterbewegung as a corpuscular analogof the traveling waves.

Emergence of the Dirac equation from gravitationalmodel of the traveling waves circulating along the closedKerr string indicates a principally new point of view thatgravitation and superstring theory may form some morefundamental theory lying beyond the Dirac theory ofelectron. On the level of the low energy string theory,the Dirac field losses the meaning of the wave functionand should be considered as a source of the charge andcurrents for the Maxwell equations. The fundamentalheterotic strings are charged, and the charges are trappedby the string. Therefore, the corresponding fermion fields

should be localized at the string core. The mechanism ofsuch localization is going beyond the low energy stringtheory and belongs to an (unknown so far) effective the-ory of more high level. There are evidences that thismechanism may be related with superconductivity of thestrings described by the Higgs mechanism of broken sym-metry, as it was suggested in the Witten model of thesuperconducting cosmic strings [38] and exploited in theKN model of gravitating soliton [6] discussed briefly inthe next section.

V. REGULARIZATION: ELECTRON ASGRAVITATING KN SOLITON

Regularization of the KN singular ring was performedin the model of gravitating soliton [6], which representsa regular version of the Lopez bubble model [12]. TheKN solution near the Kerr singular ring is replaced bya charged and rotating domain wall bubble covering theKerr singular ring. Interior of the bubble is assumed tobe flat which removes the Kerr singularity. The bubbleboundary is determined from the matching of the exter-nal KN metric with the flat interior and fixed by theequation H = 0 i n (2). It yields r = re = e2/(2m),where r is the Kerr ellipsoidal radial coordinate, and thebubble takes the form an oblate disk of the Compton

radius a = /(2m) with the thickness re = e2/(2m), cor-responding to the known classical size of the electron.The Kerr singular ring is regularized (in particular, the

vector potential is bounded, |A(str) | e/re = 2m/e,)and radius of the regularized closed string turns out tobe slightly increased, being shifted to the boundary ofthe bubble. This position of the string confirms the oldsuggestions that the heterotic strings have to be formedon the boundary of a domain wall [34].

The domain wall boundary of the bubble provides a

smooth phase transition from the external KN solutionto a flat false vacuum state inside the bubble. This phasetransition is formed by a supersymmetric system of thechiral fields i, [6] and by the potential V(r) generatedfrom the specially adapted superpotential W, which pro-vides the smooth transfer from the external KN vacuumstate, Vext = 0, to a flat internal pseudovacuum state,Vint = 0.[48] The electromagnetic KN field is regularized

by the Higgs field = || exp{i}, forming a supercon-ducting condensate inside the bubble. Interaction of theKN electromagnetic potential A with the Higgs field iscontrolled by the equations

A = I = e||2(, +eA), (25)

where the current I should be set zero inside the bubble, I = 0, while || > 0. It yields

A = 0, , +eA = 0, (26)

which shows that gradient of the Higgs phase , eatsup the field A, expelling the field strength and currents

to the string-like boundary of the bubble, where it givestwo conditions:

= ,0 = eA(str)0 , , = eA(str) . (27)

The spacelike component A(str) of the KN vector poten-

tial A(str) = (A

(str)0 ,A

(str)) is tangent to the Kerr string

and forms a regular flow in -direction, A(str) , creating

a closed Wilson line along the Kerr string. As a result of(27), the KN gravitating soliton exhibits two importantpeculiarities:

the quantum Wilson loop eA

(str) d =

4ma

results in quantization of the total angular momen-tum of the soliton, J = ma = n/2, n = 1, 2, 3,...,

the Higgs field inside the bubble turns out to be os-cillating with the frequency = 2m, and therefore,it forms a coherent vacuum state, similar to that ofthe oscillon bubble models.

The Compton size of the bubble does not contradict tothe dressed QED electron, however, there is an essen-tial difference. The dynamics of the virtual particles inQED is chaotic and can be conventionally separated fromthe bareelectron, while the vacuum state inside theKN soliton forms a coherent oscillating statejoined with

the closed Kerr string. Therefore, the bubble source ofthe KN soliton and the Kerr String represent an integralcoherent structure which cannot be separated from thebare particle.

VI. CONCLUSION.

We have showed that gravity definitely indicates pres-ence of a closed string of the Compton radius a = /(2m)


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in the electron background geometry. This string hasgravitational origin and is close related with the funda-mental closed strings of the low energy string theory.Starting from the corpuscular aspect of the underlyingmodel of traveling waves along the Kerr string, we showedemergence of the Dirac equation. The original Diractheory is modified in this case, and the wave functionsacquire the singular stringlike carriers, so the Dirac plane

wave solutions turn out to be propagating along the ax-ial singular strings. Contrary to the widespread opinionthat quantum theory predominates over all other the-ories, the considered model indicates opposite point ofview that gravity, as a basic part of the underlying su-perstring theory, may lie beyond quantum theory. There-fore, the Kerr-Newman solution and its closed string in-terpretation form a bridge between gravity, superstringtheory and the Dirac quantum theory, displaying a newway towards unification of quantum theory and gravity.

The observable parameters of the electron determineunambiguously the KN background of the electron, pres-ence of the Kerr string and its Compton radius. Thissize is very big with respect to the modern scale of theexperimental resolution, and it seems, that the stringshould easily be experimentally detected. However, thehigh-energy scattering detects only the pointlike electronstructure down to 1016cm. One of the reasons of thisfact was considered earlier in [7]. It was argued that thepointlike character of the interactions for the KN parti-cles may be caused by their interactions via the axial pp-strings, corresponding to a real intermediate photon withthe lightlike momentum transfer q212 = 0.[49] Such in-teraction corresponds to an action-at-a-distance, whenthe points of the interacting objects x1 and x2 are related

by the lightlike interval s212 (x1 x2)2 = 0, which isequivalent to the contact (pointlike) interaction. To rec-ognize the extended shape of the particle, the hard con-dition q212 = 0 should be weaken, i.e. the shape should beprobed by a virtual photon. Specifically, it is necessarytwo special conditions:

a) the scattering should be deeply virtual, which meansvery large Q2 = q212, and p

q12, while the Bjorken variable

= Q2/(P q12) is finite.b) the momentum transfer should be relatively low to

provide a coherent diffractive scattering with the wave-lengths comparable with extension of the string.

Both these conditions are satisfied in the novel ap-proach, the Deeply Virtual Compton Scattering (DVCS),or a non-forward Compton scattering, [19, 20]. In thelast decade it was suggested as a new regime of the highenergy scattering, which should be effective for tomog-raphy of the inner structure of the elementary particles[21].

If the predicted Kerr-Newman closed string will be re-

ally detected in the electron structure, it could lead toessential progress in understanding of Quantum theoryand its development towards Quantum Gravity.

Acknowledgments

Author thanks M. Bordag, A. Efremov, D. Stevens, O.Teryaev and F. Winterberg for conversations and use-ful references, and also A. Efremov, D. Galtsov, Yu.Danoyan, Yu. Obukhov, O. Selyugin and K. Stepaniantsfor useful discussions, and especially to A. Radyushkinfor clarifications concerning the GPD application.

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[40] In the units = G = c = 1[41] It is determined by the Kerr theorem in twistor terms,

[7, 8][42] The structure of function (Y) is not essential for us here.[43] Similar spinor solutions were obtained on the KN back-

ground in the frame of the coupled Einstein-Maxwell-Dirac equations, [37]. The resulting massless spinor so-lutions turn out to be singular at the Kerr ring and thefermionic wave excitations generate simultaneously thetraveling waves along the Kerr ring and the coupled ax-ial singular pp-waves.

[44] It was also pointed out superconducting nature of thesestrings and its positioning at the boundary of an axiondomain wall. The both these properties are confirmedin the regularized KN model of a gravitating soliton [6],which is briefly discussed in sec.5.

[45] Do not mix here angle with wave function.[46] It turns in the 4d Klein-Gordon equation by the corre-

sponding Lorentz transformation.[47] We use here notations of the book [36].[48] It is a supersymmetric version of the suggested by Witten

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[49] It is just the interaction via a common twistor line of theKerr congruence considered in [39] by the analysis of theKerr theorem for multiparticle Kerr-Schild solutions.
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Gravitational Strings Beyond Quantum Theory- Electron as a Closed String 1109.3872v3