could the classical relativistic electron be a strange attractor

8/13/2019 Could the Classical Relativistic Electron be a Strange Attractor

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Fig.1b

1.9

2

2.1

2.2

2.3

2.4

gam

ma2

0 0.002 0.004 0.006 0.008 0.01

t


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Fig.1c

4

4.5

5

5.5

6

6.5

gam

ma3

0 0.002 0.004 0.006 0.008 0.01

t


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Fig.1a

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

gam

ma1

0 0.002 0.004 0.006 0.008 0.01

t


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Fig.2

200000

400000

600000

gamma3

40 20 20 40 60 80 100 120

gamma2


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Fig.3

500000

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

gamma3

60 40 20 20 40 60 80

gamma2


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Fig.4

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0

2000

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6000

gamma3

0.4 0.2 0.2 0.4 0.6 0.8 1

gamma2


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Fig.5a

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0

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gamma3

0.12 0.14 0.16 0.18gamma2


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Fig.5b

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gamma3

0.1255 0.126 0.1265 0.127 0.1275

gamma2


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Fig.5c

48

46

44

42

40

38

36

34

gam

ma3

0.125 0.1251 0.1252 0.1253 0.1254

gamma2


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Fig.6

40

20

0

20

40

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gamma2

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

gamma1


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Fig.7a

60

40

20

0

20

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60

80

gamma

3

0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1

gamma2


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Fig.7b

0.4

0.2

0

0.2

0.4

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0.8

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gamma2

0.9992 0.9994 0.9996 0.9998 1

gamma1


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Fig.8

0.002

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0.006

0.008

0.01

0.012

lambda

0 2000 4000 6000 8000 10000 12000

it


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Fig.9

0.982

0.984

0.986

0.9880.99

0.992

0.994

0.996

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1

gamma1(t+2)

0.9860.988

0.990.992

0.9940.996

0.9981

gamma1(t+1)

0.9820.984

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0.990.992

0.9940.996

0.9981

gamma1(t)


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Fig.10

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4020

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gamma2(t+2)

6040

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80gamma2(t+1)

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80gamma2(t)


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Fig.11

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gamma3(t+2)

6040

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80gamma3(t+1)

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80gamma3(t)


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arXiv:physics/0401098

v1

[physics.class-ph]20Jan2004

CAPTIONS

fig.1 : , , as functions of. These are called gamma1, gamma2, and gamma3, respec-tively, on the figures. To make the second and third figures convenient to represent, wehave added 150 to the values of the second, and 5103 to the last, assuring positivity(and value greater than unity) in our data range, and taken the logarithm base 10.Recursive, approximate quasiperiodic behavior is clearly visible in the variables ,.

fig.2 : The orbit as it enters the region of the attractor, shown in the space of ,, showingthe transition from the initial conditions to the strange attractor. In , the curve

starts on the axis to the right of the origin, and ends at the lower left of the origin.fig.3 : This curve continues the evolution within the attractor, starting in the neighborhood

of the origin, winding up and around to the right and terminating almost at the samepoint (a little above), coming in from the right.

fig.4 : This curve is a magnification the motion in the neighborhood of the end point offigure 3, starting below the axis and moving to the left. It makes another loop, andends in a smaller pattern.

fig.5 : The smaller pattern approached in figure 4 is shown in detail, now displaying asomewhat different general pattern. Here, enters oscillations as approaches theneighborhood of unity. Figures 5a and 5b show enlargements of the innner loop, andfigure 5c shows the endpoint in more detail.

fig.6 : , plane in the interval represented in fig. 2.

fig7a : , plane in the interval represented in figure 3.

fig.7b : , plane in theinterval represented in figure 4, including the details corresponding

to figure 5.fig.8 : Largest averaged Lyapunov exponent (the values must be divided by the size of the

interval 107) as a function of the accumulated . We have used interval 0.4 106.Note the sharp jump at approximately 10, 400 steps, indicating the development of avery large instability in this neighborhood (the additional terms here are divided by104, the number of time steps.

fig.9 : The time series for () on a three dimensional space corresponding to shift by 20

time steps.fig.10 : The time series for () on a three dimensional space corresponding to shift by 40

time steps.

fig.11 : The time series for () on a three dimensional space corresponding to shift by 20time steps.
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arXiv:physics/0401098

v1

[physics.class-ph]20Jan2004

TAUP 2753-0325 November, 2003

Could the Classical Relativistic Electron

be a

Strange Attractor?

L.P. Horwitza,b, N. Katza, and O. Oronb

a Department of Physics

Bar Ilan University, Ramat Gan 529000, Israelb School of Physics and AstronomyRaymond and Beverly Sackler Faculty of Exact Sciences

Tel Aviv University, Ramat Aviv 69978, Israel

Abstract: We review the formulation of the problem of the electromagnetic self-interactionof a relativistic charged particle in the framework of the manifestly covariant classicalmechanics of Stueckeleberg, Horwitz and Piron. The gauge fields of this theory, in gen-

eral, cause the mass of the particle to change. We show that the non-linear Lorentz forceequation for the self-interaction resulting from the expansion of the Greens function haschaotic solutions. We study the autonomous equation for the off-shell particle mass here,for which the effective charged particle mass achieves a macroscopic average value deter-mined by what appears to be a strange attractor.
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1. Introduction

The advent of powerful computers in the second half of the twentieth century has

made it a possible and attractive possibility to investigate physical problems that involvea high level of intrinsic nonlinearity. The remarkable discovery by Lorenz in 1963[1] thatrather simple nonlinear systems are capable of displaying highly complex, unstable, and,in many cases, beautiful, systems of orbits, gave rise to the subject of chaos, now underintensive study both in specific applications and in its general properties. The instability ofthese orbits is characterized by the fact that small variations of initial conditions generatenew orbits that diverge from the original orbit exponentially.

Such studies range from nonlinear dynamical systems, both dissipative and non-dissipative, involving deterministic chaos, and hard surface type collisions (for example,billliards), to the study of fluctation phenomena in quantum field theory [2]. The re-sults have led to a deeper understanding of turbulence in hydrodynamics, the behavior ofnonlinear electrical circuits, plasmas, biological systems and chemical reactions. It has,furthermore, given deep insights into the foundations of statistical mechanics.

What appears to be a truly striking fact that has emerged from this experience, is thatthe appearance of what has become to be known as chaotic behavior is not just the property

of some special systems designed for the purpose of achieving some result depending onthe nonlinearity, such as components of an electric circuit, but seems to occur almostuniversally in our perception of the physical world. The potentialities presented by thissubject are therefore very extensive, and provide a domain for discovery that appears tobe virtually unlimited.

As an example of the occurrence of chaotic behavior in one of the most fundamentaland elementary systems in nature, we wish to discuss here the results of our study of theclassical relativistic charged particle (for example, the classical electron without takinginto account the dynamical effects of its spin, an intrinsically quantum effect).

There remains, in the area of research on chaotic systems, the question of making acorrespondence between classical chaotic behavior and the properties of the correspondingquantum systems. Much is known on the signatures of chaos for quantum systems, forexample, the occurrence of Wigner distributions in the spectra for Hamiltonian chaos. Thestudy of relativistic phenomena, of the type we shall consider here, where the explicit self-interaction problem gives rise to instability, may have a counterpart in the renormalization

program in quantum field theory. The existence of radiation due to the accelerated motionof the particle raises the question of how this radiation, acting back on the particle, affectsits motion. Rohrlich [3] has described the historical development of this problem, wherethe first steps were taken by Abraham in 1905[4], culminating in the work of Dirac[5], whoderived the equation for the ideal point electron in the form


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Here, the indices , , running over 0, 1, 2, 3, label the spacetime variables that rep-resent the action of the Lorentz group; the index raising and lowering Lorentz invariant

tensoris of the formdiag(1, +1, +1, +1). The expression for

was originally foundby Abraham in 1905[4], shortly after the discovery of special relativity, and is known asthe Abraham four-vector of radiation reaction. Diracs derivation [5] was based on a di-rect application of the Greens functions for the Maxwell fields, obtaining the form (1.1),which we shall call the Abraham-Lorentz-Dirac equation. In this calculation, Dirac usedthe difference between retarded and advanced Greens functions, so as to eliminate thesingularity carried by each. Sokolov and Terner [6], for example, give a derivation usingthe retarded Greens function alone, and show how the singular term can be absorbed into

the mass m.

The formula (1.1) contains a so-called singular perturbation problem. There is a smallcoefficient multiplying a derivative of higher order than that of the unperturbed problem;since the highest order derivative is the most important in the equation, one sees that,dividing bye2, any small deviation in the lower order terms results in a large effect on theorbit, and there are unstable solutions, often called runaway solutions. There is a largeliterature [7] on the methods of treating this instability, and there is considerable discussion

in Rohrlichs book [3] as well (he describes a method for eliminating the unstable solutionsby studying asymptotic properties).

The existence of these runaway solutions, for which the electron undergoes an ex-ponential accelaration with no external force beyond a short initial perturbation of thefree motion, is a difficulty for the point electron picture in the framework of the Maxwelltheory with the covariant Lorentz force. Rohrlich [8] has discussed the idea that the pointelectron idealization may not be really physical, based on arguments from classical and

quantum theory, and emphasized that, for the corresponding classical problem, a finitesize can eliminate this instability. Of course, this argument is valid, but it leaves openthe question of the consistency of the Maxwell-Lorentz theory which admits the conceptof point charges, as well as what is the nature of the elementary charges that make up theextended distribution.

It is quite remarkable that Gupta and Padmanabhan[9], using essentially geometri-cal arguments (solving the static problem in the frame of the accelerating particle with

a curved background metric), have shown that the description of the motion of an accel-erating charged particle mustinclude the radiation terms of the Abraham-Lorentz-Diracequation. Recognizing that the electrons acceleration precludes the use of a sequence ofinstantaneous inertial frames to describe the action of the forces on the electron [10],they carry out a Fermi-Walker transformation [11], going to an accelerating frame (as-suming constant acceleration) in which the electron is actually inertial, and there solve the


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of the electron mass), carries an implication that the electron mass may play an importantdynamical role.

Stueckelberg, in 1941[12], proposed a manifestly covariant form of classical and quan-tum mechanics in which space and time become dynamical observables. They are thereforerepresented in quantum theory by operators on a Hilbert space on square integrable func-tions in space and time. The dynamical development of the state is controlled by aninvariant parameter , which one might call the world time, coinciding with the time onthe (on mass shell)freely falling clocks of general relativity. Stueckelberg [12] started hisanalysis by considering a classical world-line, and argued that under the action of forces,the world line would not be straight, and in fact could be curved back in time. He identi-

fied the branch of the curve running backward in time with the antiparticle, a view takenalso by Feynman in his perturbative formulation of quantum electrodynamics in 1948 [11].Realizing that such a curve could not be parametrized by t (for some values of t thereare two values of the space variables), Stueckelberg introduced the parameter along thetrajectory.

This parameter is not necessarily identical to proper time, even for inertial motion forwhich proper time is a meaningful concept. Stueckelberg postulated the existence of an

invariant Hamiltonian K, which would generate Hamilton equations for the canonicalvariables x andp of the form

x = K

p(1.3)

and

p = Kx

, (1.4)

where the dot indicates differentiation with respect to . Taking, for example, the model

K0=pp

2M , (1.5)

we see that the Hamilton equations imply that

x = p

M (1.6)

It then follows that dx

dt =

p

E, (1.7)

wherep0 E, where we set the velocity of light c = 1; this is the correct definition for thevelocity of a free relativistic particle. It follows, moreover, that


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and hence the proper time is not identical to the evolution parameter . In the case thatm2 =M2, it follows that ds = d, and we say that the particle is on shell.

For example, in the case of an external potential V(x), where we write x x

, theHamiltonian becomes

K= pp

2M + V(x) (1.10)

so that, since K is a constant of the motion, m2 varies from point to point with thevariations ofV(x). It is important to recognize from this discussion that the observableparticle mass depends on the stateof the system (in the quantum theory, the expectation

value of the operator p

p provides the expected value of the mass squared).One may see, alternatively, that phenomenologically the mass of a nucleon, such as

the neutron, clearly depends on the state of the system. The free neutron is not stable,but decays spontaneously into a proton, electron and antineutrino, since it is heavier thanthe proton. However, bound in a nucleus, it may be stable (in the nucleus, the proton maydecay into neutron, positron and neutrino, since the proton may be sufficiently heavierthan the neutron). The mass of the bound electron (in interaction with the electromag-netic field), as computed in quantum electrodynamics, is different from that of the free

electron, and the difference contributes to the Lamb shift. This implies that, if one wishesto construct a covariant quantum theory, the variables E (energy) andp should be inde-pendent, and not constrained by the relationE2 =p2 +m2, where mis a fixed constant.This relation implies, moreover, that m2 is a dynamical variable. It then follows, quantummechanically, through the Fourier relation between the energy-momentum representationof a wave function and the spacetime representation, that the variable t, along with thevariable x is a dynamical variable. Classically, t and x are recognized as variables of the

phase space through the Hamilton equations.Since, in nature, particles appear with fairly sharp mass values (not necessarily with

zero spread), we may assume the existence of some mechanism which will drive the parti-cles mass back to its original mass-shell value (after the source responsible for the masschange ceases to act) so that the particles mass shell is defined. We shall not take sucha mechanism into account explicitly here in developing the dynamical equations. We shallassume that if this mechanism is working, it is a relatively smooth function (for example,a minimum in free energy which is broad enough for our off-shell driving force to workfairly freely)1

In an application of statistical mechanics to this theory[15], it has been found that ahigh temperature phase transition can be responsible for the restriction of the particlesmass (on the average, in equilibrium). In the present work, we shall see that, at least in


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the classical theory, the non-linear equations induced by radiation reaction may have asimilar effect.

A theory of Stueckelberg type, providing a framework for dynamical evolution of arelativistic charged particle, therefore appears to be a natural dynamical generalization ofthe curved space formulation of Gupta and Padmanbhan[9] and the static picture of Dirac.We shall study this in the following.

The Stueckelberg formulation implies the existence of a fifth electromagnetic poten-tial, through the requirement of gauge invariance, and there is a generalized Lorentz forcewhich contains a term that drives the particle off-shell, whereas the terms corresponding tothe electric and magnetic parts of the usual Maxwell fields do not (for the nonrelativistic

case, the electric field may change the energy of a charged particle, but not the magneticfield; the electromagnetic field tensor in our case is analogous to the magnetic field, andthe new field strengths, derived from the dependence of the fields and the additionalgauge field, are analogous to the electric field, as we shall see).

In the following, we give the structure of the field equations, and show that thestandard Maxwell theory is properly contained in this more general framework. Applyingthe Greens functions to the current source provided by the relativistic particle, and the

generalized Lorentz force, we obtain equations of motion for the relativistic particle whichis, in general off-shell. As in Diracs result, these equations are of third order in theevolution paramter, and therefore are highly unstable. However, the equations are verynonlinear, and give rise to chaotic behavior.

Our results exhibit what appears to be a strange attractor in the phase space ofthe autonomous equation for the off-mass shell deviation, This attractor may stabilizethe electrons mass in some neighborhood. We conjecture that it stabilizes the orbitsmacroscopically as well, but a detailed analysis awaits the application of more powerful

computing facilities and procedures.

2. Equations of motion

The Stueckelberg-Schrodinger equation which governs the evolution of a quantumstate state over the manifold of spacetime was postulated by Stueckelberg[10] to be, forthe free particle,

i

=pp

2M

(2.1)

where, on functions of spacetime, p is represented by /x .

Taking into account full U(1) gauge invariance, corresponding to the requirementthat the theory maintain its form under the replacement of by eie0, the Stueckelberg-Schrodinger equation (including a compensation field for the -derivative of ) is [16]


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in place of (2.1). Stueckelberg [10] did not take into account this full gauge invariancerequirement, working in the analog of what is known in the nonrelativistic case as the

Hamilton gauge (where the gauge function is restricted to be independent of time). Theequations of motion for the field variables are given (for both the classical and quantumtheories) by [16]

f(x, ) = e0j

(x, ), (2.4)

where, = 0, 1, 2, 3, 5, the last corresponding to the index, and, of dimension1, isa factor on the terms ff in the Lagrangian associated with (2.2) (with, in addition,degrees of freedom of the fields) required by dimensionality. The field strengths are

f =a a, (2.5)

and the current satisfies the conservation law [16]

j(x, ) = 0; (2.6)

integrating over on (

,

), and assuming that j5(x, ) vanishes at||

, one findsthat

J(x) = 0,

where (for some dimensionless ) [17]

J(x) =

d j(x, ). (2.7)

We identify this J(x) with the Maxwell conserved current. In ref. [18], for example, thisexpression occurs with

j(x, ) = x()4(x x()), (2.8)and is identified with the proper time of the particle (an identification which can bemade for the motion of a free particle). The conservation of the integrated current thenfollows from the fact that

j = x()

4(x x()) = dd

4(x x()),

a total derivative; we assume that the world line runs to infinity (at least in the timedimension) and therefore its integral vanishes at the end points [10, 18], in accordance


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Integrating the -components of Eq. (2.4) over (assuming f5(x, ) 0 for ), we obtain the Maxwell equations with the Maxwell charge e = e0/and the Maxwellfields given by

A(x) =

a(x, ) d. (2.10)

A Hamiltonian of the form (2.3) without dependence of the fields, and without the a5terms, as written by Stueckelberg [10], can be recovered in the limit of the zero mode ofthe fields (with a5 = 0) in a physical state for which this limit is a good approximationi.e., when the Fourier transform of the fields, defined by

a(x, ) =

dsa(x, s)eis, (2.11)

has support only in the neighborhood sofs = 0. The vector potential then takes on theforma(x, ) sa(x, 0) = (s/2)A(x), and we identify e= (s/2)e0. The zeromode therefore emerges when the inverse correlation length of the field s is sufficientlysmall, and then = 2/s. We remark that in this limit, the fifth equation obtained from(2.4) decouples. The Lorentz force obtained from this Hamiltonian, using the Hamilton

equations, coincides with the usual Lorentz force, and, as we have seen, the generalizedMaxwell equation reduce to the usual Maxwell equations. The theory therefore containsthe usual Maxwell Lorentz theory in the limit of the zero mode; for this reason we havecalled this generalized theory the pre-Maxwell theory.

If such a pre-Maxwell theory really underlies the standard Maxwell theory, then thereshould be some physical mechanism which restricts most observations in the laboratoryto be close to the zero mode. For example, in a metal there is a frequency, the plasmafrequency, above which there is no transmission of electromagnetic waves. In this case, ifthe physical universe is imbedded in a medium which does not allow high frequenciesto pass, the pre-Maxwell theory reduces to the Maxwell theory. Some study has beencarried out, for a quite different purpose (of achieving a form of analog gravity), of theproperties of the generalized fields in a medium with general dielectric tensor[19]. We shallsee in the present work that the high level of nonlinearity of this theory in interaction withmatter may itself generate an effective reduction to Maxwell-Lorentz theory, with the highfrequency chaotic behavior providing the regularization achieved by models of the type

discussed by Rohrlich[8].We remark that integration over does not bring the generalized Lorentz force into

the form of the standard Lorentz force, since it is nonlinear, and a convolution remains. Ifthe resulting convolution is trivial, i.e., in the zero mode, the two theories then coincide.Hence, we expect to see dynamical effects in the generalized theory which are not presenti th t d d M ll L t th


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Multiplying this equation by x, one obtains

Mxx

=e0xf

5; (2.14)

this equation therefore does not necessarily lead to the trivial relation between ds andddiscussed above in connection with Eq. (1.9). The f 5 term has the effect of moving theparticle off-shell (as, in the nonrelativistic case, the energy is altered by the electric field).

Let us now define

= 1 + xx= 1 ds2

d2, (2.15)

whereds2 =dt2 dx2 is the square of the proper time. Since x = (p e0a)/M, if weinterpret p e0a)(p e0a) = m2, the gauge invariant particle mass [16], then

= 1 m2

M2 (2.17)

measures the deviation from mass shell (on mass shell, ds2 =d2).

3. Derivation of the differential equations for the spacetime orbit with off-shell

corrections

We now review the derivation of the radiation reaction formula in the Stueckelbergformalism (see also [17]. Calculating the self interaction contribution one must includethe effects of the force acting upon the particle due to its own field ( fself) in addition tothe fields generated by other electromagnetic sources (fext). Therefore, the generalizedLorentz force, using Eq.(2.12) takes the form:

Mx =e0xfext

+ e0x

fself+ e0fext

5+ e0fself

5, (3.1)

where the dynamical derivatives (dot) are with respect to the universal time , and thefields are evaluated on the events trajectory. Multiplying Eq.(3.1) by x we get theprojected equation (2.14) in the form

M

2 = e0xfext

5+e0xfself

5. (3.2)

The field generated by the current is given by the pre-Maxwell equations Eq.(2 .4),and choosing for it the generalized Lorentz gauge a

= 0, we get

a(x ) = (2 2 +2)a = e j(x ) (3 3)


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Integrating this expression over all gives the Greens function for the standard Maxwellfield. Assuming that the radiation reaction acts causally in , we shall use here the -

retarded Greens function. In his calculation of the radiation corrections to the Lorentzforce, Dirac used the difference between advanced and retarded Greens functions in orderto cancel the singularities that they contain. One can, alternatively use the retardedGreens function and renormalize the mass in order to eliminate the singularity [6]. Inthis analysis, we follow the latter procedure.

The- retarded Greens function [17] is given by multiplying the principal part of theintegral Eq.(3.4) by (). Carrying out the integrations (on a complex contour in ; weconsider the case = +1 in the following), one finds (this Greens function differs from

thet-retarded Greens function, constructed on a complex contour ink0),

G(x, ) =2()

(2)3

tan1

x22

(x22)

3

2

x2(x2+2)

x2 + 2 0,(3.5)

where we have written x2 xx.With the help of this Greens function, the solutions of Eq.(3.3) for the self-fields(substituting the current from Eq.(2.9)) are

aself(x, ) =e0

d4xdG(x x, )x()4(x x())

=e0

dx()G(x x(), )

(3.6)

anda5self(x, ) =

e0

dG(x x(), ). (3.7)

The Greens function is written as a scalar, acting in the same way on all five com-ponents of the source j; to assure that the resulting field is in Lorentz gauge, however, itshould be written as a five by five matrix, with the factor kk/k2 (k5 = ) includedin the integrand. Since we compute only the gauge invariant field strengths here, this extraterm will not influence any of the results. It then follows that the generalized Lorentz force

for the self-action (the force of the fields generated by the world line on a point x() ofthe trajectory), along with the effect of external fields, is

Mx =e20

d(x()x(

) x()x())G(x x())|x=x()2


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Eq.(3.8) then becomes

Mx = 2 e20

dx()x(

)(x() x())

x()x()(x() x())} u

G(x x(), )|x=x()

+e20

d(2(x() x())

u x()}G(x x(), )|x=x()

+e0fextx

+ fext5.

(3.10)

In the self-interaction problem where ,x()x() 0, the Greens functionis very divergent. Therefore one can expand all expressions in = assuming thatthe dominant contribution is from the neighborhood of small . The divergent terms arelater absorbed into the mass and charge definitions leading to renormalization (effectivemass and charge). Expanding the integrands in Taylor series around the most singularpoint = and keeping the lowest order terms, the variable u reduces to 2 :

u = xx2 xx3 +13

x...x

4 +1

4xx

4. (3.11)

We now recall the definition of the off-shell deviation, given in Eq.(2.15), along withits derivatives:

xx

= 1 +xx

= 1

2

x...x

+ xx =

1

2.

(3.12)

Next, we define

w 1

12x...

x

+1

8

12

+ w2

.

(3.13)

Using these definitions along with those of Eq.(3.11) and Eq.(3.12) we find


29/38


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From Eq.(3.6) and Eq.(3.7), we have

fself5(x(), ) =

e

(2(x() x())

u x()}

G(x x(), )|x=x()d.()

We then expand x() x() and x() x() in power series in , and write theintegrals formally with infinite limits.

Substituting Eq.(3.19) into Eq.(3.2), we obtain (note that x and its derivatives are

evaluated at the point , and are not subject to the integration), after integrating byparts using () =

(),

M

2=

2e20(2)3

d g1

4( 1) g2

3

2+

g32

x...x

h13

( 1) + h222

2 + h32

( 1)w+

+ h482

2( 1)() + e0xfext 5,(3.20)

where we have defined

g1= f1 f2 3f3 , g2= 12

f1 f2 2f3 , g3=16

f1 12

f2 12

f3

h1= 1

2f1

1

2f2 f3 , h2 =

1

4f1

1

2f2

1

2f3 , h3 = (f

1 f2 f3)

h4=f1 f2 f3 .

(3.21)

The integrals are divergent at the lower bound = 0 imposed by the -function; wetherefore take these integrals to a cut-off >0. Eq.(3.20) then becomes

M

2=

2e20(2)3

g133

( 1) g242

+g3

x

...x

h122 ( 1) +

h22

2

+h3

( 1)w++

h48

2( 1)() +e0xfext 5.(3.22)

Following a similar procedure, we obtain from Eq.(3.8)


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Substituting Eq.(3.22) for the coefficients of the x terms in the second line of Eq.(3.23)we find:

M(, )x = 12

M()(1 )x

+ 2e20

(2)3F()

...x

+ 1

(1 )x...x

x

+e0xxfext

5

1 +e0fextx

+e0fext5,

(3.24)

where

F() = f1

3

(1

) +g3. (3.25)

Here, the coefficients of x have been grouped formally into a renormalized (off-shell)mass term, defined (as done in the standard radiation reaction problem )

M(, ) =M+ e2

2

f1(1 )2

+g2 e21

4f1(1 ) +h2

, (3.26)

where, as we shall see below,

e2 = 2e20

(2)3 (3.27)

can be identified with the Maxwell charge by studying the on-shell limit.We remark that one can change variables, with the help of (2 .15) (here, for simplicity,

assuming


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We remark that when one multiplies this equation by x, it becomes an identity (all

of the terms except for e0fextx

may be grouped to be proportional toxx1 +

); one

must use Eq.(3.22) to compute the off-shell mass shift corresponding to the longitudinaldegree of freedom in the direction of the four velocity of the particle. Eq.(3.28) determinesthe motion orthogonal to the four velocity. Equations (3.22) and (3.28) are the fundamentaldynamical equations governing the off-shell orbit.

We remark that as in [17] it can be shown that Eq.(3.28) reduces to the ordinary(Abraham-Lorentz-Dirac) radiation reaction formula for small, slowly changing and thatthat no instability, no radiation, and no acceleration of the electron occurs when it is on

shell. There is therefore no runaway solution for the exact mass shell limit of this theory;the unstable Dirac result is approximate for close to, but not precisely zero.

4. The evolution

We now derive an equation for the evolution of the off-shell deviation, , when theexternal field is removed. We then use this equation to prove that a fixed mass-shell isconsistent only if the particle is not accelerating, and therefore no runaway solution occurs.Using the definitions

F1() =g1( 1)

32

F2() =g2 2( 1)h1

4

F3() = g3+ 1

12( 1)h3

F4() =

1

2 h2+

1

8 ( 1)h4F5() =

1

8(1 )h3

(4.1)

in equation Eq.(3.22), in the absence of external fields, we write

x...x

= 1

F3()

M2e2

F1() + F2() F4()2 F5()

. (4.2)

Differentiating with respect to we find:

x....x

+ x

...x

= 1

F3

F2

2 + M

2e2+F2

F1 F4()3 2F4()+F M


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which is the derivative of the last equation in Eq.(3.11), one finds from Eq.(4.3),

x...

x

=1

4+ F52F3 ()

...

1

2 H(, , ). (4.5)

Multiplying Eq.(3.28) by x (with no external fields) and using Eq.(4.2) and Eq.(4.5), weobtain

1 + 2

F5F3

... A() +B()2 +C()D() + E()3 +I()= 0, (4.6)

where

A= 2

F3

M2e2

+F2

+ 2M()

e2F() 4M()F5

2e2F() ,

B = 2F3

F23

F2 M

2e2 2F2

F3+

2

1 1

F3

M2e2

+F2 M()

e2F()

1

1 ,

C= 4M()

e2

F()

1

F3

M

2e2

+ F2 2

F23

F1F3

2F1

(1 )F3+

2

F3F1,

D= 4M()

e2F()

F1F3

E= 2F4

(1 )F3 2F4F3

F32 F

4

F3

I= 2

F5F3

+ 2F4F3 F5F

3

F32

2F5(1

)F3

.

(4.7)

5. Dynamical Behavior of System

In this section, we present results of a preliminary study of the dynamical behavior ofthe system. The four orbit equations pose a very heavy computational problem, which weshall treat in a later publication. We remark here, however, that the quantities dx/d anddt/dappear to rise rapidly, but the ratio, corresponding to the observed velocity dx/dt

actually falls, indicating that what would be observed is a dissipative effect.We shall concentrate here on the evolution of the off-shell mass correction , and showthat there is a highly complex dynamical behavior showing strong evidence of chaos.

It appears that reaches values close to (but less than) unity, and there exhibits verycomplex behavior, with large fluctuations. The effective mass M(, ) therefore appears toreach a macroscopically steady value apparently a little less than unity In this neighbor


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where = 1, 2, 3 = , ,. We have obtained numerical solutions over a range ofvariation of. In figure 1, we show, on a logarithmic scale, these three functions; one seesrapid fluctuations in2 and3, and large variations in 1. The action of what appears tobe an attractor occurs on these graphs in the neighborhood of= 0.002 (time steps weretaken to be of the order of 106). The time that the system remains on this attractor, inour calculation about four times this interval, covers about three cycles. The characteristicsof the orbits in phase space are very complex, reflecting an evolution of the nature of theattractor.

The dynamical significance of these functions is most clearly seen in the phase spaceplots. In figure 2, we show = 3 vs. = 2. This figure shows the approach to the

apparent attractor inducing motion beginning at the lower convex portion of the unsym-metrical orbit which then turns back in a characteristic way to reach the last point visibleon this graph. We then continue to examine the motion on a larger scale, where we show,in figure 3, that this end point opens to a larger and more symmetric pattern. The orbitthen continues, as shown in figure 4 to a third loop, much smaller, which then reaches astructure that is significantly different. The end region of this cycle is shown on largerscale in figure 5, where a small cycle terminates in oscillations that appear to become very

rapid as approaches unity. The coefficients in the differential equation become very largein this neighborhood.

In figure 6 we see the approach to the attractor in the graph of = 2 vs. = 1;this curve enters a region of folding, which develops to a loop shown in figure 7a,and infig. 7b, on a larger scale, where we had to terminate the calculation due to limitations ofour computer.

We have computed the global Lyapunov exponents according to the standard proce-dures; recognizing, however, that there appear to be (at least) two levels of behavior types,we have studied the integration on phase space for somewhat less than unity, since theintegration diverges for close to one. For very close to one, in fact, due to the relationds= (1)d, this neighborhood contributes very little to the development of the observedorbits.

In this calculation, we have computed the largest Lyapunov exponent (positive) bystudying the average separation of the orbits associated with nearby initial conditions[20]. We show the develpment of the Lyapunov exponent as the average is taken over an

increasing interval in fig. 8 as data is accumulated up to 13,800 iterations (at least twocycles on the attractor). Note that there are intervals of relative stability and very stronginstability.

Starting the calulation with different initial conditions, we found very similar dynam-ical behavior, confirming the existence of the apparent attractor.


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dimensional delay phase space for the time series for each of these functions.In figs. 12a and 12b, we show the behavior of solution of the autonomous equation

for vs. and vs. . These results are very similar in form to those given above,demonstrating stability of the results.

6. Discussion

We have examined the equations of motion generated for the classical relativisticcharged particle, taking into account dynamically generated variations in the mass of theparticle, and the existence of a fifth electromagnetic type potential essentially reflectingthe gauge degree of freedom of the variable mass. We have focussed our attention on theautonomous equation for the off-shell mass deviation defined in (2.15) and (2.17) as anindication of the dynamical behavior of the system, and shown that there indeed appears tobe a highly complex strange attractor. It is our conjecture that the formation of dynamicalattractors of this type in the orbit equations as well will lead to a macroscopically smoothbehavior of the perturbed relativistic charged particle, possibly associated with an effectivenon-zero size, as discussed, for example, by Rohrlich[8]. Some evidence for such a smoothbehavior appears in the very large effective mass generated near = 1, stabilizing theeffect of the Lorentz forces, as well as our preliminary result that the larger accellerationsemerging in derivatives with respect to are strongly damped, in this neighborhood of,in the transformation to motion seen as velocity (dx/dt) in a particular frame.

Our investigation of the phase space of the autonomous equation used somewhatarbitrary initial conditions, and we find that changing these conditions somewhat does notaffect the general pattern of the results. Moreover, since depends on the third derivativeofx(), this initial condition should not be taken as arbitrary, but as determined by theresults for the orbit (initial conditions for the orbit depend only on x and its first twoderivatives). Computing the orbit for a typical choice of initial conditions, and using theresult to fix , we find that the corresponding solution of the autonomous equation indeedlies in the attractor. The results of our investigation of the autonomous equation shouldtherefore be valid for initial conditions that obey the physical constraints imposed by theequations of motion as well.

We have specified all of the dynamical equations here, but a complete investigationmust await further planned work with more powerful computing facilities and procedures.

Acknowledgement

One of us (L.P.H.) would like to thank the Institute for Advanced Study, Princeton,N.J., for partial support, and Steve Adler for his hospitality, during his visit in the Spring


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3. F. Rohrlich, Classical Charged Particles, Addison-Wesley, Reading (1965).4. M. Abraham, Theorie der Elektrizitat, vol. II, Springer. Leipzig (1905).5. P.A.M. Dirac, Proc. Roy. Soc.(London) A167, 148 (1938).6. A.A. Sokolov and I.M. Ternov, Radiation from Relativistic Electrons, Amer. Inst. of

Phys. Translation Series, New York (1986).7. For example, J.M. Aguirrebiria, J. Phys. A:Math. Gen30, 2391), D. Villarroel, Phys.

Rev. A 55, 3333 (1997),and references therein.8. F. Rohrlich, Found. Phys. 26, 1617 (1996); Phys. Rev.D60, 084017 (1999); Amer.

J. Phys. 68,1109 (2000). See also, H. Levine, E.J. Moniz and D.H. Sharp, Amer.Jour. Phys. 45, 75 (1977); A.D. Yaghjian, Relativistic Dynamics of a Charged Sphere,

Springer, Berlin (1992).9. A. Gupta and T. Padmanabhan, Phys. Rev. D57,7241 (1998).

10. B. Mashoon, Proc. VII Brazilian School of Cosmology and Gravitation, EditionsFrontieres (1994); Phys. Lett. A 145, 147 (1990); Phys. Rev. A 47, 4498 (1993).

11. C.W. Misner, K.S. Thorne and J.A. Wheeler,Gravitation, W.H. Freeman, San Fran-cisco (1973).

12. E.C.G.Stueckelberg, Helv. Phys. Acta14, 322(1941); 14, 588 (1941).13. R.P. Feynman, Rev. Mod. Phys.20, 367 (1948).14. L.P. Horwitz, Found. Phys. 25, 39 (1995).15. L. Burakovsky, L.P. Horwitz and W.C. Schieve, Phys. Rev. D 54, 4029 (1996).16. D.Saad, L.P.Horwitz and R.I.Arshansky, Found. of Phys.19, 1125 (1989); M.C. Land,

N. Shnerb and L.P. Horwitz, Jour. Math. Phys. 36, 3263 (1995); N. Shnerb and L.P.Horwitz, Phys. Rev. 48A, 4068 (1993).

17. O. Oron and L.P. Horwitz, Phys. Lett.A 280, 265 (2001).18. J.D. Jackson, Classical Electrodynamics, 2nd edition, John Wiley and Sons, New

York(1975).19. O. Oron and L.P. Horwitz, ound. Phys.33, 1323 (2003).20. J.C. Sprott, Chaos and Time-Series Analysis, Oxford Univ. Press, Oxford (2003).21. H.D.I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag, New York

(1996); G.L. Baker and J.P. Gollub, Chaotic dynamics, an introduction, CambridgeUniv. Press, Cambridge (1996).


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Fig.12a

300

200

100

0

100

200

300

gamm

a3

0.5 0.6 0.7 0.8 0.9 1

gamm2


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Fig.12b

2e+06

0

2e+06

4e+06

6e+06

8e+06

1e+07

gamma3

300 200 100 100 200 300

gamm2

could the classical relativistic electron be a strange attractor

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