Could the classical relativistic electron be a strange attractor?

We review the formulation of the problem of the electromagnetic self-interaction of a relativistic charged particle in the framework of the manifestly covariant classical mechanics of Stueckelberg, Horwitz and Piron. The gauge fields of this theory, in general, cause the mass of the particle to change. We show that the non-linear Lorentz force equation for the self-interaction resulting from the expansion of the Green's function has chaotic 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 determined by what appears to be a strange attractor.

: ε,ε,ε as functions of τ . These are called gamma1, gamma2, and gamma3, respectively, on the figures. To make the second and third figures convenient to represent, we have added 150 to the values of the second, and 5 × 10 3 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 variablesε,ε. fig.2 : The orbit as it enters the region of the attractor, shown in the space ofε,ε, showing the transition from the initial conditions to the strange attractor. In τ , the curve starts on theε axis to the right of the origin, and ends at the lower left of the origin.  fig.12. : In these figures, showingε vs. ε andε vs.ε, we have changed the initial conditions, and find a similar pattern.

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 involve a high level of intrinsic nonlinearity. The remarkable discovery by Lorenz in 1963[1] that rather 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 under intensive study both in specific applications and in its general properties. The instability of these orbits is characterized by the fact that small variations of initial conditions generate new orbits that diverge from the original orbit exponentially.
Such studies range from nonlinear dynamical systems, both dissipative and nondissipative, involving deterministic chaos, and hard surface type collisions (for example, billliards), to the study of fluctation phenomena in quantum field theory [2]. The results have led to a deeper understanding of turbulence in hydrodynamics, the behavior of nonlinear 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 that the 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 on the nonlinearity, such as components of an electric circuit, but seems to occur almost universally in our perception of the physical world. The potentialities presented by this subject are therefore very extensive, and provide a domain for discovery that appears to be virtually unlimited.
As an example of the occurrence of chaotic behavior in one of the most fundamental and elementary systems in nature, we wish to discuss here the results of our study of the classical relativistic charged particle (for example, the classical electron without taking into 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 a correspondence between classical chaotic behavior and the properties of the corresponding quantum systems. Much is known on the signatures of chaos for quantum systems, for example, the occurrence of Wigner distributions in the spectra for Hamiltonian chaos. The study of relativistic phenomena, of the type we shall consider here, where the explicit selfinteraction 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 motion of the particle raises the question of how this radiation, acting back on the particle, affects its motion. Rohrlich [3] has described the historical development of this problem, where the first steps were taken by Abraham in 1905 [4], culminating in the work of Dirac [5], who derived the equation for the ideal point electron in the form where m is the electron mass, including electromagnetic correction, s is the proper time along the trajectory x µ (s) in spacetime, F µ ν is the covariant form of the electromagnetic force tensor, e is the electron charge, and Here, the indices µ, ν, running over 0, 1, 2, 3, label the spacetime variables that represent the action of the Lorentz group; the index raising and lowering Lorentz invariant tensor η µ ν is of the form diag(−1, +1, +1, +1). The expression for Γ µ was originally found by Abraham in 1905[4], shortly after the discovery of special relativity, and is known as the Abraham four-vector of radiation reaction. Dirac's derivation [5] was based on a direct application of the Green's functions for the Maxwell fields, obtaining the form (1.1), which we shall call the Abraham-Lorentz-Dirac equation. In this calculation, Dirac used the difference between retarded and advanced Green's functions, so as to eliminate the singularity carried by each. Sokolov and Terner [6], for example, give a derivation using the retarded Green's 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 small coefficient 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 by e 2 , any small deviation in the lower order terms results in a large effect on the orbit, and there are unstable solutions, often called "runaway solutions". There is a large literature [7] on the methods of treating this instability, and there is considerable discussion in Rohrlich's book [3] as well (he describes a method for eliminating the unstable solutions by studying asymptotic properties).
The existence of these "runaway solutions", for which the electron undergoes an exponential accelaration with no external force beyond a short initial perturbation of the free motion, is a difficulty for the point electron picture in the framework of the Maxwell theory with the covariant Lorentz force. Rohrlich [8] has discussed the idea that the point electron idealization may not be really physical, based on arguments from classical and quantum theory, and emphasized that, for the corresponding classical problem, a finite size can eliminate this instability. Of course, this argument is valid, but it leaves open the question of the consistency of the Maxwell-Lorentz theory which admits the concept of point charges, as well as what is the nature of the elementary charges that make up the extended distribution.
It is quite remarkable that Gupta and Padmanabhan [9], using essentially geometrical 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 accelerating charged particle must include the radiation terms of the Abraham-Lorentz-Dirac equation. Recognizing that the electron's acceleration precludes the use of a sequence of "instantaneous" 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 (assuming constant acceleration) in which the electron is actually inertial, and there solve the Coulomb problem in the curved coordinates provided by the Fermi-Walker transformation. Transforming back to laboratory coordinates, they find the Abraham-Lorentz-Dirac equation without the direct use of the Maxwell Green's functions for the radiation field. This result, suggesting the relevance of curvature in the spacetime manifold, such as that generated by sources in general relativity, along with other, more elementary manifestations of mass renormalization (such as the contribution to the mass due to electromagnetic interactions and the identification of the Green's function singularity contribution with part of the electron mass), carries an implication that the electron mass may play an important dynamical role. Stueckelberg, in 1941[12], proposed a manifestly covariant form of classical and quantum mechanics in which space and time become dynamical observables. They are therefore represented in quantum theory by operators on a Hilbert space on square integrable functions in space and time. The dynamical development of the state is controlled by an invariant parameter τ , which one might call the world time, coinciding with the time on the (on mass shell)freely falling clocks of general relativity. Stueckelberg [12] started his analysis 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 identified the branch of the curve running backward in time with the antiparticle, a view taken also 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 there are two values of the space variables), Stueckelberg introduced the parameter τ along the trajectory.
This parameter is not necessarily identical to proper time, even for inertial motion for which proper time is a meaningful concept. Stueckelberg postulated the existence of an invariant "Hamiltonian" K, which would generate Hamilton equations for the canonical variables x µ and p µ of the formẋ where the dot indicates differentiation with respect to τ . Taking, for example, the model we see that the Hamilton equations imply thaṫ where p 0 ≡ E, where we set the velocity of light c = 1; this is the correct definition for the velocity of a free relativistic particle. It follows, moreover, thaṫ With our choice of metric, dx µ dx µ = −ds 2 , and p µ p µ = −m 2 , where m is the classical experimentally measured mass of the particle (at a given instant of τ ). We see from this that and hence the proper time is not identical to the evolution parameter τ . In the case that m 2 = M 2 , it follows that ds = dτ , 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 µ , the Hamiltonian becomes so that, since K is a constant of the motion, m 2 varies from point to point with the variations of V (x). It is important to recognize from this discussion that the observable particle mass depends on the state of the system (in the quantum theory, the expectation value of the operator p µ p µ 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 than the proton. However, bound in a nucleus, it may be stable (in the nucleus, the proton may decay into neutron, positron and neutrino, since the proton may be sufficiently heavier than the neutron). The mass of the bound electron (in interaction with the electromagnetic 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 wishes to construct a covariant quantum theory, the variables E (energy) and p µ should be independent, and not constrained by the relation E 2 = p 2 + m 2 , where m is a fixed constant. This relation implies, moreover, that m 2 is a dynamical variable. It then follows, quantum mechanically, through the Fourier relation between the energy-momentum representation of a wave function and the spacetime representation, that the variable t, along with the variable 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 particle's mass back to its original mass-shell value (after the source responsible for the mass change ceases to act) so that the particle's mass shell is defined. We shall not take such a mechanism into account explicitly here in developing the dynamical equations. We shall assume 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 work fairly freely) 1 In an application of statistical mechanics to this theory [15], it has been found that a high temperature phase transition can be responsible for the restriction of the particle's mass (on the average, in equilibrium). In the present work, we shall see that, at least in the classical theory, the non-linear equations induced by radiation reaction may have a similar effect.
A theory of Stueckelberg type, providing a framework for dynamical evolution of a relativistic charged particle, therefore appears to be a natural dynamical generalization of the 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" potential, through the requirement of gauge invariance, and there is a generalized Lorentz force which contains a term that drives the particle off-shell, whereas the terms corresponding to the 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 magnetic field; the electromagnetic field tensor in our case is analogous to the magnetic field, and the new field strengths, derived from the τ dependence of the fields and the additional gauge 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 the standard Maxwell theory is properly contained in this more general framework. Applying the Green's functions to the current source provided by the relativistic particle, and the generalized Lorentz force, we obtain equations of motion for the relativistic particle which is, in general off-shell. As in Dirac's result, these equations are of third order in the evolution paramter, and therefore are highly unstable. However, the equations are very nonlinear, and give rise to chaotic behavior.
Our results exhibit what appears to be a strange attractor in the phase space of the autonomous equation for the off-mass shell deviation, This attractor may stabilize the electron's mass in some neighborhood. We conjecture that it stabilizes the orbits macroscopically as well, but a detailed analysis awaits the application of more powerful computing facilities and procedures.

Equations of motion
The Stueckelberg-Schrödinger equation which governs the evolution of a quantum state state over the manifold of spacetime was postulated by Stueckelberg [10] to be, for the free particle, where, on functions of spacetime, p µ is represented by ∂/∂x µ ≡ ∂ µ . Taking into account full U (1) gauge invariance, corresponding to the requirement that the theory maintain its form under the replacement of ψ by e ie 0 Λ ψ, the Stueckelberg-Schrödinger equation (including a compensation field for the τ -derivative of Λ) is [16] i ∂ ∂τ where the gauge fields may depend on τ , and e 0 is a coupling constant which we shall see has the dimension ℓ −1 . The corresponding classical Hamiltonian then has the form in place of (2.1). Stueckelberg [10] did not take into account this full gauge invariance requirement, working in the analog of what is known in the nonrelativistic case as the Hamilton gauge (where the gauge function Λ is restricted to be independent of time). The equations of motion for the field variables are given (for both the classical and quantum theories) by [16] λ∂ α f βα (x, τ ) = e 0 j β (x, τ ), (2.4) where α, β = 0, 1, 2, 3, 5, the last corresponding to the τ index, and λ, of dimension ℓ −1 , is a factor on the terms f αβ f αβ in the Lagrangian associated with (2.2) (with, in addition, degrees of freedom of the fields) required by dimensionality. The field strengths are 5) and the current satisfies the conservation law [16] ∂ α j α (x, τ ) = 0; (2.6) integrating over τ on (−∞, ∞), and assuming that j 5 (x, τ ) vanishes at |τ | → ∞, one finds that We identify this J µ (x) with the Maxwell conserved current. In ref. [18], for example, this expression occurs with j µ (x, τ ) =ẋ µ (τ )δ 4 (x − x(τ )), (2.8) and τ is identified with the proper time of the particle (an identification which can be made for the motion of a free particle). The conservation of the integrated current then follows from the fact that a total derivative; we assume that the world line runs to infinity (at least in the time dimension) and therefore it's integral vanishes at the end points [10,18], in accordance with the discusssion above.
As for the Maxwell case, one can write the current formally in five-dimensional form For α = 5, the factorẋ 5 is unity, and this component therefore represents the event density in spacetime.
Integrating the µ-components of Eq. (2.4) over τ (assuming f µ5 (x, τ ) → 0 for τ → ±∞), we obtain the Maxwell equations with the Maxwell charge e = e 0 /η and the Maxwell fields given by (2.10) A Hamiltonian of the form (2.3) without τ dependence of the fields, and without the a 5 terms, as written by Stueckelberg [10], can be recovered in the limit of the zero mode of the fields (with a 5 = 0) in a physical state for which this limit is a good approximation i.e., when the Fourier transform of the fields, defined by has support only in the neighborhood ∆s of s = 0. The vector potential then takes on the form a µ (x, τ ) ∼ ∆sâ µ (x, 0) = (∆s/2πλ)A µ (x), and we identify e = (∆s/2πλ)e 0 . The zero mode therefore emerges when the inverse correlation length of the field ∆s is sufficiently small, and then η = 2πλ/∆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 generalized Maxwell equation reduce to the usual Maxwell equations. The theory therefore contains the usual Maxwell Lorentz theory in the limit of the zero mode; for this reason we have called this generalized theory the "pre-Maxwell" theory. If such a pre-Maxwell theory really underlies the standard Maxwell theory, then there should be some physical mechanism which restricts most observations in the laboratory to be close to the zero mode. For example, in a metal there is a frequency, the plasma frequency, above which there is no transmission of electromagnetic waves. In this case, if the physical universe is imbedded in a medium which does not allow high "frequencies" to pass, the pre-Maxwell theory reduces to the Maxwell theory. Some study has been carried out, for a quite different purpose (of achieving a form of analog gravity), of the properties of the generalized fields in a medium with general dielectric tensor [19]. We shall see in the present work that the high level of nonlinearity of this theory in interaction with matter may itself generate an effective reduction to Maxwell-Lorentz theory, with the high frequency chaotic behavior providing the regularization achieved by models of the type discussed by Rohrlich [8].
We remark that integration over τ does not bring the generalized Lorentz force into the form of the standard Lorentz force, since it is nonlinear, and a convolution remains. If the 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 present in the standard Maxwell-Lorentz theory.
Writing the Hamilton equationṡ for the Hamiltonian (2.3), we find the generalized Lorentz force (2.13)

Multiplying this equation byẋ µ , one obtains
Mẋ µẍ µ = e 0ẋµ f µ 5 ; (2.14) this equation therefore does not necessarily lead to the trivial relation between ds and dτ discussed above in connection with Eq. (1.9). The f µ 5 term has the effect of moving the particle off-shell (as, in the nonrelativistic case, the energy is altered by the electric field).

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 Stueckelberg formalism (see also [17]. Calculating the self interaction contribution one must include the effects of the force acting upon the particle due to its own field (f self ) in addition to the fields generated by other electromagnetic sources (f ext ). Therefore, the generalized Lorentz force, using Eq. The field generated by the current is given by the pre-Maxwell equations Eq.(2.4), and choosing for it the generalized Lorentz gauge ∂ α a α = 0, we get Integrating this expression over all τ gives the Green's function for the standard Maxwell field. Assuming that the radiation reaction acts causally in τ , we shall use here the τretarded Green's function. In his calculation of the radiation corrections to the Lorentz force, Dirac used the difference between advanced and retarded Green's functions in order to cancel the singularities that they contain. One can, alternatively use the retarded Green's function and "renormalize" the mass in order to eliminate the singularity [6]. In this analysis, we follow the latter procedure. The τ -retarded Green's function [17] is given by multiplying the principal part of the integral Eq.(3.4) by θ(τ ). Carrying out the integrations (on a complex contour in κ; we consider the case σ = +1 in the following), one finds (this Green's function differs from the t-retarded Green's function, constructed on a complex contour in k 0 ), With the help of this Green's function, the solutions of Eq.(3.3) for the self-fields (substituting the current from Eq.(2.9)) are and The Green's function is written as a scalar, acting in the same way on all five components of the source j α ; to assure that the resulting field is in Lorentz gauge, however, it should be written as a five by five matrix, with the factor δ α β − k α k β /k 2 (k 5 = κ) included in the integrand. Since we compute only the gauge invariant field strengths here, this extra term 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 µ (τ ) of the trajectory), along with the effect of external fields, is (3.9)

Eq.(3.8) then becomes
(3.10) In the self-interaction problem where τ → τ ′ , x µ (τ ′ )−x µ (τ ) → 0, the Green's function is very divergent. Therefore one can expand all expressions in τ ′′ = τ − τ ′ assuming that the dominant contribution is from the neighborhood of small τ ′′ . The divergent terms are later absorbed into the mass and charge definitions leading to renormalization (effective mass and charge). Expanding the integrands in Taylor series around the most singular point τ ′ = τ and keeping the lowest order terms, the variable u reduces to 2 : We now recall the definition of the off-shell deviation, ε given in Eq.(2.15), along with its derivatives:ẋ ..
For either sign of ε, when ε ∼ 0, One sees that the derivatives in Eq. (3.15) have no singularity in ε at ε = 0.
Here, the coefficients ofẍ µ have been grouped formally into a renormalized (off-shell) mass term, defined (as done in the standard radiation reaction problem ) where, as we shall see below, 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 ε < 1), to obtain a differential equation in which all derivatives with respect to τ are replaced by derivatives with respect to the proper time s. The coefficient of the second derivative of x µ with respect to s, and "effective mass" for the proper time equation, is then given by Note that the renormalized mass depends on ε(τ ); for this quantity to act as a mass, ε must be slowly varying on some interval on the orbit of the evolution compared to all other motions. The computer analysis we give below indeed shows that there are large intervals of almost constant ε. In case, as at some points, ε may be rapidly varying, one may consider the definition Eq.(3.26) as formal; clearly, however, if M (ε,ε) is large,ẍ µ will be suppressed (e.g., for ε close to unity, where M S (ε,ε) goes asε/(1 − ε) 2 and M (ε,ε) asε/(1 − ε) 3 ; note that F (ε) goes as (1 − ε) −3 ).
We now obtain, from Eq.(3.24), .. (3.28) We remark that when one multiplies this equation byẋ µ , it becomes an identity (all of the terms except for e 0 f ext µ νẋ ν may be grouped to be proportional to ẋ µẋ ν 1−ε + δ µ ν ); one must use Eq. (3.22) to compute the off-shell mass shift ε corresponding to the longitudinal degree of freedom in the direction of the four velocity of the particle. Eq.(3.28) determines the motion orthogonal to the four velocity. Equations (3.22) and (3.28) are the fundamental dynamical 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 ε and that that 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 ε close to, but not precisely zero.

The ε evolution
We now derive an equation for the evolution of the off-shell deviation, ε, when the external field is removed. We then use this equation to prove that a fixed mass-shell is consistent only if the particle is not accelerating, and therefore no runaway solution occurs. Using the definitions in equation Eq. (3.22), in the absence of external fields, we writė Differentiating with respect to τ we find: ε .

Dynamical Behavior of System
In this section, we present results of a preliminary study of the dynamical behavior of the system. The four orbit equations pose a very heavy computational problem, which we shall treat in a later publication. We remark here, however, that the quantities dx/dτ and dt/dτ appear 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 ε, and show that there is a highly complex dynamical behavior showing strong evidence of chaos.
It appears that ε reaches values close to (but less than) unity, and there exhibits very complex behavior, with large fluctuations. The effective mass M (ε,ε) therefore appears to reach a macroscopically steady value, apparently a little less than unity. In this neighborhood, it is easy to see from the expression (3.26) that the effective mass itself becomes very large, asε/(1 − ε) 3 , and therefore the effect of the Lorentz forces in producing acceleration becomes small.
The third order equation of ε, Eq.(4.6), can be written as a set of three first order equations of the form where γ = γ1, γ2, γ3 = ε,ε,ε. We have obtained numerical solutions over a range of variation of τ . In figure 1, we show, on a logarithmic scale, these three functions; one sees rapid fluctuations in γ2 and γ3, and large variations in γ1. The action of what appears to be an attractor occurs on these graphs in the neighborhood of τ = 0.002 (time steps were taken to be of the order of 10 −6 ). The time that the system remains on this attractor, in our calculation about four times this interval, covers about three cycles. The characteristics of the orbits in phase space are very complex, reflecting an evolution of the nature of the attractor.
The dynamical significance of these functions is most clearly seen in the phase space plots. In figure 2, we showε = γ3 vs.ε = γ2. This figure shows the approach to the apparent attractor inducing motion beginning at the lower convex portion of the unsymmetrical orbit which then turns back in a characteristic way to reach the last point visible on 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 orbit then continues, as shown in figure 4 to a third loop, much smaller, which then reaches a structure that is significantly different. The end region of this cycle is shown on larger scale in figure 5, where a small cycle terminates in oscillations that appear to become very rapid as ε approaches unity. The coefficients in the differential equation become very large in this neighborhood.
In figure 6 we see the approach to the attractor in the graph ofε = γ2 vs. ε = γ1; this curve enters a region of folding, which develops to a loop shown in figure 7a,and in fig. 7b, on a larger scale, where we had to terminate the calculation due to limitations of our computer.
We have computed the global Lyapunov exponents according to the standard procedures; recognizing, however, that there appear to be (at least) two levels of behavior types, we have studied the integration on phase space for ε somewhat less than unity, since the integration diverges for ε close to one. For ε very close to one, in fact, due to the relation ds = (1−ε)dτ , this neighborhood contributes very little to the development of the observed orbits.
In this calculation, we have computed the largest Lyapunov exponent (positive) by studying 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 two cycles on the attractor). Note that there are intervals of relative stability and very strong instability.
Starting the calulation with different initial conditions, we found very similar dynamical behavior, confirming the existence of the apparent attractor.
We have also studied the time series associated with the three variables of the phase space, shown in figures 9-11. The autocorrelation functions forε andε fall off quickly giving characteristic correlation times; the autocorrelation function for ε does not fall off faster than linearly, but the time series shows significant structure resembling the phase space structure (subtracting out the local average from the function ε generates a deviation function that does indeed show a fall off of the autocorrelation function). We have used these scales, for 20 time steps (forε and ε) and 40 time steps (forε, to contruct a three 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 forε vs. ε andε vs.ε. These results are very similar in form to those given above, demonstrating stability of the results.

Discussion
We have examined the equations of motion generated for the classical relativistic charged particle, taking into account dynamically generated variations in the mass of the particle, and the existence of a fifth electromagnetic type potential essentially reflecting the gauge degree of freedom of the variable mass. We have focussed our attention on the autonomous equation for the off-shell mass deviation defined in (2.15) and (2.17) as an indication of the dynamical behavior of the system, and shown that there indeed appears to be a highly complex strange attractor. It is our conjecture that the formation of dynamical attractors of this type in the orbit equations as well will lead to a macroscopically smooth behavior of the perturbed relativistic charged particle, possibly associated with an effective non-zero size, as discussed, for example, by Rohrlich [8]. Some evidence for such a smooth behavior appears in the very large effective mass generated near ε ∼ = 1, stabilizing the effect of the Lorentz forces, as well as our preliminary result that the larger accellerations emerging in derivatives with respect to τ are strongly damped, in this neighborhood of ε, 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 somewhat arbitrary initial conditions, and we find that changing these conditions somewhat does not affect the general pattern of the results. Moreover, sinceε depends on the third derivative of x µ (τ ), this initial condition should not be taken as arbitrary, but as determined by the results for the orbit (initial conditions for the orbit depend only on x µ and its first two derivatives). Computing the orbit for a typical choice of initial conditions, and using the result to fixε, we find that the corresponding solution of the autonomous equation indeed lies in the attractor. The results of our investigation of the autonomous equation should therefore be valid for initial conditions that obey the physical constraints imposed by the equations of motion as well.
We have specified all of the dynamical equations here, but a complete investigation must await further planned work with more powerful computing facilities and procedures.