aa r X i v : . [ qu a n t - ph ] J un On an electrodynamic origin of quantum fluctuations ´Alvaro G. L´opez ∗ Nonlinear Dynamics, Chaos and Complex Systems GroupDepartamento de F´ısica, Universidad Rey Juan Carlos,Tulip´an s/n, 28933 M´ostoles, Madrid, Spain (Dated: June 15, 2020)
Abstract
In the present work we use the Li´enard-Wiechert potential to show that very violent fluctuationsare experienced by an electromagnetic charged extended particle when it is perturbed from its reststate. The feedback interaction of Coulombian and radiative fields among the different chargedparts of the particle makes uniform motion unstable. As a consequence, we show that radiativefields and radiation reaction produce both dissipative and antidamping effects, leading to self-oscillations. Finally, we derive a series expansion of the self-potential, which in addition to rest andkinetic energy, gives rise to a new contribution that shares features with the quantum potential. Thenovelty of this potential is that it produces a symmetry breaking of the Lorentz group, triggeringthe oscillatory motion of the electrodynamic body. We propose that this contribution to self-energymight serve as a bridge between classical electromagnetism and quantum mechanics.
Keywords: Nonlinear Dynamics - Quantum Fluctuations - Electrodynamics - Relativity ∗ [email protected] . INTRODUCTION It was shown in the mid-sixties that a dynamical theory of quantum mechanics can beprovided based on a process of conservative diffusion [1]. The theory of stochastic mechanicsis a monumental mathematical achievement that has been carefully and slowly carried outalong two decades with the best of the rigors and mathematical intuition [2]. However, asfar as the authors are concerned, the grandeur of this theoretical effort is that it proposes akinematic description of the dynamics of quantum particles, based on the theory of stochasticprocesses [3]. Just as Bohmian mechanics [4, 5], it tries to offer a geometrical picture of thetrajectory of a quantum particle, which would be so very welcomed by many physicists.In the end, establishing a link between dynamical forces and kinematics is at the core ofNewton’s revolutionary work [6].Perhaps, the absence of geometrical intuition in this traditional sense, during the de-velopment of the quantum mechanical formalism, has hindered the understanding of theunderlying physical mechanism that leads to quantum fluctuations. In turn, it has con-demned the physicist to a systematic titanic effort of mathematical engineering, designingever-increasing complicated theoretical frameworks. Despite of providing a very refined ex-planation of many experimental data, which is the main purpose of any physical theory,needless to say, these frameworks entail a certain degree of obscurantism and a lack of un-derstanding. Concerning comprehension only, quantum mechanics constitutes a paradigmof these kind of paradoxical theories, which imply that the more time that it is dedicated tothe their study, the less clear that the physical picture of nature becomes. As it has beenpointed out by Bohm, this might be a consequence of renouncing to models in which allphysical objects are unambiguously related to mathematical concepts [4].On the contrary, hydrodynamical experimental models that serve as analogies to quantummechanical systems have been developed recently, which allow to clearly visualize how thedynamics of a possible quantum particle might be [7, 8]. These experimental contemporarymodels share many features with the mechanics of quantum particles [9, 10] and, fortunately,they are based on firmly established and understandable principles of nonlinear dynamicaloscillatory systems and chaos theory [11, 12]. As it is well accepted, these conceptual frame-works have shaken the grounds of the physical consciousness of many scientists by showingthe tremendous complexity of the dynamical motion of rather simple classical mechanical2ystems, and not so simple as well [12]. Doubtlessly, the development of computation hasproven to be a fundamental tool in this regard, serving as a microscope to the modern physi-cist, which allows him to unveil the complex patterns and fractal structures that explain thehidden regularities of chaotic motion [13]. Thus, even if we can not experimentally tracea particle’s path because we perturb its dynamics by the mere act of looking at it, we canalways use our powerful computers to simulate their dynamics.In the final pages of Nelson’s work, it is seductively suggested that a theory of quantummechanics based on classical fields should not be disregarded, as was originally the purpose ofAlbert Einstein [2]. This aim of providing quantum mechanics with a kinematic description,together with the desire of showing the unjustified belief of electrodynamic fields as a merelydissipative force on sources of charge, and not as an exciting self-force as well, are the twocore reasons that have spurred the authors to pursue the present goal. By using a toy modeland rather simple mathematics, we show as a main result in what follows that a finite-sizedcharged accelerated body always carries a vibrating field with it, what can convert thisparticle into a stable limit cycle oscillator by virtue of self-interactions. This implies thatthe rest state of this charged particle can be unstable, and that stillness (or uniform motion)might not the default state of matter, but also accelerated oscillatory dynamics. We closethis work by deriving an analytical expression of the self-potential. For this purpose we onlyneed to assume that inertia is of purely electromagnetic origin. As it will be demonstrated,the first order terms of this self-potential contain the relativistic energy (the rest and thekinetic energy) of the electrodynamic body, while higher order terms can be related to anew function, that can be correlated to the quantum potential. In this manner, we hopeto provide a better understanding of quantum motion or, at least, to pave the way towardssuch an understanding.
II. THE SELF-FORCE
We begin with the Li´enard-Wiechert potential [14, 15] for a body formed by two chargedpoint particles attached to a neutral rod that move transversally along the x-axis. In general,any motion with transversal field component suffices to derive the main conclusions of thiswork. However, to avoid dealing with the rotation of the dumbbell, we restrict to a one-dimensional translational motion. This allows to keep mathematics as simple as possible,3
IG. 1.
A model for an electrodynamic body . An extended electron, modeled as a dumbbelljoining two point charged particles (black dots) at a fixed distance d . The particle is shown at theretarded time t r and at a some later time t . During this time interval, the corpuscle accelerates inthe x-axis, advancing some distance l in such direction. As we can see, the particle in the upperpart emits a field perturbation at the retarded time (red photon), and this perturbation reachesthe second particle at the opposite part of the dumbbell at a later time (and vice versa). In thismanner, an extended corpuscle can feel itself in the past. The speed and the acceleration of theparticle are represented in blue and green, respectively. since the Li´enard-Wiechert potential is retarded in time, and this non-conservative characterof electrodynamics makes the computations very entangled. This elementary model waswisely designed in previous works to derive from first principles the Lorentz-Abraham force[16, 17] and also to study a possible electromagnetic origin of inertia [18, 19]. It is a toymodel of an electron, represented as an extended electrodynamic body with approximate size d , as shown in Fig. 1. Among the aforementioned virtues, we also find that some propertiesresulting from considering more complex geometries (spherical, for example) of a particle,can be derived by superposition [19]. We shall use this elementary model all along ourexposition, which is more than sufficient to illustrate the fundamental mechanism that leadsto electrodynamic fluctuations.As we can see in Fig. 1, the first particle affects the other at a later time, since the4erturbations of the field have to travel from one particle to the other. In other words,an extended body can affect itself. This sort of interaction is traditionally known as aself-interaction in the literature [18] and, as can be seen ahead, for any charged particle, itproduces an excitatory force, together with a recoil force and an elastic restoring force aswell. The complete Li´enard-Wiechert potential permits to write the electric field created bythe first particle at the point of the second as E = q πǫ r ( r · u ) (cid:18) u (1 − β ) + 1 c r × ( u × a ) (cid:19) , (1)where we have now defined the vector u = ˆ r − β , with the relative position betweenparticles r ( t r ), their velocity β ( t r ) = v ( t r ) /c and their acceleration a ( t r ) depending onthe retarded time t r = t − r/c . The retarded time appears due to the limited speed at whichelectromagnetic field perturbations travel in spacetime, according to Maxwell’s equations[20]. This restriction imposes the constraint r = c ( t − t r ) , (2)which assigns a particular time in the past from which the signals coming from one particle ofthe dumbbell affect the remaining particle. As we shall see, the fact that dynamical systemsunder electrodynamic interactions are time-delayed ( i.e. the non-Markovian character ofelectrodynamics), is at the basis of the whole mechanism. Now we follow the picture in Fig. 1and write the position, the velocity and the acceleration vectors as r = l ˆ x + d ˆ y , β = v/c ˆ x and a = a ˆ x , respectively, where the distance l = x ( t ) − x ( t r ) between the present position ofthe particle and the position at the retarded time has been introduced. Using these relations,the vector u can be computed immediately as u = ( l − rβ ) ˆ x + d ˆ y r , (3)which, in turn, allows to write the inner product r · u = r − lβ , by virtue of the Pythagoras’theorem r = ( x ( t ) − x ( t r )) + d . Concerning the radiative fields, we can express the triplecross-product as r × ( r u × a ) = − d a ˆ x + dal ˆ y . We now compute the net self-force on theparticle’s centre of mass as F self = q E + E ) = qE x ˆ x , (4)where E is the force of the second particle on the first. Note that we have assumed thatall the forces on the y-axis cancel, because we have simplified the model by using a rigid5umbbell to keep the distance of the charges fixed. This includes repulsive electric forcesand also magnetic attractive forces as well. Therefore, in the present section we do notcover the much more complicated problem of the particle’s stability, which is discussed inthe last section of the present work. Such a problem is of the greatest importance and leadto the introduction of Poincar´e’s stresses in the past [21] and, among other reasons ( e.g. atomic collapse), to the rejection of classical electrodynamics as a fundamental theory [22].If prefered, from a theoretical point of view, the reader can consider that the two pointparticles of our model are kept at a fixed distance by means of some balancing externalelectromagnetic field oriented along the y-axis.Now, we replace the value of the charge with the charge of the electron q = − e tofinally arrive at the mathematical expression describing the self-force of the particle, whichis written as F self = e πǫ r − lβ ) (cid:18) ( l − rβ )(1 − β ) − d c a (cid:19) ˆ x . (5) III. THE EQUATION OF MOTION
We are now committed to write down Newton’s second law in the non-relativistic limit F self = m a and redefine the mass of the particle since, as we show right ahead, the elec-trostatic internal interactions add a term to the inertial content of the particle. The mainpurpose of the following lines is to expand in series the self-force to show its different contri-butions to the equation of motion. The two most resounding terms are the Lorentz-Abrahamforce and the force of inertia. However, we draw attention to other relevant nonlinear terms,which are of fundamental importance. These expansions will enable a discussion about theelectromagnetic origin of mass and, based on such line of reasoning, we shall derive theappropriate and precise equation of motion.As it has been shown in previous works [18, 19], it is possible to express l as a functionof r by means of the series expansion l = x (cid:16) t r + rc (cid:17) − x ( t r ) = βr + a c r + ˙ a c r + ¨ a c r + ... (6)This trick of approximating magnitudes presenting delay differences by means of a Taylorseries has been used sometimes in the study of delayed systems along history [23]. Werecall that this simplification is not a minor issue, since by truncating this expansion we6re replacing a system with memory by a Markovian one. Nevertheless, the reader mustbe aware that delayed systems are infinite-dimensional. In fact, as we show below, anytruncation of the previous equation is mistaken since, even though the time-delay r/c issmall, the terms in the acceleration, the jerk and so on, are not of order zero in such factor.As shown in the Appendix, together with Eq. (2), the previous expansion allows to expressthe corpuscle’s size in terms of the time-delay by means of the series d = r − a c βr − (cid:18) a c + β ˙ a c (cid:19) r + ... (7)This Taylor series can be inverted to compute the expansion of r in terms of d , which canbe written to first order in β as r = d + a c βd + (cid:18) a c + β ˙ a c (cid:19) d + ... (8)Finally, by inserting Eq. (8) in the previous Eq. (6) and then both equations in Eq. (5), withthe aid of Newton’s second law, we compute, to first order in β , the identity (cid:18) m + e πǫ c d (cid:19) a = e πǫ (cid:18) c a v + 5 d c a a + 16 c ˙ a + d c ¨ a + ... (cid:19) , (9)after a great deal of algebra. These computations are enormously simplified by means ofmodern software for symbolic computation [24].We notice that the Lorentz-Abraham force has appeared in the third term of the right-hand side of this last equation, together with a few other linear and nonlinear terms. In-terestingly, we recall that the term of inertia dominates all other terms for small speedsand accelerations. We can truncate this equation up to the jerk term ˙ a , disregarding itsnonlinearity and also derivatives of higher order. We can also define the renormalized massof the electron as m e = m + e πǫ c d , (10)and recall the relation between the electron’s charge and Planck’s constant by means of thefine structure constant ~ αc = e πǫ , (11)according to Sommerfeld’s equation [25]. Then, we get the approximated solution¨ β − m e c ~ α ˙ β (cid:18) − ~ αd m e c ˙ β (cid:19) + 3 a c β + ... = 0 , (12)7hich reminds of the equation of a nonlinear oscillator.Thus, we see that the term of inertia, which is the linear term in the acceleration andwhich dominates when the particle is perturbed from rest, acts as an antidamping. Thisterm is due to radiation fields and is responsible for the amplification of fluctuations. Thisfact does not contradict Newton’s third law, since it is the addition of matter and radiationmomentum that must be conserved as a whole. In other words, the particle can propelitself for a finite time by taking energy from its “own” field. However, the nonlinear cubicterm in ˙ β in Eq. (12), which has opposite sign, limits the growth of the fluctuations. Whenthe acceleration surpasses a certain critical value, the radiation reaction and the radiativefields do not act in phase anymore, and the fluctuations are damped away. Therefore, thepathological attributes that have been predicated of this marvelous recoil force [19] areunjustified, and arise as a consequence of disregarding nonlinearities, which are responsiblefor the system’s stabilization and, as we shall demonstrate, its self-oscillatory dynamics.Importantly, at this point we notice that, if we assume that the inertia of the electronhas an exclusive electromagnetic origin and recall that the dumbbell is neutral ( m = 0), allthe mass must come from the charged points. Then, using the Eqs. (10) and (11) we canwrite the mass as m e = ~ α dc , (13)which was obtained in previous works [18] and gives an approximate radius of the particle r e = d/ . × − m. Except for a factor of eight due to the dumbbell’s geometry, thisvalue corresponds to the classical radius of the electron. In this manner, we do not need tointroduce spurious elements (artificial mechanical inertia) in the theory of electromagnetism,and simply use the D’Alembert’s principle instead of Newton’s second law [26]. If desired,and to extol Newton’s intuition, the second law of classical mechanics would be a conclusionof electromagnetism, which is the most fundamental of classical theories. What it is amazingis that Newton was capable of figuring it out without any knowledge on electrodynamics.However, this wonderment partly fades out if we bear in mind the unavoidable corollary. Forif mass is of electromagnetic origin, the gravitational field must be a residual electromagneticfield. If we are willing to accept these two inextricable facts, inertia would just be an internalresistance or self-induction force produced by the field perturbations to the motion of thecharged body, when an external field is applied. We tackle more deeply this issue in thecolophon of this work. 8n summary, we believe that it is more appropriate to simply consider Newton’s secondlaw as a static problem F ext + F self = 0. In our case, we simply have F self = 0. This wayof posing the problem can be regarded as computing the geodesic equation of motion of theparticle, as it occurs, for example, in the theory of general relativity. The resulting equationof motion reads (cid:18) − v ( t r ) c (cid:19) (cid:16) x ( t ) − x ( t r ) − rc v ( t r ) (cid:17) − d c a ( t r ) = 0 , (14)where we recall that for v = c the first term vanishes, not allowing the particle to overcomethe speed of light.We now derive two relations that shall prove of great assistance in forthcoming sectionsto compute exact results. For this purpose, we use again the Pythagoras’ theorem r =( x ( t ) − x ( t r )) + d and the equality appearing in Eq. (14). By combining these two equationsit is straightforward to derive a second order polynomial in r , which is solved yielding r = γd s γ ˙ β (cid:18) dc (cid:19) + γ cβ ˙ β (cid:18) dc (cid:19) , (15)where the Lorentz factor γ = (1 − β ) − / has been introduced and the kinematic variables areevaluated at the retarded time. Note that, contrary to the previous Eq. (8), this expressionis exact and has the virtue of suggesting that any consistent power series expansion of r should be carried out in terms of the factor d/c . We also notice that, by virtue of thisequation, the delay becomes dependent on the speed and the acceleration of the particle. Asthe corpuscle speeds up, the self-signals come from earlier times in the past. In other words,the light cone of the corpuscle is dynamically evolving, and this evolution selects differentsignals coming from the past.Finally, the insertion of this relation into the equation r = l + d leads to the obtainmentof l as a function of β and ˙ β in a closed form. Again, this avoids the use of an infinite numberof derivatives. The final result can be written as l = vuut γ c β (cid:18) dc (cid:19) + γ c ˙ β (1 + β ) (cid:18) dc (cid:19) + 2 c γ β ˙ β (cid:18) dc (cid:19) s γ ˙ β (cid:18) dc (cid:19) . (16)These two Eqs. (15) and (16) will allow us to derive analytical results in a fully relativisticmanner, specially concerning the self-potential.9 V. THE INSTABILITY OF REST
Even though we shall prove a more general statement in Sec. 5, we believe that the factthat oscillatory dynamics can be the default state of matter, instead of a stationary state,is of paramount importance. In turn, this study provides a double check of the resultspresented in such section. Therefore, we independently study the stability of the rest stateof the particle in the following lines. Our goal is to show that the rest state is unstableand to identify the magnitude that leads to the amplification of fluctuations. For thispurpose, we begin with the expansion appearing in Eqs. (6) and (8), and replace them inEq. (14), neglecting all the nonlinear terms. Such terms can be disregarded since the reststate is represented by v and all its higher derivatives are equal to zero. Thus, when slightlyperturbing the rest state of the charged particle, we only need to retain linear contributions.The resulting infinite-dimensional differential equation is − c d a + 16 c ˙ a + d c ¨ a + d c ... a + ... = 0 . (17)This equation can be more clearly written as a Laurent series in the factor d/c , as previ-ously suggested. We obtain the result − cd a + 16 ˙ a + 124 dc ¨ a + 1120 d c ... a + ... = 0 , (18)which can be generally expressed as − a + ∞ X n =1 n + 2)! d n a d t n (cid:18) dc (cid:19) n = 0 . (19)The characteristic polynomial of this equation is obtained by considering as solution a ( t ) = a e λt . We compute the relation −
12 + ∞ X n =1 n + 2)! (cid:18) λdc (cid:19) n = 0 , (20)which can be more elegantly written by using the Maclaurin series of the exponential func-tion. If we redefine it by means of the variable µ = λd/c , we get −
12 + 1 µ ∞ X n =1 µ n +2 ( n + 2)! = −
12 + 1 µ (cid:18) e µ − µ − µ − (cid:19) = 0 . (21)The solutions to this equation can be obtained numerically. Apart from zero, the onlypurely real solution can be nicely approximated as λ = 95 cd , (22)10hich is a positive value. In summary, the rest state is not stable in the Lyapunov sense [27],and this implies that the particle can not be found at rest. In fact, as can be shown in Fig. 2,the complex function f ( z ) = z + z + 1 − e z has an infinite set of zeros in the complex plane.All of them have a positive real part, while all except two of them are complex conjugatenumbers with non-zero imaginary part. It can be analytically shown that, for zeros withnegative real part to exist, they have to be confined in a small region close to the origin.Consequently, numerical simulation is enough to confirm both the instability of rest and theexistence of self-oscillations in the system.As more generally stated below, everything is jiggling because electromagnetic fluctua-tions are amplified. Consequently, motion would be the essence of being and not rest, ascould be inferred from the principle of inertia in Newtonian mechanics. More precisely, andas we are about to show, it is uniform motion that it is unstable. This notion is preciselya strong suggestion in order to assume that inertia has an electromagnetic origin. But weshall give a more compelling one below. Be that as it may, the instability of stillness can beconsidered, by far, the most fundamental finding of the present analysis. V. SELF-OSCILLATIONS
We now proceed to show the existence of limit cycle oscillations of the particle. Sincethe rest state is unstable and the speed of light can not be surpassed according to Eq. (14),the only possibilities left are uniform motion or some sort of oscillatory dynamics, weatherregular or chaotic. In the first place, we rewrite the Eq. (14) to a more amenable and familiarform. We have d c a ( t r ) + rc (cid:18) − v ( t r ) c (cid:19) v ( t r ) + (cid:18) − v ( t r ) c (cid:19) ( x ( t r ) − x ( t )) = 0 . (23)The main handicap of this equation is that it is expressed in terms of the retarded time t r = t − r/c , which it is the customary expression of the Li´enard-Wiechert potentials. Toobtain the same equation in terms of the present time t , we simply perform a time translationto the advanced time t a = t + r/c . This allows to write a ( t ) + rd cd (cid:18) − v ( t ) c (cid:19) v ( t ) + (cid:16) cd (cid:17) (cid:18) − v ( t ) c (cid:19) (cid:16) x ( t ) − x (cid:16) t + rc (cid:17)(cid:17) = 0 . (24)But now the problem is that this equation depends on the advanced time. In otherwords, Eq. (24) allows to derive the position and velocity at some time from the knowledge11 IG. 2.
The roots of the polynomial f ( z ) = z + z +1 − e z . (a) A domain coloring representationof the function. The color represents the phase of the complex function. The shiny level curvesrepresent the values for which | f ( z ) | is an integer, while the dark stripes are the curves Re f ( z )and Im f ( z ) equal to a constant integer. The roots and poles can be localized where all colorsmeet. In the present case we clearly identify the roots z = 0 and z = 9 /
5. (b) Here a zoom out ofthe function is shown, with the distribution of zeros (black dots). The coloring scheme has beensimplified. As can be seen, all of them are distributed on the positive real part of the complexplane. of such position and velocity in the past, by using the position in the future. This equationreminds of the equation of a self-oscillator [28]. Apart from the term of inertia and thelinear oscillating term representing Hooke’s law [29], we have two nonlinear contributions.On the one hand, the second contribution on the left hand side acts here as a dampingterm and it is responsible for the system’s dissipation. This term is identical to other termsappearing in traditional self-oscillating systems, as for example the oscillator introduced byLord Rayleigh’s to describe the motion of a clarinet reed [30] and, to some extent, alsoto the Van der Pol’s oscillator [31]. On the other hand, the antidamping comes from theadvanced potential. At first sight, in the limit of small velocities, the frequency of oscillation12s ω = c/d , what allows to approximate the period as T = 4 π r e c , (25)where r e = d/ T = 1 . × − s for the classical radius of the electron. Therefore, theparticle would oscillate very violently, giving rise to an apparently stochastic kind of motion.This motion and the value of the frequency should not be unfamiliar to quantum mechanicaltheorists, since they can be related to the trembling motion appearing in Dirac’s equation[32], commonly known as zitterbewegung .As we have shown in Sec. 2, the time-delay r depends on the kinematic variables. Weinsist that, in this sense, despite of the simplicity of the model at analysis, we are facing aterribly complicated dynamical system, since the delay itself depends on the speed and theacceleration of the particle. This kind of systems are formally referred in the literature asstate-dependent delayed dynamical systems [33] and, from an analytical point of view, theyare mostly intractable. Importantly, we note that for a system of particles, the dependenceof the delay of a certain particle on the kinematic variables of the others at several times inthe past and at the present as well, turn electrodynamics into a nonlocal theory [34]. Thisfunctional dependence sheds some light into the significance of entanglement, which can nowbe regarded as a process of entrainment of nonlinear oscillators [35].All this complexity notwithstanding, since we just aim at illustrating the existence of self-oscillatory dynamics, we shall have no problems concerning the integration of this system.According to Eq. (22), when the system is amplifying fluctuations from its rest state, wesee that the rate at which the amplitude of fluctuations grows is comparable to the periodof the oscillations. Therefore, averaging techniques, as for example the Krylov-Bogoliubovmethod [36], cannot be safely applied in the present situation. More simply, we consider thedifferential equation (24) and write it in the phase space as˙ x = y, ˙ y = − cd rd (cid:18) − y c (cid:19) y − (cid:16) cd (cid:17) (cid:18) − y c (cid:19) ( x − x τ ) , (26)where x τ represents the position at the advanced time t + τ = t + r/c . As we have shownin the previous section, the fixed point ˙ x = ˙ y = 0 is unstable. Apart from the rest state,asymptotically, there can be only two possibilities. Since the speed of light is unattainable13or massive particles, either the particle settles at a constant uniform motion with a lowerspeed, or its speed fluctuates around some specific value. We do not enter into the issueweather these asymptotic oscillations are periodic, quasiperiodic or chaotic. We shall justprove that uniform motion is not stable and, consequently, self-oscillatory dynamics is theonly possibility, whatever its periodicity might be. Assume that uniform motion is possibleat some speed y , which is a constant number βc . Then, we have that x ( t ) = yt and alsothat x ( t + r/c ) = yt + yr/c , which implies x − x τ = − yr/c . Substitution in Eq. (25) yields˙ x = y, ˙ y = − cd rd (cid:18) − y c (cid:19) y + cd rd (cid:18) − y c (cid:19) y = 0 . (27)Thus, certainly, any uniform motion is also an invariant solution (a fixed trajectory, so tospeak) of our state-dependent delayed dynamical system. However, it is immediate to showthat this solution is unstable as well. We prove this assertion by computing the variationalequation related to inertial observers δ ˙ x = δy,δ ˙ y = − cd δrd (cid:18) − y c (cid:19) y − cd rd (cid:18) − y c (cid:19) δy + cd rd y c δy −− cd rd y c δy − (cid:16) cd (cid:17) (cid:18) − y c (cid:19) ( δx − δx τ ) . (28)At this point, we have to compute δr at ˙ y = 0 and y = βc , with β a constant value. Usingthe formula (15), but evaluated at the present time, this calculation can be carried outwithout difficulties yielding δr ( t ) = γ β (cid:18) dc (cid:19) δ ˙ y ( t ) + dδγ ( t ) , (29)where again we notice that the variables are evaluated at the present time. Gathering termsand using the fact that r = γd for ˙ y = 0, we finally arrive at the variational problem δ ˙ x = δy,δ ˙ yγ = − cd γδy − (cid:16) cd (cid:17) (cid:0) − β (cid:1) ( δx − δx τ ) . (30)If we consider solutions of the form δx = Ae λt , the characteristic polynomial of the system(30) is found. It reads λ γ + cd γλ + (cid:16) cd (cid:17) (1 − β )(1 − e λγd/c ) = 0 . (31)14wo limiting situations appear. In the non-relativistic limit β → λ + cd λ + (cid:16) cd (cid:17) (1 − e λd/c ) = 0 . (32)which, considering µ = λd/c , can be written as µ + µ + 1 − e µ = 0 . (33)This is in conformity with previous results (see Eq. (21)). Finally, in the relativistic limit,we get µ + µ + (1 − e µ )(1 − β ) = 0 , (34)where we have now defined µ = λγd/c . Except for one eigenvalue, the real part of thesolutions to this equation are always positive and therefore unstable for any value of β , asconfirmed by numerical simulations (see Fig. 3). Again, an infinite set of frequencies areobtained, which can be written as ω n = η n cγd , (35)where the factor γ accounts for the time dilation related to Lorentz boosts. The parameters η n , according to Fig. 3, can be reasonably approximated by means of a linear dependenceon n , which is an integer greater or equal than one. From the same image we can see thatthese parameters are independent of the speed of the system.In this manner, we have proved the existence of self-oscillating motion in this dynamicalsystem for all values of β . We recall en passant that the damping term and the delayintroduce an arrow of time in the system [37]. In other words, the limit cycle can berun in one time direction, but not in the reverse. This lack of reversibility is inherent todelayed systems, which depend on their previous history functions [38] and, therefore, arefundamentally non-conservative systems. Nevertheless, we note that the violation of energyconservation should only last a small time until the invariant limit set is obtained, and thatit applies as long as long as we just look at the particle and not to the fields. This factevokes nicely the time-energy uncertainty relations, as can be noticed in the next section.Even though self-oscillations were pointed out a long time ago for a charged particle [39],the instability of “classical” geodesic motion had been unnoticed before, perhaps due to thefact that artificial inertia was assumed and because there exists a dependence of the degreeof instability on the geometry of the particle [40]. This would be simply natural, given thecomplexity of retarded fields, and justifies the use of the apparently simple present model.15 IG. 3.
The roots of the polynomial f ( z ) = z + z + (1 − e z )(1 − β ). The complex roots ofthe f ( z ) have been numerically computed using Newton’s method for different values of the speed,ranging from the rest state ( β = 0) to the ultrarelativistic limit. As we can see, the values of theimaginary part do not seem to depend on β and can be written as multiples of a fundamentalfrequency. Since z = γd/c , we get the spectrum of frequencies for the self-oscillation ω n ∝ nc/γd ,at least right after the state of uniform motion is slightly perturbed. VI. THE SELF-POTENTIAL
In the present section we obtain the relativistic expression of the potential energy of thecharged body, starting again from the Li´enard-Wiechert potential of the electromagneticfield. We denote this self-energy as U since, it can be regarded as the non-dissipative energyrequired to assemble the system and set it at a certain dynamical state. As it will be clearat the end of the section, it harbors both the rest and the kinetic energy of the particle andalso a kinematic formulation of what we suggest might be the quantum potential, which isfrequently written as Q in the literature [41].The electrodynamic energy of the dumbbell can be computed as the energy required tosettle it in a particular dynamical state. Since the magnetic fields do not perform work, we16ould have to compute the integral U = e Z rr E · d r = − e Z rr ∇ ϕ · d r − e Z rr ∂ A ∂t · d r , (36)along some specific history describing a possible journey of the dumbbell. However, it can beshown that the second term is just the dissipative contribution. Therefore, we concentrateon the irrotational part of the field. The electrodynamic potential energy of the dumbbellis just given by the Li´enard-Wiechert potential as U = e πǫ r · u , (37)where the additional one fourth factor comes from the fact that each charge brings a value q = − e/
2. This can be written by means of the Eq. (3) as U = ~ αc r − lβ ) . (38)If we now substitute the Eqs. (15) and (16), and develop them in powers of d/c , we obtainthe series expansion of the self-potential U = γ ~ αc d − γ a c ~ α (cid:18) dc (cid:19) + γ a c ~ α (cid:18) dc (cid:19) − γ a c ~ α (cid:18) dc (cid:19) + ... (39)We recall that these computations are very lengthy and again strongly recommend the useof software for symbolic computation. We arrive in this manner at the crucial point of thisexposition. If we once again simply assume the idea that inertia has an electromagneticorigin, we can write the size of the particle as d = ~ α m e c . (40)Substitution in the previous equation yields the series U = γm e c − ~ m e α c γ γ a c − γ a c (cid:18) dc (cid:19) + γ a c (cid:18) dc (cid:19) − ... ! , (41)which can be written more formally as U = γm e c + ~ m e α r e γ ∞ X n =1 q n ( − n γ n a n c n (cid:18) dc (cid:19) n , (42)where the coefficients q n = { / , / , / , / , / ... } of the expansion belong to asequence, which can be computed from the quadrature q n = Z cos n (2 πx )d x = (2 n − n n ! . (43)17e clearly identify two terms in Eq. (42). The first one is just the relativistic energy[42], which contains both the rest and the kinetic energy of the particle. But note that, inaddition to the kinetic and the rest energy of the particle, the potential Q = ~ m e α r e γ ∞ X n =1 q n ( − n γ n a n c n (cid:18) dc (cid:19) n , (44)has unveiled as a new contribution. By inserting the integral appearing in Eq. (43) intoEq. (44), we can derive, after summation of the series and one additional integration, thepotential Q = − ~ m e α r e γ − q γ ˙ β (cid:0) dc (cid:1) , (45)which vanishes for uniform motion. Again, we note how the Lorentz factor precludes trav-eling at speeds higher or equal than the speed of light.This potential evokes nicely the quantum potential appearing in Bohmian mechanics[4, 5], with the same term ~ / m e preceding it. Importantly, we notice the dependence offluctuations on the fine structure constant. Moreover, we have found a dependence of thispotential on the acceleration of the particle that, we should not forget, is evaluated at theretarded time. On the other hand, since Q = − ~ m e ∇ RR , (46)in quantum mechanics, we can settle a bridge between the square modulus of the wavefunction and the kinematics of the particle in the non-relativistic limit. In this way, wewould restore the old relationship between forces and geometrical magnitudes. Once thedynamics is constrained to the asymptotic limit cycle, a relation between the accelerationof the particle and its position can be established and replaced in Q . Then, the resultingpartial differential equation is similar to Helmholtz’s equation ∇ R + 2 m e ~ QR = 0 , (47)while we can independently write down the Hamilton-Jacobi equation for a particle immersedin an external potential V ( x, t ). In the non-relativistic limit, it is given by ∂S∂t + 12 m e ( ∇ S ) + Q + V = 0 . (48)18n principle, once the two previous Eqs. (47) and (48) have been solved using the knowl-edge of the trajectory of the particle, the wave function can be built as ψ ( x, t ) = R ( x, t ) exp (cid:18) i ~ S ( x, t ) (cid:19) , (49)even though this solution may not be easily attained in most cases, specially when an externalpotential is present. Interestingly, we can see from these relations that the wave functionis a real objective field, as claimed in the seminal works of David Bohm [4, 5], and notjust a probabilistic entity. Both its modulus and phase are related to internal and externalelectrodynamic forces.To gain some insight into the self-potential of the “free” particle, we illustrate these ideasby means of an example. For this purpose, we can invoke the oscillatory dynamics afterthe transient amplification to show the repulsive nature of electrodynamic fluctuations. Aconservative version of the potential Q c ( x ) can be derived, which should only be regardedas an illustrative approximation. If we disregard the delay and consider the approximation a = − ω x , in the non-relativistic limit, and keeping just the two first term of the series, weobtain the potential Q c ( x ) = − ~ m e α r e (cid:18) d x − d x (cid:19) . (50)This potential is very well known in the world of nonlinear dynamical systems, since itappears in the Duffing oscillator [43]. This oscillator has been a paradigmatic model in thestudy of chaotic dynamical systems and has received remarkable attention both in physicsand engineering, since it can describe many important phenomena, such as beam buckling,superconducting Josephson parametric amplifiers, or ionization waves in plasmas, amongmany others. It illustrates in a very clear manner the instability of stillness, because Q c ( x )presents a maximum at x = 0. In particular, this potential is responsible for the spontaneoussymmetry breaking of the Poincar´e group. We recall that symmetry breaking is a typicalfeature of nonlinear dynamical systems [44, 45].Interestingly, this potential can be written in a simplified form as Q c ( x ) = − ~ ω (cid:18) d x − d x (cid:19) , (51)where the frequency ω = αc/ d has been defined, which is manifestly related to the frequencyof zitterbewegung of the dumbbell. 19 IG. 4.
The quantum potential Q c ( x ). This conservative approximation of the repulsive poten-tial (blue line) has an unstable fixed point at the origin x ∗ = 0, flanked by two minima, representingstable fixed points at x ∗ = ± p /
3. The repulsive character of this potential guarantees the per-petual oscillatory motion of electrodynamic bodies. An approximation of the self-force is shown inred.
What we find of the greatest interest about this expression is that it nicely evokes Planck’srelation. Moreover, we recall that m e is proportional to ~ , as long as we are in a position toassume that mass is of electromagnetic origin. Therefore, all sorts of energy and momentumcan be ultimately written as proportional to Planck’s constant. For example, the rest energyof the electron is written as ~ ω/
2. It is then reasonable to argue that photons, which are lightpulses emitted from accelerated electron transitions between different energy states, haveenergy E = ~ ω . Furthermore, by considering the relativistic relation E = pc , it is immediateto obtain from this equality that p = ~ k , which brings in the De Broglie’s relation betweenmomentum and wavelength.As we can see, perhaps the main problem when studying the electrodynamics of extendedbodies is that it leads to very complicated state-dependent delayed differential equations.Things would get terribly complicated if continuous bodies are considered, instead of thesimple toy discrete model used here [40]. This physical phenomenon arises as a consequenceof the principle of causality, which imposes a limited speed at which information can travel20n physics, introducing an infinite number of degrees of freedom in the nonlinear Lagrangeequations. In fact, we wonder how the principle of least action can be reformulated tocover the complex time-delayed systems appearing in electrodynamics. In light of thesefacts, and from a practical point of view, the Schr¨odinger equation [46] would be surelya much more appropriate and manageable mathematical framework than the use of thecomplicated functional differential equations resulting from the Li´enard-Wiechert potentialsto treat quantum problems. Certainly, it would not be surprising that partial differentialequations, which have an infinite number of degrees of freedom, are of so much usefulnessreplacing delayed systems, which harbor an infinite number of degrees of freedom as well. VII. DISCUSSION
As we have shown, the dynamics of an extended charged moving body has resemblanceswith the dynamics of the silicon droplets experimentally found in the recent years. How-ever, in our picture, the waves travelling with the particle “belong” to the particle itself,and do not require of any medium of propagation (any aether), since they are of electromag-netic origin. In our model, the fluctuations arise as self-interactions of the particle with itsown field and have as analogy the fluctuating platform appearing in their experiments [7].Nevertheless, this analogy must be drawn with great care, since the physical phenomenonleading to fluctuations in our moving charged body is not resonance, but self-oscillation [28].In particular, we predict a simple relation of proportionality between quantum fluctuationsand the coupling electromagnetic constant α . Concerning self-oscillations, we also recallthat a nonlocal probabilistic theory equivalent to a conservative diffusion process has beendeveloped not so long ago, which is mathematically equivalent to non-relativistic quantummechanics [2]. This is in agreement with the present work since, as we have shown, ourcorpuscle exhibits very violent oscillations, as it is also suggested in other works [4, 5].The most astonishing consequence of the present work is the demonstration of the pos-sibility of an instability of natural or uniform motion, which defies common intuition andbeliefs on radiation as a purely damping field on electromagnetic extended moving sources.We believe that this misunderstanding is present in the beginning of many important intro-ductory texts on quantum theory to justify the imperious necessity of a quantum mechanicaltheory that has no basis on the classical world [47]. On the contrary, the present work sug-21ests that self-interactions provide the required repulsive force (the quantum force) to avoidthe collapse of electrodynamical systems. In particular, we predict that self-interactions andrecoil forces are enough to stabilize the hydrogen atom and prevent its collapse [48]. Thisis because radiation has, not only stabilizing dissipative effects as a whole on the system,but antidamping effects as well through self-excitation and radiation reaction on its severalcomponents. In the same way that it can excite an electron inside an atom to higher energylevels, it can self-excite an extended object by self-absorption.We also note that the wave-particle duality is immediately solved in our framework. Thewaves are just perturbations of the fields, and any charged accelerated particle can presentsuch perturbations as a consequence of its self-oscillatory dynamics. Furthermore, theredoes not exist a fundamental particle that does not participate from some fundamental in-teraction and, consequently, there can be a pilot-wave [49] attached to any charged particlein accelerated motion. Importantly, we highlight the rich dynamical feedback interactionbetween these two apparently differentiated entities. We recall that feedback is a crucialphenomenon for the understading of nonlinear dynamical systems in general, chaotic dy-namics and, specially, for control theory [50]. In light of this paragraph, it seems obviousthat nothing can travel faster than field perturbations since, any aggregate of charge, what-ever its nature is, will show resistance to acceleration due to its electromagnetic energy.This intuition brings back the concept of vis insita , as appearing in Newton’s work [6]. Aconcept that is also related to the original notion of inertia and Galileo’s resistenza interna [51], and which can be traced back to the seminal works of the dominic friar Domingo deSoto [52, 53].We now bring to discussion the most delicate point of the present work. The fact thatthe inertia of a body might be of electromagnetic origin (electroweak and strong, if desired)is and old argument in physical theories. As we have shown, it has been a sufficient andnecessary condition to derive Newton’s second law, kinetic energy, Einstein’s mass-energyrelation and what seems to be the quantum potential, just from Maxwell’s electrodynamics.In this way, the present work gives a foundation of classical and quantum mechanics in thetheory of electrodynamics [54]. Perhaps, the greatest lesson of Einstein’s relation is not thatenergy is mass, but that mass is a useful and simple way to gather the constants appearing inelectrostatic energy. Consequently, we shall not invoke Occam’s razor to defend the idea ofgravitational mass as a redundant concept in fundamental physics. Instead, we adopt a more22rudent position and focus the attention on the fact that our findings imply to reconsiderNewton’s second law as a law of statics, just as suggested by D’Alembert. In light of thesefacts, we believe that it is very natural that an electrodynamic mechanism gives mass tofundamental particles in the standard model, which is luckily known nowadays thanks tothe work of Higgs [55].Following the same line of reasoning, this idea would perfectly connect with the theoryof general relativity, since the principle of equivalence simply states that, in a non-inertialreference frame comoving with a body, any object experiences forces of inertia. In fact, theseforces are equivalent to a gravitational field. Therefore, an electromagnetic theory of thegravitational field would also be in accordance with the principle of equivalence. Moreover,the identity of inertial and gravitational mass would be the consequence of a very simplefact, i.e. , their common electromagnetic origin. However, we must be careful at this point,since electromagnetic forces create strong ripples in space-time. Thus, a free falling extendedcharged particle in a gravitational field should experience very strong tidal self-forces. Aswe have shown, these forces can lead to self-oscillations.Delving deeper into the principle of covariance, we recall that the electromagnetic stress-energy tensor can be plugged into Einstein’s equation and interpreted as a curvature ofspacetime. The Einstein-Maxwell equations are terribly nonlinear high-dimensional partialdifferential equations, which can have as solutions solitary waves [56–58]. Certainly, themodel presented in this work is far too simplistic and unrealistic, because it assumes arigid solid as a particle, which is contrary to electromagnetic theory, and whose structure isunstable. We expect particles to rotate and also to be deformable, and wonder if these twoproperties should be enough to stabilize the electron.In this framework, gravitational waves would simply emerge from light waves. As a matterof fact, if the force of gravitation had an electromagnetic origin, the gravitational field, asa residual field, would have to be much weaker, which it is well-known to be the case. Thefact that it falls with an inverse-square law should not be a priori regarded as a problem.In fact, an average inverse quadratic law can be derived from radiative fields of a system ofoscillating particles, which originally fall with the inverse of the distance. However, as far asthe author has investigated, deriving a precise relation between the gravitational constant G and the electron’s charge e from the Li´enard-Wiechert potential of a system of particleswould remain an open problem of paramount relevance.23o conclude, we would also like to evince our most radical skepticism concerning thepresent analysis. Firstly, the simplicity of the model should prevent us from drawing toogeneral conclusions. It can be shown that purely longitudinal motion of the dumbbell is dis-sipative. Although this motion by itself is unstable to transverse perturbations, the authorsrecognize to have found a dependence of instability on the geometry of an electrodynamicmoving body [40]. As the shape of the body turns from oblate to prolate, a Hopf bifurcationbefalls. Therefore, it might happen that some external electromagnetic field is necessary tounleash the oscillation for more complicated bodies. Or, perhaps, the rotational motion ofthe particle is essential to have unstable dynamics independently of its geometry. Secondly,a full correspondence between electrodynamics and the relativistic formalism of quantummechanics has not been here provided. Nevertheless, and to close this lengthy discussion,we hope that this new perspective, based on modern theories of nonlinear dynamics, mightserve to enlighten the complex dynamics of elementary classical particles and, if not, at leastto drive physics closer to the establishment of a dynamical picture of fundamental particles,if such an endeavor is allowed and possible. VIII. ACKNOWLEDGMENTS
The author wishes to thank Alexandre R. Nieto for valuable comments on the elaborationof the present manuscript and discussion on some of its ideas. He also wishes to thankAlejandro Jenkins and Juan Sabuco for introducing him to the key role of self-oscillation inopen physical systems, and for fruitful discussions on this concept as well. This work hasbeen supported by the Spanish State Research Agency (AEI) and the European RegionalDevelopment Fund (ERDF) under Project No. FIS2016-76883-P.
APPENDIX
The following lines are devoted to obtain a power series relating the size of the particle d and the magnitude of the delay r/c . This relation allows us to approximate the distance l between the dumbbell’s position at time t and at the delayed time t r , as a function ofthe mass center velocity, its derivatives and the particle’s size [18, 19]. We begin with the24elation d = r s − (cid:18) lr (cid:19) = r (cid:18) − z − z − ... (cid:19) , (52)where the variable z = l/r has been introduced. On the other hand, the Eq. (6) can berewritten as z = lr = β + a c r + ˙ a c r + ¨ a c r + ... a c r ... (53)The square of z can then be computed. If we disregard the terms of the third order andhigher orders as well, we obtain z = β + ac βr + a c r + ˙ a c βr + O ( r ) . (54)Concerning the fourth power of z we can write z = β + 2 ac β r + 3 a c β r + 2 ˙ a c β r + O ( r ) . (55)to the same approximation as before.Substitution of Eqs. (54) and (55) into equation (52), after gathering terms, yields d = (cid:18) − β − β (cid:19) r − a c β (cid:18) β (cid:19) r − (cid:18) a c (cid:18) β (cid:19) + ˙ aβ c (cid:18) β (cid:19)(cid:19) r + O ( r ) . (56)If we consider the non-relativistic limit, by just keeping terms of the first order in β , wearrive at the approximated relation d = r − a c βr − (cid:18) a c + ˙ a c β (cid:19) r . (57) CONFLICT OF INTEREST
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