Conflict Between Classical Mechanics and Electromagnetism: The Harmonic Oscillator in Equilibrium with a Bath
aa r X i v : . [ phy s i c s . c l a ss - ph ] S e p Conflict Between Classical Mechanics and Electromagnetism: TheHarmonic Oscillator in Equilibrium with a Bath
Timothy H. Boyer
Department of Physics, City College of the CityUniversity of New York, New York, New York 10031
Abstract
It is pointed out that an electric charge oscillating in a one-dimensional purely-harmonic potentialis in detailed balance at its harmonics with a radiation bath whose energy U rad per normal modeis linear in frequency ω , U rad = const × ω, and hence is Lorentz invariant, as seems appropriate forrelativistic electromagnetism. The oscillating charge is not in equilibrium with the Rayleigh-Jeansspectrum which arises from energy-sharing equipartition ideas which are valid only in nonrelativisticmechanics. Here we explore the contrasting behavior of harmonic oscillators connected to bathsin classical mechanics and electromagnetism. It is emphasized that modern physics text are inerror in suggesting that the Rayleigh-Jeans spectrum corresponds to the equilibrium spectrum ofrandom classical radiation, and in ignoring Lorentz-invariant classical zero-point radiation whichis indeed a classical equilibrium spectrum. . INTRODUCTIONA. Harmonic Oscillators as Illustrative of the Mechanics-Electomagnetism Con-flict Ask a physicist to name the spectrum of random classical radiation with which an oscillat-ing classical charge in a one-dimensional purely-harmonic potential will come to equilibriumat its harmonics, and, influenced by current modern physics textbooks, he will invariablygive the wrong answer. He will suggest “the Rayleigh-Jeans spectrum.” But this sugges-tion is untrue. The correct answer is “a Lorentz-invariant spectrum.” The Rayleigh-Jeansspectrum is associated with the energy-sharing equipartition ideas which are valid only innonrelativistic classical mechanics. On the other hand, electromagnetism is a relativistictheory, and a classical charge oscillating in one dimension is in equilibrium at its harmonicswith a relativistically-invariant spectrum of random radiation. In this article, we explore theconflict between Newtonian mechanics and electromagnetism in connection with harmonicoscillators and equilibrium baths.The mismatch between Newtonian mechanics and classical electromagnetism goes unap-preciated by many physics students and by their instructors. The mismatch is fundamentalsince Newtonian mechanics satisfies Galilean invariance while electrodynamics is relativisti-cally invariant. Yet the mismatch is usually ignored. The mismatch is usually ignored inour classes because relativity is treated as a specialty subject covered at the beginning of aclass on modern physics; only point collisions between particles are considered for relativis-tic interactions, and relativity is regarded as needed only in nuclear and elementary-particlephysics. It is usually assumed that relativity becomes significant only for situations out ofour ordinary experience when particles are traveling at velocities close to the speed of light.Here we show that relativity is central to striking contrasts in harmonic oscillator systemsinvolving low particle velocities in conjunction with ambient baths. The simple analysisexposes the profound conflict between nonrelativistic mechanical systems and relativisticelectromagnetic systems.The essential contrast is as follows. A frictionless harmonic oscillator system in Newto-nian mechanics can oscillate forever with a constant amplitude and energy. There is no needfor any bath for the oscillator. However, if the mechanical oscillator is connected by point2ollisions to a bath of particles, the total system energy is shared between the oscillator andthe particle bath so as to give energy equipartition. In contradiction with this mechanicalsituation, an electromagnetic oscillator involving the oscillation of a charged particle can-not exist on its own; the charged particle must be coupled to radiation. Within classicalelectrodynamics, there is no such thing as an oscillating charge which exists without radia-tion. It turns out that the spectrum of radiation which must provide the equilibrium bathfor an oscillating charge in a one-dimensional purely-harmonic potential is uniquely definedby electromagnetic theory, and turns out to correspond to a Lorentz-invariant spectrum ofclassical electromagnetic radiation.This last statement will come as a surprise to the many physicists who are aware of thehistorical situation involving Planck’s linear oscillator taken in dipole approximation whichwill match the energy of the radiation bath at the oscillator frequency, but will not determinethe radiation spectrum. Such an oscillator appears in many modern physics texts in con-nection with the blackbody radiation spectrum.[1] However, any electrodynamic oscillatormust involve an oscillating charge with a finite non-zero amplitude of motion in the har-monic potential. The moment the amplitude of oscillation is non-zero, the charge radiatesnot only at the fundamental frequency (as given by the dipole approximation), but also atall the harmonics. The radiation at the harmonics determines the spectrum of the ambientradiation whose absorption is needed to balance the emitted radiation. Thus by consideringthe harmonics, one arrives at a unique spectrum for the equilibrium bath for the oscillatingcharge. A recent relativistic calculation shows[2] that classical electromagnetic theory forthe one-dimensional electromagnetic harmonic oscillator uniquely determines the radiationspectrum with which the oscillator is in equilibrium, and the spectrum is that of Lorentz-invariance where the energy of a radiation normal mode is proportional to the frequency ofthe mode, U rad = const × ω . The well-known radiation spectrum U rad ( ω ) = (1 / ~ ω ofclassical electromagnetic zero-point radiation fits this Lorentz-invariance requirement. B. Outline of the Article
The outline of the presentation is as follows. First we describe the assumed mechanicaland electromagnetic oscillators and their baths. Then we consider the equilibrium situationsfor the oscillators. For the electromagnetic oscillator, we first describe the dipole harmonic3pproximation which is of historical importance in suggesting the Rayleigh-Jeans spectrum,and also the nonlinear oscillators taken in the dipole approximation which seem to confirmthe Rayleigh-Jeans conclusion. Then we sketch the extension of the oscillator analysis toquadrupole radiation at double the frequency of the oscillator and note that equilibriumrequires a linear (and hence relativistic) radiation spectrum for equilibrium. Next we turnto adiabatic changes of the oscillator’s fundamental frequency, and see how these changesfit with thermodynamic ideas of the equilibrium temperature. The final analysis notes thelimitations in the use of a one-dimensional purely-harmonic oscillator. Finally, we give someconcluding remarks.
II. THE OSCILLATORSA. Mechanical Oscillator
Here we discuss the conflict between Newtonian mechanics and electromagnetism in con-nection with equilibrium for small harmonic oscillators in a bath. In the case of classicalmechanics, the oscillator consists of a particle attached to a spring along the x -axis in theabsence of any friction. The particle of mass M is attached to a spring of constant κ sothat Newton’s second law gives the equation of motion due to the linear restoring force F = − b xκx as M ¨ x = − κx, (1)with the solution x ( t ) = x cos [ ω t + φ ] (2)where the angular frequency is ω = p κ/M and φ is a constant phase angle. B. Electromagnetic Oscillator
In the case of the electromagnetic oscillator, we are considering only a one-dimensionaloscillator consisting of a charge e with mass M confined along the x -axis by two charges q of the same sign as e. Thus we consider two charges q placed along the x -axis at x q ± = ± a with the charge e in unstable equilibrium at the origin of coordinates. If the charge e is4isplaced from the origin to the point x , then the force on the charge is F = b xqe (cid:20) − a − x ) − a − x ) (cid:21) ≈ − b x qea x. (3)Thus for small displacements x , there is a linear restoring force on the charge e . If the systemwere treated as a purely mechanical system, the (angular) frequency of of oscillation wouldbe ω = p qe/ ( a M ). Now electromagnetism is a relativistic theory. Our electromagneticoscillator can be regarded as relativistic to any degree of approximation provided that theamplitude x of oscillation and hence the maximum velocity x ω of the oscillation is takenas sufficiently small, x ω << c. III. EQUILIBRIUM OF OSCILLATORS IN BATHSA. Optional Mechanical Bath
Nonrelativistic mechanics deals only with massive particles, and accordingly, the bathwith which the mechanical oscillator can be regarded as interacting will be taken as composedof particles of mass m moving in one dimension. The oscillator particle of mass M isconnected by point collisions to a bath of non-interacting particles of mass m confined toa large one-dimensional box of length L with an elastically reflecting wall. The distantend of the box provides the elastic-rebound wall, and the oscillator particle M (attached tothe spring and wall) provides the other end of the one-dimensional box. Then the pointcollisions of the bath particles with the oscillator of mass M provides a means of transfer ofenergy and momentum between the bath particles and the oscillator system.Point collisions of massive particles lead to “energy sharing” among all the interactingparticles of the system. The work on kinetic theory carried out in the 19th century in-troduced the idea of kinetic energy equipartition among the particles. Furthermore, themechanical harmonic oscillator of mass M shares energy equally between its kinetic energyand its potential energy. Thus the “energy-sharing” idea extends to all the modes of thenonrelativistic mechanical oscillator coupled by point collisions to the bath of non-interactingparticles. This energy-sharing idea is carried over into classical statistical mechanics andprovides the basis for the Boltzmann distribution on phase space. The system can be de-scribed satisfactorily by classical statistical mechanics with the resulting Boltzmann prob-ability distribution. We note that the energy-sharing concept has no particular role for5he frequency of any oscillator nor any limit on the (finite) number of parameters whichmay enter the nonrelativistic mechanical system interacting with a finite number-density ofparticles. Equilibrium will involve the preferred inertial frame of the confining box and theaverage kinetic energy of a particle. B. Required Electromagnetic Bath
For the electromagnetic oscillator, the bath is totally different from that of the non-relativistic mechanical situation. The mechanical oscillator can be imagined to oscillatewithout friction and without any mechanical bath of particles. In contrast, the charge e ofthe electromagnetic oscillator may oscillate harmonically without mechanical friction, butthe oscillating charge must accelerate. And the accelerating oscillaor charge emits radia-tion at its natural frequency of oscillation ω . According to classical theory, the oscillatingelectromagnetic particle is always coupled to radiation. Thus within classical theory, equi-librium for an electromagnetic oscillator requires the presence of a bath of electromagneticradiation.Furthermore, electromagnetism is a relativistic theory. Although a bath of mechanicalparticles with finite particle density can be described in terms of a finite number of wavemodes, the number of normal modes for relativistic waves is infinite. Thus an electromag-netic oscillator is necessarily connected to electromagnetic radiation involving an infinitenumber of wave modes. It is natural to ask, “What is the spectrum of the radiationbath which is tied to a small (relativistic) motion of the charge e of the one-dimensionalelectromagnetic oscillator?” IV. REVIEW OF THE ELECTROMAGNETIC RADIATION BATH IN DIPOLEAPPROXIMATIONA. Random Classical Radiation
An oscillator in a closed container with perfectly reflecting walls might be in equilibriumwith coherent radiation. The far more usual case is for the oscillator to come to equilibriumwith random radiation. The problem of an electromagnetic oscillator interacting with6andom radiation is an old problem going back to Planck’s work at the end of the 19thcentury.[3] Here we will first review the historical dipole approximation calculation beforeturning to the new aspects which are unfamiliar to most physicists.Random classical radiation can be described as Planck described it at then end of the19th century in terms of plane waves with random phases. If we consider a very large cubicbox with sides of length a , then the random radiation can be written as E ( r , t ) = X k ,λ b ǫ ( k , λ ) (cid:18) πU rad ( ω ) a (cid:19) / { exp [ i k · r − iωt + iθ ( k , λ )] + cc } (4) B ( r , t ) = X k ,λ b k × b ǫ ( k , λ ) (cid:18) πU rad ( ω ) a (cid:19) / { exp [ i k · r − iωt + iθ ( k , λ )] + cc } (5)where the sum over the wave vectors k = b x πl/a + b y πm/a + b z πn/a involves integers l, m, n = 0 , ± , ± , ... running over all positive and negative values, there are two polariza-tions λ = 1 , , and the random phases θ ( k , λ ) are distributed uniformly over the interval(0 , π ] , independently for each wave vector k and polarization λ . The energy per normalmode at radiation frequency ω is given by U rad ( ω ) , and we have assumed that the radiationspectrum is isotropic. B. Oscillator Equation of Motion
The one-dimensional electromagnetic oscillator is located at the origin and oscillates alongthe x -axis. For small displacements x , the oscillator motion satisfies Newton’s second law M ¨ x = − M ω x + M τ ... x + eE x ( x, , , t ) , (6)where the time τ = 2 e / (3 M c ) , and the x -component of the electric field follows from Eq.(4). In the electric dipole approximation corresponding to a point oscillator, the electric field E x is approximated as that located at the origin (center of the oscillator), E x ( x, , , t ) ≈ E x (0 , , , t ) . Substituting from Eq. (4), we have a linear stochastic differential equationwith the steady-state solution[4] x ( t ) = eM X k ,λ ǫ x ( k , λ ) (cid:18) πU rad ( ω ) a (cid:19) / (cid:26) exp { i [ k · r − ωt + θ ( k , λ )] }− ω + ω + iτ ω + cc (cid:27) (7)7he time derivative ˙ x ( t ) follows from Eq. (7). The average values can be obtained byaveraging over the random phases as h exp[ iθ ( k , λ )] exp [ − iθ ( k ′ , λ ′ )] i = δ kk ′ δ λλ ′ . (8)Thus averaging over the random phases and then summing over the Kronecker delta, themean-square displacement is (cid:10) x ( t ) (cid:11) = X k ,λ ǫ x ( k , λ ) (cid:18) πU rad ( ω ) a (cid:19) e m h ( − ω + ω ) + ( τ ω ) i . (9)Assuming that the box for the periodic boundary conditions becomes very large, the sumover the discrete values of k can be replaced by an integral, P k → R d k [ a/ (2 π )] = R ∞ k dk R d Ω [ a/ (2 π )] . The only angular dependence appears in the polarization vectors ǫ x ( k , λ ), so that, on angular integration, each polarization will contribute a value of 1 / . We are left with the integration over k. If we assume that the charge e is small so thatthe damping is small and the integrand is sharply peaked at ω , then we may extend thelower limit of the integral to −∞ , and replace all frequencies ω by ω , except where thecombination ω − ω appears. The remaining integral is of the form Z ∞−∞ dua u + b = πab . (10)The mean-square displacement is (cid:10) x (cid:11) = U rad ( ω ) mω . (11)An analogous procedure can be followed for the evaluation of h ˙ x i . Then the average energyof the oscillator follows as U osc = (cid:28) M ˙ x + 12 M ω x (cid:29) = U rad ( ω ) , (12)so that in equilibrium (for the small-charge approximation) the point dipole oscillator hasthe same average energy as the radiation modes at the same frequency as the oscillator.[4] C. Phase Space Distribution for the Electromagnetic Oscillator
One can also obtain the averages for all products involving an arbitrary number of fac-tors of x ( t ) and ˙ x ( t ) by repeated use of the average in Eq. (8).[5] Indeed, the averages8orrespond to a probability distribution P ( x, p ) for the displacement x ( t ) and momentum p ( t ) = M ˙ x ( t ) , for −∞ < x < ∞ , −∞ < p < ∞ , given by[4][5] P osc ( x, p ) dxdp = 12 πU osc r κM exp (cid:20) − ( κx + p /M )2 U osc (cid:21) dxdp. (13)It is interesting to rewrite the probability distribution for the oscillator in random radi-ation in terms of action-angle variables[6] using x ( t ) = r Jmω cos wp ( t ) = p mJ ω cos w (14)Then the phase space distribution for the electromagnetic oscillator, for 0 ≤ w ≤ π, ≤ J < ∞ , is given by P osc ( w, J ) dwdJ = 12 π U osc ω exp (cid:20) − J ω U osc (cid:21) dwdJ. (15)As shown in earlier work,[7] the scatting of the random radiation by the point electricdipole oscillator does not change the frequency spectrum or the isotropic nature of therandom radiation. We emphasize that the energy U osc of the point electric dipole oscillatormatches the radiation energy U rad in the normal modes of the radiation field at the oscillatornatural frequency ω . However, this is merely a connection at one point ω = ω of thespectral function U rad ( ω ). There is nothing in this calculation which suggests any preferredspectrum for the random radiation in equilibrium with the electromagnetic oscillator indipole approximation. D. Historical Use of the Electromagnetic Oscillator in Dipole Approximation
It was Planck who introduced the electromagnetic oscillator taken in the dipole approx-imation in connection with the problem of blackbody radiation at the end of the 19thcentury.[3] The oscillator was regarded as a connection between the thermodynamics ofmatter and the corresponding electromagnetic thermal radiation. Initially, Planck hopedthat the scattering of radiation by the oscillator itself would force electromagnetic radiationinto equilibrium and so reveal the blackbody radiation spectrum. However, Boltzmannpointed out that electromagnetic theory is invariant under time reversal so that Planck’shope was empty.[8] It was only when Planck became convinced that the dipole oscillator9ould not change the frequencies of scattered radiation and so would not push a spectrumof random radiation towards equilibrium that Planck turned to statistical mechanical ideasfor particles. Only then did Planck seek to obtain the thermodynamic behavior of theelectromagnetic oscillator in order to determine the equilibrium spectrum of the associatedthermal radiation.[8][9]The historical use of the electromagnetic dipole oscillator still appears in the textbooksof modern physics.[1] Using the ideas of nonrelativistic classical statistical mechanics forthe oscillator, the texts claim that a classical electromagnetic oscillator must assume theequipartition energy U osc = k B T of a Newtonian mechanical oscillator, and therefore, byPlanck’s calculation showing the equality of energy between the electromagnetic dipole os-cillator and the radiation normal modes, the spectrum of thermal radiation U rad must bethe Rayleigh-Jeans spectrum U rad ( ω, T ) = U RJ ( ω, T ) = k B T. (16)Although energy equipartition is strictly a result of nonrelativistic mechanics and has nothingto do with relativistic classical electromagnetism, it is claimed that this Rayleigh-Jeans resultis the prediction of “classical physics.” And this prediction is made without any allowancefor the conflict between nonrelativistic mechanics and relativistic electromagnetism.This same result that nonrelativistic physics leads to the Rayleigh-Jeans radiation spec-trum has been obtained by other researchers without using classical statistical mechanicsbut by using nonrelativistic nonlinear scattering systems which are connected to electro-magnetic radiation by the dipole radiation approximation for the mechanical systems.[10]Whereas the analysis of the modern physics texts uses nonrelativistic classical statisticalmechanics, all of these scattering calculation use nonrelativistic nonlinear potential systemsfor the radiation scatterers. All of these derivations give a false result because (despite anyclaims to being relativistic calculations) they all exclude Coulomb potentials and involve anonrelativistic basis for the attainment of equilibrium.10 . RELATIVISTIC RADIATION EQUILIBRIUMA. Few Relativistic Scattering Systems The use of nonrelativistic mechanical systems in connection with electromagnetic radia-tion equilibrium persists because there are very few relativistic scattering systems in naturewhich are familiar to physicists and which allow tractable analytic calculations. Indeed, onlythe system of a point charge in a Coulomb potential when coupled to electromagnetic radia-tion can be regarded as fully relativistic. And analytic treatment of the Coulomb-potentialsystem seems extraordinarily difficult. To date, only numerical simulations have providedsome insight into the Coulomb system in radiation, and then only in the nonrelativisticapproximation for the particle motion.[11] The one other system which can be regarded asapproximately relativistic is that of a point charge in a one-dimensional harmonic potentialwhen the amplitude of oscillation is taken as very small so that the particle velocity is verysmall, v << c . B. Analysis of the Electromagnetic Oscillator Beyond the Dipole Approximation
Although the classical electromagnetic calculation for the point electric dipole oscillator inrandom radiation has been presented repeatedly for a century, the possibility of going beyondthe dipole approximation was emphasized only recently in work by Huang and Batelaan[12]who considered the absorption of a radiation pulse by an oscillator of non-zero amplitude.We will not consider the absorption of a radiation pulse, but, inspired by the work of Huangand Batelaan, we wish to continue the analysis of the equilibrium between an electromagneticharmonic oscillator and a radiation bath by considering the higher harmonics. We note thatby taking the amplitude of oscillation as small enough, the motion of the oscillating charge e can be regarded as relativistic to any degree of approximation, v << c, and yet, providedthat the amplitude of motion is non-zero, there is always non-zero radiation at harmonics.Thus if the amplitude of oscillation of the electromagnetic oscillator is not zero, thenthe harmonic oscillator of natural frequency ω must be emitting quadrupole radiation atfrequency 2 ω . But here indeed is an opportunity to determine the equilibrium spectrumof a very small electromagnetic oscillator. There must be energy in the random radiationspectrum at frequency 2 ω which is absorbed by the oscillator and balances the energy radi-11ted as quadrupole radiation at 2 ω . And indeed, the argument can be repeated for all thehigher radiation multipoles of the oscillating charge e. The situation here is totally differentfrom the earlier scattering calculations[10], all of which involved nonrelativistic non-linearoscillators scattering electromagnetic radiation in the dipole approximation, and all of whichled to the Rayleigh-Jeans spectrum as the equilibrium spectrum of random radiation. Herefor an electromagnetic oscillator in a purely harmonic potential, the equilibrium spectrumfor the electromagnetic oscillator is not determined by any assumption about the mechanicalstructure of the oscillator, but rather the spectrum is determined by purely electromagneticconsiderations. This is the first scattering calculation for electromagnetic radiation whichcan be regarded as fully relativistic.[2]The change from the earlier work involves the equation of motion for the oscillator givenin Eq. (6), but now not taking the dipole approximation E x ( x, , , t ) ≈ E x (0 , , , t ) , butrather, for the quadrupole term, making the next approximation E x ( x, , , t ) ≈ E x (0 , , , t ) + x ( t ) (cid:18) ddx ′ E x ( x ′ , , , t ) (cid:19) x ′ =0 (17)where on the right-hand side we will insert for x ( t ) the dipole-approximation result appearingin Eq. (7). Indeed, the whole idea is to make successive approximations in the smallamplitude of oscillator motion. Thus now for the quadrupole radiation approximation, wewrite the equation of motion for the oscillating charge e as M ¨ x ≈ − M ω x + M τ ... x + eE x (0 , , , t ) + x (0) ( t ) (cid:18) ddx ′ E x ( x ′ , , , t ) (cid:19) x ′ =0 (18)where x (0) ( t ) is the dipole approximation appearing in Eq. (7). We notice that whereas theequation of motion for the electromagnetic oscillator in the dipole approximation involvedone factor of exp [ − iωt + iθ ( k , λ )] on the right-hand side there are now two factors of thisform in Eq. (18).One can solve[2] the equation of motion (18), and one finds that the scattered radiationpreserves the frequency spectrum and the angular distribution of isotropic radiation providedthat the spectrum of random radiation has twice as much energy per normal mode atfrequency 2 ω compared to the energy per normal mode at frequency ω . [2] It is easyto see where this analysis is going. The electromagnetic oscillator in one dimension isin equilibrium only with a radiation bath U rad ( nω ) = const × nω for all multiples n ofthe fundamental oscillator frequency ω . Thus detailed balance for radiation energy holds12rovided that the spectrum of random radiation is linear in frequency U rad ( ω ) = const × ω. (19)This situation corresponds to Lorentz invariance for the spectrum.[13] Indeed, it seemsnatural that the Lorentz-invariant theory of classical electrodynamics should pick out aLorentz-invariant spectrum of random radiation for the electromagnetic oscillator which hasno structure other than the characteristic frequency ω . VI. ADIABATIC CHANGE AND TEMPERATUREA. Adiabatic Change of the Mechanical Spring Constant
Adiabatic changes in the oscillator frequency for a Newtonian mechanical oscillator inequilibrium through point collisions with a particle bath show that the temperature of thebath can never be zero unless the total system energy vanishes. If the spring constant κ of the oscillator is readjusted very slowly, then the external agent which provides thereadjustment is carrying out an adiabatic change of the system. If we imagine that thecollision interactions between the oscillator and the bath particles are removed during theadiabatic change of the oscillator, then the ratio of the energy U osc to the oscillator (angular)frequency ω is an adiabatic invariant, U osc /ω = J osc , where J osc is the action variable[15]in the action-angle treatment of the mechanics of the oscillator, with U osc = J osc ω . Thusduring the adiabatic readjustment of the spring constant, the probability distribution of theaction variable P ( J osc ) is unchanged, but the average energy of the oscillator does indeedchange since U osc = J osc ω , and ω has changed.If the mechanical oscillator is now reconnected to the bath of particles, the oscillatorwill no longer be in equilibrium with the bath because its average energy per normal modehas been changed. Rather, energy will be exchanged between the oscillator and the bathby means of point collisions, and a new equilibrium will be established with a differenttotal energy for the combined system of the oscillator and the bath. Since the adiabaticchange is not the same as a isothermal change, the temperature of the oscillator and itsparticle bath must be regarded as non-zero. Thus provided that the mechanical motion ofthe oscillator and its surrounding bath particles has not actually ceased, the oscillator-bathsystem must be regarded as at equilibrium at some positive temperature determined by the13verage kinetic energy of any of the particles. There is no such thing as particle motion atthe absolute zero of temperature in nonrelativistic classical mechanics. B. Adiabatic Change of the Frequency of the Electromagnetic Oscillator
In contrast with the Newtonian mechanical oscillator, an adiabatic change in the oscillatorfrequency of a one-dimensional electromagnetic oscillator in equilibrium with random radi-ation shows that the one-dimensional-electromagnetic-oscillator-radiation system has zerotemperature.Electromagnetism depends crucially on frequency-dependent resonances whereas pointcollisions do not have any associated frequency. Thus the frequency of the Newtonian me-chanical oscillator is largely irrelevant in connection with the particle bath. However, thefrequency of the electromagnetic oscillator determines the radiation frequencies with whichthe electromagnetic oscillator will interact. If we imagine the electromagnetic oscillator tem-porarily uncoupled from its radiation and the separation a between the confining charges q aschanged, then the adiabatic change of the oscillation frequency ω appearing in Eq. (3) leadsto a change in energy ∆ U osc = J osc ∆ ω of the oscillator while keeping the value of the actionvariable J osc fixed. For an ensemble of oscillators, the phase space distribution involving J osc is unchanged under the adiabatic change in frequency, but the average oscillator energyis still linear in frequency with the same phase space distribution as given in Eq. (15) sincethe ratio of the oscillator energy to the oscillator frequency has not changed. But now whenthe electromagnetic oscillator is reconnected with the initial Lorentz-invariant spectrum ofrandom radiation, there is no transfer of average energy between the oscillator and the bathof electromagnetic radiation.[16] The transformed oscillator is again in equilibrium with theoriginal radiation spectrum. Thus an adiabatic change is the same as an isothermal changefor this situation, and we would describe the one-dimensional electromagnetic oscillator asin equilibrium with its radiation bath at the absolute zero of temperature, T = 0 . We emphasize that the energy introduced in the adiabatic change went into the energy ofthe oscillator, and (on average) no energy was transferred to the energy-divergent radiationfield. There is “apparent decoupling” between the electromagnetic oscillator and its equi-librium radiation bath. Thus in a sense the electromagnetic oscillator acts as though it weredecoupled from the radiation field even though the radiation field is required for the stability14f the oscillating charge in classical electromagnetism. In this connection, it is noteworthythat textbooks of thermodynamics and statistical mechanics will often introduce oscillatorsand electromagnetic systems without worrying about any divergent radiation bath whichwould be required for their stability within classical physics.
VII. CLASSICAL ELECTROMAGNETIC ZERO-POINT RADIATIONA. Unique Lorentz-Invariant Spectrum
In nature, oscillating charges can not each have their own Lorentz-invariant spectrum ofrandom radiation. Thus classical electromagnetic theory demands that in order to have os-cillating electric charges, there must be one Lorentz-invariant spectrum of random classicalelectromagnetic radiation. This radiation spectrum is usually termed classical electromag-netic zero-point radiation.[17] And within a purely classical electromagnetic description ofnature, there is good experimental evidence for classical electromagnetic zero-point radiationwith an energy per radiation normal mode U rad ( ω ) = (1 / ~ ω, (20)where the constant ~ takes the same value as is given for Planck’s constant ~ = h/ (2 π ) . The existence of a spectrum of classical electromagnetic zero-point radiation provides thebasis for classical explanations of Casimir forces, van der Waals forces, the specific heats ofsolids, diamagnetism, and the absence of atomic collapse.[18]
B. Phase Space of Classical Zero-Point Radiation
An electromagnetic oscillator in zero-point radiation takes on the same phase space dis-tribution as is found for the radiation normal modes. Both the electromagnetic oscillatorand all the modes of the electromagnetic radiation field have the distribution on phase spacefollowing from Eq. (15) for 0 ≤ w ≤ π, ≤ J < ∞ ,P ( w, J ) dwdJ = 12 π ~ (cid:20) − J ~ / (cid:21) dwdJ, (21)which does not depend upon the frequency of the oscillator or of the radiation mode.[14][16]On adiabatic change of the frequency of a purely-harmonic oscillator, the phase space dis-tribution is unchanged and remains in equilibrium with the zero-point radiation spectrum.15 . Spectrum of “Least Possible Information” The Lorentz-invariant spectrum of zero-point radiation is the “radiation spectrum ofleast possible information in a relativistic theory.” The zero-point radiation spectrum in-volves one overall constant ~ and takes the same isotropic form in every inertial frame. Onthe other hand, the Rayleigh-Jeans spectrum contains more information than the Lorentz-invariant zero-point radiation spectrum. Thus in addition to one over-all parameter k B T ,the Rayleigh-Jeans spectrum has a preferred inertial frame in which the simple Rayleigh-Jeans form in Eq. (16) appears; any other inertial frame moving with constant velocityrelative to the preferred frame will find a non-isotropic spectrum of radiation. Nonrela-tivistic nonlinear oscillators scatter radiation so as to impose their own inertial frame as thepreferred inertial frame of the random radiation spectrum to which they are attached. Incontrast, the calculation[2] showing that the one-dimensional electromagnetic oscillator ina purely-harmonic potential is in equilibrium with a Lorentz-invariant radiation spectrumindeed suggests that the purely-harmonic oscillator is not imposing its own inertial frame onthe equilibrium radiation spectrum upon which it depends. Indeed, the electromagnetic os-cillator based upon the electrostatic consideration leading up to Eq. (3) will have relativistic energy transformation properties between inertial frames provided the the particle velocityis small in the oscillator frame. On the other hand, the potential energy of a nonrelativisticnonlinear oscillator will not transform in a relativistic fashion.In addition to providing a Lorentz-invariant radiation spectrum in Minkowski spacetime,zero-point radiation allows incorporation into general relativity where it again corresponds tothe “radiation spectrum of least possible information in a relativistic theory.” In the generalrelativistic situation, the correlation functions for zero-point radiation between spacetimepoints depend upon only the geodesic separations between the spacetime points.[19] VIII. LIMITATIONS OF THE ANALYSIS
The one-dimensional electromagnetic oscillator in a purely-harmonic potential scattersradiation so as to push the radiation toward its equilibrium spectrum which is a Lorentz-invariant spectrum.[2] This is indeed consistent with classical electrodynamics which isa Lorentz-invariant theory. However, zero-point radiation must be interpreted as corre-16ponding to a temperature of abslute zero, and the scattering analysis does not suggest anyequilibrium at a thermal spectrum with non-zero temperature. Also, we note that the one-dimensional oscillator shows no velocity-dependent damping, which must exist for thermalradiation at non-zero temperature. Indeed the one-dimensional analysis allows no role forthe magnetic field of the random radiation which would cause forces perpendicular to the di-rection of oscillation which are eliminated by the constraint giving one-dimensional motion.Finally, the one-dimensional oscillator is not a purely electromagnetic system because of thenon-electromagnetic constraint which gives one-dimensional motion. Indeed, Earnshaw’stheory tells us that a stable purely-harmonic potential cannot be obtained from interactingpoint charges.It is conjectured that all of these considerations are related. It is suggested that velocity-dependent damping in random radiation is possible only if the charged particle can oscillate(in a sort of zitterbewegung) in the direction perpendicular to the direction of damping.Indeed, the Einstein-Hopf analysis[20] for the motion of a classical particle in thermal ra-diation involves a dipole oscillator, internal to the particle, which oscillates in a directionperpendicular to the direction of the particle’s constrained one-dimension motion. Thus itis conjectured that the complete understanding of the thermal radiation spectrum in termsof scattering by a purely classical system must wait until there is a classical understandingof something like the classical hydrogen atom. While awaiting such a classical scatteringanalysis, one is encouraged by many points of view that suggest that thermal radiation canindeed be understood in terms of classical physics.[9]
IX. CONCLUDING COMMENTS
The biggest shock to undergraduate physics students is their encounter with the unusualideas of quantum theory. However, they are often not confronted with just how differ-ent the ideas of classical electrodynamics are from the every-day experiences of Newtonianmechanics. Like the physicists of the early 20th century, contemporary physicists still donot appreciated the conflict between Newtonian mechanics and electromagnetism. Further-more, the textbooks of modern physics continue to mislead physics students regarding thismatter by insisting that the energy-sharing ideas of nonrelativistic classical mechanics are tobe carried over to the relativistic theory of classical electromagnetic radiation equilibrium.17ewtonian mechanics and classical electrodynamics are vastly different in their assump-tions and in their descriptions of nature. A nonrelativistic mechanical harmonic oscillatorcan be assumed to have no friction, and so will oscillate indefinitely with no need for any bathassociated with the equilibrium situation. In addition, nonrelativistic mechanical systemsof particles come to equilibrium by energy-sharing throughout the mechancal system. Thisenergy-sharing idea appears in the equipartition theorem and in the Boltzmann distributionwhich are incorporated into nonrelativistic classical statistical mechanics.In contrast with the mechanical situation, an electric charge is always connected to elec-tromagnetic fields. If the electric charge oscillates, then the charge must exchange energywith the radiation field. There is no such thing as a classical electromagnetic oscillatorwhich is not coupled to electromagnetic radiation. Furthermore, a relativistic field theorywill involve an infinite number of radiation normal modes. Thus the nonrelativistic idea ofenergy-sharing across the entire system has no place in electromagnetism. Rather, resonancephenomena associated with frequencies appear crucially in electromagnetism, whereas suchfrequency-dependent resonances are of no significance in nonrelativistic mechanical equilib-rium.It is very hard to find relativistic electromagnetic systems allowing easy calculations in-volving radiation equilibrium. In this article, we have considered the simplest relativisticclassical electromagnetic system which will lead to a preferred equilibrium radiation spec-trum. The oscillation of an electric charge in a one-dimensional purely-harmonic potentialleads to a required equilibrium radiation spectrum, not through its mechancal motion whichis always simply harmonic, but through the electromagnetic aspects of the multipole radi-ation which arise in classical electromagnetism. One finds that the one-dimensional elec-tromagnetic oscillator is in equilibrium with a spectrum whose normal mode energy U rad islinear in frequency ω , U rad = const × ω, corresponding to a Lorentz-invariant radiation spec-trum. Indeed, the relativistic nature of the equilibrium spectrum fits with the relativisticnature of classical electromagnetism. X. ACKNOWLEDGEMENT
I am deeply indebted to the work of Dr. Wayne Huang and Dr. Herman Batelaan andof Dr. Daniel C. Cole who have emphasized the importance of radiation harmonics in their18nalyses of electromagnetic systems. [1] See for example, R. Eisberg and R. Resnick,
Quantum Physics of Atoms, Molecules, Solids,Nuclei, and Particles 2nd ed. (Wiley, New York 1985), Sect. 1-2,1-4; K. S. Krane,
ModernPhysics , 2n edn (Wiley, New York 1996), Sect. 3.3.[2] T. H. Boyer, “Equilibrium for classical zero-point radiation: detailed balance under scatteringby a classical charged harmonic oscillator,” J. Phys. Commun. , 105014(17) (2018).[3] See for example, M. Planck, The Theory of Heat Radiation (Dover, New York 1959).[4] For details see, for example, T. W. Marshall, “Random electrodynamics,” Proc. R. Soc.
A276 ,475-491 (1963); T. H. Boyer, “Random electrodynamics: The theory of classical electrody-namics with classical electromagnetic zero-point radiation,” Phys. Rev. D , 790-808 (1975);B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York 1991), pp.73-74.[5] T. H. Boyer, “General connection between random electrodynamics and quantum electrody-namics for free electromagnetic fields and for dipole oscillator systems,” Phys. Rev. D ,809-830 (1975).[6] See, for example, H. Goldstein, Classical Mechanics π, corresponding to thetransition from h over to ~ . [7] See the appendix of Boyer’s article in ref. 4.[8] See M. J. Klein, “Thermodynamics and Quanta in Planck’s Work,” in History of Physics edited by S. R. Weart and M. Philips (AIP, New York 1985), pp. 294-302. T. S. Kuhn,
Black-Body Theory and the Quantum Discontinuity 1894-1912 (Oxford U. Press, New York 1978),Chapter 3.[9] T. H. Boyer, “Blackbody radiation in classical physics: A historical perspective,” Am. J. Phys. , 495-509 (2018).[10] J. H. van Vleck, “The absorption of radiation by multiply periodic orbits, and its relation tothe correspondence principle and the Rayleigh-Jeans law: Part II. Calculation of absorptionby multiply periodic orbits,” Phys. Rev. , 347-365 (1924); “A correspondence principle for bsorption,” Jour. Opt. Soc. Amer. , 27-30 (1924). T. H. Boyer, “Equilibrium of randomclassical electromagnetic radiation in the presence of a nonrelativistic nonlinear electric dipoleoscillator,” Phys. Rev. D , 2832-2845 (1976). R. Blanco, L. Pesquera, and E. Santos,“Equilibrium between radiation and matter for classical relativistic multiperiodic systems.Derivation of Maxwell-Boltzmann distribution from Rayleigh-Jeans spectrum,” Phys. Rev.D , 1254-1287 (1983); “Equilibrium between radiation and matter for classical relativisticmultiperiodic systems. II. Study of radiative equilibrium with Rayleigh-Jeans radiation,” Phys.Rev. D , 2240-2254 (1984).[11] D. C. Cole and Y. Zou, “Quantum Mechanical Ground State of Hydrogen Obtained from Clas-sical Electrodynamics,” Phys. Lett. A , 14-20 (2003). D. C. Cole and Y. Zou, “Analysisof Orbital Decay Time for the Classical Hydrogen Atom Interacting with Circularly PolarizedElectromagnetic Radiation,” Physical Review E , 016601(12) (2004); “Subharmonic reso-nance behavior for the classical hydrogen atomic system,” Journal of Scientific Computing , 1-27 (2009). D. C. Cole, “Subharmonic resonance and critical eccentricity for the classicalhydrogen atomic system,” Eur. Phys. J. D , 200-214 (2018).[12] W. C-W. Huang and H. Batelaan, “Discrete Excitation Spectrum of a Classical HarmonicOscillator in Zero-Point Radiation,” Found. Phys. , 333-353 (2015).[13] T. W. Marshall, “Statistical Electrodynamics,” Proc. Camb. Phil. Soc. , 537-546 (1965). T.H. Boyer, “Derivation of the Blackbody Radiation Spectrum without Quantum Assumptions,”Phys. Rev. , 1374-1383 (1969).[14] T. H. Boyer, “Statistical equilibrium of nonrelativistic multiply periodic classical systems andrandom classical electromagnetic radiation,” Phys. Rev. A , 1228-1237 (1978).[15] See Goldstein’s discussion of adiabatic invariants in ref. 6, Section 11-7.[16] See the discussion by T. H. Boyer, “A Connection Between the Adiabatic Hypothesis of OldQuantum Theory and Classical Electrodynamics with Classical Electromagnetic Zero-PointRadiation,” Phys. Rev. A , 1238-1245 (1978).[17] See, for example, T. H. Boyer, “Understanding zero-point energy in the context of classicalelectromagnetism,” Eur. J. Phys. , 055206(14) (2016).[18] For a recent review, see T. H. Boyer, “Stochastic Electrodynamics: The Closest ClassicalApproximation to Quantum Theory,” Atoms (1), 29-39 (2019). A review of the work onclassical electromagnetic zero-point radiation up to 1996 is provided by L. de la Pena and . M. Cetto, The Quantum Dice - An Introduction to Stochastic Electrodynamics (KluwerAcademic, Dordrecht 1996).[19] T. H. Boyer, “Contrasting Classical and Quantum Vacuum States in Non-inertial Frames,”Found. Phys. , 923-947 (2013).[20] A. Einstein and L. Hopf, “Statistische Untersuchung der Bewegung eines Resonators in einemStrahlungsfeld,” Annalen der Physik (Leipzig) , 1105-1115 (1910). For calculations inmore recent notation see Boyer in ref. 13 and P. W. Milonni, The Quantum Vacuum: AnIntroduction to Quantum Electrodynamics (Academic Press, Boston1994), pp. 11-14. Recentuse of the Einstein-Hopf analysis appears in T. H. Boyer, “Particle Brownian motion due torandom classical radiation: Superfluid-like behavior in classical zero-point radiation,” Eur. J.Phys. , 055103 (17pp) (2020). https://doi.org/10.1088/1361-6404/ab988d, 055103 (17pp) (2020). https://doi.org/10.1088/1361-6404/ab988d