Classical Electron: Conclusion

DISCUSSION AND CONCLUSION

          This thesis began with a comparison of  some similarities between mass and electric charge.  The particular similarity of interest is the inertia of both mass and charge, their tendency to resist being accelerated.  The main concepts are radiation reaction, electromagnetic mass and renormalization, as applied to classical models of the electron.

          Radiation reaction can be seen as something of a fundamental electrodynamic feedback process, and as such can introduce unstable solutions to the electron’s equations of motion.  As discussed in Section 4.2, the instabilities can be avoided by an appropriate choice of radius for a spherical surface charge model of the electron.  Numerically, the minimum value of the radius for this case is 2/3 the classical electron radius,

 Although the model predicts a stable electron, it does not seem to agree with current experimental evidence.  High energy scattering experiments indicate the upper limit on the radius of the electron is on the order of 10^-18 meter.[i]   According to the extended charge delay-differential equation discussed in Section 4.2, a radius this small means the electromagnetic mass of the electron is greater than its experimental mass m, and therefore the model electron is unstable with respect to runaway solutions.

          Classical physics, however, is generally considered inapplicable for length scales close to the electron’s Compton wavelength,  on the order of 10^-12 meter.  Thus the classical electron is something of a contradiction in terms.  In the quantum realm, however, if the Compton wavelength formally replaces the electron radius, the extended charge model can be taken to the point limit without runaway solutions or pre-acceleration.   According to Sharp,[ii] “It is the fact that there is a new length scale in quantum theory which allows this to happen.”

          In classical field theory, the old dream of unifying general relativity with electromagnetism is still being pursued.  According to Cooperstock,[iii] in general relativity “the Kerr-Newman metric reveals a gyromagnetic ratio which agrees with that of the electron,” which appears promising for classically describing electron spin.  However, Cooperstock also points out that in a system of units that give charge and mass in centimeters, “whenever gravity plays a significant role, the charge-to-mass ratio e/m  is [expected to be] of order unity or less,” whereas the experimental value in these units is approximately 10²¹.

          In both the quantum and classical realms, renormalization raises the question of how much of the electron’s observed or experimental mass is mechanical or bare and how much is electromagnetic.  Classically, electrodynamics and relativity provide a way to answer the question, although the answer still contains ambiguities.[iv]

          Relativity says electromagnetic mass cannot be measured dynamically as a separate entity from mechanical mass.  When this news first arrived in the physics community in 1905, it dampened the hopes of determining whether the mass of the electron is entirely electromagnetic.  At the same time, Poincaré’s solution to the 4/3 problem gave a partial answer to that very question--although the  entire mass of the Abraham-Lorentz electron can be electromagnetic, it is not a stable charge distribution. 

          Poincaré’s solution leads to a question:  Why is the Abraham-Lorentz electromagnetic mass greater than the Einstein field-energy electromagnetic mass?  Feynman[v] points out that, because of the necessity of the Poincaré stresses--which must be added to the Abraham-Lorentz model to make it agree with the field energy model--it is “impossible to get all of the mass to be electromagnetic in the way we hoped.”

          Panofsky and Phillips[vi] say that the force accounting for stability is due to mass:  “The electromagnetic mass of the electron does not account for the entire mass;  the electron must have nonelectromagnetic mass of unknown origin to account for its stability.”

          In common with other references consulted for this thesis, Feynman and Panofsky and Phillips do not explicitly say that the “mass of unknown origin” can be considered to be responsible for the binding energy of the classical electron.  The more massive Abraham-Lorentz model is electrostatically unstable when compared to the smaller electromagnetic mass of the field energy calculation.  This is comparable to the mass deficit (or mass defect) of a nucleus, where the missing mass of the nucleus as compared with its separated nucleons is the binding energy of the nucleus.

          Since the Abraham-Lorentz electromagnetic mass is 4/3 times greater than the Einstein field-energy electromagnetic mass, the binding energy term is 1/3 the field energy electromagnetic mass,

 ,

so that the spherical shell charge has a binding energy of

This is the work required to assemble the shell of charge.[vii]  The physical meaning of this amount of energy in a realistic setting is unclear.

          What is lacking at the present time is a theory relating the mass and charge of the electron, proton, and other elementary particles.  The need for such a theory was mentioned by Oppenheimer[viii] in 1930, when he found that “it is impossible on the present theory to eliminate the interaction of a charge with its own field,” which led to predicted infinite energy level shifts in atoms. Oppenheimer concluded:  “It appears improbable that the difficulties discussed in this work will be soluble without an adequate theory of the masses of the electron and proton; nor is it certain that such a theory will be possible on the basis of the special theory of relativity.”  Although the problem of the infinite electromagnetic mass of the electron was fixed by the renormalization procedure of quantum electrodynamics, Oppenheimer and also Dirac[ix] believed that some new theory was necessary.  Such a new theory has not yet been developed.

          There are two ways to look at the problems posed by the classical electron model.  Why not just say that an electron is given to us as a point charge and there is no sense in asking what its structure is?  That is the working assumption of quantum electrodynamics.  The answer to that question--the other way to look at the classical model of the electron--is contained in the questions about the electron’s (and proton’s) charge and mass posed in the Introduction.  An improved classical relativistic theory of the electron could lead to an improved quantum theory, and that could lead to a theory that explains and relates the charges and masses of the elementary particles.

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[i] F. I. Cooperstock, “Non-linear gauge invariant field theories of the electron and other elementary particles,” in The Electron: New Theory and Experiment, ed. D. Hestenes and A. Weingartshofer (Kluwer, Dortrecht, 1991), pp. 171-181.

[ii] D. H. Sharp, p. 130.

[iii] Cooperstock,  p. 176.

[iv] F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65, 1051-1056 (1997).  The discussion of such issues is in the appendix of  this paper, although the electron is not what is being discussed.

[v] Feynman, et al, p. 28-4.

[vi] Wolfgang Panofsky and Melba Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading MA, 1955), p. 320.

[vii] T. H. Boyer, p. 3250.  Instead of using the term “work”, Boyer says this extra energy is needed “to maintain the validity of the force-momentum balance.”

[viii] J.R. Oppenheimer, p. 477.

[ix] P.A.M. Dirac, Directions in Physics, ed. H. Hora and J.R. Shepanski (Wiley, New York, 1978) p. 37.

Classical Electron: Planck's dipole resonators

APPENDIX

PLANCK’S DIPOLE RESONATOR EQUATIONS

          In the same year the electron was discovered, the first theoretical expression for radiation damping was derived by Max Planck in his research on the relation between the energy and entropy of the electromagnetic field as it interacts with small “dipole resonators”.[i]  Using the assumption of incident electromagnetic plane waves polarized parallel to the dipole axis, Planck found an equation relating the incident electric field to the electric dipole moment of the resonators:

K p(x,t)  +  L d²p(x,t)/dt² –  (2/3c³) d³p(x,t)/dt³ = E(x,t)

where, in Gaussian units, E(x,t) is the incident electric field, K and L are resonator-dependent constants,  and p(x,t) is the dipole moment of the resonator. Planck showed that when the resonator’s conditions are such that its energy changes slowly in comparison with variations in the incident field--when damping is much slower than the frequency of the incident electromagnetic wave--the third time derivative can be replaced with a first time derivative.  (This is simple harmonic oscillation approximation.)  The result is

K p(x,t)  +  L d²p(x,t)/dt²  +  (2K/3c³L) dp(x,t)/dt  =  E(x,t)

This equation is of the form used in the classical model of the atom.

          Planck was searching for an explanation via electrodynamics for the increase in entropy of closed systems, the observed macroscopic irreversibility in time not predicted by mechanical equations of motion, which are time reversal invariant.

          The appearance of the damping terms in the resonator equations implies a partial breaking of the forward and backward symmetry in time, since first derivatives and third derivatives with respect to time  change sign when t (time) changes sign.  Planck called radiation damping “conservative damping” in order to distinguish it from the dissipative effects of non-conservative damping forces such as friction.[ii]  Planck’s work, however, did not lead him to an explanation of entropy based on electrodynamics.  As Kuhn[iii] explains:

Through most of the year 1897, Planck continued to believe that he could prove irreversibility directly, without the aid of any statistical or other special hypotheses.  That proof had been his initial objective in taking up the black-body problem at all.  But by the spring of 1898 he had recognized that that goal could not possibly be achieved, and the concepts deployed in his subsequent papers came more and more to resemble those developed by Boltzmann for gas theory.

Though Planck was not successful at explaining entropy from fundamental electrodynamics, his research resulted in his postulate that a charged oscillator must emit and absorb radiation only in discrete amounts--the idea that gave birth to quantum mechanics in 1900.

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[i] T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912, (Oxford University Press, New York,  1978) p. 33.

[ii] M. S. Longair, Theoretical Concepts in Physics, (Cambridge University Press, Cambridge, 1984)

[iii] Kuhn, p. 36.

Concatenated-gaussian model of quiet sleep

1963-64: When We Moved to Pine Bluff

Published in the Jefferson County Historical Quarterly, Spring 2014.

 When We Moved to Pine Bluff

By David Trulock

Quite a few historically significant 50th anniversaries have come and gone recently. Among them, the anniversaries of President Kennedy’s assassination and of the Beatles arrival in the United States have deservedly received the most attention. One event that occurred between the JFK assassination and the Beatles debut in America has deservedly received no attention at all, until now:  My family’s move to Pine Bluff.

We weren’t particularly special in moving to Pine Bluff at that time. Between 1960 and 1970, the city’s population grew from 44,037 to 57,389, a thirty percent increase that resulted in the highest census count Pine Bluff has yet achieved.  There were numerous jobs to be had in Pine Bluff in the 1960s, jobs with the railroad, the two paper mills, the Pine Bluff Arsenal, Arkansas Power & Light Company, the newly built Jefferson Hospital, and the huge nearby dredging operations on the Arkansas River that helped to create the McClellan-Kerr Navigation System and the Port of Pine Bluff.

In contrast to people who moved from far away to take new jobs in Pine Bluff, and severed or stretched ties with family and friends in doing so, my family moved only about ten miles.  And my father didn’t change jobs—he’d done that a few years before, when he went from farming to being a stock broker and owning his own business, Trulock & Company, in downtown Pine Bluff.

Since 1955 my family had lived on my Trulock grandparents’ farm along the Arkansas River, just across the Free Bridge five miles north of town on Highway 79 (now Highway 79B). We lived in a sort of an extended-family compound, in an arrangement reminiscent of the Old South. My grandparents’ home was about a double stone’s throw from the converted brick dairy-barn house my family lived in, and my uncle, aunt and cousin (Leo, Sue and Lynne Trulock) lived within a half mile of us.

In the middle of our family compound were the “quarters,” a few basic sharecropper houses, rather old but constructed of long-lasting cypress, where some of the black families whose fathers worked on the farm lived. Two families with kids close to my age were those of Calvin Murry and Charlie Lee.  Although we weren’t supposed to go in their houses and vice-versa, the outdoor activities among the boys in the Lee, Murry, and Trulock families were a free-for-all of swimming in cattle ponds we weren’t supposed to swim in and playing in cotton trailers—at least up until a cap pistol caused a fire at the gin and my grandfather put the cotton trailers off limits.

So moving to town did change the childhood activities my brothers and I had taken for granted, but it didn’t stretch family ties or change my parents’ friendships and community activities. My maternal grandparents, Arch and Elner (King) Miller, weren’t affected by our move, either. They’d lived in Pine Bluff in the 1920s, 30s, and early 40s, but by 1964 they were living in the Pulaski Heights section of Little Rock (a frugal middle-class couple could still afford to buy a house there in the 1950s). So they were still about the same distance from us once we’d moved into town.

The address we moved to was 4006 Cherry Street, with Eden Park Country Club a few blocks to the east, and Jefferson Hospital a few blocks to the west. Although I didn’t realize it at the time, the move to town was inevitable, given the fact that my parents wanted their sons to attend school in the Pine Bluff school district, as they both had done, rather than in the Altheimer school district. Prior to buying the house on Cherry Street in the late summer of ’63, my father had been paying tuition at Trinity Episcopal Day School for three of his sons, and two more would be entering the school system in a few years.

The purchase of the house on Cherry Street meant my parents could enroll my two school-age brothers and me at Forrest Park elementary school for the 1963-64 school year. (Other cities have their Forest Parks; Pine Bluff has its Forrest Park section of town, the result of either a misspelling of forest or of someone's desire to memorialize the Confederate general of that name, or both.)  My brother Jeff was in the 5th grade, I was in the 4th grade, and Steven was in the 2nd grade.  At the start of the school year, Greg was two years old and Arch was still in the womb.

When Mother asked Greg the standard question of whether he wanted the baby to be a boy or a girl, he chose a third option:  “A turtle!”  Being the youngest of five brothers, and not having any sisters, Arch probably sometimes wished he’d had a protective shell.  He was born on October 23rd in Jefferson Hospital. We were still living on the farm at the time.

With Arch’s birth our family was complete, and the remodeling of the house on Cherry Street, under the supervision of Leslie McIntyre, was about halfway finished.  In December, we had our last Christmas on the farm, and sometime around the beginning of the New Year we moved into town.

In the world at large during 1963 and ‘64, changes were occurring that would have lasting effects.  I can think of quite a few others besides the two events mentioned at the beginning of this article, but will focus only on two more, each historically significant in its own way. I’ll also mention one other event related to my family, significant only to me.

In Pine Bluff, in February of ’63, sit-ins at Woolworth’s lunch counter by AM&N students and others signaled the beginning of the end of the city’s retail establishments’ racial discrimination in its most blatant form, the unequal segregation of "colored" from "white." In May of ’64, in New Jersey, two physicists at Bell Laboratories—trying to find the source of the constant, low-level noise picked up by their microwave antenna— accidentally discovered the “cosmic background radiation” that later provided the first incontrovertible evidence for the Big Bang cosmological model.

Also in May of 1964, a Pine Bluff Commercial photographer came to 4006 Cherry and took a photo for an article about the League of Women Voters.  The original photo is reproduced below. As is often the case in the newsprint business, something had to be cut to make the published item fit the space available.  I was cut out of the published photo, and also not mentioned along with my brothers in the caption.  After fifty years, I’m glad to be able to put myself back in the picture.

 

One of the things us Trulock boys (and our dad) had to grudgingly put up with was Mother's insistence on having "family meetings" that turned out to be lectures from her.  The photographer who snapped this picture thus captured an emblematic moment in our family history.  From left: Steven, Betty, Arch, Jeff, Greg, and me (the one who got cropped).  --D.T.