Structure and Self-Energy of the Electron

S. M. BLINDER Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055

Received 10 April 2001; accepted 2 July 2001

DOI 10.1002/qua.1806

= 2 2 ≈ × −13 ABSTRACT: The region very close to an electron (r r0 e /mc 2.8 10 cm) is, according to quantum electrodynamics, a seething maelstrom of virtual electron–positron pairs ﬂashing in and out of existence. To take account of this well-established physical reality, a phenomenological representation for vacuum polarization is introduced into the framework of classical electrodynamics. Such a model enables a consistent picture of classical point charges with ﬁnite electromagnetic self-energy. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem 90: 144–147, 2002

Key words: classical electrodynamics; electron self-energy; point charge; vacuum polarization

er-Olov Löwdin was an enthusiastic advocate on the scale of the classical electron radius r0 = P of the application of the techniques and modes e2/mc2 ≈ 2.818×10−13 cm. The electromagnetic self- of thought of quantum chemistry to a wide range energy of such a finite structure would be of the or- 2 2 of scientific problems, including material science, der of W ≈ e /r0 ≈ mc and thus implies an electron molecular biology, and even particle physics [1]. In- rest mass predominantly electromagnetic in origin. spired by such a fearless spirit of adventure, we Yet all experimental evidence implies an electron proposetogopossibly“wherenoquantumchemist radius much smaller than r0, consistent, in fact, has ever gone” to explore a different vista of the the- with a particle of point mass and point charge [5]. ory of matter—the interior of the electron. This work Recent high-energy electron–positron scattering ex- − is dedicated in fond memory to Per-Olov Löwdin periments imply an upper limit of 2 × 10 16 cm on for his intellectual inspiration, guidance, and friend- the electron size. ship. If the electron is indeed a structureless point Since the discovery of the electron by J. J. Thom- charge, as assumed in quantum theories, in partic- son over a century ago [2], a persistant puzzle has ular quantum electrodynamics, then how does one been how to account for its electromagnetic self- avoid the divergent electromagnetic self-energy? energy. This has been the subject of extensive theo- An analogous problem is dealt with very ele- retical contemplation by some of the leading figures gantly in quantum chemistry. In Hartree–Fock the- of twenty century physics [3]. The earliest models ory, the electrostatic interaction between two spin- (Thomson, Poincaré, Lorentz, Abraham, Schott) [4] orbitals φα and φβ is given by the difference between pictured the electron as a finite charged sphere, the Coulomb and exchange integrals: − = −1 − −1 Correspondence to: S. M. Blinder; e-mail: [email protected]. Jαβ Kαβ α, β r12 α, β α, β r12 β, α .(1)

This reduces to zero when α = β, thus canceling out order of magnitude of the neutron–proton mass dif- all orbital self-interactions. We should like to extend ference (1.29 MeV/c2). There is no need to invoke this idea to treat the self-energy of a single electron. any nonelectromagnetic forces within the electron— A number of ingenious schemes to avoid a diver- collectively known as Poincaré stresses. gent electromagnetic self-energy for a point electron The energy of an electromagnetic field in a rest have been proposed over the years by Dirac [6], frame is given by Wheeler and Feynman [7], Rohrlich [8], Teitel- 1 boim [9], and many others. The more recent ap- W = (E · D + B · H) d3r.(2) π proaches invoke such arcana as advanced solutions 8 of Maxwell’s equations (superposed on the conven- Thefieldproducedbyapointchargee in vacuum is tional retarded solutions) and/or renormalization represented by D = E = erˆ/r2, B = H = 0and of mass and charge infinities. This enables the diver- 1 e2 gent part of the self-interaction to be avoided while W = 4πr2 dr =∞,(3) π 4 leaving intact the radiation reaction, an effect long 8 r known and thoroughly tested. unless a lower cutoff is introduced. From a more general perspective, the singular- It was suggested a long time ago by Furry ities in fields and energies associated with point and Oppenheimer [11] that quantum electrody- charges in classical electrodynamics has been a per- namic (QED) effects could give the vacuum some vasive flaw in what has been an otherwise beauti- characteristics of a polarizable medium, which fully complete and consistent theory. An immense Weisskopf [12] represented phenomenologically by number of attempts to address this problem have an inhomogeneous dielectric constant, viz been based, roughly speaking, on one of the follow- D(r) = (r)E(r). (4) ing lines of argument: (1) Actual point charges do not exist—real particles have a finite size—hence the Constitutive relations in classical electrodynam- problem is artificial. (2) By a clever limiting proce- ics describe properties of matter which must be dure in the formalism, the radius of a charge can determined experimentally or, in favorable cases, be reduced to zero without introducing infinities. by quantum-theoretical computation. In the same (3) Point charges are quantum objects and classical sense, Eq. (4) represents a constitutive relation for electrodynamics has no business dealing with them. the vacuum, as implied by quantum electrodynam- The last point of view, espoused by Frenkel [10] ics. Using (4) in (2), and others, asserts that any classical model is fu- ∝ 1 1 e2 tile because the electron is a quantum mechanical = π 2 W 4 4 r dr (5) object with no substructure. It is nonetheless of at 8π 0 (r) r least academic interest to have a consistent classi- and equating this to the self-energy of the electron cal relativistic model that connects to macroscopic 2 ∞ electrodynamics, while remaining cognizant of its = e dr = 2 W 2 mc .(6) limitations. The purpose of the present study is 2 0 r (r) a modified theory able to handle the singularities Remarkably, the functional form of (r) need not be produced by point charges while reducing to stan- further specified, provided only that it satisfies the dard electrodynamics for r r0. limiting conditions We propose to provide possible finishing touches to Maxwell’s electromagnetism without making any (∞) = 1and(0) =∞.(7) ad hoc modifications of the fundamental equations Maxwell’s first equation ∇ · E = 4πρ applied to the of the theory. The key to our approach is the electric field physical reality of vacuum polarization in the sub- er microscopic vicinity of charged elementary parti- E = (8) 3 cles. (r)r We will proceed on the premise that the electron determines the charge density rest mass (0.511 MeV/c2)istotally electromagnetic, e (r) which was the original idea of Lorentz and Abra- ρ(r) =− .(9) π 2 2 ham. This is consistent with the (nearly, if not ex- 4 r [ (r)] actly) zero rest mass of the electron’s uncharged Note that this represents the net or total charge den- weak isodoublet partner—the neutrino—and with sity, the sum of the free and polarization densities.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 145 BLINDER

This function is appropriately normalized since ∞ ∞ (r) dr ρ π 2 =− (r)4 r dr e 2 0 0 [(r)] 1 1 = e − = e. (10) (∞) (0) An explicit functional form for (r) does follow if it is conjectured that the net charge density (9) is proportional to the field energy density from (6). For then, (r) e2 =− (11) (r) 2mc2r2 FIGURE 1. Vacuum polarization produced by a point with the solution charge. The vertical bar at r = 0representsadelta function, which cancels the free charge. e2 r (r) = exp = exp 0 . (12) 2mc2r 2r It should be emphasized for the benefit of QED The electrostatic potential corresponding to (14) theorists who might be reading this that our use is given by of the term “vacuum polarization” is intended only 2e −r /2r in a classical phenomenological context. The lead- (r) = 1 − e 0 . (15) r0 ing contribution to vacuum polarization in real life comes from the interaction of the electron with the This implies a deviation from Coulomb’s law of the transverse radiation field, which does not enter in same magnitude as the fine structure in atoms, but our model. We are thereby overlooking additional totally negligible on a macroscopic scale reality such self-energy contributions arising from fluctuations as the Lamb shift. Note that (15) reduces to e/r when → →∞ hi the vacuum radiation field. Accordingly, our rep- either r0 0orr . resentation of vacuum polarization is not to be com- An alternative evaluation of the electromagnetic self-energy follows from transformation of Eq. (2): pared with QED computations. Somewhat of a rationalization for the functional 1 3 1 3 W = E · D d r = freeρ d r (16) form of (r) is suggested by Debye–Hückel theory 8π 2 for ionic solutions and plasmas. The dielectric constant depends on a Boltzmann factor e−E/kT.Ifin using place of the average thermal energy kT,wesubsti- er D =−∇ = (17) tute the relativistic energy of pair formation 2mc2, free r3 regarding the vacuum as an effective thermal re- and assuming the requisite vanishing of integrands E = 2 servior, then Eq. (12) follows with e /r. at infinity. Thus An explicit expression for the charge density fol- ∞ 1 2 lows by substituting (12) into (9): W = free(r)ρ(r)4πr dr 2 0 er − 0 r0/2r ∞ − ρ(r) = e . (13) 2 r0/2r 8πr4 = e r0 e = 2 3 dr mc (18) 4 0 r Since ρfree(r) = eδ(r), the density from vacuum polarization must equal in agreement with the previous result, and further justification for the conjectured functional form er0 − ρ = r0/2r − δ of (r). VP(r) 4 e e (r), (14) 8πr We have also been able to obtain the result of this which is represented graphically in Figure 1. Ac- study by an alternative derivation from the view- cording to this model, the free point charge is ex- point of general relativity [13]. A modification of actly canceled by the delta function term of the the Reissner–Nordstrøm solution to the Einstein– polarization charge. This is certainly reminiscent of Maxwell equations has been shown to give a finite the self-energy cancellation that occurs in Eq. (1). electron self-energy.

146 VOL. 90, NO. 1 STRUCTURE AND SELF-ENERGY OF AN ELECTRON

6. Dirac, P. A. M. Proc Roy Soc (London) 1938, A167, 148. References 7. Wheeler, J. A.; Feynman, R. P. Rev Mod Phys 1949, 17, 157; 1945, 21, 425. 1. See Preface to any of the early issues of Advances in Quan- tum Chemistry; Academic Press: New York; for example, 8. Rohrlich, F. Phys Rev Lett 1964, 12, 375. Vol. 2, 1965. 9. Teitelboim, C. Phys Rev D 1970, 1, 1572; 1970, 3, 297; 1971, 2. Thomson, J. J. Phil Mag 1897, 44, 293. 4, 345. 3. For a retrospective on the electron centennial, see Weinberg, 10. Frenkel, J. Zeit Phys 1925, 32, 518. S. Nature 1997, 386, 213. 11. Furry, W.; Oppenheimer, J. R. Phys Rev 1934, 45, 245. 4. A deﬁnitive review of classical electron theories is given by Rohrlich, F. Classical Charged Particles; Addison-Wesley: 12. Weisskopf, V. F. Det Kgl Danske Videnskab Selskab Mat- Reading, MA, 1990. Fys Medd 1936, 14, 1; Reprinted in Schwinger, J. Quantum 5. See,forexample,Perkins,D.H.IntroductiontoHighEnergy Electrodynamics; Dover: New York, 1958. Physics; Addison-Wesley: Reading, MA, 1987. 13. Blinder, S. M. Repts Math Phys 2001, 47, 279.

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