The Stability of the Electron

1. The Stability of the Electron F. J. Himpsel, Physics Dept., Univ. Wisconsin Madison • Coulomb explosion of the electron: a century-old problem • Exchange hole and exchange electron • Force balance in the H atom (single particle) • Stability of the vacuum polarization (manybody) • Stability of an electron in the Dirac sea (the real deal) • Application to insulators and semiconductors • Connection to the fine structure constant

2. A really bad hair day Equal charges repel each other Coulomb explosion of the electron ?

3. The self-energy of the electron 1934 V. F. Weisskopf, Zeits. f. Physik 89, 27; Phys. Rev. 56, 72 (1939) The stability of the electron: a century-old problem 1897 Discovery of the electron by J. J. Thomson,Phil. Mag. 44, 293 1905 H. Poincaré,Comptes Rendus 140, 1505 1909H. A. Lorentz,The Theory of Electrons,Columbia University Press 1922 E. Fermi, Z. Physik 23, 340 1938 P. A. M. Dirac, Proc. R. Soc. Lond. A 167,148 (1938); A 268,57(1962)

4. Sommerfeld’s successor, my Diplom thesis advisor

5. Considerations for a solution 1. Can magnetic attraction compensate electric repulsion? Requiresacharge rotatingwiththespeedof light at thereduced Compton wavelength, where classical physics loses its validity. 2. Can gravity compensate repulsion, forming a black hole? The Schwarzschild radius RS of the electron corresponds to an energyof1040GeVviatheuncertaintyrelationpħ/RS andE(p). That generates an astronomical number of extra ee+ pairs. • Compensate Coulomb repulsion with exchange attraction? • The self-Coulomb and self-exchange terms cancel each other. • An exchange hole forms around the electron due to the exclu- sion principle. The positive hole threatens to collapse. • What can prevent exchange collapse? • a) Compressing the exchangehole generates a repulsive force. • b) Adding anexchangeelectron to thehole preserves neutrality.

6. Eel positrons (holes) electrons (Epos) The Dirac sea The Dirac equation for electrons admits solutions with both positive andnegative energy inorderto satisfyspecial relativity (E2=p2+m2). In the vacuum of quantum electrodynamics the states with negative energy arealloccupied and those with positive energy areallempty. To satisfy particle-antiparticle symmetry and to cancel the infinite negative charge of the vacuum electrons one has to assign empty states topositrons (=holes) withnegative energy. The energy dia-gram is similar to that of an insulator with a band gap of 2m1MeV.

7. The exchange hole The exclusion principleforbids two electrons with the same spin to occupythesamelocation.Asaresult,nearbyelectronswiththesame spin are pushed away from a reference electron, forming a positive hole with oppositespin.Thisexchangehole hasbeendefinedmathe-matically for an electrongas,such as the Fermi seaformed by the electrons in a metal(Slater1951,Gunnarson andLundqvist 1976). Weisskopf’s workin1934 can be viewed in retrospect asanattempt to define the exchange hole for the Dirac sea (see the next slide). Generalizing the definition from the Fermisea tothe Diracsea givesa slightlydifferent picture (two slides ahead). Weisskopf’s displaced electron becomes the exchange hole. But both aredescribed bythe same Bessel function: -1/22K1(r)/r The size of the exchange hole in the Fermi sea is givenbytheFermi wavelength F. That becomes the reduced Compton wavelength C in the Dirac sea (C=1/m , with the electron mass minunits ofh,c).

8. Weisskopf’s picture Apoint-like electron is compensated bya point-like hole. The electron displacedby the hole spreadsout over C. An electron addedto the Dirac sea Phys. Rev. 56, 72 (1939)

9. exchange electron sum An electron insidethe Dirac sea Vacuum electron density4r2 exchange hole arXiv:1701.08080 [quant-ph] (2017) Distance r (in reduced Compton wavelengths) The new picture A point-like electron is surrounded by a spread-out exchange hole. The exchange hole is surrounded by the displaced electron. Itiscreated by two sequential exchanges (“exchange electron“).

10. The response of the Dirac sea to an electron: exchange hole + displaced electron The exchange hole is defined via the pair correlation,i.e.,the proba- bility of finding a hole at r2 if there is an electron at r1 . The additional displacedelectron canberemoved toinfinity in the Fermi sea (using a long metallic wire).This is not possible in the Dirac sea,becauseit is an insulator with an energy barrier of 1MeV. Thus,the displaced electronstays near the exchangehole,forming an exchange exciton. Including the displacedelectron requiresa three-fermion correlation, i.e.,the probability of finding ahole at r2 andanelectronatr3, if there is an electron at r1. This three-body system resembles the negative positronium ion. arXiv:1701.08080[quant-ph] (2017)

11. Force balance in a simple system: the H atom The Dirac wave function is equivalent to a classical field. The Lagrangian formalism defines the two force densitiesacting on:electrostaticattraction and confinement repulsion. They cancel each other at every point in space. Such a local force balance between force densities goes beyond the usual stability criteria. Those rely on a global energy minimum. arXiv: 1511.07782 [physics.atom-ph] (2015)

12. Force densities in the singlet ground state of H. The hyperfine interaction adds the contribution fE,tothe elec-trostatic force density fE,C . Both are nearly balanced by the correspon-dingconfinementforcedensities(not shown). The residual f is givenwith an amplification factor 106. The electrostatic force density fVP acts on the vacuumpolarization sur-rounding the proton. It is balanced by a confinement force density,too. Adding the magnetic hyperfine interaction to the Coulomb potential leads beyond classical field theory.The magnetic field is generated by the quantum-mechanical angular momentum operator. The ground state wave function remains isotropic, since the proton spin has equalprobabilityof pointing up or down in the entangled singlet spinwave function (p e–pe)/2.Theeffectof thehyperfineinteraction ismainlyelectrostatic.The electron density becomescompressed near the proton and thereby enhances both electrostatic attraction and confinement repulsion. arXiv: 1702.05844 [physics.atom-ph] (2017)

13. Vacuum electron density 4r2 r (in reduced Compton wavelengths) A manybody system: vacuum polarization The charge induced inthe Dirac sea by the Coulombpotential is attracted toward the inducing point charge. This attractionis compensated by the confinementre-pulsion (as in the H atom). This is shown explicitly using hydrogenic solutions for the vacuum electrons/positrons and summing their force densities over all radial and angular momenta (see the figure).Theforce balance remains valid for each filled shell in the sum (i.e.,all mj for each j), and thus for the complete sum. arXiv:1512.08257 [quant-ph] (2015)

14. Force densities between holes and electrons Electrostatic force densities are obtained easily from the products of charge densities and electric fields (see the plot below).Butthe properdefinition of confinement force densities remains unsolved. Force density4r2

15. Displaced electron (-1) Exchangehole Electron density Distance from the reference electron (in Fermi wavelengths) Application to insulators and semiconductors The concept of an exchange exciton might have applications in solid state physics forcharacterizing the exchangeinteraction in insulators and semiconductors. The neutral exchange exciton is a better matchfor them, since the electron displaced by the exchangehole cannot delocalize. The exchange exciton can be calculated from the three-electron correlation. It contains the standard exchange hole.

16. Can the fine structure constant beobtained froma forcebalance? (another century-old problem) There are two possible outcomes of any force balance: If the opposing forces scale the same way with , theforcebalance doesnot provide any information about .That is thecase for theH atom and the vacuum polarization. If the opposingforces scaledifferentlywith , a balance exists only for a particular value of .That might be the case foranelectronin the Diracsea.The electromagentic interaction scales like,but theexchange interaction is independent of .It isgivenbyfree-electron wave functions. As a result, the charge of the electric vacuum pola-rization is about afactorof  smaller than that of the exchange hole.

17. “…was die Welt im Innersten zusammenhält“ Goethe’s Faust makes a pact with the devil to learn about the fundamental mechanism holding everything together.