1 The Zitterbewegung and the de Broglie wavelength Jean Louis Van Belle, Drs, MAEc, BAEc, Bphil 12 February 2019 Summary: This paper explores some more philosophical aspects of the Zitterbewegung model of an electron, including a geometric interpretation of the de Broglie wavelength. Contents Zitterbewegung basics .................................................................................................................................. 1 Applying wavefunction math ........................................................................................................................ 4 The quantization of space ............................................................................................................................. 5 A black-hole model for an electron?............................................................................................................. 7 How to interpret the de Broglie wavelength? .............................................................................................. 8 Zitterbewegung basics The Zitterbewegung model of an electron 1 is intuitive because it combines John Wheeler’s ‘mass without mass’ idea with the idea of a pointlike electric charge. Indeed, the most basic Zitterbewegung model is the one Paul A.M. Dirac described: “a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us.” He added: “As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light. This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory motion is so high and its amplitude is so small. But one must believe in this consequence of the theory, since other consequences of the theory which are inseparably bound up with this one, such as the law of scattering of light by an electron, are confirmed by experiment.” 2 The reference to light scattering – photon scattering – is important because we do get the Compton radius of an electron when combining the E = mc 2 , E = ma 2 ω 2 and E = ħω equations: E = m 2 ω 2 = m 2 E 2 ℏ 2 ⟺ℏ 2 = m 2 E = m 2 m 2 =m 2  2  2 ⟺= ℏ m = λ  2π ≈ 0.386 × 10 −12 m The idea is visualized in the illustration below (for which credit goes to the modern zbw theorists Celani, Vassallo and Di Tommaso). The illustration shows a rather particular implication of relativity: the radius 1 We are hesitant to generalize to matter-particles (fermions) in general. Current theory suggests all elementary matter-particles (apart from electrons, we have quarks, and more) have some electric charge. However, all of our thinking has been focused on mainstream theory in the QED sector of the Standard Model only. Hence, that is the theory that deals with electrons and photons, and their interactions, basically. We always thought some basic upgrade of classical mechanics and electromagnetism, based on the idea of the integrity of a cycle, would do to deal with the ‘quantum’ in quantum electrodynamics, and we feel the Zitterbewegung hypothesis does exactly that. 2 See: Paul A.M. Dirac, Theory of Electrons and Positrons, Nobel Lecture, December 12, 1933