The Zitterbewegung or “Trembling-Along-The-Way” Electron

^{By Dr. Inés Urdaneta, Physicist at Resonance Science Foundation}

The German word zitterbewegung -zitter- means “trembling along the way” and it was coined for the first time by Erwin Schrödinger in 1930 when studying the solutions of Dirac equations for free relativistic electrons. When analyzing the behavior of the wave packets, Schrödinger found an oscillatory term with an amplitude of the magnitude of the Compton wavelength and frequency of 2m_ec²/ħ., where m_e is the mass of the electron, c is the speed of light and ħ is the reduced Planck constant. This zitter character of the electron found by Schrodinger suggested that the electron performed an extremely-high-frequency circular motion.

As it stands, Dirac’s fully relativistic equation in the form originally proposed by Dirac [1] for a free electron, describes all spin-1/2 massive particles, and it is consistent with the principles of quantum mechanics and the theory of special relativity. The relevance of this equation relies upon three critical aspects: its accounts for the fine structure of the hydrogen spectrum, it predicts the existence of antimatter, and it provides a theoretical justification for Pauli’s phenomenological theory of spin. The only controversial feature of this equation is that it depicts subatomic particles oscillating at the speed of light c; a movement that superposes to its translational motion.

Since no mass could travel at or faster than the speed of light, this physical oscillation of mass would violate special relativity principles. Therefore, this fluctuation in Dirac’s theory was interpreted instead as a fluctuation between positive and negative energies in the Dirac sea. The Dirac sea establishes that all negative energy states are occupied and Pauli’s exclusion principle forces any additional electron to occupy the positive states, which are also supposed to be all occupied. Dynamically, the interaction of both energy states happens via the quanta of the electromagnetic fields; photons that are continuously absorbed and emitted by electrons leaving holes behind or falling into the Dirac sea and seemingly annihilating. Virtual particles that pop in and out of existence as a particle-antiparticle pairs creation/annihilation process, are supposed to be the origin of vacuum fluctuations as well, becoming the most widely accepted mechanism of the quantum vacuum state and its zero-point field (for a detailed explanation, don't miss our article Spacetime Engineering & Harnessing Zero-Point Energy of the Quantum Vacuum), 

In such a scenario there should be a direct link between zitter and the zero-point energy fluctuations of the quantum vacuum -zpe-, which are always present. In modern terms, the Dirac sea can be understood via quantum field theory -QFT- as a sum of creation and annihilation operators for the Dirac spinor. And yet, zitter is traditionally associated only to the Dirac equation at the level of quantum mechanics, and very little has been done in quantum electrodynamics (QED) to address it [2], while it introduces the concept of vacuum fluctuations following the quantization of the electromagnetic field, i.e., an EM field composed of quantum harmonic oscillators. The quantum vacuum state is the fundamental state of the electromagnetic field, and the complex interaction of the Dirac sea with the quanta of the EM fields produces a variety of phenomena, such as the spontaneous emission of an electron interacting with this fundamental field. 

The Dirac sea predicted the existence of antimatter because an electron in the Dirac sea absorbing energy would leave a hole with the same mass as the electron but with a positive charge, a positron. Antimatter was later confirmed by Carl Anderson [3] in 1930, validating Dirac's sea model. Hence, the depiction of zitter as a real mechanical oscillation has no place in the modern version of Dirac's theory, Quantum Electrodynamics (QED); zitter is only considered in relativistic scenarios as the energy fluctuation of the Dirac sea, being neglected elsewhere. This last raises concerns among the authors who consider zitter a real oscillation not of mass but of charge -thus removing the violation of special relativity-, responsible for the spin and magnetic moment of particles, which are present all the time and not just in relativistic domains. Meanwhile, the extremely energetic zpe fluctuations are always present, not just in relativistic domains, and this should raise deep concerns as well because since they are assumed to be random (their mean value being zero), they are commonly discarded under the assumption that the play a negligible role at macroscopic scale. Such a strong assumption has no experimental proof.

In summary, on one hand we have the zitter fluctuations happening at light speed c and associated only to particles as predicted by Dirac’s equation, and on the other hand we have the very energetic zpe fluctuations associated to the quantum vacuum state of the electromagnetic field as addressed by QED/QFT and experimentally proven by the Casimir effect.. A very important link seems to be missing in the current state of particle physics and QED, since one could reasonably suppose there should be an unambiguous connection between zitter and the quantum vacuum fluctuations, regardless of the discussion on their nature; if real oscillations/rotations of particles and of the quantum vacuum, or not.

As strange as it may sound, zitter and zpe are not yet unified in a consented framework showing their relationship. Additionally, if zitter was a real physical rotation, it would imply that the electron has inner structure, while QFT considers the electron a fundamental particle because it has no inner structure; it is a point-like particle as established by the Standard model.

Due to the extremely high speed of the oscillation, mainly the speed of light, the zitter motion of a free electron would be impossible to detect experimentally and various experimental simulations of the effect have given insight. Recently it has been shown theoretically that electrons in III-V semiconductors are governed by similar equations in the presence of spin-orbit coupling, where a small energy splitting up to 1meV result in zitter at much smaller frequencies, being experimentally accessible as an AC current that would demonstrate the zitter of electrons in a solid [4]. Zitter has also been used to explain the nontrivial behavior of the conductivity at zero temperature in graphene [5] where the importance of considering this electron effect has been pointed out. Among other consequences, the zitter behavior of the electron is used to produce the Darwin term for the hydrogen atom, which plays the role in the fine structure as a small correction of the energy level of the s-orbitals [6]. Also, the electron channeling and internal clock experiments are explained in terms of zitter [7, 8]. One then wonders what criteria defines when zitter should be considered, or discarded.

Many interesting features emerge from the zitter behavior that are worth-like noticing. The electron would have to be massless at the level where these fluctuations occur, since zitter happens at the speed of light. Therefore, mass would have to appear at some outer, external frame or level of motion. The fluctuations could also explain a smearing out of the average position over a Compton radius volume, which would give a physical interpretation to the wave function and the associated probability density. And this is somehow supported by scattering experiments which indicate that the electron is far smaller than its Compton size, being more of a point-like charge [9].

As Wilczek pointed out [10]:

“An electron’s structure is revealed only when one supplies enough energy […] at least 1 MeV, which corresponds to the unearthly temperature of 1010 kelvin” below which it ‘appears’ point-like and structure-less" .

Numerical simulations have shown that if a massless fluctuating charge is accelerated in an electric field, zitter acquires a helical motion suggestive of spin [11]. Therefore, it has been claimed by some researchers that zitter is the origin of spin, fact that could be supported in the relativistic regime through Schrodinger’s original circular motion interpretation of Dirac´s solution for the velocity operator, if it's applied to the electron's charge instead. However, such a circular motion is considered by Barut and Zanghi  [12, 13], Hestenes [14], Daywitt [15], Huang [11], Gauthier [16], Consa [17], Vassallo [18], Knuth [19], Wilson [20], Puthof, Rueda and Haisch [21], among many others, as the origin of the spin feature of the particle, becoming a real physical motion and not just a quantum mechanical component, and suggesting, therefore, that there must be a more fundamental explanation for the zitter motion, since spin is not restricted to the relativistic domain. Spin is always present. 

As these authors propose, if the electron had inner structure, many incongruent features of quantum mechanics would be removed, and the Copenhagen interpretation would no longer hold. The particles’ oscillation would no longer be virtual, but a real mechanical feature, a physical rotation responsible for the spin (therefore, a real spin) and magnetic moment of the electron. In this case, then, what is the mass of the electron?

Hestenes states [22]:

“it implies that Schrödinger’s original wave packet oscillation is merely an epiphenomenon revealing the zbw (zitter) periodicity which was already inherent in the complex phase factors of both electron and positron plane wave states. The essential feature of the zbw idea is the association of the spin with a local circulatory motion characterized by the phase factor. Since the complex phase factor is the main feature which the Dirac wave function shares with its nonrelativistic limit, it follows that the Schrödinger equation for an electron inherits a zbw interpretation from the Dirac theory. It follows that such familiar consequences of the Schrödinger theory as barrier penetration can be interpreted as manifestations of the zbw.”

If the electron was really spinning, the proton would be as well, and since everything else in the universe is spinning (planets, stars, solar systems, galaxies, all made of atoms), one may wonder: is it matter what’s spinning, or is it space (that connects all scales and composes more than 99,999999% of atoms) what spins, creating a vortex which curvature we understand as gravity thanks to Einstein, while that coherently spinning region of space would be what we understand as mass?

These mysteries are addressed in The Zitterbewegung Electron Puzzle, our latest published paper [23].

As it is stated in [23]:

"whether it is charge and not mass what is oscillating at zitter frequency can only be solved unambiguously when the relation between mass and charge, that requires unification of gravity and electromagnetism, is achieved."

RSF in perspective:

The prevalent misunderstanding regarding the zero-point energy of free space is explained in [24]: “the vacuum fluctuations are described as occurring as a result of the Heisenberg uncertainty principle; this is in error. In fact, contrary to popular belief, the Heisenberg uncertainty principle is a consequence—not the source of—vacuum fluctuations of the zero-point energy field (Haramein et al., the origin of mass and nature of gravity, pre-print, 2023). In the QFT formulation (e.g., for a dipole oscillator) the position and momentum operators are non-commutative, which roughly means they do not give the same results. Zero-point energy is required within this formalism to maintain the non-commutativity of the position and momentum operators. It then follows that the foundations of quantum mechanics and the uncertainty principle are firmly rooted in the dynamics of ZPE vacuum fluctuations that define the bath (or field) in which particles appear, evolve, and interact.”

Our next free technical seminar (Sep. 28/2023) will explain in detail the upcoming paper The Origin of Mass and the Nature of Gravity, where we extend further the holographic approach by using the correlation functions on the zero point energy to show that when the correlation time determining the coherency of the oscillating quantum vacuum fluctuations is chosen as the characteristic time of the proton, we obtain precisely the energy density of the proton rest-mass. This clearly demonstrates that the mass emerges from the quantum vacuum fluctuations, where their decoherence process defines a screening mechanism that determines the mass-energy of a system. Like QFT in which the infinite bare-mass and the bare-charge of particles are shielded or screened by the quantum vacuum fluctuations “dressing” the particle, only that in our case the vacuum fluctuations are the origin of the mass. This would prove as well that the vacuum fluctuations and zitter are real rotations in the quantum vacuum plasma. One could say that zitter are the zpe fluctuations in that region of space that we call a "particle".

Following the holographic approach, the macroscopic space is granular at the Planck scale. The Planck Spherical Units (PSUs) quantize space and organize creating different phases; they are real mechanical rotors, which is why the Planck constant is associated to the quantum of action or angular momentum. These PSU represent the zpe fluctuations of the quantum vacuum and because each PSU has the correct (and humongous) energy density of 10⁹³ gr/cm³ (Planck mass in the volume of a Planck sphere of diameter Planck length), the model works to compute the real energy content of a system (proton, electron, black holes ...).

One of the most striking results of the model is that the strong force keeping protons together in the nucleus of an atom becomes gravity acting at nucleonic scale. This is easy to understand once we see that using the energy density unit PSU we can calculate the real mass-energy content in the volume of a proton, which is of the order of 10⁵⁵ gr. This order of magnitude is the same as the total baryonic mass of the universe, which is why we consider the proton as the holographic unit of the universe.

Just as in water when opening the drain, the circulation of its quantized constituents (atoms of oxygen and hydrogen) creates a vortex which curvature we understand as gravity thanks to Einstein. In this analogy, water is the space at macroscopic scale, and its atoms are the PSUs.

It's all about the fluid dynamics and circulation of theses PSU ...

References

[1] P. Dirac, Principles of Quantum Mechanics. International Series of Monographs on Physics, 4th ed. (Oxford University Press, Oxford, 1982), p. 255.

[2] W. Zhi-Yong and X. Cai-Dong, Zitterbewegung in quantum field theory, Chinese Phys. B, 17, 4170 (2008).

[3] C. D. Anderson, The apparent existence of easily deflective positives, Science 76, 238 (1967).

[4] I. Stepanov, M. Ersfeld, A. V. Poshakinskiy, M. Lepsa, E. L. Ivchenko, S.A.Tarasenko and B. Beschoten, Coherent Electron Zitterbewegung, e-print arXiv:1612.06190 [condmat.mes-hall] (2016).

[5] M. Katsnelson, Zitterbewegung, chirality, and minimal conductivity in graphene, Eur. Phys. J. B 51, 157 (2006).

[6] P. R. Berman, Hydrogen Atom with Spin in External Fields, Introductory Quantum Mechanics. UNITEXT for Physics (Springer, Cham, 2018). [10.1007/978-3-319-68598-4_21]

[7] P. Catillon, N. Cue, M. J. Gaillard, R. Genre, M. Gouane`re, R. G. Kirsch, J.-C. Poizat, J. Remillieux, L. Roussel, and M. Spighel, A Search for the de Broglie Particle Internal Clock by Means of Electron Channeling, Found. Phys. 38, 659 (2008).

[8] G. R. Osche, Electron channeling resonance and de Broglie’s internal clock, Ann. Fond. Louis Broglie 36, 61 (2011).

[9] J. L. van Belle, The electron as a harmonic electromagnetic oscillator, e-print https://vixra.org/pdf/1905.0521v4.pdf (2019).

[10] F. Wilczek, The enigmatic electron, Nature 498, 31 (2013).

[11] K. Huang, On the zitterbewegung of the Dirac Electron, Am. J. Phys. 20, 479 (1952).

[12] A. O. Barut and N. Zanghi, Classical Model of the Dirac electron, Phys. Rev. Lett. 52, 2010 (1984).

[13] A. O. Barut and A. J. Bracken, Zitterbewegung and the internal geometry of the electron, Phys. Rev. D 23, 2454 (1981).

[14] D. Hestenes, Zitterwegung modeling, Found. Phys. 23, 365 (1993).

[15] W. Daywitt, The Electron-Vacuum Coupling Force in the Dirac Electron Theory and its Relation to the Zitterbewegung, Prog. Phys., 3, 25 (2013).

[16] R. Gauthier, Quantum-entangled superluminal double-helix photon produces a relativistic superluminal quantum-vortex zitterbewegung electron and positron, J. Phys.: Conf. Ser. 1251, 012016 (2019).

[17] O. Consa, Helical Solenoid Model of the Electron, Prog. Phys. 14, 80 (2018).

[18] F. Celani, A. O. Di Tommaso, and G. Vassallo, The Electron and Occam’s Razor, J. Condens. Matter Nucl. Sci. 25, 76 (2017).

[19] K. H. Knuth, The problem of motion: The statistical mechanics of Zitterbewegung, AIP Conf. Proc. 1641, 588 (2015).

[20] J. H. Wilson, The Dirac Electron Discrete Internal Structure and Its Rapidly Oscillating Charge Shell Phys. Essays 28, 1 (2015).

[21] B. Haisch, A. Rueda, and H. E. Puthoff, Inertia as a zero-point-field Lorentz force, Phys. Rev. A 49, 678 (1994).

[22] D. Hestenes, The Zitterbewegung Interpretation of Quantum Mechanics, Found. Phys. 20, 1213 (1990).

[23] I. Urdaneta, The zitterbewegung electron puzzle, Phys. Essays 36, 299 (2023).

[24] W. Brown, Spacetime Engineering & Harnessing Zero-Point Energy of the Quantum Vacuum. RSF Science Publishing, (2023).