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American Journal of Physics

Relativistic invariance and Zitterbewegung

James A. Lock

Published Mar 1, 1984• 1 citation
Publisher

Abstract

We examine the question of what it is about the structure of relativistic quantum mechanics that causes the nonintuitive phenomenon of the Zitterbewegung of particle position to exist. Further, we examine various assumptions concerning the measurement process which are implicit in the observation of Zitterbewegung.
DOI
10.1119/1.13697

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•Oct 28, 2026•0 citations
In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in ℏ by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows us to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac–Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity.
The fact that the Dirac equation predicts Zitterbewegung (ZBW) for a free electron is well known from Schrodinger [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] from the very early days of Dirac's relativistic quantum theory. Schrodingerdescribed ZBW qualitatively as a persistent interference between positive and negative energy states, and the objective of this and previous papers by this author [Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] is to describe ways to measure the presence or absence ofZBW for a free (or nearly free) electron. If ZBW measurements show that it is not present for a free electron, then the Dirac Equation and Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] are inaccurate. If ZBW ispresent as Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] of the Dirac Equation asserts, then a description of a free electron emerges that is significantly different than the present very small, static point electronwith “intrinsic” properties of spin and magnetic moment accepted by most physicists [Eides et al., Phys. Reports 342, 63 (2001).] today. This paper develops a quantitative theory of the free electron directly from the Dirac Equation and Schrodinger's analyses[E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] without any other assumptions. The fact that ZBW exists for the electron in the hydrogen atom is confirmed by the Darwin term [J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley,Reading, MA, 1967)]. It seems unnatural to the author that ZBW would vanish when the electron is removed from the hydrogen atom, and becomes free. The unanswered question is whether ZBW can be confirmed or refuted by measurement for a free electron, and possible ways to measure its presenceor absence are described in this paper. The remainder of the paper describes the free electron, assuming ZBW exists. The Dirac Equation [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] produces a physical description of a free electron's ZBW as rapidly oscillating,quantized Center of Charge (CoC) point rotating about a fictitious Center of Mass (CoM) at a radius of 3/ 2 times the electron's Compton wavelength. The results in the remainder of this paper, as well as Schrodinger's analyses [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.)24, 418 (1930).], are only true if ZBW can be confirmed by measurement. In case measurements show no ZBW exists for a free electron, the Dirac Equation is not a correct description of a single free electron, and more serious problems arise. The static point like electron deduced fromscattering experiments [Eides et al., Phys. Rep. 342, 63 (2001)] that give estimates of the free electron's internal structure based on far field measurements can be misleading in estimating the free electron's internal structure. In Classical Electrodynamics, when one observesthe electric field from a point electron and from a sphere with the same charge spread uniformly over its surface from the far field, Gauss' Law shows that the electric fields are identical. When one tries to “observe” or “measure” the free electron's internal structurewith the high-energy photons required, the single particle free electron is completely destroyed by the measurement process. In fact, any attempt to observe or measure a free electron's structure ensures that the electron is no longer a totally free single particle, and one must be carefulto use only elastic observation methods [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] for this very difficult measurement. Observing a free Quark, much less its internal structure, has proven very difficult, if not impossible.
Book of Physics — On Difference Tractatus Principio, Volume II What if the fundamental entities of physics are not the foundations of reality, but the results of the relations through which reality becomes measurable? Book of Physics, the second volume of Tractatus Principio — On Difference by C.H.F. Warnke, extends the relational foundation developed in Volume I into the domain of physical reality. The work challenges a fundamental assumption underlying many physical descriptions: that mathematical abstractions such as points, particles, fields, and measurable quantities must correspond to independently existing constituents of the world. Its central distinction is between the mathematical representation of a physical phenomenon and the ontological interpretation of that representation. Beginning with a critical examination of classical mechanics, Cartesian geometry, Newtonian point-mass models, relativity, and quantum physics, the book investigates how relational descriptions can become interpreted as properties of self-contained physical objects. The work proposes a reversal of this perspective: physical objects are not the original bearers of relations. Rather, their determinacy emerges through relational structures, interactions, and the conditions under which they are observed and measured. Building upon the mathematical language of difference, represented by |, and its self-referential unfolding through Δ, the volume develops alternative interpretations of space, time, energy, mass, gravitation, and light. It examines dimensional measurement, relational invariance, the proposed significance of 39.19 within its own formal framework, and the organization of physical appearances through recursive structures and degrees of freedom. Gravitation is reconsidered through a relational shell model. Motion and time are investigated through the concept of complete relational states, or Frozen Frames, whose differences become readable as temporal succession. A central part of the work concerns measurement itself. Rather than treating measurement as the passive discovery of an already fully determined property, the book examines the relationship between observer, apparatus, registration, and interpretation. A measuring apparatus is understood as a materially realized question: its construction determines which differences can become detectable and how their results can be interpreted. From this perspective, particles and other physical entities are investigated as stable forms of relational determination rather than necessarily self-standing objects independent of every measurement context. The later chapters explore the uniformity of relational processes across different scales, including a reinterpretation of Bose–Einstein condensation, before returning to the distinction between scientific explanation and the conceptual images through which reality is understood. These proposals constitute an alternative theoretical interpretation within the framework of Tractatus Principio. Their physical validity is not established merely by the mathematical or conceptual arguments presented. The investigation ultimately reaches the observer himself. If every measurement is relational, the question arises of how the observer becomes capable of recognizing his own position within those relations. This question forms the transition to Volume III, the Book of Self-Knowledge. The central question of Volume II is not merely what physics measures, but what makes a physical reality measurable as something determinate in the first place.