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arXiv:physics/0312107

Classical elementary particles, spin, zitterbewegung and all that

M. Rivas

Published Dec 17, 2003• 3 citations
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Abstract

After a revision of the main features of the structure of the Dirac electron a plausible definition of elementary particle is stated. It is shown that this definition leads in the classical case to a picture which produces a very clear correspondence between the classical and quantum mechanical features of the electron. It is analyzed how the classical spin structure and zitterbewegung are related to the classical variables that define the kinematical state of the particle.

DOI
10.48550/arxiv.physics/0312107

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After a revision of the main features of the structure of the Dirac electron a plausible definition of elementary particle is stated. It is shown that this definition leads in the classical case to a picture which produces a very clear correspondence between the classical and quantum mechanical features of the electron. It is analyzed how the classical spin structure and zitterbewegung are related to the classical variables that define the kinematical state of the particle.
The concept of elementary particle rests on the idea that it is a physical system with no excited states, so that all possible kinematical states of the particle are just kinematical modifications of any one of them. The way of describing the particle attributes is equivalent to describe the collection of consecutive inertial observers who describe the particle in the same kinematical state. The kinematical state space of an elementary particle is a homogeneous space of the kinematical group. By considering the largest homogeneous spaces of both, Galilei and Poincaré groups, it is shown how the spin structure is related to the different degrees of freedom. The formalism is quantized by means of Feynman's path integral approach and special attention is paid to the classical model which satisfies Dirac's equation. Dirac's algebra is related to the classical observables, in particular to the orientation variables. Several spin effects are also analyzed.
This paper contains the Hamiltonian theory of a classical particle displaying all the features of the Dirac electron (spin, Zitterbewegung, etc.) except, of course, for the superposition principle. The particle is described by eight internal canonical variables, of which three are the spin angular momentum vector. The five other variables have no simple physical meaning, but are nevertheless necessary for a consistent theory. The equations of motion are not manifestly covariant, and the Lorentz invariance of the theory is proved by constructing the ten generators of the inhomogeneous Lorentz group.
A covariant, symplectic, classical dynamical system is presented whose quantization, by replacement of the Poisson brackets with commutators, gives precisely the Dirac electron theory. For the classical system the velocity and momentum are independent dynamical variables, as in Dirac electron; it undergoes a real Zitterbewegung so that the spin is the orbital angular momentum of the Zitterbewegung. Thus the longstanding problem of representing the configuration space of quantum spin by a classical model is solved. Three different forms of the dynamical equations are given.
The description of the Zitterbewegung at a classical level indicates possible predictions of the electron quantum properties even before quantization: this quivering motion is restricted to a plane, which leads us to only two possible orientations of the corresponding angular momentum, even in the absence of a quantized theory or external fields. Besides, the angular momentum associated with the Zitterbewegung turns out to be proportional to $\ensuremath{\hbar}$. Namely, using a standard constraint analysis we have proven this last result. Thus, assuming that this is an observable phenomenon, we recognize this oscillatory motion as a classical signature of the spin of the electron. We also propose here an interpretation of the Zitterbewegung based on geometrical grounds: it can be seen as the physical degrees of freedom of position variables constrained to a sphere, which enforces the hypothesis of assuming an electron internal structure.
A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.
With respect to the principal operators including the sixteen 4-4 matrices of the Dirac electron, two covariant forms of the extended Heisenberg picture can be presented with the “two covariant Hamiltonians HI and HII” of the first and second order, respectively. One form with HII involving the proper electromagnetic moment corresponds perfectly to the Frenkel-Kramers model which was recently treated by the author; the other form with HI containing the Zitterbewegung corresponds partly to the so-called Weyssenhoff-like model. That an amalgamation of the former form and the Zitterbewegung brings immediately the latter is explicitly shown by virtue of a decomposition of the operator of the Dirac electron into two corresponding parts. This decomposition is derived from the author's extension of the Feynman-Bunge-Corben formula for the position operator of the electron and, in case without external fields, it corresponds to the covariant form of the so-called ‘gerade-ungerade’ decomposition. In the case of the free state mixed with positive and negative energy, the Zitterbewegung of operators derived from HI can properly be realized by the expectation values of the operators. In this process, the author's complex hyperbolic metric of velocity which represents a new negative energy state plays an essential role. These results give an important clue to solve the puzzling problem of the construction of a consistent classical model for the Dirac electron, and also may be useful for investigation of ‘internal structure’ of elementary particles as well as their spin.