We propose a Hamiltonian for a nonrelativistic spin 1/2 \QTR{it}{free} particle (e.g. an electron) and find that it contains information of its internal degrees of freedom in the rest coordinate system. We comment on the dynamical symmetry associated with the electron \QTR{it}{Zitterbewegung}.
Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field U, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when U satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are obtained. As a result, the center of mass follows along a complicated trajectory round a direction of motion of the center of inertia. This motion can be interpreted as Zitterbewegung. It is shown that equation of moments is a consequence of the equation of motion. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electrical charge, sign of which corresponds to a sign of spin polarization.
The formal mathematical structure of the truly Hamiltonian formulations of nonrelativistic and relativistic quantum mechanics is critically examined for both spin 0 and spin 12 particles with nonzero mass. It is shown that the relativistic quantum mechanics can, in principle, at least be fitted into a truly covariant Hamiltonian procedure where the Hamiltonian is a Lorentz-invariant world scalar (and is the negative of 12 the rest energy of the particle, regardless of its spin). In particular, a truly covariant Hamiltonian theory of the electron is presented. It will be seen that the Lorentz-invariant Dirac theory of the electron does not fall within the framework of a truly covariant Hamiltonian procedure. Further, the dynamical variables associated with the translational degree of freedom of the particle (such as the position, the canonical momentum, and the velocity four-vectors of the relativistic quantum mechanics) correspond precisely to the same variables of the truly covariant formulation of relativistic classical mechanics. The mathematical structure of relativistic quantum mechanics is seen to be rather similar to that of nonrelativistic quantum mechanics for both spin 0 and spin 12 particles with nonzero mass.
Schr\"odinger's work on the Zitterbewegung of the free electron is reexamined. His proposed "microscopic momentum" vector for the Zitterbewegung is rejected in favor of a "relative momentum" vector, with the value $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}=mc\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}$ in the rest frame of the center of mass. His oscillatory "microscopic coordinate" vector is retained. In the rest frame, it takes the form $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}=\ensuremath{-}i(\frac{\ensuremath{\hbar}}{2mc})\ensuremath{\beta}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}$, and the Zitterbewegung is described in this frame in terms of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$, and the Hamiltonian $m{c}^{2}\ensuremath{\beta}$, as a finite three-dimensional harmonic oscillator with a compact phase space. The Lie algebra generated by $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$ is that of SO(5), and in particular $[{Q}_{i},{P}_{j}]=\ensuremath{-}i\ensuremath{\hbar}{\ensuremath{\delta}}_{\mathrm{ij}}\ensuremath{\beta}$. It is argued that the simplest possible finite, three-dimensional, isotropic, quantum-mechanical system requires such an SO(5) structure, incorporates a fundamental length, and has harmonic-oscillator dynamics. Dirac's equation is derived as the wave equation appropriate to the description of such a finite quantum system in an arbitrary moving frame of reference, using a dynamical group SO(3,2) which can be extended to SO(4,2). Spin appears here as the orbital angular momentum associated with the internal system, and rest-mass energy appears as the internal energy in the rest frame. Possible generalizations of these ideas are indicated, in particular those involving higher-dimensional representations of SO(5).
In this paper, we examine quantum systems with relativistic dynamics. We show that for a successful description of these systems, the application of Galilei invariant nonrelativistic Hamiltonian is necessary. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. Finally, we introduce and investigate the Schrödinger equation with relativistic dynamics for two-particle systems with harmonic oscillator and Coulomb potentials.
A single-particle Hamiltonian independent of the particle's coordinate ensures the conserves momentum, i.e., is free. Lorentz-covariance of that Hamiltonian's energy-momentum specifies it up to the particle's rest energy; the it describes has speed below c and constant velocity parallel to its conserved momentum. Dirac took his free-particle Hamiltonian to have the same squared value as that relativistic one, but unwittingly blocked Lorentz-covariance of his Hamiltonian's energy-momentum by requiring it to be inhomogeneously linear in momentum. The Dirac free particle badly flouts relativity and even physical cogency; its velocity direction is extremely nonconstant, while its speed is fixed to c times the square root of three even when it interacts electromagnetically. Both its rest energy and total energy can be negative, and its velocity components and rest energy are artificially correlated by being mutually anticommuting; its alleged spin is an artifact of the anticommutation of its velocity components. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is apparently physically sensible for speed far below c. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero velocity, and is obtained in closed form if there is no applied magnetic field; a successive approximation scheme must otherwise be used.
Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field U, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when U satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are found. As a result, the center of mass follows along helical and more complicated trajectories round a direction of motion of the center of inertia. This motion can be interpreted as trembling movement (Zitterbewegung). On this basis a conclusion is done that Zitterbewegung has purely classical origin, arising even in a nonrelativistic case if internal degrees of freedom are taken into account. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electric charge, sign of which corresponds to a sign of spin polarization.