The trajectory of a charged-massive point-particle in presence of an external magnetic field, at the classical level, is here obtained by using spinorial equations of motion. Within this representation the solution is given by means of two 2-component Weyl-spinors which contain global phase-factors coinciding with the quantum representation of a 1/2 spin particle in its rest-frame. Each individual spinor may be then associated to the up- and -down state respectively. The formalism then appears to reveal classical properties of the spin as well as quantum ones.
The trajectory of a charged-massive point-particle in presence of an external magnetic field, at the classical level, is here obtained by using spinorial equations of motion. Within this representation the solution is given by means of two 2-component Weyl-spinors which contain global phase-factors coinciding with the quantum representation of a 1/2 spin particle in its rest-frame. Each individual spinor may be then associated to the up- and -down state respectively. The formalism then appears to reveal classical properties of the spin as well as quantum ones.
A set of two first-order equations for classical spinor fields are used to formulate the relativistic quantum mechanics (RQM) of a single massive spin-1/2 particle. These equations are generalized Weyl equations and they are not equivalent to the Dirac equation. They lead to a conserved charge that is not positive definite, thus allowing for a probabilistic interpretation of the wave function in RQM. The charge is also conserved when the fields interact with a scalar field through the charge-current densities.
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass M of the particle differs from the mass m which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of M is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass M and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.
Although intrinsic spin is usually viewed as a purely quantum property with no classical analog, we present evidence here that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise naturally in the Clifford's geometric algebra of physical space and not only provide powerful tools for solving problems in classical electrodynamics, but also reproduce a number of quantum results. In particular, many properites of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, and its magnetic moment to Zitterbewegung and de Broglie waves. The relationship is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and creation operators. The approach resolves Pauli's argument against treating time as an operator by recognizing phase factors as projected rotation operators.
Introduction The claim that it is impossible to construct a realistic physical model of quantum matter is often presented as definitively and finally proved in the case of systems with spin ½, or internal angular momentum of magnitude ħ/2, which exhibit a characteristic ‘nonclassical two-valuedness’. Nevertheless, the extension of the causal interpretation to embrace nonrelativistic spin ½ systems was successfully carried through in the 1950s (Bohm, Schiller & Tiomno, 1955; Bohm & Schiller, 1955; Takabayasi, 1955; Takabayasi & Vigier, 1957) and developed more recently (Dewdney, Holland & Kyprianidis, 1986, 1987a, b; Dewdney, Holland, Kyprianidis & Vigier, 1988). The idea is to assume that the spinor wave represents a new type of physical field propagating in spacetime that exerts an influence on a corpuscle moving within it. The additional information conveyed by the spinor beyond that contained in the spinless Schrödinger wave implies that the state of the particle is not exhausted by specifying its position and momentum, but that it is endowed with further properties such as an internal angular momentum. The equation governing the dynamics of the spinor wavefunction is the Pauli equation: where the Hamiltonian operator is given by Here e, m and μ are respectively the charge, mass and magnetic moment of the particle, A 0 and A are the electromagnetic potentials, B = ∇ × A is the magnetic field, V is any other external scalar potential, and σ is a vector whose components are the 2 × 2 Pauli matrices. The Hamiltonian (9.1.2) istherefore a 2 × 2 matrix operator and the wavefunction ψ( x , t ) has two complex components.
This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.
The incorporation of spin within classical mechanics suggests the following revision: the Euler equations, or the concept of a time-derivative operator relative to different reference frames, should be our new theoretical paradigm. From here, the existence of formal developments valid both in classical and in quantum mechanics are explored through the analysis of topics such as: the dynamics of a particle with spin acted upon by a torque, the Thomas precession, the equations of evolution for spin, the concept of quantization, the anomalous Zeeman effect and the energy of the spin-orbit interaction. The obtained results coincide for already-known concepts as well as those found within quantum theories. For example, when considering the evolution of particles with spin within magnetic fields, classical equations of motion present equivalent results as those derived using equations of evolution for expected quantum values. This means that there are similarities supporting and justifying the point of view adopted in this book. The author hopes that the reader may benefit from and enjoy reading this text. For all whom read and enjoy reading this publication, he pass on the words Virgil used when talking about Lucretius, the author of De Rerum Natura, “Fortunate is he who is able to know the causes of things.”