We derive a modified Hamilton-Jacobi theory for a classical spinning dipole and show that this classical theory can be put in a form almost identical with the Pauli spin theory. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single-valued, and that it satisfy the usual energy eigenvalue equation. In this manner we deduce the correct energy levels for a hydrogen atom in an external magnetic field. We derive the integral kernel for the Pauli equation from this classical model and discuss the properties of our spinor solutions under canonical transformation. We exhibit the charge-conjugate solution to the classical spin equation.
We derive a modified Hamilton-Jacobi theory for a classical spinning dipole and show that this classical theory can be put in a form almost identical with the Pauli spin theory. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single-valued, and that it satisfy the usual energy eigenvalue equation. In this manner we deduce the correct energy levels for a hydrogen atom in an external magnetic field. We derive the integral kernel for the Pauli equation from this classical model and discuss the properties of our spinor solutions under canonical transformation. We exhibit the charge-conjugate solution to the classical spin equation.
The quasi-classical spin theory is a modified Hamilton-Jacobi theory for a classical relativistic dipole whose space-time trajectory is coupled to its spin motion. The equations of this quasi-classical theory are almost identical with the second-order (squared) Dirac equation, and, in fact, form a new WKB approximation to the second-order Dirac equation. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single valued, and that it satisfy an eigenvalue equation. We find for a particle in a uniform magnetic field, and in a Coulomb field, that this quasi-classical theory predicts the same energy and angular momentum eigenvalues as the Dirac theory. The quasi-classical theory is invariant under Lorentz transformation, spin rotations, and charge conjugation. The spinor wave functions are form invariant with respect to arbitrary canonical transformations.
We develop a modified Hamiltonian-Jacobi theory of classical mechanics following the early work of Van Vleck. This modified Hamiltonian-Jacobi theory, or quasi-classical theory, permits us to exhibit in classical mechanics many features that in the past have been exclusively associated with quantum mechanics. We deal with classical wave functions, classical operators, classical "eigenvalue" equations, a classical "sum over paths" formulation of classical mechanics, and with classical creation and destruction operators. Following Van Vleck, one can derive the WKB approximate solutions to the Schr\"odinger equation from the solutions of the classical Hamilton-Jacobi equation. If we apply the methods of Keller to the nonrelativistic and relativistic Kepler problem, we derive eigenvalues from the requirement of single-valuedness imposed on the WKB solutions. It turns out that the energy eigenvalues are those given by the Schr\"odinger equation and the Klein-Gordon equation, respectively. In the particular case of the harmonic oscillator there exists a canonical transformation which transforms the quasi-classical equation into an exact equation of quantum mechanics. We conjecture that if the WKB approximation and the Schr\"odinger equation predict the same eigenvalues, then there always exists a canonical transformation which transforms the quasi-classical equation into the corresponding Schr\"odinger equation. Finally we derive the quasi-classical equations in momentum space.
The functional form of the wave function with respect to variables appropriate for the WKB approximation is chosen by taking into account the transformation properties of the spinor wave-function.This representation makes it possible, using only the wave equation, to pursue directly the time changes in the variables, i.e. the spin direction, velocity and density, etc., characterizing an electron.These changes correspond straightforwardly to a classical model of the non-relativistic Pauli electron with the proper magnetic moment, and the covariant form of this model, which is just "the Frenkel-Kramers model of the Dirac electron" introduced recently by the author.In this derivation, the existence of freedom of • rotation around the .spinaxis and certain definite angular velocity of this rotation are required.These results and the correspondence obtained may clarify the classical-mechanical significance of the spinor wave-function and also of the spin itself.An explanation is appended of the hyperbolic functions of the velocity metric, which help in the complicated calculations for the Dirac electron. § I. Introduction*) The main part of this paper appeared previously in Rep. Seisanken.No. 22 (1957), 55, in Japanese.
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.
One of the most satisfactory pictures for spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic extended-like electron, that relates spin to zitterbewegung (zbw). The BZ motion equations constituted the starting point for recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. This language results to be actually suited and fruitful for a hydrodynamical re-formulation of the BZ theory. Working out, in such a way, a ``probabilistic fluid'', we are allowed to re-interpret the original classical spinors as quantum wave-functions for the electron. Thus, we can pass to ``quantize" the BZ theory employing this time the tensorial language, more popular in first-quantization. ``Quantizing'' the BZ theory, however, does not lead to the Dirac equation, but rather to a non-linear, Dirac--like equation, which can be regarded as the actual ``quantum limit'' of the BZ classical theory. Moreover, an original variational approach to the the BZ probabilistic fluid shows that it is a typical ``Weyssenhoff fluid'', while the Hamilton-Jacobi equation (linking together mass, spin and zbw frequency) appears to be nothing but a special case of de Broglie's famous energy-frequency relation. Finally, after having discussed the remarkable correlation between the gauge transformation U(1) and a general rotation on the spin plan, we clarify and comment on the two-valuedness nature of the fermionic wave-function, and on the parity and charge conjugation transformations.
Abstract The anomalous Zeeman effect made it clear that charged particles like the electron possess a magnetic dipole moment. Classically, this could be understood if the charged particle possesses an eigenrotation, that is, spin. Within Nelson's stochastic mechanics, it was shown that the model of a rotating charged ball is able to reproduce the well‐known spin expectation values. This classically motivated model of intrinsic rotation described in terms of a stochastic process is revisited here within the formalism of stochastic optimal control theory. Quantum Hamilton equations (QHE) for spinning particles are derived, which reduce to their classical counterpart in the zero quantum noise limit. These equations enable the calculation of the common spin expectation values without the use of the wave function. They also offer information on the orientation dynamics of the magnetic moment of charged particles beyond the behavior of the spin averages.