We study the interaction of a nonrelativistic particle with a scalar field, with particular application to the theory of polarons. The approach is based on a general classical method for the integration of equations of motion. The Hamiltonian is transformed by successive canonical transformations, the first corresponding to describing the motion relative to special solutions of the equations of motion. This stage as applied to suitably ordered Heisenberg equations of motion is identical with intermediate coupling theory. The second transformation treats the coupled small oscillations of particle and field oscillators about the chosen special solution. This affords a natural extension of intermediate coupling theory for this problem. Differences between the classical and quantum theories arise in the ordering of operators; the differences play a crucial role in determining the effective cutoff in wave vector space.
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.
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.