Five-Dimensional Hamilton–Jacobi Approach to Relativistic Quantum Mechanics

Anastasios Kyprianidis

Velimir Jurdjevic

Harald Rose

A novel theory is outlined for describing the dynamics of relativistic electrons and positrons.By introducing the Lorentz-invariant universal time as a fifth independent variable, the Hamilton-Jacobi formalism of classical mechanics is extended from three to four spatial dimensions.This approach allows one to incorporate gravitation and spin interactions in the extended fivedimensional Lagrangian in a covariant form.The universal time has the function of a hidden Bell parameter.By employing the method of variation with respect to the four coordinates of the particle and the components of the electromagnetic field, the path equation and the electromagnetic field produced by the charge and the spin of the moving particle are derived.In addition the covariant equations for the dynamics of the components of the spin tensor are obtained.These equations can be transformed to the familiar BMT equation in the case of homogeneous electromagnetic fields.The quantization of the five-dimensional Hamilton-Jacobi equation yields a five-dimensional spinor wave equation, which degenerates to the Dirac equation in the stationary case if we neglect gravitation.The quantity which corresponds to the probability density of standard quantum mechanics is the four-dimensional mass density which has a real physical meaning.By means of the Green method the wave equation is transformed into an integral equation enabling a covariant relativistic path integral formulation.Using this approach a very accurate approximation for the four-dimensional propagator is derived.The proposed formalism makes Dirac's hole theory obsolete and can readily be extended to many particles.

A. Pareek

Oliver Davis Johns

Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.

Luis Grave de Peralta, Arquímedes Ruiz-Columbié

Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained by abduction. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.