An approximate Lorentz-invariant Hamilton-Jacobi equation

L. Motz

Jacques Chauveheid, Francis X. Vacanti

M.-J. Kim, C.-K. Choi

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.

Ziping Li

In order to study the symmetry of a singular system, based on phase-space generation of Green function of a singular system with finite degrees of freedom, an integral invariant of quantum Poincare-Cartan (excluding ground state sign |0) is established. The relationship between the Poincare-Cartan integral invariant and the Hamilton-Jacobi equation is discussed at the quantum level. It is pointed that the Hamilton-Jacobi equation can be deduced from integral invariant of the quantum Poincare-Cartan.

J.D. Bulnes, M.A.I. Travassos, Davron Aslonqulovich Juraev +1 more

In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: ``"Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum mechanics for a function ψ through an exact procedure. This element is the linear independence of the functions ψ and ψ* (their complex conjugate). Their independence is demonstrated for physical systems where the acting physical potential does not explicitly depend on time. Proceeding in reverse, from the Schrödinger equation, we obtain the Hamilton-Jacobi equation, exactly, without additional terms.