Path Integral for Relativistic Equations of Motion

H. G�r

Gian Fabrizio De Angelis, Maurizio Serva

Sh. M. Nagiyev, E. I. Jafarov, Messoud Efendiyev

The SU(1,1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU(1,1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr–Sommerfeld quantization rule yields the exact expression for the energy spectrum.

Ted Jacobson A, Lawrence S. Schulman

Feynman's path integral for the one-dimensional Dirac particle uses paths for which Delta x approximately Delta t. For non-relativistic path integrals, typical paths satisfy ( Delta x) 2 approximately Delta t (as in Brownian motion). The authors demonstrate the consistency of these two stochastic schemes and show how the relativistic formalism contains within itself the scale of the transition regime, namely Delta x approximately h(cross)/mc, the Compton wavelength.

Tal Hendel

Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.