From the Old Quantum Theory to symplectic quantization

D. J. Simms, L. Castel

Steven Kenneth Kauffmann

A carefully motivated symmetric variant of the Poisson bracket in ordinary (not Grassmann) phase space variables is shown to satisfy identities which are in algebraic correspondence with the anticommutation postulates for quantized Fermion systems. "Symplecticity" in terms of this symmetric Poisson bracket implies generalized Hamilton's equations that can only be of Schroedinger type (e.g., the Dirac equation but not the Klein-Gordon or Maxwell equations). This restriction also excludes the old "four-Fermion" theory of beta decay.

Jerzy Kijowski

David C. Latimer

In Phys. Rev. A 70, 032104 (2004), M. Montesinos and G. F. Torres del Castillo consider various symplectic structures on the classical phase-space of the two-dimensional isotropic harmonic oscillator. Using Dirac's quantization condition, the authors investigate how these alternative symplectic forms affect this system's quantization. They claim that these symplectic structures result in mutually inequivalent quantum theories. In fact, we show here that there exists a unitary map between the two representation spaces so that the various quantizations are equivalent.

Paul Lescot, Jean-Claude Zambrini

Abstract The study of Riemannian stochastic differential geometry is almost as old as the theory of stochastic differential equations themselves. But almost nothing is known about stochastic symplectic geometry. Using only the geometrical content of the heat equation (in a sense inspired by E. Cartan) and its relations with quantum dynamics, we present here the first steps toward such a stochastic version of symplectic geometry.

M. D. de Oliveira, M.C.B. Fernandes, F. C. Khanna +2 more