Symplectic structure and quantization

Jerzy Kijowski

Maurice A. de Gosson

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.

Tom McClain

Abstract Though quantization is—from an ontological standpoint—a very strange operation, it seems unavoidable in the actual practice of physics. From a mathematical standpoint, canonical quantization was superseded decades ago by more elegant constructions, yet among practicing physicists it remains the de-facto champion among the many alternatives. Despite this fact, there is to this day no mathematically well-defined, coordinate independent construction that reproduces the results of canonical quantization for the most physically important phase space functions: position, momentum, angular momentum, and (quadratic) Hamiltonian functions. In this paper, I construct such a quantization map using standard structures from symplectic and Riemannian geometry in non-standard ways.

Kay-Michael Voit