aqora

Equations of motion, commutation relations and ambiguities in the Lagrangian formalism

Marc Henneaux A

Annals of PhysicsApr 1, 198296 citations

No Lagrangian? No quantization!

Sergio A. Hojman, Lawrence C. Shepley

This work starts with classical equations of motion and sets very general quantization conditions (commutation relations). It is proved that these conditions imply that the equations of motion are equivalent to the Euler–Lagrange equations of a Lagrangian L. The result is a generalization of work by Feynman, recently reported by Dyson [Am. J. Phys. 58, 209–211 (1990)]. The Lagrangian L need not be unique. Examples are given, including classical equations that do not come from a Lagrangian and therefore cannot be quantized consistently.This work starts with classical equations of motion and sets very general quantization conditions (commutation relations). It is proved that these conditions imply that the equations of motion are equivalent to the Euler–Lagrange equations of a Lagrangian L. The result is a generalization of work by Feynman, recently reported by Dyson [Am. J. Phys. 58, 209–211 (1990)]. The Lagrangian L need not be unique. Examples are given, including classical equations that do not come from a Lagrangian and therefore cannot be quantized consistently.

Journal of Mathematical PhysicsJan 1, 199177 citations

Quantization of classical lagrangian mechanics

E. I. Bogdanov

Theoretical and Mathematical PhysicsJun 1, 19923 citations

Ambiguities in Quantizing a Classical System

Ian H. Redmount, Wai-Mo Suen, K. Young

One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.

arXiv:gr-qc/9904042Apr 19, 19991 citation

On the quantization of classical systems

J.-P. Amiet

Il Nuovo CimentoJul 1, 19634 citations

The quantization of classical systems

P. B. Guest

Reports on Mathematical PhysicsSep 1, 19744 citations

Direct quantization of classical systems using the equation of motion and removing the degeneracy of Lagrangian formalism

Abstract

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Direct quantization of classical systems using the equation of motion and removing the degeneracy of Lagrangian formalism

Abstract

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