Canonical quantization of so-called non-Lagrangian systems

Abstract

We present an approach to the canonical quantization of systems with equations of motion that are historically called non-Lagrangian equations. Our viewpoint of this problem is the following: despite the fact that a set of differential equations cannot be directly identified with a set of Euler-Lagrange equations, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. It turns out that in the general case the hamiltonization and canonical quantization of such an action are non-trivial problems, since the theory involves time-dependent constraints. We adopt the general approach of hamiltonization and canonical quantization for such theories (Gitman, Tyutin, 1990) to the case under consideration. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The proposed scheme is applied to the quantization of a general quadratic theory. In addition, we consider the quantization of a damped oscillator and of a radiating point-like charge.

13 pages

References in corpus (13)

• Wave Function Evolution of a Dissipative System
• Exact Diagonalization of Two Quantum Models for the Damped Harmonic Oscillator
• A Canonical Approach to the Quantization of the Damped Harmonic Oscillator
• Quantum damped oscillator I: dissipation and resonances
• One-Parameter Squeezed Gaussian States of Time-Dependent Harmonic Oscillator and Selection Rule for Vacuum States
• Bateman's dual system revisited: I. Quantization, geometric phase and relation with the ground-state energy of the linear harmonic oscillator
• Quantum damped oscillator II: Bateman's Hamiltonian vs. 2D Parabolic Potential Barrier
• Hamiltonization of theories with degenerate coordinates
• Quantum Dissipation in Open Harmonic Systems: Operator Solution
• Deformation quantization of linear dissipative systems
• To what extent do the Classical Equations of Motion Determine the Quantization Scheme?
• Quantum models related to fouled Hamiltonians of the harmonic oscillator
• Time-dependent driven anharmonic oscillator and the adiabaticity

Cited by in corpus (15)

• Position-dependent noncommutativity in quantum mechanics
• Quantization of the Damped Harmonic Oscillator Revisited
• Noncommutativity due to spin
• On the action principle for a system of differential equations
• On the lagrangian description of dissipative systems
• Functional integral for non-Lagrangian systems
• Analytical Rayleigh potential for the general relativistic Poynting-Robertson effect
• Poynting-Robertson effect as a dissipative system in general relativity
• Gauge invariance and classical dynamics of noncommutative particle theory
• Dynamical invariants and quantization of the one-dimensional time-dependent, damped, and driven harmonic oscillator
• Deformation Quantization of a Certain Type of Open Systems
• Invariant variational principle for Hamiltonian mechanics
• An approach to the quantization of black-hole quasi-normal modes
• N=2 supersymmetric radiation damping problem on a noncommutative plane
• The variational formulation of the theory of non-stationary propagation of a femtosecond laser radiation