Canonical quantization of so-called non-Lagrangian systems

TL;DR

This paper introduces a method for the canonical quantization of non-Lagrangian systems by reformulating their equations into an equivalent action, addressing challenges like time-dependent constraints and ambiguities in Lagrangian formulation.

Contribution

It provides a systematic approach to quantize non-Lagrangian systems through explicit action construction and explores applications to quadratic theories, damped oscillators, and radiating charges.

Findings

Successfully reformulated non-Lagrangian equations into an equivalent action.

Addressed the ambiguity in associating Lagrangians with equations of motion.

Applied the scheme to quantize damped oscillators and radiating charges.

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.