Quantization of systems with time-dependent constraints. Example of relativistic particle in plane wave

TL;DR

This paper develops a modified canonical quantization method for systems with time-dependent constraints, applying it to a relativistic particle in a plane wave, and demonstrates the equivalence of different approaches to Klein-Gordon theory.

Contribution

It introduces a new quantization procedure for time-dependent second-class constraints and compares two approaches, showing their equivalence in describing a relativistic particle in a plane wave.

Findings

Both quantization methods yield equivalent quantum mechanics.

The constructed quantum theory aligns with Klein-Gordon theory.

Conditions for unitarity of the physical sector are discussed.

Abstract

A modification of the canonical quantization procedure for systems with time-dependent second-class constraints is discussed and applied to the quantization of the relativistic particle in a plane wave. The time dependence of constraints appears in the problem in two ways. The Lagrangian depends on time explicitly by origin, and a special time-dependent gauge is used. Two possible approaches to the quantization are demonstrated in this case. One is to solve directly a system of operator equations, proposed by Tyutin and one of the authors (Gitman) as a generalization of Dirac canonical quantization in nonstationary case, and another to find first a canonical transformation, which makes it possible to discribe the dynamics in the physical sactor by means of some effective Hamiltonian. Quantum mechanics constructed in both cases proves to be equivalent to Klein-Gordon theory of the relativistic particle in a plane wave. The general conditions of unitarity of the dynamics in the physical sector are discussed.