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

A modification of the canonical quantization procedure for systems with time-dependent second-class constraints is discussed and applied to the quantization of a 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 Gitman (1990) as a generalization of Dirac canonical quantization in the non-stationary case, and another is first to find a canonical transformation, which makes it possible to describe the dynamics in the physical sector by means of some effective Hamiltonian. Quantum mechanics in both cases proves to be equivalent to the Klein-Gordon theory of a relativistic particle in a plane wave. The general conditions for unitarity of the dynamics in the physical sector are discussed.