Canonical quantization for effective theories with perturbations altering degrees of freedom: a covariant phase space approach

The standard approach to canonical quantization encounters difficulties in dealing with perturbations that alter the kinetic structure of unperturbed theories. We show that the covariant phase space formalism provides a natural and technically efficient way to circumvent this obstruction. We illustrate the method with an exactly solvable model: a two-dimensional non-relativistic charged particle moving in a magnetic field and a harmonic confining potential, with its kinetic energy viewed as a perturbation. We quantize this model with covariant phase space formalism by constructing the solution perturbatively. We then calculate the energy spectrum and the unequal-time commutators of this model, and obtain the results that agree with the expansion of the exact theory. The procedure developed here is intended to serve as a systematic framework for the canonical quantization of more complex effective theories with higher-derivative or velocity-dependent perturbations.