Covariant quantizations in plane and curved spaces

Abstract

Abstract We present covariant quantization rules for nonsingular finite-dimensional classical theories with flat and curved configuration spaces. In the beginning, we construct a family of covariant quantizations in flat spaces and Cartesian coordinates. This family is parametrized by a function $$\omega (\theta )$$ ω ( θ ) , $$\theta \in (1,0)$$ θ ∈ ( 1 , 0 ) , which describes an ambiguity of the quantization. We generalize this construction presenting covariant quantizations of theories with flat configuration spaces but already with arbitrary curvilinear coordinates. Then we construct a so-called minimal family of covariant quantizations for theories with curved configuration spaces. This family of quantizations is parametrized by the same function $$\omega (\theta ).$$ ω ( θ ) . Finally, we describe a more wide family of covariant quantizations in curved spaces. This family is already parametrized by two functions, the previous one $$\omega (\theta )$$ ω ( θ ) and by an additional function $$\varTheta (x,\xi )$$ Θ ( x , ξ ) . The above mentioned minimal family is a part at $$\varTheta =1$$ Θ = 1 of the wide family of quantizations. We study constructed quantizations in detail, proving their consistency and covariance. As a physical application, we consider a quantization of a non-relativistic particle moving in a curved space, discussing the problem of a quantum potential. Applying the covariant quantizations in flat spaces to an old problem of constructing quantum Hamiltonian in polar coordinates, we directly obtain a correct result.