Covariant quantizations in plane and curved spaces

TL;DR

This paper develops covariant quantization methods for classical theories in flat and curved spaces, introducing parametrized families of quantizations that ensure covariance and consistency, with applications to particles in curved spaces and polar coordinates.

Contribution

It introduces new parametrized covariant quantization schemes for flat and curved configuration spaces, generalizing previous approaches and addressing the quantum potential problem.

Findings

Constructed covariant quantizations in flat and curved spaces.

Proved the consistency and covariance of the quantization schemes.

Applied methods to quantize particles in curved spaces and polar coordinates.

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 $\omega(\theta)$, $\theta\in\left( 1,0\right)$, 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 $\omega \left( \theta \right)$. Finally, we describe a more wide family of covariant quantizations in curved spaces. This family is already parametrized by two functions, the previous one $\omega(\theta)$ and by an additional function $\Theta \left( x,\xi \right)$. The above mentioned minimal family is a part at $\Theta =1$ 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.