Ziemowit Domański
Quantization in arbitrary coordinate system and transformations of coordinates in quantum mechanics
The quantization is the process of constructing a quantum system from a classical one. The basic ingredient in the usual quantization procedure is a way to construct operators on a certain Hilbert space from functions defined on the phase space of the system. A standard way to do this is to write a function in Cartesian coordinates and replace position and momentum variables with appropriate operators. Since the position and momentum operators do not commute it is also necessary to appropriately order them. The problem arises when one would like to repeat this procedure in different coordinates, because the received operators corresponding to the same phase space function will not be unitarily equivalent. The usual way of resolving this inconsistency is to quantize only in Cartesian coordinates or in a coordinate independent way. However, it should be possible to quantize in any coordinate system in a consistent way.
We resolve this problem by adding to the phase space function $\hbar$-dependent correction terms and then constructing an operator from such modified function in a usual way. We find a systematic way of constructing such corrections by employing the phase space formalism of quantum mechanics. This formalism also allows for introduction of transformation of coordinates in a natural way. In particular we define quantum canonical transformations of coordinates.
