https://doi.org/10.1140/epjc/s10052-017-5041-0
Regular Article - Theoretical Physics
Covariant quantizations in plane and curved spaces
1
Institute of Physics, University of São Paulo, São Paulo, Brazil
2
Department of Physics, Tomsk State University, Tomsk, Russia
3
P.N.Lebedev Physical Institute, Moscow, Russia
4
Institute of Physics, University of São Paulo, São Paulo, Brazil
* e-mail: gitman@if.usp.br
Received:
10
June
2017
Accepted:
29
June
2017
Published online:
17
July
2017
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 
, 
, 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 
Finally, we describe a more wide family of covariant quantizations in curved spaces. This family is already parametrized by two functions, the previous one and by an additional function 
. The above mentioned minimal family is a part at 
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.
© The Author(s), 2017
