Quantization in Curved Spaces

J. M. Domingos, Maria Helena Caldeira

João Luis Meloni Assirati, Dmitri M Gitman

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 $$\omega (\theta )$$ , $$\theta \in (1,0)$$ , 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 $$\omega (\theta ).$$ 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 $$\varTheta (x,\xi )$$ . The above mentioned minimal family is a part at $$\varTheta =1$$ 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.

P. V. Elyutin, V. D. Krivchenkov

Takahiro Homma, Tadashi Inamoto, Tadashi Miyazaki

Dirac-bracket quantization of the nonrelativistic particle whose motion is constrained on the hypersurface f(x)=const embedded in a general curved space is discussed. The noncanonical nature of commutation relations makes it difficult to obtain the coordinate representation. Then the system with the derivative-type constraint df(x)/dt=0 is alternatively quantized, treating carefully the operator-ordering problem. In this system it is shown that there exist no constraints on coordinates and momenta and that one can thus have a straightforward representation, which leads, in turn, to the representation and the Schr\"odinger equation in the former system.

Norman E. Hurt

A. G. Chirkov, Alexander Berdnikov

New quantization rules for classical systems are obtained using the Titchmarsh expansion. These rules generalize the conventional ones and are reduced to them when a transition to Cartesian coordinates exists. An equation generalizing the Schrödinger equation to arbitrary natural systems is found. The principle of minimal constraint (strong equivalence principle) makes it possible to extend this equation to any curved spaces.