Eur. Phys. J. C 21, 369-381 (2001)
DOI: 10.1007/s100520100712

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach I

N.M. Chepilko¹ and A.V. Romanenko<sup>2

1</sup>  Physics Institute of the Ukrainian Academy of Sciences, Kyiv-03 028, Ukraine
²  Kyiv Taras Shevchenko University, Department of Physics, Kyiv-03 022, Ukraine

(Received: 27 June 2000 / Revised version: 10 May 2001 / Published online: 19 July 2001 -© Springer-Verlag / Società Italiana di Fisica 2001 )

Abstract
Schwinger's quantization scheme is extended in order to solve the problem of the formulation of quantum mechanics on a space with a group structure. The importance of Killing vectors in the quantization scheme is shown. Usage of these vectors makes the algebraic properties of the operators consistent with the geometrical structure of the manifold. The procedure of the definition of the quantum Lagrangian of a free particle and the norm of the velocity (momentum) operators is given. These constructions are invariant under a general coordinate transformation. The unified procedure for constructing the quantum theory on a space with a group structure is developed. Using this, quantum mechanics on a Riemannian manifold with a simply transitive group acting on it is investigated.

© Società Italiana di Fisica, Springer-Verlag 2001