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. Romanenko21 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