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Particles and Fields

Eur. Phys. J. C 21, 757-767 (2001)
DOI: 10.1007/s100520100711

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

N.M. Chepilko1 and A.V. Romanenko2

1  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: 19 February 2001 / Revised version: 10 May 2001 / Published online: 6 July 2001 -© Springer-Verlag / Società Italiana di Fisica 2001 )

Using the extended Schwinger quantization approach, quantum mechanics on a Riemannian manifold M with the given action of an intransitive group of isometries is developed. It was shown that quantum mechanics can be determined unequivocally only on submanifolds of M where G acts simply transitively (orbits of G action). The remaining part of the degrees of freedom can be described unequivocally after introducing some additional assumptions. Being logically unmotivated, these assumptions are similar to the canonical quantization postulates. Besides this ambiguity which is of a geometrical nature there is an undetermined gauge field of the order of $\hbar$ (or higher), vanishing in the classical limit $\hbar\to 0$.

© Società Italiana di Fisica, Springer-Verlag 2001