Abstract
The purpose of this presentation is to construct a discrete quantum mechanics in which differential quotients appear instead of differentials, because these can be measured directly in experiments. This means to derive a theory which is based exclusively on the following objects (x i =x i (x), i= 1, 2, x ∈ R):
where real C ∞-functions f should be defined on a discrete set. We want this set to be a discretization of a manifold M with non-trivial topology (e. g., with non-vanishing first fundamental group), because in this case, the structure of the position and momentum operators given by Borel quantization and the corresponding dynamics derived from it is richer, i. e., it contains not only an additional constant but also a topological potential which is related to the topology of M.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
T. Kobayashi and T. Suzuki, Phys. Lett. B317 (1993), 359.
M. Gerstenhaber, Ann. Math. 79 (1964), 59.
V. K. Dobrev, H.-D. Doebner, and C. Mrugalla, J. Phys. A: Math. Gen. 29 (1996), 5909.
B. Angermann, H.-D. Doebner, and J. Tolar, in: Lecture Notes in Mathematics 1037, Springer, Berlin, 1983.
H.-D. Doebner and J. Tolar, Ann. Phys. Leipzig 47 (1990), 116.
R. Kemmoku and S. Saito, J. Phys. A: Math. Gen. 29 (1996), 4141.
H.-D. Doebner and J. D. Hennig, in: Symmetry in Science VIII, B. Gruber (ed.), Plenum, New York, 1995, 85.
H.-D. Doebner and G. A. Goldin, Phys. Lett. A162 (1992), 397.
H.-D. Doebner and G. A. Goldin, J. Phys. A: Math. Gen. 27 (1994), 1771.
V. K. Dobrev, H.-D. Doebner, and R. Twarock, A discrete, nonlinear q-Schrödinger equation via Borel quantization and q-deformation of the Witt algebra. ASI-TPA 19/96, INRNE-TH 12/96, submitted to Phys. Rev A.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Twarock, R. (1997). q-Quantum Mechanics on S1. In: Gruber, B., Ramek, M. (eds) Symmetries in Science IX. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5921-4_25
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5921-4_25
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7715-3
Online ISBN: 978-1-4615-5921-4
eBook Packages: Springer Book Archive