Abstract
We introduce a smooth mapping of some discrete space-time symmetries into quasi-continuous ones. Such transformations are related with q-deformations of the dilations of the Euclidean space and with the noncommutative space. We work out two examples of Hamiltonian invariance under such symmetries. The Schrödinger equation for a free particle is investigated in such a noncommutative plane and a connection with anyonic statistics is found.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. G. Drinfeld,Proceedings Intern. Congress of Mathematicians, Berkeley, California, A. M. Gleason, ed. (AMS, Providence, Rhode Island, 1986), p. 798. M. Jimbo,Lett. Math. Phys. 10, 63 (1985).
E. K. Sklyanin,Funct. Anal. Appl. 16, 262 (1982); P. P. Kulish and N. Reshetikhin,J. Sov. Math. 23, 2435 (1983).
L. C. Biedenharn,J. Phys. A: Math. Gen. 22, L873 (1990). A. J. Macfarlane,J. Phys. A: Math. Gen. 22, 4581 (1990). Y. J. Ng,J. Phys. A: Math. Gen. 23, 1023–1027 (1990).
Z. Chang,Phys. Rep. 262, 137–225 (1995). C. de Concini and C. Procesi, inD-modules, Representation Theory and Quantum Groups, G. Zampieri and A. D'Agnolo, eds. (Springer, New York, 1993).
L. H. Kauffman,Knots and Physics, (World Scientific, Singapore, 1991).
J. Wess and B. Zumino, “Covariant differential calculus on the quantum hyperplane,” CERN-TH-5697/90, LAPP-TH-284/90. S. L. Woronowicz,Commun. Math. Phys. 111, 613 (1987);122, 125 (1989).
S. Chaturvedi, R. Jagannathan, R. Sridhar, and V. Srinivasan,J. Phys. A: Math. Gen. 26, L105 (1993).
H. Exton,q-Hypergeometric Functions and Applications (Horwood, Chichester, 1983).
A. Connes,Geometrie Non-Commutative (inter Editions, 1990). Yu. I. Manin,Commun. Math. Phys. 123, 163 (1989).
A. Lerda and S. Sciuto,Nucl. Phys. B 401, 613 (1993).
F. Bagarello,J. Phys. A: Math. Gen. 29, 565 (1996).
Author information
Authors and Affiliations
Additional information
This article is dedicated to the memory of Professor Larry C. Biedenharn. Larry was a friend for over 30 years. We met first in 1962 when I (W.G.) was a young assistant Professor at the University of Maryland. In his book on “Coulomb Excitation” (with P. Brussard) he devoted a whole chapter to the “Rotation Vibration Model” which had just been put forward by A. Faessler and Walter Greiner. He invited me down to Duke University where I was offered a tenured Associate Professorship and a couple of years later a full Professorship. I turned down this offer, because meanwhile I had become a full professor and director of the Institut für Theoretische Physik at the Johann-Wolfgang-Goethe University in Frankfurt am Main. However, a life-long friendship and collaboration (e.g., on Eigenchannel Theory of Nuclear Reactions, with M. Danos) began. Larry was frequently visiting professor in Frankfurt, as I was at Duke. In 1976 he received the A.v. Humboldt award and subsequently spent an extended period at our Institute. Some of my former students (H. J. Weber, now Professor at the University of Virginia, M. G. Huber, now rector of the University of Bonn, and others) came to Duke University. Another former student of mine, Berndt Müller, became Larry Biedenharn's successor at Duke University. Also some of his students, e.g., Louis Wright (now Professor at Ohio State University at Athens/Ohio), and former colleague R. Cusson have spent extended periods of time in Frankfurt. Larry was an outstanding mathematical physicist and a friend. I treasure his memory.
Rights and permissions
About this article
Cite this article
Ludu, A., Greiner, W. Quasi-continuous symmetries of non-lie type. Found Phys 27, 1123–1138 (1997). https://doi.org/10.1007/BF02551437
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02551437