Abstract
Because the various topological effects (θ-structures) such as the Aharonov-Bohm effect, the anyon system, and non-Abelian statistics are pure quantum effects and should emerge naturally in a quantization procedure, we systematically discuss a general quantization scheme in a geometric formalism where wave-functions are smooth sections of some vector bundles over configuration space. Following ideas of L. Schulman, M. Laidlaw, J. S. Dowker, and others, we choose those vector bundles to be the associated bundles of the universal covering space of configuration space. Theθ-structures are shown to result from the fact that various vector bundles can be built over the universal covering space, which are labeled by the nonequivalent irreducible unitary representations of the fundamental group of configuration space. A flat connection description ofθ-structures is also possible, owing to Milnor's theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aharonov, Y., and Bohm, D. (1959).Physical Review,115, 485.
Bao, D., and Zhu, Z. Y. (1991).Communications in Theoretical Physics,16, 201.
Chern, S. S. (1967).Complex Manifolds without Potential Theory, Van Nostrand.
Date, G., Govindarajan, T., Sankavan, P., and Shankar, R. (1990).Communications in Mathematical Physics,132, 293.
Dieudonne, J. (1972).Treatise on Analysis, Vol. III, Academic Press, New York.
Dirac, P. A. M. (1958).The Principles of Quantum Mechanics, 4th ed., Oxford University Press, Oxford.
Dowker, J. S. (1972).Journal of Physics A,5, 936.
Eguchi, T., Gilkey, P., and Hanson, A. J. (1980).Physics Reports,66, 213.
Felder, G. (1989).Nuclear Physics B,317, 215.
Frohlich, J., and Marchetti, P. A. (1991).Nuclear Physics B,356, 533.
Gottlieb, D. H. (1978). InGeometric Applications of Homotopy Theory, Springer, Berlin.
Isham, C. J. (1984). InRelativity, Groups and Topology II (B. S. DeWitt and R. Stora, eds.), New York.
Laidlaw, M., and DeWitt, C. M. (1971).Physical Review D,3, 1375.
Milnor, J. (1975).Commentarii Mathematici Helvetici,32, 215.
Schulman, L. (1968).Physical Review,176, 1558.
Sniatycki, J. (1980).Geometric Quantization and Quantum Mechanics, Springer-Verlag, New York.
Sorkin, R. D. (1986). InTopological Properties and Global Structure of Space-Time (P. G. Bergmann and V. de Sabbata, eds.), Plenum Press, New York.
Sudarshan, E., Imbo, T., and Govindarajan, T. (1988).Physics Letters B,213, 471.
Wallach, N. R. (1973).Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, New York.
Wen, X. G., Wilczeck, F., and Zee, A. (1989).Physical Review B,39, 11413.
Wilczeck, F. (1982).Physical Review Letters,49, 957.
Wilczeck, F. (1990).Fractional Statistics and Anyonic Superconductivity, World Scientific, Singapore.
Witten, E. (1979).Physics Letters,86B, 283.
Witten, E. (1983).Nuclear Physics B,223, 422.
Woodhouse, N. (1980).Geometric Quantization, Oxford University Press, Oxford.
Yu, Y., Zhu, Z. Y., and Lee, H. C. (1991).Communications in Theoretical Physics,15, 339.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bao, D., Zhu, Zy. θ-structures in quantum theory in view of geometric quantization. Int J Theor Phys 32, 51–62 (1993). https://doi.org/10.1007/BF00674396
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00674396