Abstract
I exhibit a prequantization of the torus which is actually a “full” quantization in the sense that a certain minimal complete set of classical observables is irreducibly represented. Thus in this instance there is no Groenewold-Van Hove obstruction to quantization.
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References
A. A. Kirillov, Geometric quantization, in: “Dynamical Systems IV: Symplectic Geometry and Its Applications”, V.I. Arnol’d and S.P. Novikov (eds.), Encyclopaedia Math. Sci. IV, Springer, New York 137–172 (1990)
H. J. Groenewold, On the principles of elementary quantum mechanics, Physics 12: 405–460 (1946)
L. Van Hove, Sur le problème des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques de la mécanique classique, Acad. Roy. Belgique Bull. Cl. Sci. (5) 37: 610–620 (1951)
L. Van Hove, Sur certaines représentations unitaires d’un groupe infini de transformations, Mém. Acad. Roy. Belgique Cl. Sci. 26: 61–102 (1951)
P. Chernoff, Mathematical obstructions to quantization, Hadronic J. 4: 879–898 (1981)
G. B. Folland, “Harmonic Analysis in Phase Space”, Ann. Math. Studies 122, Princeton Univ. Press, Princeton (1989)
A. Joseph, Derivations of Lie brackets and canonical quantization, Comm. Math. Phys. 17: 210–232 (1970)
M. J. Gotay, H. Grundling and C. A. Hurst, A Groenewold-Van Hove theorem for S 2, dg-ga/9502008. To appear in Trans. Amer. Math. Soc. (1995)
P. Chernoff, Irreducible representations of infinite-dimensional transformation groups and Lie algebras, I (1994). To appear in J. Funct. Anal.
J. Zak, Dynamics of electrons in solids in external fields, Phys. Rev. 168: 686–695 (1968)
N. S. Manton, The Schwinger model and its axial anomaly, Ann. Phys. 159: 220–251 (1985)
M. Fernández, M. J. Gotay and A. Gray, Compact parallelizable four dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc. 103: 1209–1212 (1988)
M. A. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math. Phys. 122: 531–562 (1989)
M. Asorey, Classical and quantum anomalies in the quantum Hall effect, in: “Proc. on the Fall Workshop on Differential Geometry and Its Applications”, X. Gràcia, M.C. Muñoz and N. Román (eds.), CPET, Barcelona 75–80 (1994)
A. Avez, Symplectic group, quantum mechanics and Anosov’s systems, in: “Dynamical Systems and Microphysics”, A. Blaquière et al. (eds.), Springer, New York, 301–324 (1980)
V. Aldaya, M. Calixto and J. Guerrero, Algebraic quantization, good operators and fractional quantum numbers, hep-th/9507016 (1995).
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Gotay, M.J. (1995). On a Full Quantization of the Torus. In: Antoine, JP., Ali, S.T., Lisiecki, W., Mladenov, I.M., Odzijewicz, A. (eds) Quantization, Coherent States, and Complex Structures. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1060-8_6
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DOI: https://doi.org/10.1007/978-1-4899-1060-8_6
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