caa a22 4500 477037070 CHVBK 20180405111308.0 cr unu---uuuuu 170330e19960901xx s 000 0 eng 10.1007/BF02440163 doi (NATIONALLICENCE)springer-10.1007/BF02440163 Obstruction results in quantization theory [Elektronische Daten] [M. Gotay, H. Grundling, G. Tuynman] Summary: Quantization is not a straightforward proposition, as demonstrated by Groenewold's and Van Hove's discovery, exactly fifty years ago, of an "obstruction” to quantization. Their "no-go theorems” assert that it is in principle impossible to consistently quantize every classical observable on the phase spaceR 2n in a physically meaningful way. A similar obstruction was recently found forS 2, buttressing the common belief that no-go theoremss should hold in some generality. Surprisingly, this is not so—it has also just been proven that there is no obstruction to quantizing a torus. In this paper we take first steps towards delineating the circumstances under which such obstructions will appear and understanding the mechanisms which produce them. Our objectives are to conjecture a generalized Groenewold-Van Hove theorem and to determine the maximal subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of classical systems and their representations. To these ends we include an exposition of both prequantization (in an extended sense) and quantization theory—formulated in terms of "basic sets of observables”—and review in detail the known results forR 2n,S 2, andT 2. Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert space-based quantization techniques. Springer-Verlag New York Inc., 1996 Gotay M. Department of Mathematics, University of Hawai'i, 2565 The Mall, 96822, Honolulu, HI, USA aut Grundling H. Department of Pure Mathematics, University of New South Wales, P.O. Box 1, NSW 2033, Kensington, Australia aut Tuynman G. URA 751 au CNRS & UFR de Mathématiques, Université de Lille I, F-59655, Villeneuve d'Ascq Cedex, France aut Journal of Nonlinear Science Springer-Verlag 6/5(1996-09-01), 469-498 0938-8974 6:5<469 1996 6 332 https://doi.org/10.1007/BF02440163 text/html Onlinezugriff via DOI 1 review-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02440163 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gotay M. Department of Mathematics, University of Hawai'i, 2565 The Mall, 96822, Honolulu, HI, USA aut NATIONALLICENCE 700 1- Grundling H. Department of Pure Mathematics, University of New South Wales, P.O. Box 1, NSW 2033, Kensington, Australia aut NATIONALLICENCE 700 1- Tuynman G. URA 751 au CNRS & UFR de Mathématiques, Université de Lille I, F-59655, Villeneuve d'Ascq Cedex, France aut NATIONALLICENCE 773 0- Journal of Nonlinear Science Springer-Verlag 6/5(1996-09-01), 469-498 0938-8974 6:5<469 1996 6 332 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer