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.
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Communicated by Jerrold Marsden and Stephen Wiggins
This paper is dedicated to the memory of Juan C. Simo
Supported in part by NSF Grants DMS 92-22241 and 96-23083 (M.J.G.).
This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.
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Gotay, M.J., Grundling, H.B. & Tuynman, G.M. Obstruction results in quantization theory. J Nonlinear Sci 6, 469–498 (1996). https://doi.org/10.1007/BF02440163
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DOI: https://doi.org/10.1007/BF02440163