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 space R^{2n} in a physically meaningful way. A similar obstruction was recently found for S^2, buttressing the common belief that no-go theorems 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 for R^{2n}, S^2, and T^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.
34 pps. 2/12/96 (Revised 4/20/96) Published in: J. Nonlinear Sci. 6, 469--498 (1996).