Mathematical Sciences: Studies in Quantization Theory

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9623083 Gotay The process of constructing a quantum formulation of a system from a knowledge of a classical approximation to it is called "quantization," and over the years many different quantization schemes have been developed. Unfortunately, quantization is not a straightforward proposition, as evidenced by the discovery, exactly fifty years ago, by Groenewold and Van Hove of an ``obstruction'' to quantization. Their ``no-go theorem'' asserts that in principle it is impossible to consistently quantize every classical observable on a Euclidean phase space, regardless of which quantization procedure is employed. Just this past year, the principal investigator proved that a similar result holds for the sphere. But no-go theorems are not universally valid; the principal investigator has recently shown that the torus admits a consistent full quantization. The goals of this proposal are to delineate the circumstances under which such obstructions will appear, and to study the underlying mechanisms which produce them. Another problem, when an obstruction does exist, is to determine the maximal subalgebras of observables that can be consistently quantized. Solutions to these problems might be used to refine extant quantization procedures, or design new ones, which are adapted to the obstruction in that they will be able to quantize these maximal subalgebras. From a mathematical standpoint, this research will lead to structural insights into the Poisson algebras of classical systems and their representations. %%% Although the universe is quantum mechanical in nature, our perceptions of it are rooted in classical physics. Thus one is often confronted with the problem of constructing a quantum formulation of a system from a knowledge of a classical approximation to it. This process is called "quantization,'' and over the years many different quantization schemes have been developed. Unfortunately, quantization is not a straightforward proposition, as ev idenced by the discovery, exactly fifty years ago, of an "obstruction'' to quantization. This "no-go theorem'' asserts that in principle it is impossible to consistently quantize a (nonrelativistic) particle, regardless of which quantization procedure is employed. Just this past year, the principal investigator proved a similar result for a spinning particle. But no-go theorems are not universally valid; the principal investigator recently found a classical system which admits a consistent quantization. The goals of this proposal are to delineate the circumstances under which such obstructions will appear, and to study the mechanisms which produce them. A solution to this problem might be used to refine extant quantization procedures, or design new ones, which are "optimal" in that they will be able to quantize systems to the extent permitted by the obstruction. ***