Obstructions in Quantization Theory

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NSF Award Abstract - DMS-0072434Mathematical Sciences: Obstructions in Quantization TheoryAbstract0072434 GotayThe 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, over 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. Similar results hold under a wide variety of circumstances. But no-go theorems are not universal; the principal investigator and collaborators have recently constructed examples of phase spaces which admit consistent full quantizations. The goals of this project are to delineate the circumstances under which such obstructions will appear and to study the underlying mechanisms that 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 will be used to refine extant quantization procedures, or design new ones, to adapt to the obstruction and 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. Physically, this research will aid in clarifying the correspondence between classical and quantum mechanics in general, and in particular will enhance our understanding of quantizations of specific classical systems.Although the universe is quantum mechanical in nature, our perceptions of it are rooted in classical physics. Thus it is often desirable to construct a quantum formulation of a system from knowledge of a classical approximation to it. This process is called "quantization," and many different quantization schemes have been developed. Unfortunately, quantization is not a straightforward proposition, as evidenced by the discovery, over fifty years ago, of an "obstruction" to quantization: in principle it is impossible to consistently quantize a (nonrelativistic) particle. But it is now known that no-go theorems are not universal; there are classical systems which admit consistent quantizations. The goals of this project are to delineate the circumstances under which such obstructions appear and to study the mechanisms which produce them. This research will aid in clarifying the correspondence between classical and quantum mechanics.