Some ideas about quantization,☆☆

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Abstract

Recent work on generalizations of Weyl quantization and of the Moyal formulation of quantum mechanics is developed. In conventional quantization schemes the Heisenberg algebra plays a fundamental role. The main idea of the generalization is to replace the Heisenberg algebra by any Lie algebra that has a canonical realization on phase space. Special cases are analyzed in detail, including the algebras E(2), SL(2, R), and so(3). The concept of ∗-polarization is introduced and related to the method of induced representations. It turns out that the ,,Exp transform” intertwines between the phase space realization and the quantum mechanical Hilbert space realization. The generalized Weyl correspondence now appears as a ,,pullback” to the linear operators of the Exp transform. The relationship between ∗-products and representation theory is investigated, as well as the question of equivalence between different ∗-products. Both problems are intimately related to the theory of characters.

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Based in part on four lectures given at Collége de France, June 1977.

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Work supported in part by the National Science Foundation.

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