COVARIANT QUANTIZATION : SPECTRAL ANALYSIS VERSUS DEFORMATION THEORY
Résumé
The theory of quantization was mathematically conceptualized in the late 1920's by Heisenberg and Weyl and nowadays it constitutes a branch of Mathematics on its own right with ramifications going from algebraic topology to pseudo-differential calculus, from number theory to non-commutative geometry, from representation theory to symplectic geometry. Since the very beginning the interaction between quantization theory and non-commutative harmonic analysis has been particularly strong and mutually fruitful. Indeed, on the one hand we might quote the well-known formalism that permits to think of elementary particles as of Hilbert spaces of unitary irreducible representations of appropriate Lie groups and hence to study their interactions by use of branching laws of tensor products of such representations. On the other hand, representation theory was considerably influenced by innovating physical ideas: these led, for instance, to the development in the mid 1960's by Kirillov and Kostant of the celebrated orbit method, or more recently to the application by Kontsevich of perturbative methods of the topological quantum field theory to the Lie theory. In this survey note we discuss some aspects of this longstanding “cooperation” between quantization theory of systems with symmetries and the representation theory of underlying Lie groups. We concentrate our considerations on the case of semi-simple Lie groups, with emphasis on subjects in which we have been particularly interested; we also leave some open questions.
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