Topological Quantum Field Theory Seminar   RSS

12/10/2006, 16:00 — 17:00 — Room P3.10, Mathematics Building
Mark Gotay, Univ. of Hawai at Manoa

Obstructions to Quantization 1

Quantization is not a straightforward proposition, as demonstrated by Groenewold's and Van Hove's discovery, sixty years ago, of an "obstruction" to quantization. Their "no-go theorems" assert that it is in principle impossible to consistently quantize every classical polynomial observable on the phase space R2n in a physically meaningful way. Similar obstructions have been recently found for S2 and T* S1 , buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so-it has just been proven that there are no obstructions to quantizing either T2 or T* R+ . In this talk we conjecture-and in some cases prove-generalized Groenewold-Van Hove theorems, and determine the maximal Lie subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of symplectic manifolds and their representations. To these ends we review known results as well as recent theoretical work. 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. (This is joint work with J. Grabowski, H. Grundling and A. Hurst.)

  1. Gotay, M. J. [2000], Obstructions to Quantization, in: Mechanics: From Theory to Computation. (Essays in Honor of Juan-Carlos Simo), J. Nonlinear Science Editors, 271-316 (Springer, New York).
  2. Gotay, M. J. [2002], On Quantizing Non-nilpotent Coadjoint Orbits of Semisimple Lie Groups. Lett. Math. Phys. 62, 47-50.

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Current organizers: José MourãoRoger Picken, Marko Stošić


FCT Projects PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry, PTDC/MAT-PUR/31089/2017, Higher Structures and Applications.