PhD Defense Philipp L. Schmitt

Title: Strict quantization of certain classes of analytic functions

Abstract:

This thesis studies strict quantizations in a Fréchet-algebraic setting.

In the Introduction, we review the quantization problem and different approaches to its solution: formal deformation quantization, strict deformation quantization in the sense of Rieffel, Berezin-Toeplitz quantization, and strict quantization in a Fréchet-algebraic setting.

In Paper I, which is joint with M. Schötz, we study strict Fréchet-algebraic quantizations of a family of manifolds $M_{red}$ that can be obtained via phase space reduction from $C^{1+n}$ with the Wick product in different signatures. In particular, we show how to reduce the formal Wick star product to $M_{red}$, compute its defining bidifferential operators explicitly, and prove that it restricts to a strict product on a subalgebra of polynomial functions. We prove that this product extends to a continuous product on the Fréchet algebra of certain analytic functions that admit a holomorphic extension to a larger space. We obtain an isomorphism between the Fréchet-algebraic quantizations for different signatures, which is similar to a Wick rotation.

In Paper II, we obtain strict quantizations for semisimple coadjoint orbits $O$ of semisimple connected Lie groups $G$. We give an explicit formula for the canonical element of the Shapovalov pairing, which was used by Alekseev-Lachowska to define a formal $G$-invariant star product on $O$. We show that the formal star product converges on polynomials, and, using the explicit formula for the canonical element, we show that it extends to a strict $G$-invariant product on the Fréchet algebra of all functions that admit a holomorphic extension to the complexification of $O$. In this setting, we also have an analogue of a Wick rotation.

In the Appendix, we show that all the reduced manifolds $M_{red}$ are coadjoint orbits, and that the strict star products obtained for $M_{red}$ via phase space reduction as in Paper I coincide with the strict star products obtained in Paper II.

PhD thesis:

http://web.math.ku.dk/noter/filer/phd.htm

Zoom meeting:

https://ucph-ku.zoom.us/j/69570494857?pwd=SGI1c1RkUmdDTWc1MTMybUU3TzhDZz09

Adviser:

Ryszard Nest (University of Copenhagen)

Assessment committee:

Anton Alekseev (University of Geneva)

Pierre Bieliavsky (Université catholique de Louvain)

Henrik Schlichtkrull (University of Copenhagen) Chair