Quantization is known as a way of constructing (noncommutative) quantum mechanical system from (commutative) classical mechanical system, mathematically forming operators from ordinary functions. The word `` quantize'' means ``discretize'' and we discuss where it came from by reviewing the motivation and background of quantum mechanics. Then we discuss mathematical formulation of (deformation) quantizations and introduce two -algebras, noncommutative two tori and quantum Heisenberg manifolds, which are main examples of Rieffel's strict deformation quantizations. We also introduce noncommutative (or quantum) Yang-Mills theory developed by Connes and Rieffel on noncommutative tori, and then review some of the results of Yang-Mills connections on noncommutative two tori and quantum Heisenberg manifolds if we have time.
Introduction to deformation quantizations and noncommutative Yang-Mills theory
The study of representations of higher rank-graph algebras, and related non-commutative structures, is of interest in its own right. It is also part of wider themes of analysis of non-commuting geometries. The talk will focus on systems that arise in such applications as representation theory, symbolic dynamics, sub-band filters from signal processing and wavelet analysis. We shall give an account of joint results, with many co-authors, on the use of representations of the Cuntz relations O_N, and their generalizations, Cuntz-Krieger, higher-rank, etc. Related applications include a class of filter problems used in the analysis of fractals, and in geometric measure theory.
By their nature, these representations reflect intrinsic self-similarity; and thus serve ideally to encode sub-bands, and more generally iterated function systems (IFSs), their dynamics, and their associated measures. At the same time, these representations offer a new harmonic analysis of signals. A key analytic tool will be classes of “transfer operators,” arising in such diverse areas as symbolic dynamics, analysis of Markov processes, and their associated path-space measures.
The role of path-space and path-space measures in symbolic dynamics, in multi-resolutions (such as generalized wavelets), and in representation theory Sponsored by the Meyer Fund
Introduction to deformation quantizations and noncommutative Yang-Mills theory