This project is designed to investigate and deepen the relationship between the operator algebraic E-theory of Alain Connes and Nigel Higson, the index theory of elliptic differential operators on manifolds, and quantum physics. First, we will investigate the correspondence between (positive) asymptotic morphisms (which form the basic cycles in E-theory) and "asymptotic" projection-valued measures. This will help to understand the fundamental use of E-theory groups as receptacles for invariants of strict (physical) quantization schemes. This should also give novel formulas for writing asymptotic morphisms as operator-valued integrals, and for computing the pairing between K-homology and K-theory, e.g., in computing the index of a Dirac type operator coupled to a gauge connection. The second part deals with extending the work of Erik Guentner in using E-theory groups to understand the relationship between elliptic differential operators and strict quantization schemes, e.g., the relationship between the (E-theory elements of) the Dolbeault operator on a Kahler manifold and the Berezin-Toeplitz quantization on the associated Bergmann-Fock space. The third part is a long-term project to develop an E-theoretic classification method for strict quantization schemes that parallels the cohomological classification in the formally algebraic (nonphysical) deformation quantization of star products.
The purpose of this project is to more thoroughly investigate the relationship between quantum physics and the operator algebraic E-theory of the Fields medalist Alain Connes and Nigel Higson. This mathematical theory associates classifying groups to pairs of operator algebras. Group elements are determined by objects called asymptotic morphisms between the two operator algebras. The relationship to quantum theory is as follows. Different types of structures are used in quantum theory to model subatomic and atomic systems, for example, Dirac operators coupled to a gauge field in quantum field theory, and operator-valued measures in operational quantum physics and quantum computing, etc. Under appropriate conditions, these structures have asymptotic morphisms associated to them (between the algebras of the classical observables and quantum observables, respectively). Hence, they define elements in an E-theory group. By fully understanding this correspondence, we want to show that computing these E-theory invariants provides a natural procedure for classifying, and defining obstructions for, diverse types of quantum-mechanical systems.
