Tautological Tuning of the Kostant-Souriau Quantization Map with Differential Geometric Structures

Abstract

For decades, mathematical physicists have searched for a coordinate independent quantization procedure to replace the ad hoc process of canonical quantization. This effort has largely coalesced into two distinct research programs: geometric quantization and deformation quantization. Though both of these programs can claim numerous successes, neither has found mainstream acceptance within the more experimentally minded quantum physics community, owing both to their mathematical complexities and their practical failures as empirical models. This paper introduces an alternative approach to coordinate-independent quantization called tautologically tuned quantization. This approach uses only differential geometric structures from symplectic and Riemannian geometry, especially the tautological one form and vector field (hence the name). In its focus on physically important functions, tautologically tuned quantization hews much more closely to the ad hoc approach of canonical quantization than either traditional geometric quantization or deformation quantization and thereby avoid some of the mathematical challenges faced by those methods. Given its focus on standard differential geometric structures, tautologically tuned quantization is also a better candidate than either traditional geometric or deformation quantization for application to covariant Hamiltonian field theories, and therefore may pave the way for the geometric quantization of classical fields.