Canonical Quantization of Singular Configuration Spaces: Symmetry Reduction, Mass Gap Bounds, and Non-smooth Calculus

In this work-in-progress, we study the quantization of singular configuration spaces, generalizing the canonical structures of smooth quantum mechanics. We develop a framework for constructing a Hilbert space of quantum states, generalized position and momentum operators, a kinetic-plus-potential Hamiltonian, unitary quantum dynamics, and a C*-algebra of observables under suitable assumptions. Our approach is based on the non-smooth calculus of metric-measure spaces formalized in recent years. One of our central contributions is a new technique to bound the mass gap of certain singular systems obtained via symplectic reduction by analyzing the curvature of the non-reduced configuration space. We illustrate our method for the harmonic oscillator constrained to zero angular momentum, lattice Yang--Mills theory, and fractional-dimensional Laakso spaces. To conclude, we speculate about future directions pertaining to the Yang--Mills mass gap problem and quantum gravity.