Topic: Singular configuration spaces

Luca Mrini presents work in progress on quantising singular configuration spaces using the non-smooth calculus of metric-measure spaces. The framework generalises smooth quantum mechanics through Hilbert spaces, position and momentum operators, Hamiltonians, unitary dynamics, and observable algebras. A method for bounding mass gaps after symplectic reduction uses the curvature of the unreduced configuration space. Examples include the harmonic oscillator constrained to zero angular momentum, lattice Yang–Mills theory, and fractional-dimensional Laakso spaces. The talk closes with prospective

Singular configuration spaces require extensions of the usual operators and dynamics of quantum mechanics. Using metric-measure calculus, this work constructs states, observables and unitary evolution under suitable assumptions. A curvature-based technique bounds spectral gaps after symmetry reduction. Examples include a constrained oscillator, lattice Yang–Mills theory and fractional-dimensional spaces; the talk also considers prospective connections with the Yang–Mills mass-gap problem and gravity.