Rough Quantization: Roughness-Dependent Canonical Commutation Relations and Geometric Renormalization

The unification of Quantum Mechanics and General Relativity is hindered by ultraviolet divergences in Quantum Field Theory (QFT) and singularities in gravity. In this paper, we propose a fundamental resolution based on Rough Operator Algebra (ROA). We introduce the Rough Canonical Commutation Relation (RCCR), where the commutator x, p evolves with the spacetime roughness index α. We derive a Generalized Uncertainty Principle (GUP) that necessitates a minimal measurable length Lₘin proportional to LP * sqrt (1-α) /α at the Planck scale. Furthermore, we extend this roughness to the time domain, formulating the Rough Schrödinger Equation with the Sunggil Derivative Dₜ^α. This framework proves that the roughness of spacetime acts as an intrinsic Geometric Cutoff, rendering QFT loop integrals finite without artificial renormalization.