Discrete--Continuous Duality in Geometric Quantization:A Unified Framework via Projection

Geometric Quantization (GQ) has traditionally been formulated on smoothsymplectic manifolds, relying fundamentally on continuous geometric structures.This paper reconstructs GQ as a dual theory consisting of discrete andcontinuous geometries. Discrete geometric frameworks, parameterized by latticespacing, cell decompositions, discrete groupoids, and discrete symplecticforms, are interpreted as generative structures underlying continuousgeometries. A projection map is introduced to unify discrete frameworks intoeffective continuous manifolds, which are then quantized using classical GQ.This establishes a two-path quantization scheme: the classical continuous pathand the discrete-to-continuous path. The resulting unified theory preserves theclassical formulation while extending it to incorporate discrete structures.