Geometric Origin of Quantum Waves from Finite Action

Quantum mechanics introduces wave–particle duality as a postulate, yet the geometric origin of wave behavior has never been derived from first principles. Here we show that a finite quantum of action, ℏgeom, compactifies the classical action manifold into a periodic U(1) phase space. Physical observables then depend only on the modular action S mod 2πℏgeom, making interference a direct geometric necessity rather than an independent assumption. We formalize this as a theorem: any system possessing finite ℏgeom must exhibit wave interference, while the classical limit corresponds to decompactification ℏgeom→0. Chronon Field Theory (ChFT) provides the physical substrate for this geometry—its causal field Φμ carries quantized symplectic flux ∮ω=ℏgeom, thereby establishing Planck’s constant as a geometric invariant of causal alignment. This unified framework links modular action, quantization, and spacetime geometry, revealing the wave nature of matter as a necessary consequence of finite causal curvature. It further predicts quantized phase discontinuities in mesoscopic interferometry, offering a concrete path toward experimental validation.