Quantum Mechanics from Broken Translation Symmetry: A Spectral–Geometric Framework

We present a structural reformulation of non-relativistic quantum mechanics in which quantum phenomena arise from the breakdown of translation symmetry of the vacuum wave operator. The vacuum is modeled as a spectrally inhomogeneous propagation medium, encoded kinematically by a scalar tempo field S(x), which de- forms the geometric Laplacian into a weighted operator ∆S = ∇·(e−2S ∇) that does not commute with spatial translations. We provide explicit derivations showing that inertia, localization, discrete spectra, the quantum potential, and the Schr¨odinger equation emerge as mathematical consequences of this symmetry breaking. Iner- tial mass is defined operationally as the Hilbert–Schmidt norm of the commutator [∆S , Tϵ], where Tϵ generates translations. We derive the Schr¨odinger equation rig- orously from hydrodynamic variables via Madelung decomposition, demonstrating that it is the unique linear closure of wave dynamics under the weighted Laplacian. The quantum potential is shown to emerge directly from the curvature penalty encoded in ∆S . A universal scaling relation R = −3 connecting inertial mass and frequency follows from Weyl’s law in three spatial dimensions. This framework does not replace quantum mechanics but provides a geometric explanation for its mathematical structure