Derivations from Action-Wave Space III: The Quantum Harmonic Oscillator

This paper derives the quantized energy spectrum of the quantum harmonic oscillator from first principles within the Action-Wave Space (AWS) framework. AWS is a Lorentzian symplectic geometry in which wavefunctions evolve on a four-dimensional manifold of integrated action variables. Quantization arises from topological phase closure over closed geodesics, without invoking operator algebra, Hilbert space structure, or canonical commutation relations. By enforcing interference stability and global coherence of the phase field, we reproduce the exact spectrum En=ℏω(n+12)E_n = \hbar \omega(n + \frac{1}{2})En=ℏω(n+21). This derivation demonstrates that AWS can recover foundational results of quantum mechanics from geometric and topological constraints alone, and provides a clear pedagogical bridge between standard quantization methods and geometric quantization via action space. Keywords: quantum harmonic oscillator, geometric quantization, action-wave space, phase coherence, symplectic geometry, operator-free quantization, interference stability, metric–symplectic compatibility