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Wave Mechanics

A mix of human-written and AI-generated textHuman understanding: all partsmath.MP — Mathematical Physicsmath.AP — Analysis of PDEsmath-ph — Mathematical Physics

Contributed by Rajesh Dachiraju ↗

Version 1 / Oct 09, 2026 / CC BY 4.0

Abstract

We introduce a complex Hilbert space structure on L2(Tm;C)L^2(\mathbb{T}^m;\mathbb{C}) in which the Hilbert transform on the torus is incorporated directly into the metric and inner product. The resulting geometry is defined via a bounded linear embedding that couples each function to its Hilbert transform, yielding an inner product with both symmetric and symplectic components. Within this framework, we redefine weak differentiation intrinsically using the complex Hilbert-space inner product rather than the ambient L2L^2 pairing. We show that this intrinsic weak derivative coincides with the classical weak derivative on its natural domain, while remaining geometrically well defined on the full space. The derivative operator is shown to be densely defined and skew-adjoint, with a purely imaginary, discrete spectrum. Its eigenfunctions are given by the Fourier modes on the torus, which induce a complete orthonormal basis adapted to the Hilbert-transform metric. This construction yields a spectral representation in which differentiation is diagonal and the Hilbert transform is absorbed into the notion of differentiability itself, providing a unified analytic and geometric framework. We formulate wave mechanics intrinsically in this complex Hilbert space, define wave function, scalar and vector potentials and associated energy functional, derive space time wave equation, derive its solution, prove existence and uniqueness and establish regularity of the solution.

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Version history

  1. v1Submitted by Rajesh DachirajuInitial depositCurrentOct 09, 2026