Wave Mechanics
We introduce a complex Hilbert space structure on $L^2(\mathbb{T}^m;\mathbb{C})$ 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 $L^2$ pairing. We show that this intrinsic weak derivative coincides with the…
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