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Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

K. Mahesh Krishna

Published Feb 6, 2024• 0 citations
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Abstract

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.
DOI
10.20944/preprints202402.0342.v1

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We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of Lp\mathcal{L}^pLp spaces for 1<p<∞1< p<\infty1<p<∞. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.
We develop a fractal-spectral reformulation of the Heisenberg uncertainty principle within a nonlinear phase-space framework motivated by the Quantum Fractal-Logical Unified Field architecture. Instead of working on the standard Hilbert space L 2 (R) with Lebesgue measure and linear position/momentum operators, we consider a fractal Hilbert space H F = L 2 (F, µ H) built on a Cantor-type metric-measure substrate with Hausdorff measure µ H and a self-adjoint fractal Hamiltonian. Within this setting we define effective position and momentum observables (X F , P F) adapted to the fractal geometry and derive a generalized uncertainty relation