Finite-dimensional position–momentum realization of GUP kinematics

Construct a finite-dimensional position–momentum transformation and complete digital simulation framework for generalized-uncertainty-principle kinematics that is consistent with the modified Hilbert-space measure, the nonuniform wave-number lattice induced by the relation k_β(p), and the associated localization structure.

Background

The paper analyzes the diagonal operator k_β(\hat p) on a fixed Jordan–Lee–Preskill momentum register and derives its Walsh–Pauli representation and compressibility. This construction does not provide a complete finite-dimensional realization of generalized-uncertainty-principle quantum mechanics because applying k_β to a uniformly discretized momentum lattice produces a nonuniform wave-number lattice.

A complete treatment would require constructing a position–momentum transformation compatible with the modified Hilbert-space measure and localization structure, rather than using the standard discrete Fourier transform, and then using that transformation to simulate full position–momentum dynamics. The paper identifies this as a separate unresolved problem and leaves it for future work.

References

Constructing a finite-dimensional transform consistent with the modified Hilbert-space measure and localization structure, and using it to simulate complete position--momentum dynamics, is a separate problem that we leave for future work.

— From Analytic Structure to Quantum Complexity: Walsh-Pauli Representations of Continuum Operators  (2609.11350 - Sousa et al., 10 Sep 2026) in Section VII, subsection “Towards a Full Digital Realisation of GUP Kinematics”