Restoring the Isometry of Position and Momentum Space in Phenomenological Quantum Gravity and Implications for Quantum Field Theory

Many phenomenological models of quantum gravity predict a minimal observable length. A common implementation is the Generalized Uncertainty Principle (GUP), which modifies the canonical commutator between position and momentum. In many formulations, the modified position operator is not symmetric with respect to the standard quantum-mechanical inner product, requiring a modified Hilbert-space measure that breaks the natural isometry between momentum and position space. We review an alternative formulation in which the position operator is symmetrized instead of the inner product. This preserves the standard Fourier relationship between momentum and position space while retaining the same generalized uncertainty relation and minimal length. The resulting position eigenstates contain an intrinsic suppression of large momenta, suggesting a possible route toward constructing GUP-modified quantum field theories with improved ultraviolet behavior.