Generalized Uncertainty Principle from deformed Angular momentum Algebra

This work establishes a link between deformed angular momentum operators and modifications of the quantum uncertainty principle. Starting from an algebra in which the standard angular momentum operators are deformed by two parameters, Formula: see text and Formula: see text, we construct the corresponding momentum operators via Formula: see text where the radial operator Formula: see text is determined by the requirement that the standard momentum operator is recovered in the limit Formula: see text, Formula: see text. The resulting commutation relations take the form Formula: see text where Formula: see text is a position-dependent deformation matrix. This leads to an uncertainty relation Formula: see text which is state-dependent, anisotropic and position-dependent. We use the term Generalized Uncertainty Principle (GUP) in its broad sense referring to any modification of the Heisenberg relation motivated by quantum-gravity ideas. However, the model does not yield a fixed minimal length of the kind familiar from conventional GUP models (e.g. Formula: see text). In the isotropic, limit the trace of the deformation matrix simplifies the bound to Formula: see text which can be considered as a rescaling of the uncertainty constant. Our work thus provides a geometric origin for modified uncertainty relations from symmetry breaking in the rotation group, and offers a framework in which quantum-gravity effects manifest themselves through state- and direction-dependent measurement limits, without necessarily introducing a universal minimal length.