Generalized Uncertainty Principle theories and their classical interpretation

TL;DR

This paper establishes a classical framework for Generalized Uncertainty Principle (GUP) theories using symplectic structures, clarifying their classical dynamics and the conditions for consistent quantum-classical correspondence.

Contribution

It introduces a classical Hamiltonian formulation for GUP theories, characterizes non-commutativity functions via angular momentum algebra, and clarifies the classical interpretation criteria based on quantum Jacobi identities.

Findings

Classical systems associated with GUP can be formulated with a consistent symplectic structure.

Rotation generators in GUP theories can be characterized without superselection at the classical level.

Proper classical interpretation of GUP requires quantum commutators to satisfy Jacobi identities.

Abstract

In this work, we show that it is possible to define a classical system associated with a Generalized Uncertainty Principle (GUP) theory via the implementation of a consistent symplectic structure. This provides a solid framework for the classical Hamiltonian formulation of such theories and the study of the dynamics of physical systems in the corresponding deformed phase space. By further characterizing the functions that govern non-commutativity in the configuration space using the algebra of angular momentum, we determine a general form for the rotation generator in these theories and crucially, we show that, under these conditions, unlike what has been previously found in the literature at the quantum level, this requirement does not lead to the superselection of GUP models at the classical level. Finally, we postulate that a properly defined GUP theory can be correctly interpreted classically if and only if the corresponding quantum commutators satisfy the Jacobi identities, identifying those quantization prescriptions for which this holds true.