Generalized Uncertainty Principle theory with a single constraint
Matteo Bruno, Sebastiano Segreto
TL;DR
This work develops a geometric framework for classical Generalized Uncertainty Principle (GUP) theories in the presence of constraints. It connects deformed Poisson brackets, encoded by a deformed symplectic form, to constrained dynamics via symplectic reduction (for Lie-group actions) and a projection via an external-time construction (for a single Hamiltonian constraint). The authors show that, under suitable regularity conditions, the reduced phase space inherits the deformed symplectic structure, preserving the deformation's form and the dynamics, with explicit results for $SO(2)$ and $SO(3)$ reductions and a cosmological Misner-variable example. They demonstrate that time-space noncommutativity is forbidden in their construction to maintain Hamiltonian consistency, and they justify using reduced forms in cosmology as a rigorous outcome of the reduction procedure. The results provide a robust prescription for analyzing GUP-deformed constrained systems, with concrete applications to early-universe cosmology and Maggiore-type algebras.
Abstract
We aim to analyze the consistency of the deformation of the Heisenberg algebra in the setting of constrained Hamiltonian systems, providing a procedure to induce the deformation on the Poisson algebra after symplectic reduction. We investigate this in the context of the classical interpretation of Generalized Uncertainty Principle theories, treating two cases separately. For the first case, we consider a group action on the phase space together with a set of first-class constraints that can be interpreted as a momentum map. We furnish an explicit example in the case of rotational invariant deformed algebras. In the second case, we consider a single constraint provided by the Hamiltonian, which is a common instance in General Relativity, with straightforward application in cosmology.