Generalized Uncertainty Principle theory with a single constraint

TL;DR

This paper explores how to consistently deform the Heisenberg algebra within constrained Hamiltonian systems, applying the approach to classical Generalized Uncertainty Principle theories in both symmetry and cosmological contexts.

Contribution

It introduces a procedure to induce algebra deformation after symplectic reduction, analyzing cases with group actions and Hamiltonian constraints relevant to gravity and cosmology.

Findings

Explicit example of rotationally invariant deformed algebras

Method for deformation in Hamiltonian systems with constraints

Application to classical GUP in cosmological models

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.