GUP-based and Snyder Non-Commutative Algebras, Relativistic Particle models and Deformed Symmetries: A Unified Approach

TL;DR

This paper presents a unified framework for analyzing non-commutative algebras derived from GUP and Snyder form, exploring their symmetries, constraints, and dynamics in relativistic particle models.

Contribution

It introduces a unified Lagrangian approach linking GUP and Snyder algebras and studies their deformed symmetries and dynamics, including external fields.

Findings

Snyder algebra emerges as an approximation of GUP algebra

Deformed Poincare generators preserve space-time symmetries

Models exhibit rich constrained dynamical structures

Abstract

We have developed a unified scheme for studying Non-Commutative algebras based on Generalized Uncertainty Principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincare generators for the systems that keep space-time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspective the models studied here are very rich and provide insights on how to consistently construct approximate models from the exact ones when non-linear constraints are present in the system. We also study dynamics of the GUP particle in presence of external electromagnetic field.