Particle Dynamics on Snyder space

TL;DR

This paper explores the Hamiltonian formalism on Snyder space, introducing a dynamical time operator for particles and constructing symmetry generators, advancing understanding of quantum lattice structures.

Contribution

It develops a dynamical time operator for particles on Snyder space and constructs the associated Poincaré and Galilei group generators, a novel approach in this context.

Findings

Dynamical time operator formulated for Snyder space.

Generators of Poincaré and Galilei groups constructed.

Insights into quantum lattice dynamics provided.

Abstract

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincar\'e (Galilei) group on Snyder space.