Derivation of the deformed Heisenberg algebra from discrete spacetime

TL;DR

This paper rigorously derives the deformed Heisenberg algebra from a discrete spacetime model inspired by quantum gravity, confirming heuristic results and exploring higher-order corrections and symmetry breaking effects.

Contribution

It provides a rigorous derivation of the deformed Heisenberg algebra from a discrete spacetime model, including higher-order corrections and symmetry considerations.

Findings

Leading order corrections match heuristic quantum gravity predictions

Higher-order corrections are model-dependent

Graphene can serve as an analogue system for quantum gravity studies

Abstract

Although the deformation of the Heisenberg algebra by a minimal length has become a central tool in quantum gravity phenomenology, it has never been rigorously obtained and is often derived using heuristic reasoning. In this study, we move beyond the heuristic derivation of the deformed Heisenberg algebra and explicitly derive it using a model of discrete spacetime, which is motivated by quantum gravity. Initially, we investigate the effects of the leading order Planckian lattice corrections and demonstrate that they precisely match those suggested by the heuristic arguments commonly used in quantum gravity phenomenology. Furthermore, we rigorously obtain deformations from the higher-order Planckian lattice corrections. In contrast to the leading-order corrections, these higher-order corrections are model dependent. We select a specific model that breaks the rotational symmetry, as the importance of such rotational symmetry breaking lies in the relationship between CMB anisotropies and quantum gravitational effects. Based on the mathematical similarity of the Planckian lattice used here with the graphene lattice, we propose that graphene can serve as an analogue system for the study of quantum gravity. Finally, we examine the deformation of the covariant form of the Heisenberg algebra using a four-dimensional Euclidean lattice.