Space and time transformations with a minimal length

TL;DR

This paper explores how a minimal measurable length in quantum gravity affects fundamental quantities like energy, momentum, and Hamiltonian, leading to the implication of a minimal time due to the bounded Hamiltonian.

Contribution

It demonstrates that minimal length considerations extend beyond dynamics, influencing the definitions of core physical quantities and introducing a minimal time scale.

Findings

Minimal length modifies definitions of energy, momentum, and Hamiltonian.

A bounded Hamiltonian implies the existence of a minimal measurable time.

Quantum gravity phenomenology impacts fundamental quantum mechanics concepts.

Abstract

Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics so to introduce a minimal uncertainty in position in quantum mechanics. Such a minimal uncertainty and the consequent minimal measurable length have important consequences on the dynamics of quantum systems. In the present work, we show that such consequences go beyond dynamics, reaching the definition of quantities such as energy, momentum, and the same Hamiltonian. Furthermore, since the Hamiltonian, defined as the generator of time evolution, results to be bounded, a minimal length implies a minimal time.