The fate of Galilean relativity in minimal-length theories

TL;DR

This paper investigates how minimal-length theories, modeled via a generalized uncertainty principle, affect Galilean invariance and finds that only Hamiltonians related by canonical transformations preserve a deformed Galilean symmetry, leading to deformed dynamics.

Contribution

It demonstrates that GUP-inspired Hamiltonians must be related by canonical transformations to preserve deformed Galilean invariance, revealing the structure of such deformed symmetries.

Findings

Standard GUP-Hamiltonians lack invariance under deformed Galilean transformations.

Hamiltonians with deformed Galilean invariance are connected to ordinary ones via canonical transformations.

Deformed dynamics are exemplified through a harmonic interaction model.

Abstract

A number of arguments at the interplay of general relativity and quantum theory suggest an operational limit to spatial resolution, conventionally modelled as a generalized uncertainty principle (GUP). Recently, it has been demonstrated that the dynamics postulated as a part of these models are only loosely related to the existence of the minimal-length scale. In this paper, we intend to make a more informed choice on the Hamiltonian by demanding, among other properties, that the model be invariant under (possibly) deformed Galilean transformations in one dimension. In this vein, we study a two-particle system with general interaction potential under the condition that the composition as well as the action of Galilean boosts on wave numbers be deformed so as to comply with the cut-off. We find that the customary GUP-Hamiltonian does not allow for invariance under (any kind of) generalised Galilean transformations. Those Hamiltonians which allow for a deformed relativity principle have to be related to the ordinary Galilean ones by virtue of a momentum-space diffeomorphism, i.e. a canonical transformation. Far from being trivial, the resulting dynamics is deformed, as we show at the example of the harmonic interaction.