Nilpotence and the generalized uncertainty principle(s)

TL;DR

This paper explores how certain generalized uncertainty principles are linked to solvable or nilpotent Lie algebra deformations, discussing their mathematical properties and potential implications for classical mechanics and quantum gravity.

Contribution

It reveals the algebraic structures underlying generalized uncertainty principles and discusses their formal aspects and possible connections to classical and quantum theories.

Findings

Some generalized uncertainty principles originate from solvable or nilpotent Lie algebra deformations.

Potential relation between these algebras and classical mechanics via the symplectic non-squeezing theorem.

Connection to a hierarchy of generalized measure theories in quantum gravity formalism.

Abstract

We point out that some of the proposed generalized/modified uncertainty principles originate from solvable, or nilpotent at appropriate limits, "deformations" of Lie algebras. We briefly comment on formal aspects related to the well-posedness of one of these algebras. We point out a potential relation of such algebras with Classical Mechanics in the spirit of the symplectic non-squeezing theorem. We also point out their relation to a hierarchy of generalized measure theories emerging in a covariant formalism of quantum gravity.