Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

TL;DR

This paper develops a consistent Hamiltonian and Lagrangian framework for classical and quantum systems with a minimal length scale, resolving formal inconsistencies in their descriptions.

Contribution

It introduces a precise Legendre transformation and two canonical variable sets to unify classical and quantum descriptions under GUP.

Findings

Resolved inconsistencies between classical and quantum formalisms.

Defined a clear Legendre transformation for minimal length theories.

Established two canonical variable sets with a mapping between them.

Abstract

GUP is a phenomenological model aimed for a description of a minimal length in quantum and classical systems. However, the analysis of problems in classical physics is usually approached preferring a different formalism than the one used for quantum systems, and vice versa. Potentially, the two approaches can result in inconsistencies. Here, we eliminate such inconsistencies proposing particular meanings and relations between the variables used to describe physical systems, resulting in a precise form of the Legendre transformation. Furthermore, we introduce two different sets of canonical variables and the relative map between them. These two sets allow for a complete and unambiguous description of classical and quantum systems.