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Generalized Uncertainty Principle theories and their classical interpretation

Matteo Bruno, Sebastiano Segreto, Giovanni Montani

TL;DR

This work develops a rigorous classical counterpart to generalized uncertainty principle theories by constructing a symplectic form on a $2d$-dimensional phase space and enforcing Jacobi identities on the deformed Poisson brackets $\{q_i,q_j\}=L_{ij}(q,p)$, $\{q_i,p_j\}=f(q,p)\delta_{ij}$, $\{p_i,p_j\}=0$. It derives general constraints relating the deformation functions $f$ and $L_{ij}$ and shows how rotation generators arise as angular-momentum–like objects across dimensions, including explicit Kempf–Mangano–Mann (KMM) and Maggiore-type realizations. A central conjecture proposes that a GUP theory admits a consistent classical interpretation if and only if the quantum commutators satisfy Jacobi identities, with several concrete examples supporting the link and guiding quantum ordering choices. The findings provide a coherent semiclassical framework for GUP models, clarifying how classical consistency constrains viable quantizations and suggesting directions for relativistic extensions and Snyder-type spaces.

Abstract

In this work, we show that it is possible to define a classical system associated with a Generalized Uncertainty Principle (GUP) theory via the implementation of a consistent symplectic structure. This provides a solid framework for the classical Hamiltonian formulation of such theories and the study of the dynamics of physical systems in the corresponding deformed phase space. By further characterizing the functions that govern non-commutativity in the configuration space using the algebra of angular momentum, we determine a general form for the rotation generator in these theories and crucially, we show that, under these conditions, unlike what has been previously found in the literature at the quantum level, this requirement does not lead to the superselection of GUP models at the classical level. Finally, we postulate that a properly defined GUP theory can be correctly interpreted classically if and only if the corresponding quantum commutators satisfy the Jacobi identities, identifying those quantization prescriptions for which this holds true.

Generalized Uncertainty Principle theories and their classical interpretation

TL;DR

This work develops a rigorous classical counterpart to generalized uncertainty principle theories by constructing a symplectic form on a -dimensional phase space and enforcing Jacobi identities on the deformed Poisson brackets , , . It derives general constraints relating the deformation functions and and shows how rotation generators arise as angular-momentum–like objects across dimensions, including explicit Kempf–Mangano–Mann (KMM) and Maggiore-type realizations. A central conjecture proposes that a GUP theory admits a consistent classical interpretation if and only if the quantum commutators satisfy Jacobi identities, with several concrete examples supporting the link and guiding quantum ordering choices. The findings provide a coherent semiclassical framework for GUP models, clarifying how classical consistency constrains viable quantizations and suggesting directions for relativistic extensions and Snyder-type spaces.

Abstract

In this work, we show that it is possible to define a classical system associated with a Generalized Uncertainty Principle (GUP) theory via the implementation of a consistent symplectic structure. This provides a solid framework for the classical Hamiltonian formulation of such theories and the study of the dynamics of physical systems in the corresponding deformed phase space. By further characterizing the functions that govern non-commutativity in the configuration space using the algebra of angular momentum, we determine a general form for the rotation generator in these theories and crucially, we show that, under these conditions, unlike what has been previously found in the literature at the quantum level, this requirement does not lead to the superselection of GUP models at the classical level. Finally, we postulate that a properly defined GUP theory can be correctly interpreted classically if and only if the corresponding quantum commutators satisfy the Jacobi identities, identifying those quantization prescriptions for which this holds true.
Paper Structure (17 sections, 2 theorems, 84 equations)

This paper contains 17 sections, 2 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Classical interpretation and Jacobi identity
  3. Building the function f(p)
  4. 1-dimensional GUP theories
  5. 2-dimensional GUP theories
  6. 2D classical mechanics and 2D non-commutative classical mechanics
  7. Interesting example: Kempf-Mangano-Mann 2D-GUP
  8. Explicit computation: Polynomial functions
  9. d-dimensional GUP theories
  10. The 3-dimensional case
  11. Interesting example: Kempf-Mangano-Mann 3D-GUP
  12. Maggiore GUP as an application
  13. Closure condition and Maggiore algebra
  14. Kempf-Mangano-Mann case in Maggiore's scheme
  15. Verifying the Conjecture
  16. ...and 2 more sections

Key Result

Proposition 5.1

A symplectic form on $\mathcal{M}$ is equivalent to a non-degenerate Poisson structure.

Theorems & Definitions (5)

  • Definition 2.1
  • Remark 2.2
  • Conjecture 2.3
  • Proposition 5.1
  • Lemma 5.2