# On deformations of classical mechanics due to Planck-scale physics

## TL;DR

This paper explores how Planck-scale physics may modify classical mechanics by inducing non-commuting operators, suggesting a quantum-like behavior emerges at the classical level due to quantum gravity effects.

## Contribution

It proposes an alternative approach to modifying classical mechanics via Koopman-Von Neumann formalism, showing Planck-scale effects can induce non-commutativity and hidden variables.

## Key findings

- Classical position and momentum operators become non-commuting at the Planck scale.
- Planck-scale modifications can induce quantum-like features in classical dynamics.
- Classicality may be destroyed by quantum gravity effects, leading to hidden variables.

## Abstract

Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman-Von Neumann's Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.