# Interaction from Geometry, Classical and Quantum

## TL;DR

This paper explores the geometric description of classical and quantum interactions, revealing the necessity of non-commutative positions for canonical formulations and discussing implications for Non-Commutative Geometry and Quantum Gravity.

## Contribution

It introduces a geometric reduction approach to describe fundamental interactions and highlights the role of non-commutative positions in canonical formulations.

## Key findings

- Non-commutative positions are essential for canonical descriptions of simple systems.
- The approach links geometric reduction to fundamental interactions.
- Implications for Non-Commutative Geometry and Quantum Gravity are discussed.

## Abstract

In this work, we have studied classical and quantum systems in interaction by means of geometric reduction procedure. The main target is the description in these terms of fundamental interactions. We have shown that, to describe in a canonical way (i.e., in terms of Poisson/Dirac Brackets) even a simple system like the relativistic free particle, one has to deal with non-commutative positions. We have explored some consequences of this result in the framework of Non-Commutative Geometry and in Quantum Gravity.