# Observables and Dynamics, Quantum to Classical, from a Relativity Symmetry and Noncommutative Geometric Perspective

## TL;DR

This paper reformulates quantum dynamics using a noncommutative geometric approach rooted in relativity symmetry, connecting quantum and classical descriptions through a unified algebraic framework inspired by the WWGM formalism.

## Contribution

It introduces a coherent reformulation of quantum mechanics based on the underlying relativity symmetry, linking quantum and classical models via noncommutative geometry and the WWGM formalism.

## Key findings

- Derives a unified quantum and classical description from the same algebraic structure.
- Shows how the classical limit emerges through relativity symmetry contraction.
- Provides a framework connecting quantum mechanics to noncommutative spacetime models.

## Abstract

With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate here the WWGM formalism starting from the canonical coherent states and taking wavefunctions as expansion coefficients in terms of this basis. This provides us with a transparent and coherent story of simple quantum dynamics where both the wavefunctions for the pure states and operators acting on them arise from the single space/algebra, which exactly includes the WWGM observable algebra. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics. It also helps to inspire and illustrate our perspective on looking at quantum mechanics and quantum physics in general in direct connection to the notion of quantum (deformed) relativity symmetries and the corresponding quantum/noncommutative models of spacetime as various levels of approximations all the way down to the Newtonian.