Hidden invariance of the free classical particle

TL;DR

This paper explores the group-theoretical symmetries of a classical free particle, revealing a hidden invariance through a central extension of the Galileo group, which also connects to quantum mechanics.

Contribution

It introduces a formalism based on group theory to analyze classical and quantum systems, highlighting a hidden invariance in the free particle case.

Findings

Identification of a larger symmetry group as a central extension of Galileo group

Characterization of Noether invariants via group geometric objects

Extension of symmetry to quantum mechanics through U(1)

Abstract

A formalism describing the dynamics of classical and quantum systems from a group theoretical point of view is presented. We apply it to the simple example of the classical free particle. The Galileo group $G$ is the symmetry group of the free equations of motion. Consideration of the free particle Lagrangian semi-invariance under $G$ leads to a larger symmetry group, which is a central extension of the Galileo group by the real numbers. We study the dynamics associated with this group, and characterize quantities like Noether invariants and evolution equations in terms of group geometric objects. An extension of the Galileo group by U(1) leads to quantum mechanics.