Galilean One-Particle Kinematics from a Smooth Family of Reference States

TL;DR

This paper develops a framework for Galilean one-particle kinematics using a smooth family of reference states, establishing a duality between observables and generators, and characterizing the structure of irreducible sectors.

Contribution

It introduces a continuous-variable approach to Galilean kinematics, deriving a local observable-generator duality and characterizing the Hilbert space structure of irreducible sectors.

Findings

Established a local observable-generator duality for Galilean kinematics.

Characterized the Hilbert space as L2(R3) tensor spin space with specific generators.

Demonstrated sharp localization and the structure of boosts, translations, and angular momentum.

Abstract

Giannelli and Chiribella derived an observable-generator duality for energy from a collision model of informational nonequilibrium. We study a continuous-variable version aimed at the Galilean one-particle sector. A smooth family of reference states around an isotropic equilibrium supplies time, translation, rotation, and boost directions. The local observable-generator correspondence is obtained by differentiating a smooth extension of the single-state duality map, and the norm-one property of localization is obtained from a fiducial focusing assumption together with covariance. Combined with the standard smearing form of covariant localization observables, this yields sharp localization. With local inertial composition, the spin-cover action of rotations, and a central boost-translation holonomy, every irreducible sector is unitarily equivalent to the Hilbert space L2(R3) tensored with a (2s+1)-dimensional spin space. In that representation translations are generated by the canonical momentum, the holonomy is a scalar mass m > 0, boosts at t = 0 are generated by m times the position observable, the Hamiltonian is the free-particle kinetic term plus a constant E0, and the total angular momentum is orbital plus spin.