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

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.