Relativistic Hamiltonian as an emergent structure from information geometry

TL;DR

This paper demonstrates that the relativistic energy-momentum relation can emerge naturally from an information geometric framework involving maximum entropy inference and Fisher-Rao geometry, without initially assuming Lorentz symmetry.

Contribution

It introduces a novel perspective where relativistic relations arise as emergent structures from statistical and geometric principles rather than being fundamental assumptions.

Findings

Relativistic dispersion relation emerges from maximum entropy inference.

Fisher-Rao geometry naturally leads to scale-invariant constraints.

Lorentz symmetry appears as a consequence of statistical averaging and geometric invariance.

Abstract

We show that the relativistic energy-momentum relation can emerge as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints, which are shown to arise naturally from the Fisher-Rao geometry of the associated statistical manifold. Within this information-geometric framework, the relativistic dispersion relation appears without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance.