Physical implications of Fisher-information’s scaling symmetry

Abstract

Abstract:We study the scaling properties of Fisher's information measure (FIM) and show that from these one can straightforwardly deduce significant quantum-mechanical results. Specifically, we investigate the scaling properties of Fisher's measure I and encounter that, from the concomitant operating rules, several interesting, albeit known, results can be derived. This entails that such results can be regarded as pre-configured by the conjunction of scaling and information theory. The central notion to be arrived at is that scaling entails that I must obey a certain partial differential equation (PDE). These PDE-solutions have properties that enable the application of a Legendre-transform (LT). The conjunction PDE+LT leads one to obtain several quantum results without recourse to the Schrödinger's equation.PACS (2008)

Legendre-transform structure derived from quantum theorems

“…Curiously enough, it can be shown that the HF theorem can be looked at as a reciprocity relation of the type ( 18) [35].…”

Some topics in thermodynamics and quantum mechanics

“…REMARK 6.1. We mention here for example the papers [3,17,19,25,39,47,50,54,55,64,67,66,68,78,79], involving q-entropy, q-information, q-TD, Fisher information, the Frieden-Soffer idea of extreme physical information, the SE, non-equilibrium TD, Legendre structure, etc. They seem to signal new portals between TD, QM, and gravitational physics to go along with the cosmological input of Padmanabhan [61,62,63] and others connecting gravity and entropy (see e.g.…”