Distributivity and deformation of the reals from Tsallis entropy

TL;DR

This paper introduces a one-parameter family of deformed real number systems inspired by Tsallis entropy, defining a generalized multiplication and addition that form a field isomorphism with the real numbers, with implications for metric and measure theory.

Contribution

It constructs a novel deformation of the real numbers based on Tsallis entropy, establishing a new algebraic framework with generalized operations and isomorphisms.

Findings

Defined a family of deformed real number systems r_q.

Introduced a generalized multiplication compatible with Tsallis entropy.

Explored metric and measure-theoretic properties of the deformed systems.

Abstract

We propose a one-parameter family \ $\mathbb{R}_q$ \ of deformations of the reals, which is motivated by the generalized additivity of the Tsallis entropy. We introduce a generalized multiplication which is distributive with respect to the generalized addition of the Tsallis entropy. These operations establish a one-parameter family of field isomorphisms \ $\tau_q$ \ between \ $\mathbb{R}$ \ and \ $\mathbb{R}_q$ \ through which an absolute value on \ $\mathbb{R}_q$ \ is introduced. This turns out to be a quasisymmetric map, whose metric and measure-theoretical implications are pointed out.