Nilpotence in Physics: the case of Tsallis entropy

TL;DR

This paper investigates the mathematical structure of Tsallis entropy by embedding real numbers into a matrix group, revealing polynomial phase space growth and offering a geometric framework for its properties.

Contribution

It introduces a novel embedding of reals into 3x3 matrices to analyze Tsallis entropy, connecting algebraic and geometric perspectives.

Findings

Establishes polynomial growth of phase space volume for Tsallis entropy

Links Tsallis entropy properties to the geometry of the Heisenberg group

Provides a framework for understanding Abe's formula via Pansu derivative

Abstract

In an attempt to understand the Tsallis entropy composition property, we construct an embedding of the reals into the set of $3\times 3$ upper triangular matrices with real entries. We explore consequences of this embedding and of the geometry of the ambient $3\times 3$ Heisenberg group. This approach establishes the polynomial growth of the volume of phase space of systems described by the Tsallis entropy and provides a general framework for understanding Abe's formula in terms of the Pansu derivative between Riemannian spaces.