Nilpotence in physics: Generalized uncertainty principles and Tsallis entropy

Abstract

Background and Objectives: The objective of this work is to illustrate concepts and applications of nilpotent structures in physics. To that end, we will use two examples, both of which have attracted considerable attention recently: the generalized uncertainty principle, in its various forms, and the Tsallis entropy. The former has been motivated and extensively used in string theory and black hole physics during the last two decades. There are almost 4000 papers written on the latter during the last 20 years, reflecting a considerable interest in Tsallis entropy as an alternative to the Boltzmann-Gibbs-Shannon (BGS) entropic form. The Tsallis entropy has been having a considerable impact in re-examining the foundations of statistical mechanics for both equilibrium and non-equilibrium processes. Nilpotent structures have a long presence in various branches of mathematics, especially in group theory and geometry with the works of Malcev and Gromov standing out as particularly pertinent for our purposes. We also consider the sub-Riemannian aspects of nilpotent structures which have applications in a variety of fields ranging from examining how a cat falls to analyzing human vision. Results: Both the generalized uncertainty principles and the Tsallis entropy indicate that the dynamical structure of spacetime and the statistical methods used to quantize it may benefit considerably if one uses general nilpotent structures instead of the two-stage Heisenberg model or the abelian group of the BGS entropy. Conclusions: Nilpotent structures are flexible enough to generalize aspects of quantum theory and statistical mechanics. At the same time, they are understood well enough to allow us to obtain results of potential physical significance. Hence they are worth taking a look at and exploring their consequences. Notes: Partial results have already been published by the author recently and have appeared in the arXiv.org repository and are being presented at various international conferences. Parts of this work were done in collaboration with A. J. Creaco of the City University of New York.