Finite Number of States, de Sitter Space and Quantum Groups at Roots of Unity

TL;DR

This paper investigates quantum group deformations at roots of unity to model finite state spaces relevant to quantum gravity and de Sitter space, highlighting limitations and analogies with black hole properties.

Contribution

It introduces observations on supersymmetry with q-bosons, draws analogies between de Sitter black holes and quantum groups, and explores a noncommutative quantum mechanics model with finite and infinite spectra.

Findings

Finite state spectrum for positive parameters

Infinite spectrum for zero or negative parameters

Analogies between black holes and quantum groups

Abstract

This paper explores the use of a deformation by a root of unity as a tool to build models with a finite number of states for applications to quantum gravity. The initial motivation for this work was cosmological breaking of supersymmetry. We explain why the project was unsuccessful. What is left are some observations on supersymmetry for q-bosons, an analogy between black holes in de Sitter and properties of quantum groups, and an observation on a noncommutative quantum mechanics model with two degrees of freedom, depending on one parameter. When this parameter is positive, the spectrum has a finite number of states; when it is negative or zero, the spectrum has an infinite number of states. This exhibits a desirable feature of quantum physics in de Sitter space, albeit in a very simple, non-gravitational context.