Quantum Entropy for the Fuzzy Sphere and its Monopoles

TL;DR

This paper constructs fuzzy spheres and monopoles using generalized bosons, analyzes their quantum states with GNS-construction, and explores the entropy and gauge symmetries arising in this non-commutative geometric setting.

Contribution

It introduces a novel approach to modeling fuzzy spheres and monopoles via reducible representations and examines the associated quantum states and entropy behavior.

Findings

Quantum states are inherently mixed with non-zero von Neumann entropy.

Entropy increases monotonically under bistochastic Markov maps.

Maximum entropy relates to the degeneracy of irreducible representations.

Abstract

Using generalized bosons, we construct the fuzzy sphere $S_F^2$ and monopoles on $S_F^2$ in a reducible representation of $SU(2)$. The corresponding quantum states are naturally obtained using the GNS-construction. We show that there is an emergent non-abelian unitary gauge symmetry which is in the commutant of the algebra of observables. The quantum states are necessarily mixed and have non-vanishing von Neumann entropy, which increases monotonically under a bistochastic Markov map. The maximum value of the entropy has a simple relation to the degeneracy of the irreps that constitute the reducible representation that underlies the fuzzy sphere.