Quantum entropy for the fuzzy sphere and its monopoles

Abstract

Using generalized bosons, we construct the fuzzy sphere S 2 F and monopoles on S 2 F 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 nonabelian 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.

New fuzzy extra dimensions fromSU(N)gauge theories

“…We discuss the stability of our vacuum solutions using the recent novel approach developed in [43] which addresses the mixed state nature of configurations with several fuzzy spheres and their quantum entropy, relying on the broader considerations of quantum entropy and its ambiguities recently discussed in [44,45]. We show that, our vacuum configurations which are direct sums of fuzzy spheres, do indeed form mixed states with non-zero von Neumann entropy, while single fuzzy sphere solutions form pure states with vanishing entropy.…”

“…In this section we follow the novel developments and reasoning given in [43] to argue the stability of vacuum solutions, in the form of direct sums of fuzzy spheres given in (2.27). For matrix models, such as the one considered in this paper and also for other string theory related matrix models (for instance those discussed in [9,10,50]), potentials may be minimized by choosing the matrix fields as the generators of su(2) Lie algebra, which are in irreducible or reducible representations.…”

Matrix model of QCD: Edge localized glueballs and phase transitions

“…One can also obtain the quantum states of these fuzzy spheres (both reducible and irreducible) by Gelfand-Naimark-Segal construction and compute the von Neumann entropy associated with the fuzzy spheres (as in [38]) to study the evolution of the system.…”

Monopoles, Dirac operator, and index theory for fuzzySU(3)/(U(1)×UAcharyya1, Díez2 2014Phys. Rev. DSelf Cite3020View full textAdd to dashboardCite

“…Among other things, M 6 F as the vacua of the matrix model (1) can be in an irreducible or reducible representation. It is easy to generalize the Schwinger construction of M 6 F by using BrandtGreenberg oscillators and obtain reducible algebras of M 6 F , as in [18]. We can obtain the quantum states for those reducible M 6 F 's using the prescription of GelfandNaimark-Segal.…”

“…The monopoles can be characterized by linear maps from one representation space of SUð3Þ to another [14][15][16][17][18]. Such sections are rectangular matrices that map a M 6 F of a given size to another of a different size.…”