The Groenewold-Moyal plane is the algebra A θ (R d+1 ) of functions on R d+1 with the * -product as the multiplication law, and the commutator [x µ ,x ν ] = iθ µν (µ, ν = 0, 1, ..., d) between the coordinate functions. Chaichian et al. [1] and Aschieri et al. [2] have proved that the Poincaré group acts as automorphisms on A θ (R d+1 ) if the coproduct is deformed. (See also the prior work of Majid [3], Oeckl [4] and Grosse et al [5]). In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of [2]. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.PACS numbers: 11.10. Nx, 11.30.Cp Dedicated to Rafael Sorkin, our friend and teacher, and a true and creative seeker of knowledge. *

Dedicated to Rafael Sorkin, our friend and teacher, and a true and creative seeker of knowledge. PrefaceOne of us (Balachandran) gave a course of lectures on "Fuzzy Physics" during spring, 2002 for students of Syracuse and Brown Universities. The course which used video conferencing technology was also put on the websites [1]. Subsequently A.P. Balachandran, S. Kürkçüoǧlu and S.Vaidya decided to edit the material and publish them as lecture notes. The present book is the outcome of that effort.The recent interest in fuzzy physics begins from the work of Madore [2, 3] and others even though the basic mathematical ideas are older and go back at least to Kostant and Kirillov [4] and Berezin [5]. It is based on the fundamental observation that coadjoint orbits of Lie groups are symplectic manifolds which can therefore be quantized under favorable circumstances. When that can be done, we get a quantum representation of the manifold. It is the fuzzy manifold for the underlying "classical manifold". It is fuzzy because no precise localization of points thereon is possible. The fuzzy manifold approaches its classical version when the effective Planck's constant of quantization goes to zero.Our interest will be in compact simple and semi-simple Lie groups for which coadjoint and adjoint orbits can be identified and are compact as well. In such a case these fuzzy manifold is a finite-dimensional matrix algebra on which the Lie group acts in simple ways. Such fuzzy spaces are therefore very simple and also retain the symmetries of their classical spaces. These are some of the reasons for their attraction.There are several reasons to study fuzzy manifolds. Our interest has its roots in quantum field theory (qft). Qft's require regularization and the conventional nonperturbative regularization is lattice regularization. It has been extensively studied for over thirty years. It fails to preserve space-time symmetries of quantum fields. It also has problems in dealing with topological subtleties like instantons, and can deal with index theory and axial anomaly only approximately. Instead fuzzy physics does not have these problems. So it merits investigation as an alternative tool to investigate qft's.A related positive feature of fuzzy physics, is its ability to deal with supersymmetry(SUSY) in a precise manner [6,7,8,9]. (See however,[10]). Fuzzy SUSY models are also finite-dimensional matrix models amenable to numerical work, so this is another reason for our attraction to this field.Interest in fuzzy physics need not just be utilitarian. Physicists have long speculated that space-time in the small has a discrete structure. Fuzzy space-time gives a very concrete and interesting method to model this speculation and test its consequences. There are many generic consequences of discrete space-time, like CPT and causality violations, and distortions of the Planck spectrum. Among these must be characteristic signals for fuzzy physics, but they remain to be identified.iii iv PREFACE These lecture notes are not exhaustive, and ref...

We show that topological superfluid strings/vortices with flux tubes exist in the color-flavor locked (CFL) phase of color superconductors. Using a Ginzburg-Landau free energy we find the configurations of these strings. These strings can form during the transition from the normal phase to the CFL phase at the core of very dense stars. We discuss an interesting scenario for a network of strings and its evolution at the core of dense stars.

We derive an explicit expression for an associative * -product on the fuzzy complex projective space, CP N−1 F . This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on CP N−1 F , represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU (N ). In the limit of infinite dimensional matrices we recover the commutative algebra of functions on CP N−1 . Derivatives on CP N−1 F are also expressed as matrix commutators.

Nuclear Physics, Section B 461 (1996) 581-596. doi:10.1016/0550-3213(95)00622-2Received by publisher: 1995-10-04Harvest Date: 2016-01-04 12:22:36DOI: 10.1016/0550-3213(95)00622-2Page Range: 581-59