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 elaborate on the role of quantum statistics in twisted Poincaré invariant theories. It is shown that, in order to have twisted Poincaré group as the symmetry of a quantum theory, statistics must be twisted. It is also confirmed that the removal of UV-IR mixing (in the absence of gauge fields) in such theories is a natural consequence.

We investigate the thermodynamics of a four-dimensional charged black hole in a finite cavity in asymptotically flat and asymptotically de Sitter space. In each case, we find a Hawking-Page-like phase transition between a black hole and a thermal gas very much like the known transition in asymptotically anti-de Sitter space. For a "supercooled" black hole-a thermodynamically unstable black hole below the critical temperature for the Hawking-Page phase transitionthe phase diagram has a line of first-order phase transitions that terminates in a second order point. For the asymptotically flat case, we calculate the critical exponents at the second order phase transition and find that they exactly match the known results for a charged black hole in anti-de Sitter space. We find strong evidence for similar phase transitions for the de Sitter black hole as well. Thus many of the thermodynamic features of charged anti-de Sitter black holes do not really depend on asymptotically anti-de Sitter boundary conditions; the thermodynamics of charged black holes is surprisingly universal.

Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of discrete physics and matrix models and discuss mathematically coherent discretizations of monopoles and solitons using fuzzy physics and noncommutative geometry. A fuzzy sigma-model action for the two-sphere fulfilling a fuzzy Belavin-Polyakov bound is also put forth.Comment: 17 pages, Latex. Uses amstex, amssymb.Spelling of the name of one Author corrected. To appear in Commun.Math.Phy

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.

We study the time-independent modes of a massless scalar field in various black hole backgrounds, and show that for these black holes, the time-independent mode is localized at the horizon. A similar analysis is done for time-independent, equilibrium modes of the five-dimensional plane AdS black hole. A self-adjointness analysis of this problem reveals that in addition to the modes corresponding to the usual glueball states, there is a discrete infinity of other equilibrium modes with imaginary mass for the glueball. We suggest these modes may be related to a Savvidy-Nielsen-Olesen-like vacuum instability in QCD.

Recent work [1,2] indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qft's) on the Groenewold-Moyal (GM) plane. In this approach to the qft's, statistics gets twisted and the S-matrix in the non-gauge qft's become independent of the noncommutativity parameter θ µν . Here we show that the noncommutative algebra has a commutative spacetime algebra as a substructure: the Poincaré, diffeomorphism and gauge groups are based on this algebra in the twisted approach as is known already from the earlier work of [1]. It is natural to base covariant derivatives for gauge and gravity fields as well on this algebra. Such an approach will in particular introduce no additional gauge fields as compared to the commutative case and also enable us to treat any gauge group (and not just U (N )). Then classical gravity and gauge sectors are the same as those for θ µν = 0, but their interactions with matter fields are sensitive to θ µν . We construct quantum noncommutative gauge theories (for arbitrary gauge groups) by requiring consistency of twisted statistics and gauge invariance. In a subsequent paper (whose results are summarized here), the locality and Lorentz invariance properties of the S-matrices of these theories will be analyzed, and new non-trivial effects coming from noncommutativity will be elaborated. This paper contains further developments of [3] and a new formulation based on its approach.