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...

LECTURES ON FUZZY AND FUZZY SUSY PHYSICS

The fuzzy supersphere S (2,2) F is a finite-dimensional matrix approximation to the supersphere S (2,2) incorporating supersymmetry exactly. Here the ⋆-product of functions on S (2,2) F is obtained by utilizing the OSp(2, 1) coherent states. We check its graded commutative limit to S (2,2) and extend it to fuzzy versions of sections of bundles using the methods of [1]. A brief discussion of the geometric structure of our ⋆-product completes our work.

This paper discusses the formulation of the non-commutative Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard *products are not correct for such manifolds, the standard non-commutative CS theory is not also appropriate here. Instead we formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work which has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approach are operators supported on a subspace of finite dimension N + η of the Hilbert space of eigenstates of a simple harmonic oscillator with N , η ∈ Z + and N = 0. This oscillator is associated with the underlying Moyal plane. The resultant matrix CS model has a fuzzy edge. It becomes the required sharp edge when N and η → ∞ in a suitable sense. The non-commutative CS theory on the strip is defined by this limiting procedure. After performing the canonical constraint analysis of the matrix theory, we find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Moody algebra. Our study shows that there are (η + 1) 2 abelian charges(observables) given by the matrix elements (Â i ) N −1 N −1 and (Â i ) nm (where n or m ≥ N ) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the N th level. We observe that all non-abelian edge observables except these three can be constructed from the (η+1) 2 abelian charges above. Using some of the results of this analysis we discuss in detail the limit where this matrix model approximates the CS theory on the infinite strip. Finally, we include a short section containing our comments on the commutative limit of our model, where we also give a closed formula for the central charge of the Kac-Moody-like algebra of the non-commutative CS theory in terms of the diagonal coherent state matrix elements of operators and star products.

We consider a U(2) Yang-Mills theory on M × S 2 F where M is a Riemannian manifold and S 2 F is the fuzzy sphere. Using essentially the representation theory of SU(2) we determine the most general SU(2)-equivariant gauge field on M×S 2 F . This allows us to reduce the Yang-Mills theory on M × S 2 F down to an abelian Higgs-type model over M. Depending on the enforcement (or non-enforcement) of a "constraint" term, the latter may (or may not) lead to the standard criticallycoupled abelian Higgs model in the commutative limit, S 2 F → S 2 . For M = R 2 , we find that the abelian Higgs-type model admits vortex solutions corresponding to instantons in the original Yang-Mills theory. Vortices are in general no longer BPS, but may attract or repel according to the values of parameters.

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We give a simplified formula for the star product on P n L , which enables us to define a twist element suited for discussing a Drinfeld twist like structure on fuzzy complex projective spaces. The existence of such a twist will have several consequences for field theories on fuzzy spaces, some of which we discuss in the present paper. As expected, we find that the twist of the coproduct is trivial for the generators of isometries on P n L . Furthermore, the twist allows us to define a covariant tensor calculus on P n L from the perspective of the standard embedding of P n in flat Euclidean space. That is, we can -in principlefind a representation of a truncated subgroup of the diffeomorphisms on P n on the algebra of functions on P n L . Using this calculus, we eventually write down an Einstein-Hilbert action on the fuzzy sphere, which is invariant under twisted diffeomorphisms.

It is by now clear that infrared sector of QED has an intriguingly complex structure. Based on earlier pioneering works on this subject, two of us recently proposed a simple modification of QED by constructing a generalization of the U (1) charge group of QED to the "Sky" group incorporating the known spontaneous Lorentz violation due to infrared photons, but still compatible in particular with locality [1]. There it was shown that the "Sky" group is generated by the algebra of angle dependent charges and a study of its superselection sectors has revealed a manifest description of spontaneous breaking of Lorentz symmetry. We further elaborate this approach here and investigate in some detail the properties of charged particles dressed by the infrared photons. We find that Lorentz violation due to soft photons may be manifestly codified in an angle dependent fermion mass modifying therefore the fermion dispersion relations. The fact that the masses of the charged particles are not Lorentz invariant affects their spin content too.Time dilation formulae for decays should also get corrections. We speculate that these effects could be measured possibly in muon decay experiments. 1