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Quantum field theory on noncommutative spaces
"... A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommuta ..."
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Cited by 400 (26 self)
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A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative YangMills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an indepth study of the gauge group of noncommutative YangMills theory. Some of the more mathematical ideas and
General properties of noncommutative field theories,” Nucl
 Phys. B
, 2000
"... In this paper we study general properties of noncommutative field theories obtained from the SeibergWitten limit of string theories in the presence of an external Bfield. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular we present a proo ..."
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Cited by 76 (3 self)
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In this paper we study general properties of noncommutative field theories obtained from the SeibergWitten limit of string theories in the presence of an external Bfield. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular we present a proof of the CPT theorem for theories with spacespace noncommutativity. We analyze as well questions associated to the spinstatistics connections, and show that noncommutative N = 4, U(1) gauge theory can be softly broken to N = 0 satisfying the axioms and providing an example where the Wilsonian low energy effective action can be constructed without UV/IR problems, after a judicious choice of soft breaking parameters is made. We also assess the phenomenological prospects of such a theory, which are in fact rather negative. 1
Exact solution of quantum field theory on noncommutative phase spaces
 017, 2004, hepth/0308043. – 43
"... We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the GroenewoldMoyal starproduct. Explicit results are presented for all Green’s fun ..."
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Cited by 59 (6 self)
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We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the GroenewoldMoyal starproduct. Explicit results are presented for all Green’s functions in arbitrary even spacetime dimensionality. Various scaling limits of the field theory are analysed nonperturbatively and the renormalizability of each limit examined. A supersymmetric extension of the field theory is also constructed in which the supersymmetry transformations are parametrized by differential operators in an infinitedimensional noncommutative algebra.
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 49 (5 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Noncommutative geometry of finite groups
 J. Phys. A
, 1996
"... A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left, right and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A dif ..."
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Cited by 32 (3 self)
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A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left, right and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of ‘extensible connections’. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a ‘dual connection ’ which exists on the dual bimodule (as defined in this work).2 1
String geometry and the noncommutative torus
 Commun. Math. Phys
, 1999
"... We describe an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra A and the noncommutative torus. We show that the tachyon subalgebra of A is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra o ..."
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Cited by 30 (8 self)
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We describe an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra A and the noncommutative torus. We show that the tachyon subalgebra of A is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding even real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d, d; Z) Morita equivalences between ddimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly dualitysymmetric. The dualityinvariant gauge theory is manifestly covariant but contains highly nonlocal interactions. We show that it also admits a new sort of particleantiparticle duality which enables the construction of instanton field configurations in any dimension. The duality nonsymmetric onshell projection of the field theory is shown to coincide with the standard nonabelian YangMills gauge theory minimally coupled to massive Dirac fermion fields. 1
Discrete Differential Calculus, Graphs, Topologies and Gauge Theory
 TO APPEAR IN J. MATH. PHYS.
, 1994
"... Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘reduction ’ of the ‘universal differential algebra ’ and this allows a systematic exploration of differential algebras on a given set. Assoc ..."
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Cited by 28 (4 self)
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Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘reduction ’ of the ‘universal differential algebra ’ and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a ‘Hasse diagram’ determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the twopoint space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an ‘internal’ discrete space (à la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a ‘symmetric lattice’ is studied which (in a certain continuum limit) turns out to be related to a ‘noncommutative differential calculus’ on manifolds.
The Small Scale Structure of SpaceTime: A Bibliographical Review
, 1995
"... This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1 ..."
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Cited by 22 (0 self)
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This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1