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Duality and Braiding in Twisted Quantum Field Theory,” arXiv:0711.1525 [hepth
"... We reexamine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green’s functions i ..."
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We reexamine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green’s functions in the noncommutative and commutative theories, causality, and the absence of UV/IR mixing. We use these fields to define the functional integral formulation of twisted quantum field theory. We exploit techniques from braided tensor algebra to argue that the twisted Fock space states of these free fields obey conventional statistics. We support our claims with a detailed analysis of the modifications induced in the presence of background magnetic fields, which induces additional twists by magnetic translation operators and alters the effective noncommutative geometry seen by the twisted quantum fields. When two such field theories are dual to one another, we demonstrate that only our braided physical states Twisted quantum field theory is a modification of the traditional approach to noncommutative field theory [19, 42] aimed at restoring the symmetries of spacetime which are broken by noncommutativity.
Cosmological and black hole spacetimes in twisted noncommutative gravity
 arXiv:0906.2730. A. Schenkel and C.F. Uhlemann
"... Abstract: We derive noncommutative Einstein equations for abelian twists and their solutions in consistently symmetry reduced sectors, corresponding to twisted FRW cosmology and Schwarzschild black holes. While some of these solutions must be rejected as models for physical spacetimes because they c ..."
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Abstract: We derive noncommutative Einstein equations for abelian twists and their solutions in consistently symmetry reduced sectors, corresponding to twisted FRW cosmology and Schwarzschild black holes. While some of these solutions must be rejected as models for physical spacetimes because they contradict observations, we find also solutions that can be made compatible with low energy phenomenology, while exhibiting strong
Noncommutative Gravity Solutions
, 2009
"... We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented. Inspired by [1, 2], we obtain solutions of noncommutative Einstein equations by considering twists that a ..."
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We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented. Inspired by [1, 2], we obtain solutions of noncommutative Einstein equations by considering twists that are compatible with
Twisted noncommutative field theory with the Wick–Voros and Moyal products, Phys
 Rev. D
"... We present a comparison of the noncommutative field theories built using two different star products: Moyal and WickVoros (or normally ordered). For the latter we discuss both the classical and the quantum field theory in the quartic potential case, and calculate the Green’s functions up to one loo ..."
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We present a comparison of the noncommutative field theories built using two different star products: Moyal and WickVoros (or normally ordered). For the latter we discuss both the classical and the quantum field theory in the quartic potential case, and calculate the Green’s functions up to one loop, for the two and four points cases. We compare the two theories in the context of the noncommutative geometry determined by a Drinfeld twist, and the comparison is made at the level of Green’s functions and Smatrix. We find that while the Green’s functions are different for the two theories, the Smatrix is It is likely that at short distances spacetime has to be described by different geometrical structures, and that the very concept of point and localizability may no longer be adequate. This is one of the main motivations for the introduction of noncommutative geometry [1, 2, 3]. The simplest kind of
The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space ⋆
"... Abstract. This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θ µν) is a variable of th ..."
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Abstract. This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θ µν) is a variable of the NC system and has a canonical conjugate momentum. Namely, for instance, in NC quantum mechanics we will show that θ ij (i, j = 1, 2, 3) is an operator in Hilbert space and we will explore the consequences of this socalled “operationalization”. The DFRA formalism is constructed in an extended spacetime with independent degrees of freedom associated with the object of noncommutativity θ µν. We will study the symmetry properties of an extended x+θ spacetime, given by the group P ′ , which has the Poincaré group P as a subgroup. The Noether formalism adapted to such extended x + θ (D = 4 + 6) spacetime is depicted. A consistent algebra involving the enlarged set of canonical operators is described, which permits one to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the NC operator sector, resulting in new features. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC
NONCOMMUTATIVE DIFFERENTIAL FORMS ON THE KAPPADEFORMED SPACE
, 812
"... Abstract. We construct a differential algebra of forms on the kappadeformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of oneforms and nilpotent exterior derivatives. We derive explicit expressions for the ..."
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Abstract. We construct a differential algebra of forms on the kappadeformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of oneforms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and oneforms in covariant and noncovariant realizations. We also introduce higherorder forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are not gradedcommutative, but they satisfy the graded Jacobi identity. The starproduct of classical differential forms is also defined. It is shown that the product depends on the realizations of both the noncommutative coordinates and oneforms on the kappadeformed space. 1.
Symmetry, Integrability and Geometry: Methods and Applications Twist Quantization of String and Hopf Algebraic Symmetry ⋆
"... doi:10.3842/SIGMA.2010.068 Abstract. We describe the twist quantization of string worldsheet theory, which unifies the description of quantization and the target space symmetry, based on the twisting of Hopf and module algebras. We formulate a method of decomposing a twist into successive twists to ..."
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doi:10.3842/SIGMA.2010.068 Abstract. We describe the twist quantization of string worldsheet theory, which unifies the description of quantization and the target space symmetry, based on the twisting of Hopf and module algebras. We formulate a method of decomposing a twist into successive twists to analyze the twisted Hopf and module algebra structure, and apply it to several examples, including finite twisted diffeomorphism and extra treatment for zero modes. Key words: string theory; qunatization; Hopf algebra; Drinfeld twist
Symmetry, Integrability and Geometry: Methods and Applications Exponential Formulas and Lie Algebra Type Star Products
"... Abstract. Given formal differential operators Fi on polynomial algebra in several variables x1,..., xn, we discuss finding expressions Kl determined by the equation exp ( ∑ i xiFi)(exp( j qjxj)) = exp ( ∑ l Klxl) and their applications. The expressions for Kl are related to the coproducts for defor ..."
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Abstract. Given formal differential operators Fi on polynomial algebra in several variables x1,..., xn, we discuss finding expressions Kl determined by the equation exp ( ∑ i xiFi)(exp( j qjxj)) = exp ( ∑ l Klxl) and their applications. The expressions for Kl are related to the coproducts for deformed momenta for the noncommutative spacetimes of Lie algebra type and also appear in the computations with a class of star products. We find combinatorial recursions and derive formal differential equations for finding Kl. We elaborate an example for a Lie algebra su(2), related to a quantum gravity application from the literature. Key words: star product; exponential expression; formal differential operator 2010 Mathematics Subject Classification: 81R60; 16S30; 16S32; 16A58 1
The Structure of Spacetime and Noncommutative Geometry
, 2008
"... We give a general and nontechnical review of some aspects of noncommutative geometry as a tool to understand the structure of spacetime. We discuss the motivations for the constructions of a noncommutative geometry, and the passage from commutative to noncommutative spaces. We then give a brief desc ..."
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We give a general and nontechnical review of some aspects of noncommutative geometry as a tool to understand the structure of spacetime. We discuss the motivations for the constructions of a noncommutative geometry, and the passage from commutative to noncommutative spaces. We then give a brief description of Connes approach to the standard model, of the noncommutative geometry of strings and of field theory on noncommutative spaces. We also discuss the role of symmetries and some possible consequences for cosmology. Talk given at the workshop: Geometry, Topology, QFT and Cosmology, Paris, 2830 In this contribution I will give a general, and personal, overview of some attempts that physicists and mathematicians are making to understand the structure of spacetime at extremely small distances. The tool used for the description of spacetime at the Planck length scale is what is called Noncommutative