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Derivations of the Moyal Algebra and Noncommutative Gauge Theories
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2009
"... The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital a ..."
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Cited by 12 (3 self)
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The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z2graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z2graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang–Mills–Higgs type models on Moyal (or related) algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC ϕ4model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.
Noncommutative εgraded connections
, 2012
"... We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differ ..."
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Cited by 10 (2 self)
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We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of εgraded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also to interpret mathematically a noncommutative gauge field theory.
Examples of derivationbased differential calculi related to noncommutative gauge theories
 INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
, 2008
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Gauge theories in noncommutative geometry
, 2011
"... In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommut ..."
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In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features.
On the Development of NonCommutative TranslationInvariant Quantum Gauge Field Models
, 2009
"... Aiming to understand the most fundamental principles of nature one has to approach the highest possible energy scales corresponding to the smallest possible distances – the Planck scale. Historically, three different theoretical fields have been developed to treat the problems appearing in this ende ..."
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Aiming to understand the most fundamental principles of nature one has to approach the highest possible energy scales corresponding to the smallest possible distances – the Planck scale. Historically, three different theoretical fields have been developed to treat the problems appearing in this endeavor: string theory, quantum gravity, and noncommutative (NC) quantum field theory (QFT). The latter was originally motivated by the conjecture that the introduction of uncertainty relations between spacetime coordinates introduces a natural energy cutoff, which should render the resulting computations well defined and finite. Despite failing to fulfill this expectation, NC physics is a challenging field of research, which has proved to be a fruitful source for new ideas and methods. Mathematically, noncommutativity is implemented by the so called Weyl quantization, giving rise to a modified product — the GroenewoldMoyal product. It realizes an operator ordering, and allows to work within the well established framework of QFT on noncommutative spaces. The main obstacle of NCQFT is the appearance of singularities being shifted from high to low energies. This effect, being referred to as ‘UV/IR mixing’, is a direct consequence of the de
On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... The harmonic term in the scalar field theory on the Moyal space removes the UVIR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann–Szabo duality, the superalgebraic approach and the noncommuta ..."
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The harmonic term in the scalar field theory on the Moyal space removes the UVIR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann–Szabo duality, the superalgebraic approach and the noncommutative scalar curvature interpretation. Then, we show some deep relationship between these interpretations.