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On the Problem of Renormalizability in NonCommutative Gauge Field Models — A Critical Review
"... When considering quantum field theories on noncommutative spaces one inevitably encounters the infamous UV/IR mixing problem. So far, only very few renormalizable models exist and all of them describe noncommutative scalar field theories on fourdimensional Euclidean GroenewoldMoyal deformed space ..."
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When considering quantum field theories on noncommutative spaces one inevitably encounters the infamous UV/IR mixing problem. So far, only very few renormalizable models exist and all of them describe noncommutative scalar field theories on fourdimensional Euclidean GroenewoldMoyal deformed space, also known as ‘θdeformed space ’ R4 θ. In this work we discuss some major obstacles of constructing a renormalizable noncommutative gauge field model and sketch some possible ways out. 1
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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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.
Multiorientable Group Field Theory
 J. Phys. A
, 2012
"... Group Field Theories (GFT) are quantum field theories over group manifolds; they can be seen as a generalization of matrix models. GFT Feynman graphs are tensor graphs generalizing ribbon graphs (or combinatorial maps); these graphs are dual not only to manifolds. In order to simplify the topologica ..."
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Group Field Theories (GFT) are quantum field theories over group manifolds; they can be seen as a generalization of matrix models. GFT Feynman graphs are tensor graphs generalizing ribbon graphs (or combinatorial maps); these graphs are dual not only to manifolds. In order to simplify the topological structure of these various singularities, colored GFT was recently introduced and intensively studied since. We propose here a different simplification of GFT, which we call multiorientable GFT. We study the relation between multiorientable GFT Feynman graphs and colorable graphs. We prove that tadfaces and some generalized tadpoles are absent. Some Feynman amplitude computations are performed. A few remarks on the renormalizability of both multiorientable and colorable GFT are made. A generalization from threedimensional to fourdimensional theories is also proposed. Key words: combinatorics, group field theory, Feynman graphs, orientability ar X iv
Magnetic fields in noncommutative quantum mechanics
"... We discuss various descriptions of a quantum particle on noncommutative space in a (possibly nonconstant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the lite ..."
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We discuss various descriptions of a quantum particle on noncommutative space in a (possibly nonconstant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature. 1 We dedicate these notes to the memory of Julius Wess whose scientific work was largely devoted to the study of gauge fields and whose latest interests concerned physical theories on noncommutative space. The fundamental and inspiring contributions of Julius to Theoretical Physics will always bear with us, but his great kindness, his clear and enthusiastic presentations, and his precious advice will be missed by all those who had the chance to meet him. 2
The Jacobian conjecture as a problem of perturbative quantum field theory, math.CO/0208173, preprint
, 2002
"... The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory. ..."
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The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory.
Tree Quantum Field Theory
 Annales Henri Poincare 10 (2009) 867 [arXiv:0807.4122 [hepth
"... We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermio ..."
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We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define nonperturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the GrosseWulkenhaar model. Perhaps most importantly it removes the spacetime background from its central place in QFT, paving the way for a nonperturbative definition of field theory in noninteger dimension. I
The two dimensional Hubbard Model at halffilling: I. Convergent Contributions, Journ. Stat. Phys. Vol 106
 Ann. Henri Poincaré
, 2002
"... We prove analyticity theorems in the coupling constant for the Hubbard model at halffilling. The model in a single renormalization group slice of index i is proved to be analytic in λ for λ  ≤ c/i for some constant c, and the skeleton part of the model at temperature T (the sum of all graphs wit ..."
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We prove analyticity theorems in the coupling constant for the Hubbard model at halffilling. The model in a single renormalization group slice of index i is proved to be analytic in λ for λ  ≤ c/i for some constant c, and the skeleton part of the model at temperature T (the sum of all graphs without two point insertions) is proved to be analytic in λ for λ  ≤ c/log T  2. These theorems are necessary steps towards proving that the Hubbard model at halffilling is not a Fermi liquid (in the mathematically precise sense of Salmhofer). I
Interacting Fermi liquid in three dimensions at finite temperature: Part I: Convergent Contributions
, 2008
"... In this paper we complete the first step, namely the uniform bound on completely convergent contributions, towards proving that a three dimensional interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The analysis relies on a direct space decomposition of the propagator, on a ..."
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In this paper we complete the first step, namely the uniform bound on completely convergent contributions, towards proving that a three dimensional interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The analysis relies on a direct space decomposition of the propagator, on a bosonic multiscale cluster expansion and on the Hadamard inequality, rather than on a Fermionic expansion and an angular analysis in momentum space, as was used in the recent proof by two of us of Salmhofer’s criterion in two dimensions.