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Adrian Tanasa

Publications and source records attributed to Adrian Tanasa.

At least 55 records · Page 3Linked to original sources
arXiv2012

Some combinatorial aspects of quantum field theory

In this short survey we present the appearance of some combinatorial notions in quantum field theory. We first focus on topological graph polynomials (the Tutte polynomial and its multivariate version) and their relation with the parametric representation of the commutative $Φ^4$ field theory. We then generalize this to ribbon graphs and present the relation of the Bollobás-Riordan polynomial with the parametric representation of some $Φ^4$ field theory on the non-commutative Moyal space. We also review the rôle played by the Connes-Kreimer Hopf algebra as the combinatorial backbone of the renormalization process in field theories. We then show how this generalizes to the scalar $Φ^4$ field theory implemented on the non-commutative Moyal space. Finally, some perspectives for the further generalization of these tools to quantum gravity tensor models are briefly sketched.

math.CO↗
arXiv2012

Combinatorics of random tensor models

In this short review we introduce group field theory, a particular class of random tensor models, which represents nowadays one of the candidates for a fundamental theory of quantum gravity. We insist on the combinatorial richness of associated structures, namely tensor graphs, natural generalization of ribbon graph (or combinatorial maps).

math.CO↗
arXiv2012

Multi-orientable Group Field Theory

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 multi-orientable GFT. We study the relation between multi-orientable 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 multi-orientable and colorable GFT are made. A generalization from three-dimensional to four-dimensional theories is also proposed.

math.CO↗
arXiv2011

Generalization of the Bollobás-Riordan polynomial for tensor graphs

Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial $\mathcal T$ encoding the supplementary topological information. This polynomial is a natural generalization of the Bollobás-Riordan polynomial (used to characterize matrix graphs) and is different of the Gur\uau polynomial, (R. Gur\uau, "Topological Graph Polynomials in Colored Group Field Theory", Annales Henri Poincare {\bf 11}, 565-584 (2010)) defined for a particular class of tensor graphs, the colorable ones. The polynomial $\mathcal T$ is defined for both colorable and non-colorable graphs and it is proved to satisfy the contraction/deletion relation. A non-trivial example of a non-colorable graphs is analyzed.

math.CO↗
arXiv2010

Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models

We investigate the group field theory formulation of the EPRL/FK spin foam models. These models aim at a dynamical, i.e. non-topological formulation of 4D quantum gravity. We introduce a saddle point method for general group field theory amplitudes and compare it with existing results, in particular for a second order correction to the EPRL/FK propagator.

gr-qc↗
arXiv2010

Translation-Invariant Noncommutative Renormalization

We review here the construction of a translation-invariant scalar model which was proved to be perturbatively renormalizable on Moyal space. Some general considerations on nonlocal renormalizability are given. Finally, we present perspectives for generalizing these quantum field theoretical techniques to group field theory, a new setting for quantum gravity.

hep-th↗
arXiv2010

Algebraic structures in quantum gravity

Starting from a recently-introduced algebraic structure on spin foam models, we define a Hopf algebra by dividing with an appropriate quotient. The structure, thus defined, naturally allows for a mirror analysis of spin foam models with quantum field theory, from a combinatorial point of view. A grafting operator is introduced allowing for the equivalent of a Dyson-Schwinger equation to be written. Non-trivial examples are explicitly worked out. Finally, the physical significance of the results is discussed.

gr-qc↗
arXiv2010

Curing the UV/IR mixing for field theories with translation-invariant $\star$ products

The ultraviolet/infrared (UV/IR) mixing of noncommutative field theories has been recently shown to be a generic feature of translation- invariant associative products. In this paper we propose to take into account the quantum corrections of the model to modify in this way the noncommutative action. This idea was already used to cure the UV/IR mixing for theories on Moyal space. We show that in the present framework also, this proposal proves successful for curing the mixing. We achieve this task by explicit calculations of one and higher loops Feynman amplitudes. For the sake of completeness, we compute the form of the new action in the matrix base for the Wick-Voros product.

hep-th↗
arXiv2009

Parametric representation of a translation-invariant renormalizable noncommutative model

We construct here the parametric representation of a translation-invariant renormalizable scalar model on the noncommutative Moyal space of even dimension $D$. This representation of the Feynman amplitudes is based on some integral form of the noncommutative propagator. All types of graphs (planar and non-planar) are analyzed. The rôle played by noncommutativity is explicitly shown. This parametric representation established allows to calculate the power counting of the model. Furthermore, the space dimension $D$ is just a parameter in the formulas obtained. This paves the road for the dimensional regularization of this noncommutative model.

math-ph↗
arXiv2009

Combinatorial Dyson-Schwinger equations in noncommutative field theory

We give here the Hopf algebra structure describing the noncommutative renormalization of a recently introduced translation-invariant model on Moyal space. We define Hochschild one-cocyles $B_+^γ$ which allows us to write down the combinatorial Dyson-Schwinger equations for noncommutative quantum field theory. One- and two-loops examples are explicitly worked out.

hep-th↗
arXiv2009

Commutative limit of a renormalizable noncommutative model

Renormalizable $ϕ^{\star 4}_4$ models on Moyal space have been obtained by modifying the commutative propagator. But these models have a divergent "naive" commutative limit. We explain here how to obtain a coherent such commutative limit for a recently proposed translation-invariant model. The mechanism relies on the analysis of the uv/ir mixing in general Feynman graphs.

hep-th↗
arXiv2008

One-loop $β$ functions of a translation-invariant renormalizable noncommutative scalar model

Recently, a new type of renormalizable $ϕ^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(θ^2p^2)$ term. We calculate here the $β$ and $γ$ functions at one-loop level for this model. The coupling constant $β_λ$ function is proved to have the same behaviour as the one of the $ϕ^4$ model on the commutative $\mathbb{R}^4$. The $β_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done.

math-ph↗
arXiv2008

Scalar and gauge translation-invariant noncommutative models

We make here a short overview of the recent developments regarding translation-invariant models on the noncommutative Moyal space. A scalar model was first proposed and proved renormalizable. Its one-loop renormalization group flow and parametric representation were calculated. Furthermore, a mechanism to take its commutative limit was recently given. Finally, a proposition for a renormalizable, translation-invariant gauge model was made.

hep-th↗
arXiv2007

Parametric representation of "critical" noncommutative QFT models

We extend the parametric representation of renormalizable non commutative quantum field theories to a class of theories which we call "critical", because their power counting is definitely more difficult to obtain. This class of theories is important since it includes gauge theories, which should be relevant for the quantum Hall effect.

math-ph↗