Combinatorial Dyson-Schwinger equations in noncommutative field theory. (English) Zbl 1273.81215
In this paper, the authors introduce the Hopf algebra structure which describes the renormalization of this noncommutative model (the translation-invariant model introduced in [R. Gurau et al., Commun. Math. Phys. 287, No. 1, 275–290 (2009; Zbl 1170.81041)]). Moreover, they present the pre-Lie and Lie algebra structures associated to graphs. Also several differences are exhibited when the ribbon graph representation of noncommutative quantum field theory (briefly NCQFT) are utilized instead of the usual Feynman graphs of commutative quantum field theory. Further, they offer the definition of Hochschild one-cocyles \(B_{+}^\gamma\) which make it possible to write down the combinatorial Dyson-Schwinger equations in NCQFT. They give the corresponding theorems and work out completely the one- and two-loop implementations of these theorems.
Reviewer: Saeid Jafari (Copenhagen)
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 81R60 | Noncommutative geometry in quantum theory |
| 81T18 | Feynman diagrams |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
Keywords:
noncommutative quantum field theory; combinatorial Dyson-Schwinger equation; Moyal space; renormalization; Hopf algebras; pre-Lie and Lie algebrasCitations:
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