La renormalisation dans la theorie non commutative des champs

TL;DR

This paper investigates the renormalization of the non-commutative $\,\phi^4$ quantum field theory on the Moyal plane, demonstrating its renormalizability and bounded coupling flow, with implications for quantum gravity and string theory.

Contribution

It proves the all-order renormalizability of the Grosse-Wulkenhaar model and introduces new representations, advancing understanding of non-commutative quantum field theories.

Findings

Proved renormalizability at all orders in position space.

Established bounded flow of the coupling constant.

Developed parametric and Mellin representations for the model.

Abstract

Non commutative quantum field theory is a possible candidate for the quantization of gravity. In our thesis we study in detail the $\phi 4$ model on the Moyal plane with an harmonic potential. Introduced by Grosse and Wulkenhaar, this model exhibits the Langmann-Szabo duality not only for the vertex but also for the propagator. We have obtained several results concerning this model. We have proved the renormalisability of this theory at all orders in the position space. We have introduced the parametric and Complete Mellin representation for the model. Furthermore we have proved that the coupling constant has a bounded flow at all orders in perturbation theory. Finally we have achieved the dimensional regularization and renormalization of the model. Further possible studies include the study of gauge theory on the Moyal plane and there possible applications for the quantization of gravity. The connections with string theory and loop quantum gravity should also be investigated.