Renormalizable Deformed Quantum Field Theory

In the last years quantum field theoretical models on deformed spacetime geometries have been studied in order to cope with both, the problems of quantum field theory and the quantization of quantum gravity. In this context a new difficulty, which is referred to as ultraviolet/ infrared mixing, was discovered six years ago. Unfortunately this property destroys the renormalizability, i.e. the perturbation series looses its significance. In order to cure this problem, Raimar Wulkenhaar and the applicant have accomplished an analysis of a canonically deformed scalar field theory. Eventually four operators turned out to be relevant/ marginal, whereas in the commutative three operators are sufficient. The collaboration of Prof. Rivasseau (Paris) was able to establish two alternative proofs, which led in particular to a renormalizable fermionic field theory. In this project we will continue applying the above method for renormalization. If the deformation matrix in four- dimensional spacetime turns out to be of rank two, a similar trick could lead to a well-defined field theory. Thereafter the analytic continuation of Euclidean geometry to a Lorentzian one will be studied. Moreover we will investigate analogous computation steps for matter fields on various deformed spaces. The vanishing of the beta- function at the self-dual point of the scalar field theory gives rise to the hope, that a nontrivial scalar field in four deformed spacetime dimensions can be constructed. However, the real challange is posed by deformed gauge fields. During the last months several proposals using formal deformations and twists, respectively, have been made. So far no quantum corrections whatsoever have been taken into account. In collaboration with Münster and the Technical University of Vienna we will study one- loop-calculations of such models initially. The idea is that a preferred model results via consistency checks and the demand for renormalizability. We attempt a full proof of renormalizability by means of the background field method. A major goal would be the formulation of a mathematically consistent deformed standard model, which includes electromagnetic, weak and strong interactions under the influence of a fluctuating spacetime. Ultimately we will also try to investigate deformed models of gravity.

In the last years quantum field theoretical models on deformed spacetime geometries have been studied in order to cope with both, the problems of quantum field theory and the quantization of quantum gravity. In this context a new difficulty, which is referred to as ultraviolet/ infrared mixing, was discovered six years ago. Unfortunately this property destroys the renormalizability, i.e. the perturbation series looses its significance. In order to cure this problem, Raimar Wulkenhaar and the applicant have accomplished an analysis of a canonically deformed scalar field theory. Eventually four operators turned out to be relevant/ marginal, whereas in the commutative three operators are sufficient. The collaboration of Prof. Rivasseau (Paris) was able to establish two alternative proofs, which led in particular to a renormalizable fermionic field theory. In this project we will continue applying the above method for renormalization. If the deformation matrix in four- dimensional spacetime turns out to be of rank two, a similar trick could lead to a well-defined field theory. Thereafter the analytic continuation of Euclidean geometry to a Lorentzian one will be studied. Moreover we will investigate analogous computation steps for matter fields on various deformed spaces. The vanishing of the beta- function at the self-dual point of the scalar field theory gives rise to the hope, that a nontrivial scalar field in four deformed spacetime dimensions can be constructed. However, the real challange is posed by deformed gauge fields. During the last months several proposals using formal deformations and twists, respectively, have been made. So far no quantum corrections whatsoever have been taken into account. In collaboration with Münster and the Technical University of Vienna we will study one-loop-calculations of such models initially. The idea is that a preferred model results via consistency checks and the demand for renormalizability. We attempt a full proof of renormalizability by means of the background field method. A major goal would be the formulation of a mathematically consistent deformed standard model, which includes electromagnetic, weak and strong interactions under the influence of a fluctuating spacetime. Ultimately we will also try to investigate deformed models of gravity.