We describe the self-interacting scalar eld on the truncated sphere and we perform the quantization using the functional (path) integral approach. The theory posseses a full symmetry with respect to the isometries of the sphere. We explicitely show that the model is nite and the UV-regularization automatically takes place. CERN-TH/95-138

We present a systematic framework for noncommutative (NC) QFT within the new concept of relativistic invariance based on the notion of twisted Poincaré symmetry (with all 10 generators), as proposed in ref. [7]. This allows to formulate and investigate all fundamental issues of relativistic QFT and offers a firm frame for the classification of particles according to the representation theory of the twisted Poincaré symmetry and as a result for the NC versions of CPT and spin-statistics theorems, among others, discussed earlier in the literature. As a further application of this new concept of relativism we prove the NC analog of Haag's theorem.

The truncated 4-dimensional sphere S 4 and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping the SO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent.

A noncommutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: 1) although we can have noncommutative U (n) (which we denote by U ⋆ (n)) gauge theory we cannot have noncommutative SU (n) and 2) the charges in noncommutative QED are quantized to just 0, ±1. We show how the latter problem with charge quantization, as well as with the gauge group, can be resolved by taking U ⋆ (3) × U ⋆ (2) × U ⋆ (1) gauge group and reducing the extra U (1) factors in an appropriate way. Then we proceed with building the noncommutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the noncommutative Standard Model (NCSM). In addition, among several peculiar features of our model, we address the inherent CP violation and new neutrino interactions.