On the renormalizability of noncommutative U(1) Gauge Theory: an algebraic approach

Full text: The year of 1999 witnessed two major developments in the noncommutative quantum field theory program. In the first one, Seiberg and Witten, inspired by the previously known result that the low energy limit of open strings could lead both to a gauge theory defined on a noncommutative space as well as to an usual commutative gauge theory, depending only on gauge choices, announced the existence of what became called the Seiberg-Witten map between noncommutative and commutative gauge theories.This achievement was then fully tested and confirmed by several authors both in the general structure of gauge transformations as in specific examples of gauge theories. It opened a window to an alternative approach to the quantum properties of the noncommutative theories. The second development just revealed the kind of difficulties one has to face when tackling the renormalization of field theories in the noncommutative space. An intrinsic mixing between high and low energy scales was associated to the noncommutativity of space-time, generating divergence which in the general case make these theories not renormalizable as they stand, the case of noncommutative gauge theories being no exception. Recently, it was finally understood that this infrared-ultraviolet (IR-UV) mix is still present even after a Seiberg- Witten map, showing that the commutative theories generated by their noncommutative counterparts suffer from the same non renormalizability. It took some time until the first proposal appeared in order to cure a noncommutative scalar theory from this IR divergence. The basic idea was to alter the free propagator of the theory through the introduction of an harmonic potential, then changing its low energy behavior. This in fact made the theory convergent in the infrared region, but at the cost of explicitly breaking translation invariance. In the reference Commun. Math. Phys. 287 : 275-290, (2009), this problem was circumvented now by the introduction of a nonlocal term, again assuring that the IR-UV mixing would be cured for a scalar theory. Soon, this proposal was generalized to the case of a noncommutative gauge theory. The main idea was still the same, to change the low energy pattern of the theory, and this was obtained through the introduction of a nonlocal term one more time. The practical effect of this term is to modify the free propagator of the gauge field, which acquires a k-4 pole, consistently defined in Euclidean space-time. Our intention here will be to present an alternative scenario of localization, noncommutativity the way to a renormalizable noncommutative gauge field theory, but avoiding to introduce any extra degree of freedom. (author)