Quantum field theory on non-commutative space-times and the persistence of ultraviolet divergences

Abstract

We study properties of a scalar quantum field theory on two-dimensional noncommutative space-times. Contrary to the common belief that noncommutativity of space-time would be a key to remove the ultraviolet divergences, we show that field theories on a noncommutative plane with the most natural Heisenberg-like commutation relations among coordinates or even on a noncommutative quantum plane with E q (2)-symmetry have ultraviolet divergences, while the theory on a noncommutative cylinder is ultraviolet finite. Thus, ultraviolet behaviour of a field theory on noncommutative spaces is sensitive to the topology of the space-time, namely to its compactness. We present general arguments for the case of higher space-time dimensions and as well discuss the symmetry transformations of physical states on noncommutative space-times.

Noncommutative Φ4 theory at two loops

“…Using Schwinger parameterization, the integral over q can be done explicitly and we remain with: 24) where Λ ef f is given by Λ −2 ef f = k • k + 1 Λ 2 . The integral over α can be exactly computed, yielding a modified Bessel function K 1 and the final result is:…”

From noncommutative bosonization to S-duality

“…In this section we derive the bosonization rules of the free fermion action on a twodimensional noncommutative space [18], using the path-integral approach described in [13,14]. Our derivation is carried out for the abelian case (and will largely follow [14]) but as we shall see it generalizes immediately to the non-abelian case [15].…”

The shape of nonabelian D-branes

“…Among various fuzzy objects, one of the interesting configurations is a fuzzy cylinder [36,37] whose shape is nontrivial. The object itself is non-compact and elongated to spacetime infinities, while the fuzzy spheres considered in the previous sections are compact objects.…”