Thursday, May 3, 2012

1205.0465 (Harald Grosse et al.)

Self-dual noncommutative φ^4-theory in four dimensions is a
non-perturbatively solvable and non-trivial quantum field theory

Harald Grosse, Raimar Wulkenhaar
We reduce the rigorous non-perturbative construction of Euclidean \phi^4-theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the limit of large noncommutativity, to the problem to prove that a non-linear equation for a function G on the unit interval [0,1] is solvable. Although we cannot prove this rigorously at the moment, the equation itself allows us to deduce several non-perturbative properties of the quantum field theory. One of them is a drastic difference between positive and negative coupling constant \lambda. Whereas for \lambda>0 the solution seems feasible, a divergent behaviour for \lambda<0 makes this more difficult or even impossible in the case \lambda<0. Provided that the solution G exists, we are able to express the planar two- and four-point functions entirely in terms of G. As result, the four-point function is non-trivial, and bare and effective coupling constants differ only by a finite ratio. This means that the beta-function is non-perturbatively zero. We implement our equation numerically and show in this way that the iteration of the equation converges rapidly in H\"older norm, and independently of the starting point, to the fixed-point solution G for coupling constants 0< \lambda <= 1/\pi. In principle, all correlation functions of the model and their properties can be computed with sufficient precision. For \lambda> 1/\pi{} our solution becomes inconsistent, but by refined discussion of a winding number it might be possible to extend the solution to larger \lambda. The numerical investigation leaves no doubt that model is solvable. More details are given in an extended abstract.
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