Renormalisation of ?4-Theory on Noncommutative ?4 in the Matrix Base

Abstract

We prove that the real four-dimensional Euclidean noncommutative φ 4 -model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R 4 . The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

Non-commutative Renormalization

“…where x = 2Θ −1 x and the metric is Euclidean, the model, in four dimensions, is renormalizable at all orders of perturbation [40]. We will see in section 7 that this additional term give rise to an infrared cut-off and allows to decouple the different scales of the theory.…”

“…H. Grosse and R. Wulkenhaar in a brilliant series of papers discovered how to renormalize φ 4 4 [38,39,40]. This "revolution" happened quietly without mediatic fanfare, but it might turn out to develop into a good Ariane's thread at the entrance of the maze.…”

“…Moyal interactions were noticed to obey a certain Langmann-Szabo duality [41], which exchanges space and momentum variables. Grosse and Wulkenhaar realized that the propagator should be modified to also respect this symmetry [40]. This means that NCQFT on Moyal spaces has to be based on the Mehler kernel, which governs propagation in a harmonic potential, rather than on the heat kernel, which governs ordinary propagation in commutative space.…”

“…By replacing these cutoffs by smoother cutoffs which cut directly the Mehler parameter into slices, we could derive rigorously the estimates that were only numerically checked in [40] hence close the last gaps in the BPHZ theorem for vulcanized non commutative φ 4 4 [42]. We could also replace the somewhat cumbersome recursive use of the Polchinski equation [46] by more direct and explicit bounds in a multiscale analysis.…”

“…Nevertheless they either are super-renormalizable (Φ 4 2 [39]) or (and) studied at a special point in the parameter space where they are solvable (Φ 3 2 , Φ 3 4 [64,65], the LSZ models [43,44,45]). Although only logarithmically divergent for parity reasons, the non-commutative Gross-Neveu model is a just renormalizable quantum field theory as Φ 4 4 .…”