Noncommutative QFT and renormalization

Abstract

It was a great pleasure for me (Harald Grosse) to be invited to talk at the meeting celebrating the 70th birthday of Prof. Julius Wess. I remember various interactions with Julius during the last years: At the time of my studies at Vienna with Walter Thirring, Julius left already Vienna, I learned from his work on effective chiral Lagrangians. Next we met at various conferences and places like CERN (were I worked with Andre Martin, an old friend of Julius), and we all learned from Julius' and Bruno's creation of supersymmetry, next we realized our common interests in noncommutative quantum field theory and did have an intensive exchange. Julius influenced our perturbative approach to gauge field theories were we used the Seiberg-Witten map after his advice. And finally I lively remember the sad days when during my invitation to Vienna Julius did have the serious heart attack. So we are very happy, that you recovered so well, and we wish you all the best for the forthcoming years. Many happy recurrences.

8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory

“…We finished the computation almost simultaneously in position space [35] and in the matrix base [36]. See also [37,38]. As a result, there are two additional terms to the Yang-Mills action, namely the integral over Xµ ⋆ Xµ and over its square, where Xµ (x) = (Θ −1 ) µν x ν + A µ (x) is a covariant coordinate [39].…”

Noncommutative induced gauge theories on Moyal spaces

“…We computed at the one-loop order the effective gauge theory action Γ(A) [7] given in (5.15), obtained by integrating over the scalar field φ, for any value of the harmonic parameter Ω ∈ [0, 1] in S(φ, A). Independently of our analysis, H. Grosse and M. Wohlgennant have recently carried out a nice calculation within the matrix-base [8] of the one-loop effective gauge action obtained from a scalar theory with harmonic term different from ours, extending a previous work [46] dealing with the limiting case Ω = 1. The scalar theory considered in [8] is somehow similar to the one that one would obtain by considering "covariant derivative" as given in (3.18) with symmetry transformations as those given by (3.16).…”

Examples of Derivation-Based Differential Calculi Related to Noncommutative Gauge Theories

“…This situation is abviously also at the heart of gauge fields theories on Moyal spaces (see [41], [42], [43], [44] and [38] and references therein). But in this situation, the fact that S(R 2 ) is an ideal of A Θ implies that it can be used as a right A Θmodule.…”

“…In that case, introducing models via noncommutative connections on the right module A Θ itself, there are two gauge potentials {A µ } µ=1,2 , defined by ∇ ∂µ 1l = A µ . Models using this structure have been considered in [41], [42], [43], [44] and [38] (for a review). Two mains problems remain to be solved in this context: to find renormalizable gauge field theories and to describe (some of) their vacuum configurations.…”