Braided structure in 4-dimensional conformal quantum field theory

Abstract

Higher dimensional conformal QFT possesses an interesting braided structure which, different from the d=1+1 models, is restricted to the timelike region and therefore easily escapes euclidean action methods. It lies behind the spectrum of anomalous dimensions which may be viewed as a kind of substitute for a missing particle interpretation in the presence of interactions.1 A formulation in the real-time setting of Wightman (i.e. noncommutative in the oldfashioned sense) which probably requires the use of more subtle conditional expectations on operator algebras has never been achieved. On the other hand there are also several real-time properties whose statistical mechanics interpretation is unknown to this date, e.g. the issue of the timelike structure in this article.2 It was foreshadowed by Kadanoff's ideas on "Coulomb Gas Representations".

The paradigm of the area law and the structure of transversal and longitudinal lightfront degrees of freedom

“…In local quantum physics one encounters several situations in which the notions of "causal disjoint" becomes synonymous with "nonoverlapping". For standard chiral theories on S 1 this is well-known, but it also holds for conformal observables living in the Dirac-Weyl compactified Minkowski spacetime if "nonoverlapping" is interpreted as "nonoverlapping of the lightlike prolongation" (Huygens principle) [28] [29]. Via the CQFT-AdS isomorphism [6] this simplified causality picture is inherited by observables in anti deSitter spacetime.…”

Algebraic Conformal Quantum Field Theory in Perspective

“…It is worth a remark that also in four dimensions, conformal partial waves -which are akin to but not exactly the same as conformal blocks -exhibit a factorization into "chiral" factors [24]. This feature has been exploited for the study of Wightman positivity (positivity of the Hilbert space inner product defined by 2n-point correlation functions) [60,59]; but the algebraic counterpart of an underlying exchange algebra, as suggested in [66], has not been identified. 4.…”

Elliptic Thermal Correlation Functions and Modular Forms in a Globally Conformal Invariant QFT

“…The primary fields, which may well have anomalous dimensions and non-trivial braiding properties (as the magnetization field in the Ising model), appear in this framework as intertwiners between the vacuum and other positive energy representations of the GCI algebra. It is then expected that only the 4-dimensional counterpart of such intertwining primary fields may display anomalous time-like braiding relations of the type discussed in [31].…”