Rationality of conformally invariant local correlation functions on compactified Minkowski space

Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D ≥ 2 of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification M of Minkowski space M and yields a strong form of locality valid for all non-isotropic intervals if assumed true for space-like separations. Introduction The study of conformal (quantum) field theory (CFT) in four (or, in fact, any number of) space time dimensions (see, e.g., [3] [22] [10] [8] [9] [11] [5] [6] [7] [12] [14] as well as the reviews [19] [16] [17] and references therein) preceded the continuing excitement with 2dimensional (2D) CFT (for a modern textbook and references to original work see [2]). Interest in higher dimensional CFT was revived (starting in late 1997) by the discovery of the AdSCFT correspondence in the context of string theory and supergravity (recent advances in this crowded field can be traced back from [1]). The present paper is more conservative in scope: we try to revitalize the old program of combining conformal invariance and operator product expansion with the general principles of quantum field theory using some insight gained in the study of 2D CFT models. The main result of the paper (Theorems 3.1 and 4.1) can be formulated (omitting technicalities) as follows. We say that a QFT (obeying Wightman axioms [15]) satisfies global conformal invariance (GCI) if for any conformal transformation g and for any set of (different) points (x1, ..., xn) in Minkowski space M such that their images (g x1 , ... , g xn) also lie in M the Wightman function W (x1, ..., xn) stays invariant under g . We note that this requirement (stated more precisely in Sec.2.1) is stronger than the one used (under the same name) in [5]. Together with local commutativity for space like separations this implies the vanishing of the commutator [φ1 (x1) , φ2 (x2) ] whenever the difference x12 = x1 − x2 is non-isotropic (this follows from Lemma 3.2). We deduce from this strong locality property combined with the energy positivity