Causality, Crossing and Analyticity in Conformal Field Theories

TL;DR

This paper rigorously investigates the analyticity and crossing properties of four-point functions in conformal field theories using Wightman axioms, emphasizing microcausality and deriving fundamental theorems to establish a conformal bootstrap equation.

Contribution

It provides a rigorous axiomatic derivation of analyticity, crossing, and bootstrap equations in conformal field theories, highlighting the role of microcausality and fundamental theorems.

Findings

Domains of analyticity are explicitly characterized.

Crossing property is derived via analytic completion techniques.

A conformal bootstrap equation is rigorously established.

Abstract

Analyticity and crossing properties of four point function are investigated in conformal field theories in the frameworks of Wightman axioms. A Hermitian scalar conformal field, satisfying the Wightman axioms, is considered. The crucial role of microcausality in deriving analyticity domains is discussed and domains of analyticity are presented. A pair of permuted Wightman functions are envisaged. The crossing property is derived by appealing to the technique of analytic completion for the pair of permuted Wightman functions. The operator product expansion of a pair of scalar fields is studied and analyticity property of the matrix elements of composite fields, appearing in the operator product expansion, is investigated. An integral representation is presented for the commutator of composite fields where microcausality is a key ingredient. Three fundamental theorems of axiomatic local field theories; namely, PCT theorem, the theorem proving equivalence between PCT theorem and weak local commutativity and the edge-of-the-wedge theorem are invoked to derive a conformal bootstrap equation rigorously.