Harmonic bilocal fields generated by globally conformal invariant scalar fields. (English) Zbl 1144.81028

Summary: The twist two contribution in the operator product expansion of \(\phi_{1} (x_{1})\phi_{2}(x_{2})\) for a pair of globally conformal invariant, scalar fields of equal scaling dimension \(d\) in four space-time dimensions is a field \(V_{1} (x_{1}, x_{2})\) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of \(V_{1}\) can be equivalently characterized by a “single-pole property” concerning the pole structure of the (rational) correlation functions involving the product \(\phi_{1}(x_{1})\phi_{2}(x_{2})\). This property is established for the dimension \(d = 2\) of \(\phi_{1}, \phi_{2}\). As an application we prove that any system of GCI scalar fields of conformal dimension 2 (in four space-time dimensions) can be presented as a (possibly infinite) superposition of products of free massless fields.


81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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