Rational Conformal Field Theory In Four Dimensions
Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four space-time dimensions. We outline the results for a model of a neutral scalar field ø of dimension 2 . It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of ø is shown to coincide with a simpler one, generated by a bilocal scalar field V (x 1, x 2) of dimension (1, 1). The modes of V together with the unit operator span an infinite dimensional Lie algebra Lv whose vacuum (i.e. zero energy lowest weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of Lv) is proven to be equivalent to c ∈ N.
KeywordsScalar Field Operator Product Expansion Free Field Stress Energy Tensor Positive Energy Representation
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