In a conformal invariant quantum field theory (in 4 space time dimensions) Wilson operator product expansions converge on the vacuum, because they are closely related to conformal partial wave expansions.

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Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory

Starting from a chiral conformal Haag-Kastler net on 2 dimensional Minkowski space we construct associated pointlike localized fields. This amounts to a proof of the existence of operator product expansions. We derive the result in two ways. One is based on the geometrical identification of the modular structure, the other depends on a "conformal cluster theorem" of the conformal two-point-functions in algebraic quantum field theory. The existence of the fields then implies important structural properties of the theory, as PCT-invarianc e, the Bisognano-Wichmann identification of modular op- erators, Haag duality and additivity.

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Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansions

Recent developments in conformal invariant quantum field theory

Let ϕ1(x) and ϕ2(y) be two local fields in a conformal quantum field theory (CQFT) in two dimensional spacetime. It is then shown that the vector-valued distribution ϕ1(x)ϕ2(y)|0〉 is a boundary value of a vectorvalued holomorphic function which is defined on a large conformally invariant domain. By group theoretical arguments alone it is proved that ϕ1(x)ϕ2(y)|0〉 can be expanded into conformal partial waves. These have all the properties of a global version of Wilson's operator product expansions when applied to the vacuum state |0〉. Finally, the corresponding calculations are carried out more explicitly in the Thirring model. Here, a complete set of local conformally covariant fields is found, which is closed under vacuum expansion of any two it its elements (a vacuum expansion is an operator product expansion applied to the vacuum).

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Operator product expansions on the vacuum in conformal quantum field theory in two spacetime dimensions

Introduction to Conformal Invariant Quantum Field Theory in Two and More Dimensions

Operator products in quantum field theory on two-dimensional Minkowski space are expanded into a series of local operators by means of the tensor product decomposition theorem for representations of the conformal group. The Thirring model is used as an explicit example. Two types of expansions result. If the operator product acts on the vacuum state, we obtain strictly covariant expansions. In general however, each term in the expansion is only semicovariant.

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B. C. Yunn

Papers

Operator product expansions in conformally covariant quantum field theory