Wightman Functions, Retarded Functions and Their Analytic Continuations
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Recent studies on the analyticity of the general n-point Wightman functions and the retarded functions in quantum field theory are reviewed. According to Wightman, the conventional axiom of quantum field theory is transcribed into an equivalent set of properties of the Wightman functions. It is then shown that the Wightman functions are boundary values of analytic functions in complex coordinate spaces and the linear properties of the Wightman functions are transcribed into an equivalent requirement on domains of analyticity of these analytic functions. For the study of retarded functions, the contributions of vacuum intermediate states are systematically subtracted from the Wightman functions. The truncated Wightman functions thus obtained have the same linear properties as those of the Wightman functions except for a smaller support in the momentum space. A generalization of θ-function(the step function of Heaviside)to the characteristic function of a convex polyhedral cone in an n-dimensional vector space is defined and is used to define the generalized retarded functions. The properties of the latter which are equivalent to the linear properties of the Wightman functions are stated. It is then shown that the generalized retarded functions are boundary values of analytic functions in complex momentum spaces and a part of the properties of the former are transcribed into an equivalent requirement on domains of analyticity of the latter. The whole discussion is carried out regardless of the choice between local commutativity and anticommutativity. The problem of obtaining the envelope of holomorphie remains unsolved.
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関連論文
- Quantum Field Theory of Unstable Particles
- Wightman Functions, Retarded Functions and Their Analytic Continuations