On the Asymptotic Conditions in Quantum Field Theory

Abstract

In this paper the so-called asymptotic conditions are introduced in a different way than usually done, thereby making use of Heisenberg operators alone. These conditions are regarded as the defining equations for a complete set of state vectors, either incoming or outgoing wave states. By closely examining the self-consistency of these defining equations, we get a set of integral equations for the vacuum expectation values of retarded products of operators. The unitary condition of the S matrix is one of the inevitable consequences of these equations. These integral equations are essentially equivalent to those given by Chew and Low, and also independently by Lehmann, Symanzik and Zimmermann. The advantage of the new integral equations over the older ones of the above mentioned authors consists in that the new ones are manifestly covariant in form and also in that all quantities appearing in the equations are related only to connected Feynman diagrams in contrast to the T products. The possibility to extend the present formalism as so to include the bound states is also discussed. Finally it is shown in the perturbation theory that these integral equations are satisfied only by the renormalized solutions of the renormalizable field theories provided that the microscopic causality condition is imposed as the boundary condition.