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 close[ y examining rhe self ·consistency of these defining equations, we get a set of integral equations for the vacuum expectation values of retarded products of operators. The unitarity 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 so as 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. § I. Introduction Recently the importance of the asymptotic conditions has been recognized by many authors/-5 > especially in connection with the use of Heisenberg operators in quantum field theory. In quantum field theory the relation between individual particle states and interacting states are settled through these conditions, and important is that only through them the Heisenberg operators can be related to observable or semi-observable quantities such as the S matrix. In particle mechanics the asymptotic behaviour of the wave functions can easily be seen by simply dividing the total Hamiltonian into two parts, one corresponding to the kinetic energies of individual particles and the other to the interaction among them, and the latter must be dropped for the investigation of the asymptotic behaviour of the wave functions. In quantum field theory, however, such a separation of the total Hamiltonian can no more be taken for merited, since individual particles interact with their own self-fields even when they are separated at long distances from each other. Consequently the description of the asymptotic behaviour of a system can no more be achieved in the Hamiltonian formulation, but instead one is obliged to stand on the overall space time point of view. 6 > The reason in favour of this standpoint of view was intuitively discussed