Asymptotic field operators in quantum field theory

Abstract

The formulation of the asymptotic condition in quantum field theory is viewed as a problem in asymptotic particle interpretation of a field theory and is solved in terms of a natural limiting process involving the retarded vacuum expectation values. Starting with the Haag-Brenig theorem translated into operator language we obtain the asymptotic fields for a nonrelativistic self-coupled field. Transcription of this expression for the asymptotic field in terms of a retarded two-point Green's function motivates the more general definition of the asymptotic field in the presence of external forces. The equations of motion as well as the commutation relations for these asymptotic fields are derived, and it is shown that in sufficiently simple cases this definition coincides with the usual definition. In addition to this conventional asymptotic field, in general, there are asymptotic fields which create particles in bound states. The definition generalizes in a natural manner to many-particle asymptotic fields and, in particular, to asymptotic fields which create and annihilate multiparticle bound states. This construction is facilitated by the standard analysis of the Bethe-Salpeter amplitudes. The familiar "contraction rules" are deduced within the present formalism. These developments are extended to relativistic field theory, again in terms of a limiting procedure involving the retarded Green's functions. As a particular application it is shown that the asymptotic electron field describes electrons which react to an external electromagnetic disturbance in the same manner as physical electrons, i.e., with their anomalous moment and vacuum polarization effects etc., in addition to the "normal" gauge invariant interaction. We briefly discuss the implications for theories with unstable quanta, and draw attention to the possibilites offered for a consistent theory of interacting particles involving unstable ghost fields.