A criterion to characterize interacting theories in the Wightman framework

Abstract

We propose that a criterion to characterize interacting theories in a suitable Wightman framework of relativistic quantum field theories which incorporates a “singularity hypothesis”, which has been conjectured for a long time, is supported by renormalization group theory, but has never been formulated mathematically. The (nonperturbative) wave function renormalization Z occurring in these theories is shown not to be necessarily equal to zero, except if the equal time commutation relations (ETCR) are assumed. Since the ETCR are not justified in general (because the interacting fields cannot in general be restricted to sharp times, as is known from model studies), the condition $$Z=0$$ is not of general validity in interacting theories. We conjecture that it characterizes either unstable (composite) particles or the charge-carrying particles, which become infraparticles in the presence of massless particles. In the case of QED, such “dressed” electrons are not expected to be confined, but in QCD, we propose a quark confinement criterion, which follows naturally from lines suggested by the works of Casher, Kogut and Susskind, and Lowenstein and Swieca.