AN ANALYSIS ON THE CONVERGENCE OF EQUAL-TIME COMMUTATORS AND THE CLOSURE OF THE BRST ALGEBRA IN YANG-MILLS THEORIES

In renormalizable theories, we define equal-time commutators (ETCs) in terms of the equal-time limit and investigate their convergence in pe rturbation theory. We find that the equal-time limit vanishes for ampl itudes with the effective dimension d(eff) less-than-or-equal-to -2 an d is finite for tree-like amplitudes with d(eff) = -1. Otherwise we ex pect divergent equal-time limits. We also find that, if the ETCs invol ved in verifying a Jacobi identity exist, then the identity is satisfi ed. Under these circumstances, we show in the Yang-Mills theory that t he ETC of the 0-component of the BRST current with itself vanishes to all orders in perturbation theory if the theory is free from the chira l anomaly, from which we conclude that [Q, Q] = 0, where Q is the BRST charge. For the case that the chiral anomaly is not canceled, we use various broken Ward identities to show that [Q, Q] is finite and [Q, [ Q, Q]] vanishes at the one-loop level and that they both diverge start ing at the two-loop level.