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 (ETC’S) in terms of the equal-time limit and investigate its convergence in perturbation theory. We find that the equal-time limit vanishes for amplitudes with the effective dimension deff ≤ −2 and is finite for those with deff = −1 but without nontrivial discontinuity. Otherwise we expect divergent equal-time limits. We also find that, if the ETC’s involved in verifying an Jacobi identity exist, the identity is satisfied. Under these circumstances, we show in the Yang-Mills theory that the ETC of the 0 component of the BRST current with each other vanishes to all orders in perturbation theory if the theory is free from the chiral 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 start to diverge at the two-loop level unless there is some unexpected cancellation mechanism that improves the degree of convergence. † E-mail address: [email protected] and [email protected] On leave of absence from College of Liberal Arts, Kanazawa University, Japan