J. Kubo, AN ANALYSIS ON THE CONVERGENCE OF EQUAL-TIME COMMUTATORS AND THE CLOSURE OF THE BRST ALGEBRA IN YANG-MILLS THEORIES, Nuclear physics. B, 427(1-2), 1994, pp. 398-424
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.