An analysis on the convergence of equal-time commutators and the closure of the BRST algebra in Yang-Mills theories

Abstract

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 d eff ≤ −2 and is finite for those with d eff = −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. †

An Alternative to the Lagrangian and Hamiltonian Formulations of Relativistic Field Theories Based on the Energy-Momentum Tensor

“…Therefore, we introduce the additional Lorentz scalar fields E a into Yang-Mills theories which, in our fixed reference frame, can serve as the conjugate momenta associated with the basic fields A 0 a . We introduce these additional variables by modifying the energymomentum tensor (10). Of course, this modification cannot be of the type (6) because we need additional equations for the new variables and moreover expect a modification of the other field equations by the new fields.…”

“…which implies a modification of the auxiliary fields (10) and of the corresponding field equations according to…”

“…Further commutators can be inferred from these relations by taking spatial derivatives (however, inconsistencies may arise from the inherent subtleties of quantum field theories as is well-known, for example, from the problem of the Schwinger term [8][9][10][11]).…”

“…In the present approach, the effect of charged matter on the electromagnetic field is obtained by simply adding the electric current density four-vector to the auxiliary field (10),…”

Quantum Field Theory as a Faithful Image of Nature

“…It has been found to be a complex number, not an operator [87]. Construction of the Schwinger term requires subtle limiting procedures so that even the Jacobi identity for nested commutators needs to be verified explicitly [70].…”