Non-Abelian Gauge Symmetry in the Causal Epstein–Glaser Approach

Abstract

Non-Abelian gauge symmetry in (3 + 1)-dimensional space–time is analyzed in the causal Epstein–Glaser framework. In this formalism, the technical details concerning the well-known UV and IR problem in quantum field theory are separated and reduced to well-defined problems, namely the causal splitting and the adiabatic switching of operator-valued distributions. Non-Abelian gauge invariance in perturbation theory is completely discussed in the well-defined Fock space of free asymptotic fields. The LSZ formalism is not used in this construction. The linear operator condition of asymptotic gauge invariance is sufficient for the unitarity of the S matrix in the physical subspace and the usual Slavnov–Taylor identities. We explicitly derive the most general specific coupling compatible with this condition. By analyzing only tree graphs in the second order of perturbation theory we show that the well-known Yang–Mills couplings with anticommuting ghosts are the only ones which are compatible with asymptotic gauge invariance. The required generalizations for linear gauges are given.