Cohomology Methods in Causal Perturbation Theory

Abstract

Various problems in perturbation theory of (quantum) gauge models can be rephrased in the language of cohomology theory. This was already noticed in the functional formulation of perturbative gauge theories. Causal perturbation theory is a fully quantum approach: is works only with the chronological products which are defined as operator‐valued distributions in the Fock space of the model. The use of causal perturbation theory leads to similar cohomology problems; the main difference with respect to the functional methods comes from the fact that the gauge transformation of the causal approach is, essentially, the linear part of the non‐linear BRST transformation.Using these methods it is possible to give a nice determination of the interaction Lagrangians for gauge models (Yang‐Mills and gravitation in the linear approximation); one obtains with this method the unicity of the interaction Lagrangian up to trivial terms. The case of quantum gravity is highly non‐trivial and can be generalized with this method to the massive graviton case. Going to higher orders of perturbation theory one finds quantum anomalies. Again the cohomological methods can be used to determine the generic form of these anomalies. Finally, one can investigate the arbitrariness of the chronological products in higher orders and reduce this problem to cohomology methods also.