Perturbative gravity in the causal approach

Abstract

Quantum theory of the gravitation in the causal approach is studied up to the second order of perturbation theory in the causal approach. We emphasize the use of cohomology methods in this framework. After describing in detail the mathematical structure of the cohomology method we apply it in three different situations: (a) the determination of the most general expression of the interaction Lagrangian; (b) the proof of gauge invariance in the second order of perturbation theory for the pure gravity system—massless and massive; (c) the investigation of the arbitrariness of the second-order chronological products compatible with renormalization principles and gauge invariance (i.e. the renormalization problem in the second order of perturbation theory). In case (a) we investigate pure gravity systems and the interaction of massless gravity with matter (described by scalars and spinors) and massless Yang–Mills fields. We obtain a difference with respect to the classical field theory due to the fact that in quantum field theory one cannot enforce the divergenceless property on the vector potential and this spoils the divergenceless property of the usual energy–momentum tensor. To correct this one needs a supplementary ghost term in the interaction Lagrangian. In all three case, the computations are more simple than by the usual methods.