Conservation of the Stress Tensor in Perturbative Interacting Quantum Field Theory in Curved Spacetimes
Abstract
We propose additional conditions (beyond those considered in our previous papers) that should be imposed on Wick products and timeordered products of a free quantum scalar field in curved spacetime. These conditions arise from a simple "Principle of Perturbative Agreement": for interaction Lagrangians L_{1} that are such that the interacting field theory can be constructed exactly — as occurs when L_{1} is a "pure divergence" or when L_{1} is at most quadratic in the field and contains no more than two derivatives — then timeordered products must be defined so that the perturbative solution for interacting fields obtained from the Bogoliubov formula agrees with the exact solution. The conditions derived from this principle include a version of the Leibniz rule (or "action Ward identity") and a condition on timeordered products that contain a factor of the free field φ or the free stressenergy tensor T_{ab}. The main results of our paper are (1) a proof that in spacetime dimensions greater than 2, our new conditions can be consistently imposed in addition to our previously considered conditions and (2) a proof that, if they are imposed, then for any polynomial interaction Lagrangian L_{1} (with no restriction on the number of derivatives appearing in L_{1}), the stressenergy tensor Θ_{ab} of the interacting theory will be conserved. Our work thereby establishes (in the context of perturbation theory) the conservation of stressenergy for an arbitrary interacting scalar field in curved spacetimes of dimension greater than 2. Our approach requires us to view timeordered products as maps taking classical field expressions into the quantum field algebra rather than as maps taking Wick polynomials of the quantum field into the quantum field algebra.
 Publication:

Reviews in Mathematical Physics
 Pub Date:
 2005
 DOI:
 10.1142/S0129055X05002340
 arXiv:
 arXiv:grqc/0404074
 Bibcode:
 2005RvMaP..17..227H
 Keywords:

 Quantum field theory on curved space;
 renormalization theory;
 stress tensor;
 perturbation theory;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 88 pages, latex, no figures, v2: changes in the proof of proposition 3.1