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Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. (English) Zbl 1078.81062
Perturbation theory for quantum fields on curved backgrounds is discussed, particularly conditions that should be imposed on Wick products and time-ordered products of a free quantum scalar field in a curved space-time. The conditions arise from the “Principle of Perturbative Agreement” requiring that the perturbative solution agree with the exact solution if the exact solution exists. A list of such conditions (axioms) has been proposed in the previous papers of the authors [Commun. Math. Phys. 223, 289–326 (2001; Zbl 0989.81081); ibid. 231, 309–345 (2002; Zbl 1015.81043)]. However, the previous analysis was restricted to the case where the Wick powers and the factors appearing in the time-ordered products do not contain derivatives of the scalar field $$\varphi$$. In the paper under review additional conditions are formulated that are necessary in case of derivatives present. One of the additional conditions is “Leibniz’s rule” that provide validity, in the context of the proposed theory, of the equations similar to the following one: $$\nabla_a\varphi^2 = 2\varphi\nabla_a\varphi$$.

##### MSC:
 81T20 Quantum field theory on curved space or space-time backgrounds 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T18 Feynman diagrams
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