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On the renormalization group in curved spacetime. (English) Zbl 1059.81138
This work is a continuation of the authors previous research on interacting quantum fields in curved spacetimes [cf. “Existence of local covariant time ordered products of quantum fields in curved spacetime”, Commun. Math. Phys. 231, No. 2, 309–345 (2002; Zbl 1015.81043) and the references therein]. Its central subject is the analysis of the renormalization group flow for a scalar quantum field theory with a quartic self interaction, propagating in a curved spacetime.
The paper starts with a review of some previous results like the definition of Wick powers and time ordered products of the free field, which together generate an algebra \({\mathcal W}(M,g)\). Then the interacting fields, its Wick powers and time ordered products are constructed as formal power series in \({\mathcal W}(M,g)\) and the dependence of the interacting theory on the renormalization prescription is analyzed. The central result of the paper states that the interacting field algebra for the spacetime \((M,\lambda^2 g)\) with coupling parameters \(p\) is isomorphic to the interacting field algebra for the space time \((M,g)\) but different values \(p(\lambda)\) of the coupling parameters. The map \(p \mapsto p(\lambda)\) yields the renormalization group flow of the theory.

81T20 Quantum field theory on curved space or space-time backgrounds
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
83C47 Methods of quantum field theory in general relativity and gravitational theory
46N50 Applications of functional analysis in quantum physics
81T17 Renormalization group methods applied to problems in quantum field theory
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