Abstract
The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field \({\varphi}\) on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field \({\phi}\) on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS4 by differential renormalization, and calculate the anomalous dimension of the composite boundary field \({\varphi^2}\) with bulk interaction \({\kappa \phi^4}\).
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Acknowledgements
MD profitted from discussions with Günter Scharf and Raymond Stora during an early stage of this work. Extensive discussions with Klaus Fredenhagen clarified many conceptual issues. We thank the anonymous referee for insisting, by his very detailed and qualified inquiries, onmore detailed explanations in Sect. 2.3, and for raising the interesting issue of the structure of the OPE.
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Communicated by M. Salmhofer
Dedicated to Raymond Stora on the occasion of his 80th birthday
Supported in part by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen, and DFG Grant RE 1208/2-1.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Dütsch, M., Rehren, KH. Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space. Commun. Math. Phys. 307, 315–350 (2011). https://doi.org/10.1007/s00220-011-1311-0
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DOI: https://doi.org/10.1007/s00220-011-1311-0