Abstract
This chapter gives a short self-contained and coordinate-free presentation of causal perturbative quantum field theory in the spirit of the Bogomolov-Epstein-Glaser renormalization method, in the version developed on curved spacetime by Brunetti and Fredenhagen. We start by explaining the relation between Moyal type star products and Hopf algebras. We then describe causal star products for field theories on a Lorentzian or Euclidean spacetime in a unified setting. We proceed by explaining Fredenhagen and Rejzner’s causal approach to the renormalization of gauge theories. We finish by explaining the relation between deformation quantization and Hopf algebraic structures, and in particular Borcherds’ quantization theorem for 4-dimensional Lorentzian field theories.
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References
Barnich, G.: Classical and quantum aspects of the extended antifield formalism. arXiv High Energy Physics—Theory e-prints (2000). arXiv:hep-th/0011120
Bahns, D., Brouder, C., Dang, N.V., Fabretti, A., Paugam, F.: Introduction to Borcherds’ renormalization and quantum field theory (2012, work in progress)
Bergbauer, C.: Epstein-Glaser renormalization, the Hopf algebra of rooted trees and the Fulton-MacPherson compactification of configuration spaces. Diplomarbeit, Freie Universität Berlin (2004)
Brunetti, R., Fredenhagen, K.: Quantum field theory on curved backgrounds. arXiv (2009). arXiv:0901.2063
Brouder, C., Fauser, B., Frabetti, A., Oeckl, R.: Quantum field theory and Hopf algebra cohomology. J. Phys. A A37, 5895–5927 (2004). doi:10.1088/0305-4470/37/22/014. arXiv:hep-th/0311253
Borcherds, R.E.: Renormalization and quantum field theory. arXiv e-prints (2010). arXiv:1008.0129
Brouder, C.: Quantum field theory meets Hopf algebra. Math. Nachr. 282(12), 1664–1690 (2009). doi:10.1002/mana.200610828
DeWitt, B., DeWitt-Morette, C.: From the Peierls bracket to the Feynman functional integral. Ann. Phys. 314(2), 448–463 (2004)
DeWitt, B.: The Global Approach to Quantum Field Theory. Vol. 1, 2. International Series of Monographs on Physics, vol. 114. Clarendon/Oxford University Press, Oxford/New York (2003). ISBN 0-19-851093-4
Epstein, H., Glaser, V.J.: The role of locality in perturbation theory. Ann. Henri Poincaré 19(3), 211–295 (1973). oai:cds.cern.ch:880480
Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory. arXiv e-prints (2011). arXiv:1110.5232
Keller, K.: Dimensional regularization in position space and a forest formula for regularized Epstein-Glaser renormalization. Universität Hamburg, Hamburg (2007). Dissertation zur Erlangung des Doktorgrades des Department Physik des Universität Hamburg
Keller, K.J.: Euclidean Epstein-Glaser renormalization. J. Math. Phys. 50(10), 103503 (2009). doi:10.1063/1.3202415. arXiv:0902.4789
Peierls, R.E.: The commutation laws of relativistic field theory. Proc. R. Soc. Lond. Ser. A 214, 143–157 (1952)
Rejzner, K.: Batalin-Vilkovisky formalism in locally covariant field theory. arXiv e-prints (2011). arXiv:1111.5130
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Paugam, F. (2014). Causal Perturbative Quantum Field Theory. In: Towards the Mathematics of Quantum Field Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-04564-1_22
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DOI: https://doi.org/10.1007/978-3-319-04564-1_22
Publisher Name: Springer, Cham
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