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Communications in Mathematical Physics

, Volume 317, Issue 3, pp 697–725 | Cite as

Batalin-Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory

  • Klaus Fredenhagen
  • Katarzyna Rejzner
Article

Abstract

On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is an extended notion of the renormalized time-ordered product as a binary product equivalent to the pointwise product of classical field theory. Originally, in causal perturbation theory, the time-ordered product is understood merely as a sequence of multilinear maps on the space of local functionals. Our extended notion of the renormalized time-ordered product (denoted by \({\cdot_{{}^{\mathcal{T}_{\rm r}}}}\)) is consistent with the old one and we found a subspace of the quantum algebra which is closed with respect to \({\cdot_{{}^{\mathcal{T}_{\rm r}}}}\) . On this space the renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and Dütsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.

Keywords

Natural Transformation Local Functional Ghost Number Quantum Algebra Quantum Master Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Albert C., Bleile B., Fröhlich J.: Batalin-Vilkovisky lntegrals in finite dimensions. J. Math. Phys. 51(1), 31 (2010)CrossRefGoogle Scholar
  2. 2.
    Barnich, G.: Classical and quantum aspects of the extended antifield formalism. ULB-TH-00-28, May 2000; These d’agregation ULB (June 2000); Proceedings of the Spring School “QFT and Hamiltonian Systems”, Calimanesti, Romania, May 2–7, 2000; avialble at http://arxiv.org/abs/hepth/0011120v1, 2000
  3. 3.
    Barnich G., Brandt F., Henneaux M.: Local BRST cohomology in the antifield formalism: I. General theorems. Commun. Math. Phys. 174, 57–92 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Barnich G., Henneaux M.: Consistent couplings between fields with a gauge freedom and deformations of the master equation. Phys. Lett. B 311, 123 (1993)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Barnich G., Henneaux M., Hurth T., Skenderis K.: Cohomological analysis of gauge-fixed gauge theories. Phys. Lett. B 492, 376 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Barnich G., Brandt F., Henneaux M.: Local BRST cohomology in gauge theories. Phys. Rept. 338, 439 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Batalin I.A., Vilkovisky G.A.: Relativistic S matrix of dynamical systems with Boson and Fermion constraints. Phys. Lett. 69, 309 (1977)Google Scholar
  8. 8.
    Batalin I.A., Vilkovisky G.A.: Gauge algebra and quantization. Phys. Lett. 102, 27 (1981)MathSciNetGoogle Scholar
  9. 9.
    Batalin I.A., Vilkovisky G.A.: Feynman rules for reducible Gauge Theories. Phys. Lett. B 120, 166 (1983)ADSCrossRefGoogle Scholar
  10. 10.
    Batalin I.A., Vilkovisky G.A.: Quantization of Gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567 (1983)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Becchi C., Rouet A., Stora R.: Renormalization of the Abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Becchi C., Rouet A., Stora R.: Renormalization of Gauge theories. Ann. Phys. 98, 287 (1976)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. New York: Interscience Publishers, Inc., 1959Google Scholar
  14. 14.
    Bonora L., Cotta-Ramusino P.: Some remarks on BRS transformations, anomalies and the cohomoloy of the Lie algebra of the group of Gauge transformations. Commun. Math. Phys. 87, 589–603 (1983)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Brennecke F., Dütsch M.: Removal of violations of the Master Ward Identity in perturbative QFT. Rev. Math. Phys. 20, 119–172 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on Physical Backgrounds. Commun. Math. Phys. 208, 623–661 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Brunetti, R., Fredenhagen, K.: Towards a Background Independent Formulation of Perturbative Quantum Gravity. In: Proceedings of Workshop on Mathematical and Physical Aspects of Quantum Gravity, Blaubeuren, Germany, 28 Jul - 1 Aug 2005. Fauser, B. (ed.) et al.: Quantum gravity, Basel: Birkhäuser, 2006, pp. 151–157Google Scholar
  18. 18.
    Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996)ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle - A new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003)MathSciNetADSzbMATHGoogle Scholar
  20. 20.
    Brunetti R., Dütsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Costello, K.: Renormalization and effective field theory. Mathematical Surveys and Monographs, Volume 170, Providence, RI: Amer. Math. Soc., 2011Google Scholar
  22. 22.
    Costello, K.: Renormalisation and the Batalin-Vilkovisky formalism. http://arxiv.org/abs/0706.1533v3 [math.QA], 2007
  23. 23.
    Costello, K.: Factorization algebras in perturbative quantum field theory. http://www.math.northwestern.edu/~costello/factorization_public.html
  24. 24.
    Dütsch M., Boas F.-M.: The master Ward identity. Rev. Math. Phys 14, 977–1049 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Dütsch, M., Fredenhagen, K.: Perturbative algebraic field theory, and deformation quantization. In: Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20–25 2000, available at http://arxiv.org/abs/hep-th/0101079v1, 2001
  26. 26.
    Dütsch M., Fredenhagen K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16(10), 1291–1348 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Dütsch M., Fredenhagen K.: The master Ward identity and generalized Schwinger-Dyson equation in classical field theory. Commun. Math. Phys. 243, 275 (2003)ADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Epstein H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré A 19, 211 (1973)MathSciNetGoogle Scholar
  29. 29.
    Fredenhagen, K.: Locally Covariant Quantum Field Theory. In: Proceedings of the XIVth International Congress on Mathematical Physics, Lisbon 2003, available at http://arxiv.org/abs/hep-th/0403007v1, 2009
  30. 30.
    Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in the functional approach to classical field theory. http://arxiv/org.abs/1101.5112v4 [math-ph], 2011
  31. 31.
    Fredenhagen, K., Rejzner, K.: Local covariance and background independence. to be published in the Proceedings of the conference “Quantum field theory and gravity”, Regensburg (28 Sep - 1 Oct 2010), http://arxiv.org/abs/1102.2376v1 [math-ph], 2011
  32. 32.
    Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    Haag, R.: Discussion des “axiomes” et des propriétés asymptotiques d’une théorie des champs locales avec particules composées. In: Les Problèmes Mathématiques de la Théorie Quantique des Champs, Colloque Internationaux du CNRS LXXV (Lille 1957), Paris: CNRS 1959, pp. 151–157Google Scholar
  34. 34.
    Haag, R.: Local Quantum Physics. 2nd ed., Berlin-Heidleberg-NewYork: Springer, 1996Google Scholar
  35. 35.
    Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7(1), 65–222 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Henneaux, M., Teitelboim, C.: Quantization of gauge systems. Princeton, NJ: Princeton Univ. Pr., 1992Google Scholar
  37. 37.
    Henneaux, M.: Lectures On The Antifield - BRST Formalism For Gauge Theories. Lectures given at 20th GIFT Int. Seminar on Theoretical Physics, Jaca, Spain, Jun 5–9, 1989, and at CECS, Santiago, Chile, June/July 1989, Nucl. Phys. B (Proc. Suppl.) A18, 47 (1990)Google Scholar
  38. 38.
    Hollands S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Hollands S., Wald R.M.: Existence of local covariant time-ordered-products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Hollands S., Wald R.M.: On the renormalization group in curved spacetime. Commun. Math. Phys. 237, 123–160 (2003)MathSciNetADSzbMATHGoogle Scholar
  42. 42.
    Hollands S., Wald R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Hörmander, L.: The analysis of linear partial differential operators I: Distribution theory and Fourier analysis. Berlin-Heidleberg-New York: Springer, 2003Google Scholar
  44. 44.
    De Jonghe F., París J., Troost W.: The BPHZ renormalised BV master equation and two-loop anomalies in chiral gravities. Nucl. Phys. B 476, 559 (1996)ADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Kriegl, A., Michor, P.: Convenient setting of global analysis, Mathematical Surveys and Monographs 53, Providence, RI: Amer. Math. Soc., 1997Google Scholar
  46. 46.
    Lee B.W., Zinn-Justin J.: Spontaneously Broken Gauge symmetries I,II,III. Phys. Rev D 5, 3121–3160 (1972)ADSCrossRefGoogle Scholar
  47. 47.
    Neeb, K.-H.: Monastir Lecture Notes on Infinite-Dimensional Lie Groups, http://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf
  48. 48.
    Osterwalder K., Schrader R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973)MathSciNetADSzbMATHCrossRefGoogle Scholar
  49. 49.
    París J.: Non-locally regularized antibracket-antifield formalism and anomalies in chiral W 3 gravity. Nucl. Phys. B 450, 357 (1995)ADSzbMATHCrossRefGoogle Scholar
  50. 50.
    Piguet, O., Sorella, S.P.: Algebraic renormalization: Perturbative renormalization, symmetries and anomalies, Vol. M28 of Lect. Notes Phys., Berlin-Heidleberg-New York: Springer, 1995Google Scholar
  51. 51.
    Polchinski J.: Renormalization and Effective Lagrangians. Nucl. Phys. B 231, 269–295 (1984)ADSCrossRefGoogle Scholar
  52. 52.
    Radzikowski M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529–553 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. 53.
    Rejzner K.: Fermionic fields in the functional approach to classical field theory. Rev. math. Phys. 23(9), 1009–1033 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Salmhofer, M.: Renormalization. An introduction. Texts and Monographs in Physics. Berlin: Springer-Verlag, 1999Google Scholar
  55. 55.
    Schreiber, U.: On Lie ∞-modules and the BV complex. http://www.math.uni-hamburg.de/home/schreiber/chain.pdf, 2008
  56. 56.
    Stasheff, J.: The (secret?) homological algebra of the Batalin-Vilkovisky approach. In: Proceedings of the Conference Secondary Calculus and Cohomological Physics, Moscow, August 24–31, 1997, http://arxiv.org/abs/hep-th/9712157v1, 1997
  57. 57.
    Stückelberg E.C.G., Rivier D.: A propos des divergences en théorie des champs quantifiés. Helv. Phys. Acta 23, 236–239 (1950)zbMATHGoogle Scholar
  58. 58.
    Stückelberg E.C.G., Petermann A.: La normalisation des constantes dans la théorie des quanta. Helv. Phys. Acta 26, 499–520 (1953)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Tonin M.: Dimensional regularization and anomalies in chiral gauge theories. Nucl. Phys. (Proc. Suppl.) B 29, 137 (1992)ADSCrossRefGoogle Scholar
  60. 60.
    Troost W., van Nieuwenhuizen P., Van Proeyen A.: Anomalies and the Batalin-Vilkovisky Lagrangian formalism. Nucl. Phys. B 333, 727 (1990)MathSciNetADSCrossRefGoogle Scholar
  61. 61.
    Tyutin, I.V.: Gauge invariance in field theory and statistical physics in operator formalism (In Russian), Lebedev preprint (1975) 75-39, available at http://arxiv.org/abs/0812.0580 [hep-th], 2008
  62. 62.
    Wess J., Zumino B.: Consequences of anomalous Ward identities. Phys. Lett. B 37, 95 (1971)MathSciNetADSGoogle Scholar
  63. 63.
    Wightman A.S., Gårding L.: Fields as operator valued distributions in relativistic quantum theory. Ark. Fys. 28, 129–189 (1964)Google Scholar
  64. 64.
    Zinn-Justin, J.: Renormalization of Gauge Theories. In: Trends in Elementary Particle Theory. edited by H. Rollnik, K. Dietz, Lecture Notes in Physics 37, Berlin: Springer-Verlag, 1975Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.II Inst. f. Theoretische PhysikUniversität HamburgHamburgGermany

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