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Remarks on Local Symmetry Invariance in Perturbative Algebraic Quantum Field Theory

Abstract

We investigate various aspects of invariance under local symmetries in the framework of perturbative algebraic quantum field theory (pAQFT). Our main result is the proof that the quantum Batalin–Vilkovisky operator, on-shell, can be written as the commutator with the interacting BRST charge. Up to now, this was proven only for a certain class of fields in quantum electrodynamics and in Yang–Mills theory. Our result is more general and it holds in a wide class of theories with local symmetries, including general relativity and the bosonic string. We also comment on other issues related to local gauge invariance and, using the language of homological algebra, we compare different approaches to quantization of gauge theories in the pAQFT framework.

References

  1. 1

    Bahns, D., Rejzner, K., Zahn, J.: The effective theory of strings. arXiv.org:math-ph/1204.6263v2

  2. 2

    Barnich, G., Henneaux, M., Hurth, T., Skenderis, K.: Cohomological analysis of gauge-fixed gauge theories. Phys. Lett. B 492, 376 (2000). (arXiv:hep-th/9910201)

    Google Scholar 

  3. 3

    Barnich, G., Brandt, F., Henneaux, M.: Local BRST cohomology in gauge theories. Phys. Rept. 338, 439 (2000). (arXiv:hep-th/0002245)

    Google Scholar 

  4. 4

    Batalin I.A., Vilkovisky G.A.: Relativistic S matrix of dynamical systems with Boson and Fermion constraints. Phys. Lett. B 69, 309 (1977)

    ADS  Article  Google Scholar 

  5. 5

    Batalin I.A., Vilkovisky G.A.: Gauge algebra and quantization. Phys. Lett. B 102, 27 (1981)

    ADS  Article  MathSciNet  Google Scholar 

  6. 6

    Batalin I.A., Vilkovisky G.A.: Feynman rules for reducible gauge theories. Phys. Lett. B 120, 166 (1983)

    ADS  Article  Google Scholar 

  7. 7

    Batalin I.A., Vilkovisky G.A.: Quantization Of gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567 (1983)

    ADS  Article  MathSciNet  Google Scholar 

  8. 8

    Battle C., Gomis J., Paris J., Roca J.: Field-antifield formalism and Hamiltonian BRST approach. Nucl. Phys. B 329, 139–154 (1990)

    ADS  Article  Google Scholar 

  9. 9

    Baulieu L., Thierry-Mieg J.: Algebraic structure of quantum gravity and the classification of the gravitational anomalies. Elsevier 145, 53–60 (1984)

    MathSciNet  Google Scholar 

  10. 10

    Becchi C., Rouet A., Stora R.: Renormalization of the Abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975)

    ADS  Article  MathSciNet  Google Scholar 

  11. 11

    Becchi C., Rouet A., Stora R.: Renormalization Of gauge theories. Ann. Phys. 98, 287 (1976)

    ADS  Article  MathSciNet  Google Scholar 

  12. 12

    Boas, F.-M.: Gauge Theories in Local Causal Perturbation Theory. Ph.D. thesis, Hamburg (1999), Hamburg DESY-THESIS-1999-032, ISSN 1435-808

  13. 13

    Bogoliubov N.N., Shirkov D.V.: Introduction to the Theory of Quantized Fields. Interscience Publishers, Inc., New York (1959)

    Google Scholar 

  14. 14

    Brennecke F., Dütsch M.: Removal of violations of the Master Ward Identity in perturbative QFT. Rev. Math. Phys. 20, 119–172 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15

    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  16. 16

    Brunetti, R., Fredenhagen, K.: Towards a background independent formulation of perturbative quantum gravity. In: Fauser, B., et al. (eds.) Quantum gravity, pp. 151–159. Proceedings of Workshop on Mathematical and Physical Aspects of Quantum Gravity, Blaubeuren, Germany, 28 Jul–1 Aug 2005. (arXiv:gr-qc/0603079v3)

  17. 17

    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). (arXiv:gr-qc/9510056)

    Google Scholar 

  18. 18

    Brunetti, R., Fredenhagen, K., Rejzner, K.: Locally covariant quantum field theory as a way to quantum gravity. (arXiv:math-ph/1306.1058)

  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)

    ADS  MATH  MathSciNet  Google 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). (arXiv:math-ph/0901.2038v2)

    Google Scholar 

  21. 21

    Chevalley C., Eilenberg S.: Cohomology theory of lie groups and lie algebras. Trans. Am. Math. Soc. (Providence: American Mathematical Society) 63, 85–124 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22

    Dütsch M., Boas F.-M.: The master ward identity. Rev. Math. Phys 14, 977–1049 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23

    Dütsch M., Fredenhagen K.: A local (perturbative) construction of observables in gauge theories: the example of QED. Commun. Math. Phys. 203, 71–105 (1999)

    ADS  Article  MATH  Google Scholar 

  24. 24

    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. (arXiv:hep-th/0101079)

  25. 25

    Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2001). (arXiv:hep-th/0001129)

    Google Scholar 

  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). (arXiv:hep-th/0403213)

    Google 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). (arXiv:hep-th/0211242)

    Google Scholar 

  28. 28

    Epstein H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré A 19, 211 (1973)

    MathSciNet  Google Scholar 

  29. 29

    Fisch J.M.L., Henneaux M.: Antibracket–antifield formalism for constrained hamiltonian systems. Phys. Lett. B 226, 80–88 (1989)

    ADS  Article  MathSciNet  Google Scholar 

  30. 30

    Fradkin E.S., Vasilev M.A.: Hamiltonian formalism, quantization and S matrix for supergravity. Phys. Lett. B 72, 70 (1977)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  31. 31

    Fradkin E.S., Vilkovisky G.A.: Quantization Of relativistic systems with constraints. Phys. Lett. B 55, 224 (1975)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  32. 32

    Fradkin, E.S., Vilkovisky, G.A.: Quantization of relativistic systems with constraints: equivalence of canonical and covariant formalisms in quantum theory of gravitational field. CERN-TH-2332

  33. 33

    Fradkin E.S., Fradkina T.E.: Quantization of relativistic systems with Boson and Fermion first and second class constraints. Phys. Lett. B 72, 343 (1978)

    ADS  Article  Google Scholar 

  34. 34

    Friedrich H.: Is general relativity “essentially understood”?. Ann. Phys. (Leipzig) 15, 84–108 (2006)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  35. 35

    Fredenhagen, K.: Locally covariant quantum field theory. In: Proceedings of the XIVth International Congress on Mathematical Physics, Lisbon 2003, (hep-th/0403007)

  36. 36

    Fredenhagen, K.: Algebraic structures in perturbative quantum field theory. A talk given at the CMTP Workshop “Two days in QFT” dedicated to the memory of Claudio D’Antoni, Rome, January 10–11, 2011

  37. 37

    Fredenhagen, K., Rejzner, K.: Batalin–Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314, 93–127 (2012). (arXiv:math-ph/1101.5112)

    Google Scholar 

  38. 38

    Fredenhagen, K., Rejzner, K.: Batalin–Vilkovisky formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697–725 (2013). (arXiv:math-ph/1110.5232)

    Google Scholar 

  39. 39

    Fulling S.A., Narcowich F.J., Wald R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime. II. Ann. Phys. 136, 243–272 (1981)

    ADS  Article  MathSciNet  Google Scholar 

  40. 40

    Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  41. 41

    Henneaux, M., Teitelboim, C.: Quantization of gauge systems, p 520. Princeton University Press, Princeton (1992)

  42. 42

    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)

  43. 43

    Hollands, S.: Renormalized quantum Yang–Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008). (arXiv:gr-qc/0705.3340v3)

    Google Scholar 

  44. 44

    Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  45. 45

    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)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  46. 46

    Hollands S., Wald R.M.: On the renormalization group in curved spacetime. Commun. Math. Phys. 237, 123–160 (2003)

    ADS  MATH  MathSciNet  Google Scholar 

  47. 47

    Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005). (arXiv:gr-qc/0404074)

    Google Scholar 

  48. 48

    Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Springer, Berin (2003)

  49. 49

    Keller, K.J.: Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization. Ph.D thesis, Hamburg (2010). (arXiv:math-ph/1006.2148v1)

  50. 50

    Kugo T., Ojima I.: Subsidiary conditions and physical S-matrix unitarity in indefinite metric quantum gravitational theory. Nucl. Phys. 144, 234 (1978)

    ADS  Article  MathSciNet  Google Scholar 

  51. 51

    Kugo T., Ojima I.: Manifestly covariant canonical formulation of Yang–Mills theories physical state subsidiary conditions and physical S-matrix unitarity. Phys. Lett. B 73, 459–462 (1978)

    ADS  Article  Google Scholar 

  52. 52

    Kugo, T., Ojima, I.: Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Suppl. Prog. Theor. Phys. 66, 1 (1979) (Prog. Theor. Phys. 71, 1121 (1984) (Erratum))

  53. 53

    Neeb, K.-H.: Monastir Lecture Notes on Infinite-Dimensional Lie Groups. http://www.math.uni-hamburg.de/home/wockel/data/monastir1

  54. 54

    Rejzner, K.: Fermionic fields in the functional approach to classical field theory. Rev. Math. Phys. 23, 1009–1033 (2011). (arXiv:math-ph/1101.5126v1)

    Google Scholar 

  55. 55

    Rejzner, K.: Batalin–Vilkovisky formalism in locally covariant field theory. Ph.D. thesis, DESY-THESIS-2011-041, Hamburg. (arXiv:math-ph/1110.5130)

  56. 56

    Tyutin, I.V.: Gauge invariance in field theory and statistical physics in operator formalism. LEBEDEV-75-39 preprint (In Russian) p 62 (1975)

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Correspondence to Katarzyna Rejzner.

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Communicated by Karl-Henning Rehren.

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Rejzner, K. Remarks on Local Symmetry Invariance in Perturbative Algebraic Quantum Field Theory. Ann. Henri Poincaré 16, 205–238 (2015). https://doi.org/10.1007/s00023-014-0312-x

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Keywords

  • Gauge Theory
  • Free Action
  • Star Product
  • Mill Theory
  • Free Theory