General Relativity and Gravitation

, Volume 45, Issue 5, pp 877–910 | Cite as

Linear bosonic and fermionic quantum gauge theories on curved spacetimes

  • Thomas-Paul HackEmail author
  • Alexander Schenkel
Editor’s Choice (Research Article)


We develop a general setting for the quantization of linear bosonic and fermionic field theories subject to local gauge invariance and show how standard examples such as linearised Yang-Mills theory and linearised general relativity fit into this framework. Our construction always leads to a well-defined and gauge-invariant quantum field algebra, the centre and representations of this algebra, however, have to be analysed on a case-by-case basis. We discuss an example of a fermionic gauge field theory where the necessary conditions for the existence of Hilbert space representations are not met on any spacetime. On the other hand, we prove that these conditions are met for the Rarita-Schwinger gauge field in linearised pure \(N=1\) supergravity on certain spacetimes, including asymptotically flat spacetimes and classes of spacetimes with compact Cauchy surfaces. We also present an explicit example of a supergravity background on which the Rarita-Schwinger gauge field can not be consistently quantized.


Quantum field theory on curved spacetimes Gauge theories Supergravity Algebraic quantum field theory 



We would like to thank Claudio Dappiaggi, Klaus Fredenhagen, Hanno Gottschalk, Katarzyna Rejzner, Ko Sanders, Christoph Stephan and Christoph F. Uhlemann for useful discussions and comments. T.P.H. gratefully acknowledges financial support from the Hamburg research cluster LEXI “Connecting Particles with the Cosmos”.


  1. 1.
    Bär, C., Fredenhagen, K. (eds.): Quantum field theory on curved spacetimes . Lecture Notes Physics, Vol. 786, p. 1 (2009)Google Scholar
  2. 2.
    Bär, C., Ginoux, N.: Classical and quantum fields on Lorentzian manifolds. Springer Proc. Math. 17, 359 (2011) [arXiv:1104.1158 [math-ph]]Google Scholar
  3. 3.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorenzian manifolds and quantization. Zuerich, Switzerland: Eur. Math. Soc. (2007) p. 194 [arXiv:0806.1036 [math.DG]]Google Scholar
  4. 4.
    Bernal, A.N., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic space-times. Commun. Math. Phys. 257, 43 (2005) [arXiv:gr-qc/0401112]Google Scholar
  5. 5.
    Bär, C.: Dependence on the spin structure of the Dirac spectrum. In: Bourguignon, J.P., Branson, T., Hijazi, O. (eds.) Seminaires et Congres 4, Global Analysis and Harmonic Analysis, pp. 17–33 (2000) [arXiv:math/0007131]Google Scholar
  6. 6.
    Bernal, A.N., Sánchez, M.: Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77, 183 (2006) [arXiv:gr-qc/0512095]Google Scholar
  7. 7.
    Binz, E., Honegger, R., Rieckers, A.: Construction and uniqueness of the \(C^\ast \)-Weyl algebra over a general pre-symplectic space. J. Math. Phys. 45, 2885 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Springer, New York (1995)Google Scholar
  9. 9.
    Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003) [arXiv:math-ph/0112041]Google Scholar
  10. 10.
    Dappiaggi, C., Hack, T.-P., Pinamonti, N.: The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor. Rev. Math. Phys. 21, 1241 (2009) [arXiv:0904.0612 [math-ph]]Google Scholar
  11. 11.
    Dappiaggi, C., Lang, B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance. [arXiv:1104.1374 [gr-qc]]Google Scholar
  12. 12.
    Dappiaggi, C., Siemssen, D.: Hadamard states for the vector potential on asymptotically flat spacetimes. [arXiv:1106.5575 [gr-qc]]Google Scholar
  13. 13.
    Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219–228 (1980)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Dimock, J.: Dirac quantum fields on a manifold. Trans. Am. Math. Soc. 269, 133–147 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dimock, J.: Quantized Electromagnetic Field on a Manifold. Rev. Math. Phys. 4, 223–233 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Donaldson, S.: Lecture Notes for TCC Course Geometric Analysis. (2008)
  17. 17.
    Fewster, C.J., Pfenning, M.J.: A quantum weak energy inequality for spin one fields in curved space-time. J. Math. Phys. 44, 4480 (2003) [arXiv:gr-qc/0303106]Google Scholar
  18. 18.
    Fewster, C.J., Hunt, D.S.: Quantization of linearized gravity in cosmological vacuum spacetimes. [arXiv:1203.0261 [math-ph]]Google Scholar
  19. 19.
    Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory. [arXiv:1110.5232 [math-ph]]Google Scholar
  20. 20.
    Furlani, E.P.: Quantization of massive vector fields in curved space-time. J. Math. Phys. 40, 2611 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Hack, T.-P., Makedonski, M.: A No-Go Theorem for the consistent quantization of spin 3/2 fields on general curved spacetimes. Phys. Lett. B. 718, 1465–1470 (2013) [arXiv:1106.6327 [hep-th]]Google Scholar
  22. 22.
    Hollands, S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008) [arXiv:0705.3340 [gr-qc]]Google Scholar
  23. 23.
    Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  24. 24.
    Muehlhoff, R.: Cauchy problem and Green’s functions for first order differential operators and algebraic quantization. J. Math. Phys. 52, 022303 (2011) [arXiv:1001.4091 [math-ph]]Google Scholar
  25. 25.
    Nilles, H.P.: Supersymmetry, supergravity and particle physics. Phys. Rep. 110, 1 (1984)ADSCrossRefGoogle Scholar
  26. 26.
    Parker, T., Taubes, C.H.: On witten’s proof of the positive energy theorem. Commun. Math. Phys. 84, 223 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Pfenning, M.J.: Quantization of the Maxwell field in curved spacetimes of arbitrary dimension. Class. Quant. Gravit. 26, 135017 (2009) [arXiv:0902.4887 [math-ph]]Google Scholar
  28. 28.
    Sanders, J.A.: Aspects of locally covariant quantum field theory. arXiv:0809.4828 [math-ph]Google Scholar
  29. 29.
    Sanders, K., Dappiaggi, C., Hack, T.-P.: Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss’ law. [arXiv:1211.6420 [math-ph]]Google Scholar
  30. 30.
    Schenkel, A., Uhlemann, C.F.: Quantization of the massive gravitino on FRW spacetimes. Phys. Rev. D 85, 024011 (2012) [arXiv:1109.2951 [hep-th]]Google Scholar
  31. 31.
    Bernal, A.N., Sánchez, M.: Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77, 183 (2006) [arXiv:gr-qc/0512095]Google Scholar
  32. 32.
    Stewart, J.M., Walker, M.: Perturbations of spacetimes in general relativity. Proc. R. Soc. Lond. A 341, 49 (1974)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Taylor, M.E.: Partial Differential Equations I—Basic Theory. Springer, New York (1996)Google Scholar
  34. 34.
    Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189 (1981)MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Van Proeyen, A.: Tools for supersymmetry. hep-th/9910030Google Scholar
  36. 36.
    Wald, R.M.: General Relativity. Chicago University Press, Chicago (1984)zbMATHCrossRefGoogle Scholar
  37. 37.
    Wess, J., Bagger, J.: Supersymmetry and Supergravity. Princeton University Press, Princeton (1992)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany
  2. 2.Fachgruppe MathematikBergische Universität WuppertalWuppertalGermany

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