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Communications in Mathematical Physics

, Volume 347, Issue 3, pp 703–721 | Cite as

A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds

  • Christian BärEmail author
  • Alexander Strohmaier
Article

Abstract

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived directly in Lorentzian signature and in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah–Singer index theorem and another term involving the \({\eta}\)-invariant of the Cauchy hypersurfaces.

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References

  1. 1.
    Adler S.L.: Axial–vector vertex in spinor electrodynamics. Phys. Rev. 177(5), 2426–2438 (1969)ADSCrossRefGoogle Scholar
  2. 2.
    Álvarez-Gaumé L., Ginsparg P.: The structure of gauge and gravitational anomalies. Ann. Phys. 161(2), 423–490 (1985)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Araki, H.: On quasifree states of CAR and Bogoliubov automorphisms. Publ. Res. Inst. Math. Sci. 6, 385–442 (1970/71)Google Scholar
  4. 4.
    Avron J., Seiler R., Simon B.: The index of a pair of projections. J. Funct. Anal. 120(1), 220–237 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bär C.: Green-hyperbolic operators on globally hyperbolic spacetimes. Commun. Math. Phys. 333(3), 1585–1615 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bär, C., Ginoux, N.: Classical and quantum fields on Lorentzian manifolds. In: Global Differential Geometry. Springer Proc. Math., vol. 17, pp. 359–400. Springer, Heidelberg (2012)Google Scholar
  7. 7.
    Bär, C., Strohmaier, A.: An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary (2015). arXiv:1506.00959
  8. 8.
    Bell J.S., Jackiw R.: A PCAC puzzle: \({\pi_0 \to \gamma \gamma}\) in the \({\sigma}\)-model. Il Nuovo Cimento A 60(1), 47–61 (1969)ADSCrossRefGoogle Scholar
  9. 9.
    Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Springer, Berlin (2004). (Corrected reprint of the 1992 original) Google Scholar
  10. 10.
    Bertlmann R.A.: Anomalies in Quantum Field Theory. Oxford University Press, Oxford (2000)zbMATHCrossRefGoogle Scholar
  11. 11.
    Bongaarts P.J.M.: The electron–positron field, coupled to external electromagnetic potentials, as an elementary C* algebra theory. Ann. Phys. 56, 108–139 (1970)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bunke U., Hirschmann T.: The index of the scattering operator on the positive spectral subspace. Commun. Math. Phys. 148(3), 487–502 (1992)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Carey A.L., Hurst C.A., O’Brien D.M.: Automorphisms of the canonical anticommutation relations and index theory. J. Funct. Anal. 48(3), 360–393 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Christensen S.M., Duff M.J.: New gravitational index theorems and super theorems. Nucl. Phys. B 154(2), 301–342 (1979)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    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(10), 1241–1312 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dawson S.P., Fewster C.J.: An explicit quantum weak energy inequality for Dirac fields in curved spacetimes. Class. Quant. Grav. 23(23), 6659–6681 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Delbourgo R., Salam A.: The gravitational correction to PCAC. Phys. Lett. B 40(3), 381–382 (1972)ADSCrossRefGoogle Scholar
  18. 18.
    Dimock J.: Dirac quantum fields on a manifold. Trans. Am. Math. Soc. 269(1), 133–147 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Dowker J.S.: Another discussion of the axial vector anomaly and the index theorem. J. Phys. A 11(2), 347–360 (1978)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Fujikawa K.: Evaluation of the chiral anomaly in gauge theories with \({\gamma_{5}}\) couplings. Phys. Rev. D (3) 29(2), 285–292 (1984)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Fujikawa K., Ojima S., Yajima S.: Simple evaluation of chiral anomalies in the path-integral approach. Phys. Rev. D (3) 34(10), 3223–3228 (1986)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gibbons G.W.: Cosmological fermion-number non-conservation. Phys. Lett. B 84(4), 431–434 (1979)ADSCrossRefGoogle Scholar
  23. 23.
    Gibbons G.W.: Spectral asymmetry and quantum field theory in curved spacetime. Ann. Phys. 125(1), 98–116 (1980)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Gibbons G.W., Richer J: Gravitational creation of odd numbers of fermions. Phys. Lett. B 89(3), 338–340 (1980)ADSCrossRefGoogle Scholar
  25. 25.
    Gornet, R., Richarson, K.: The eta invariant of two-step nilmanifolds (2012). arXiv:1210.8070
  26. 26.
    Hollands S.: The Hadamard condition for Dirac fields and adiabatic states on Robertson–Walker spacetimes. Commun. Math. Phys. 216(3), 635–661 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kimura T.: Divergence of axial-vector current in the gravitational field. Progr. Theor. Phys. 42(5), 1191–1205 (1969)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Klaus M., Scharf G.: The regular external field problem in quantum electrodynamics. Helv. Phys. Acta 50(6), 779–802 (1977)MathSciNetGoogle Scholar
  29. 29.
    Klaus M., Scharf G.: Vacuum polarization in Fock space. Helv. Phys. Acta 50(6), 803–814 (1977)MathSciNetGoogle Scholar
  30. 30.
    Lohiya D.: Anomalous production of chiral charge from black holes. Ann. Phys. 145(1), 116–130 (1983)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Lundberg L.-E.: Quasi-free “second quantization”. Commun. Math. Phys. 50(2), 103–112 (1976)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Matsui T.: The index of scattering operators of dirac equations. Commun. Math. Phys. 110(4), 553–571 (1987)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Matsui T.: The index of scattering operators of dirac equations, ii. J. Funct. Anal. 94(1), 93–109 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Bech Nielsen H., Ninomiya M.: The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130(6), 389–396 (1983)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Nielsen N.K., Römer H., Schroer B.: Anomalous currents in curved space. Nucl. Phys. B 136(3), 475–492 (1978)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Plymen R.J., Robinson P.L.: Spinors in Hilbert Space. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  37. 37.
    Sahlmann H., Verch R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13(10), 1203–1246 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Wald R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, Chicago (1994)zbMATHGoogle Scholar
  39. 39.
    Xiong, J., Kushwaha, S.K., Liang, T., Krizan, J.W., Wang, W., Cava, R.J., Ong, N.P.: Signature of the chiral anomaly in a Dirac semimetal: a current plume steered by a magnetic field (2015). arXiv:1503.08179
  40. 40.
    Zahn J.: Locally covariant chiral fermions and anomalies. Nucl. Phys. B 890, 1–16 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute for MathematicsPotsdam UniversityPotsdamGermany
  2. 2.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK

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