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Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime

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Abstract

We construct Hadamard states for the Yang–Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal \({\mathbb{R}^d}\).

We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein–Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators that contain microlocal or spectral type low-energy cutoffs.

The general problem is reduced to the ultra-static spacetime case using an extension of the deformation argument of Fulling, Narcowich and Wald.

As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

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References

  1. Araki H., Shiraishi M.: On quasi-free states of canonical commutation relations I. Publ. RIMS Kyoto Univ. 7, 105–120 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  2. Benini, M.: Optimal Space of Linear Classical Observables for Maxwell k-Forms Via Spacelike and Timelike Compact de Rham Cohomologies. arXiv:1401.7563 (2014)

  3. Bär, C., Ginoux, N.: Classical and Quantum Fields on Lorentzian Manifolds. Global Differential Geometry. Springer, Berlin (2012)

  4. Bär, C., Ginoux, N., Pfäffle, F.: Wave equation on Lorentzian manifolds and quantization. In: ESI Lectures in Mathematics and Physics, EMS (2007)

  5. Choquet-Bruhat, Y.: Yang–Mills Fields on Lorentzian Manifolds. Mechanics, Analysis and Geometry: 200 Years After Lagrange. North-Holland Delta Series, North Holland (1991)

  6. Choquet-Bruhat Y., Christodoulou D.: Existence of global solutions of the Yang–Mills, Higgs and spinor fields equations in 3 + 1 dimensions. Ann. Sci. École Norm. Sup. 14, 481–506 (1981)

    MATH  MathSciNet  Google Scholar 

  7. Cycon H.L., Froese R., Kirsch W., Simon B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer, New York (1987)

    Google Scholar 

  8. Chruściel P.T., Shatah J.: Global existence of solutions of the Yang–Mills equations on globally hyperbolic four dimensional Lorentzian manifolds. Asian J. Math. 1, 530–548 (1997)

    MATH  MathSciNet  Google Scholar 

  9. Dereziński, J.: Quantum Fields with Classical Perturbations. arXiv:1307.1162 (2013)

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

    Article  ADS  MATH  Google Scholar 

  11. Dereziński J., Gérard C.: Mathematics of quantization and quantum fields. Cambridge Monographs in Mathematical Physics. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  12. Dimock J.: Dirac quantum fields on a manifold. Trans. Am. Math. Soc. 269(1), 133–147 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dimock J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4(02), 223–233 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dappiaggi C., Hack T.-P., Sanders K.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law. Commun. Math. Phys. 328, 625–667 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Dappiaggi C., Siemssen D.: Hadamard states for the vector potential on asymptotically flat spacetimes. Rev. Math. Phys. 25, 1350002 (2013)

    Article  MathSciNet  Google Scholar 

  16. 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)

    Article  ADS  MathSciNet  Google Scholar 

  17. Fewster C.J., Pfenning M.J.: A quantum weak energy inequality for spin-one fields in curved spacetime. J. Math. Phys. 44, 4480 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Finster, F., Strohmaier, A.: Gupta–Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space–Times. arXiv:1307.1632 (2013)

  19. Furlani E.P.: Quantization of the electromagnetic field on static spacetimes. J. Math. Phys. 36(3), 1063–1079 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Furlani E.P.: Quantization of massive vector fields in curved spacetime. J. Math. Phys. 40, 2611(1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Gérard C., Wrochna M.: Construction of Hadamard states by pseudo-differential calculus. Commun. Math. Phys. 325(2), 713–755 (2014)

    Article  ADS  MATH  Google Scholar 

  22. Hack, T.-P., Schenkel, A.: Linear bosonic and fermionic quantum gauge theories on curved spacetimes. Gen. Rel. Grav. 1–34 (2012)

  23. Hollands, S.: The hadamard condition for dirac fields and adiabatic states on Robertson–Walker spacetimes. Commun. Math. Phys. 216, 635–661 (2001)

  24. Hollands S.: Renormalized quantum Yang–Mills fields in curved spacetime. Rev. Math. Phys. 20(09), 1033–1172 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hörmander L.: The analysis of linear partial differential operators I. Distribution Theory and Fourier Analysis. Springer, Berlin (1985)

    Google Scholar 

  26. Junker, W.: Adiabatic Vacua and Hadamard States for Scalar Quantum Fields on Curved Spacetime. PhD thesis, University of Hamburg 1995

  27. Khavkine, I.: Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory. arXiv:1211.1914 (2012)

  28. Marathe K.B., Martucci G.: Mathematical foundations of gauge theories. Studies in Mathematical Physics, vol. 5. North-Holland, Amsterdam (1992)

    Google Scholar 

  29. Mühlhoff R.: Cauchy problem and green’s functions for first order differential operators and algebraic quantization. J. Math. Phys. 52, 022303 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  30. Pfenning M.J.: Quantization of the maxwell field in curved spacetimes of arbitrary dimension. Class. Quantum Grav. 26(13), 135017 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  31. Rejzner, K.: Remarks on Local Gauge Invariance in Perturbative Algebraic Quantum Field Theory. arXiv:1301.7037 (2013)

  32. Segal I.: The cauchy problem for the Yang–Mills equations. J. Funct. Anal. 33(2), 175–194 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  33. 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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Taylor M.: Pseudo-Differential Operators and Nonlinear PDE. Birkhäuser, Basel (1991)

    Google Scholar 

  35. Wrochna M.: Quantum field theory in static external potentials and hadamard states. Ann. Henri Poincaré 13(8), 1841–1871 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Wrochna, M.: Singularities of Two-Point Functions in Quantum Field Theory. PhD thesis, University of Göttingen (2013)

  37. Wrochna, M., Zahn, J.: Classical Phase Space and Hadamard States in the BRST Formalism for Gauge Field Theories on Curved Spacetime. arXiv:1407.8079 (2014)

  38. Zahn J.: The renormalized locally covariant dirac field. Rev. Math. Phys. 26, 1330012 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris-Sud XI, 91405, Orsay Cedex, France

    C. Gérard

  2. UMR 5582 CNRS, Institut Fourier, Université Joseph Fourier (Grenoble 1), BP 74, 38402, Saint-Martin d’Hères Cedex, France

    M. Wrochna

Authors
  1. C. Gérard
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  2. M. Wrochna
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Correspondence to C. Gérard.

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Communicated by Y. Kawahigashi

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Gérard, C., Wrochna, M. Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime. Commun. Math. Phys. 337, 253–320 (2015). https://doi.org/10.1007/s00220-015-2305-0

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  • DOI: https://doi.org/10.1007/s00220-015-2305-0

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