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Conformal Generally Covariant Quantum Field Theory: The Scalar Field and its Wick Products

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Abstract

In this paper we generalize the construction of generally covariant quantum theories given in [BFV03] to encompass the conformal covariant case. After introducing the abstract framework, we discuss the massless conformally coupled Klein Gordon field theory, showing that its quantization corresponds to a functor between two certain categories. At the abstract level, the ordinary fields, could be thought of as natural transformations in the sense of category theory. We show that the Wick monomials without derivatives (Wick powers) can be interpreted as fields in this generalized sense, provided a non-trivial choice of the renormalization constants is given. A careful analysis shows that the transformation law of Wick powers is characterized by a weight, and it turns out that the sum of fields with different weights breaks the conformal covariance. At this point there is a difference between the previously given picture due to the presence of a bigger group of covariance. It is furthermore shown that the construction does not depend upon the scale μ appearing in the Hadamard parametrix, used to regularize the fields. Finally, we briefly discuss some further examples of more involved fields.

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References

  1. Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization. Zuerich: Eur. Math. Soc., 2007, 194 p

  2. Brunetti, R.: Locally Covariant Quantum Field Theories. Contribution to the Proceedings of the Symposium “Rigorous Quantum Field Theory” in honor of the 70th birthday of Prof. Jacques Bros (SPhT - CEA-Saclay, Paris, France, 19-21 July 2004). Progress in Mathematics 251, Basel-Boston: Birkhäuser, (2007), pp. 39–47

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Brunetti, R., Fredenhagen, K.: Towards a background independent formulation of perturbative quantum gravity. Proceedings of Workshop on Mathematical and Physical Aspects of Quantum Gravity, available at http://arxiv.org/abs/gr-qc/06030793, 2006

  5. 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)

    Article  MATH  ADS  Google Scholar 

  6. Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  7. Buchholz D., Ojima I., Roos H.: Thermodynamic properties of non-equilibrium states in quantum field theory. Ann. Phys. 297, 219 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Buchholz D., Schlemmer J.: Local temperature in curved spacetime. Class. Quant. Grav. 24, F25 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. DeWitt B.S., Brehme R.W.: Radiation damping in a gravitational field. Ann. Phys. 9, 220 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  10. Di Francesco P., Mathieu P., Senechal D.: Conformal Field Theory, p. 890. Springer, New York (1997)

    MATH  Google Scholar 

  11. Dimock J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219 (1980)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Duetsch M., Fredenhagen K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2001)

    Article  MATH  ADS  Google Scholar 

  13. Fewster C.J.: Quantum energy inequalities and local covariance. II: Categorical formulation. Gen. Rel. Grav. 39, 1855 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Friedlander F.G.: The Wave Equation on a Curved Space-Time. Cambridge University Press, Cambridge (1975)

    MATH  Google Scholar 

  15. Fulling S.A.: Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Kay B.S., Wald R.M.: Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon. Phys. Rept. 207, 49 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  20. Moretti V.: Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt’s coefficients in C(infinity) Lorentzian manifolds by a ‘local Wick rotation’. Commun. Math. Phys. 212, 165 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Moretti V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Radzikowski M.J.: Micro-Local Approach To The Hadamard Condition In Quantum Field Theory On Curved Space-Time. Commun. Math. Phys. 179, 529 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Schlemmer J., Verch R.: Local Thermal Equilibrium States and Quantum Energy Inequalities. Ann. Henri Poincaré 9, 945–978 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tadaki S.: Hadamard Regularization and Conformal Transformation. Prog. Theor. Phys. 81, 891 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  25. Wald R.M.: General Relativity. University of Chicago Press, Chicago (1984)

    MATH  Google Scholar 

  26. Wald R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, Chicago (1994)

    MATH  Google Scholar 

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Authors and Affiliations

  1. II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761, Hamburg, Germany

    Nicola Pinamonti

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  1. Nicola Pinamonti
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Correspondence to Nicola Pinamonti.

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Communicated by Y. Kawahigashi

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Pinamonti, N. Conformal Generally Covariant Quantum Field Theory: The Scalar Field and its Wick Products. Commun. Math. Phys. 288, 1117–1135 (2009). https://doi.org/10.1007/s00220-009-0780-x

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