Communications in Mathematical Physics

, Volume 288, Issue 3, pp 1117–1135 | Cite as

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

  • Nicola PinamontiEmail author


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.


Transformation Rule Conformal Transformation Natural Transformation Renormalization Constant Covariant Quantum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. BGP07.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization. Zuerich: Eur. Math. Soc., 2007, 194 pGoogle Scholar
  2. Br04.
    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. 39–47Google Scholar
  3. BF00.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000)zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. BF06.
    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, 2006
  5. BFK96.
    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)zbMATHCrossRefADSGoogle Scholar
  6. BFV03.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003)zbMATHADSMathSciNetGoogle Scholar
  7. BOR02.
    Buchholz D., Ojima I., Roos H.: Thermodynamic properties of non-equilibrium states in quantum field theory. Ann. Phys. 297, 219 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. BS07.
    Buchholz D., Schlemmer J.: Local temperature in curved spacetime. Class. Quant. Grav. 24, F25 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. DB60.
    DeWitt B.S., Brehme R.W.: Radiation damping in a gravitational field. Ann. Phys. 9, 220 (1960)CrossRefADSMathSciNetGoogle Scholar
  10. DMS97.
    Di Francesco P., Mathieu P., Senechal D.: Conformal Field Theory, p. 890. Springer, New York (1997)zbMATHGoogle Scholar
  11. Di80.
    Dimock J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219 (1980)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. DF01.
    Duetsch M., Fredenhagen K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2001)zbMATHCrossRefADSGoogle Scholar
  13. Fe07.
    Fewster C.J.: Quantum energy inequalities and local covariance. II: Categorical formulation. Gen. Rel. Grav. 39, 1855 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. Fr75.
    Friedlander F.G.: The Wave Equation on a Curved Space-Time. Cambridge University Press, Cambridge (1975)zbMATHGoogle Scholar
  15. Fu89.
    Fulling S.A.: Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  16. HW01.
    Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001)zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. HW02.
    Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. HW05.
    Hollands S., Wald R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. KW91.
    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)CrossRefADSMathSciNetGoogle Scholar
  20. Mo00.
    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)zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. Mo03.
    Moretti V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189 (2003)zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. Ra96.
    Radzikowski M.J.: Micro-Local Approach To The Hadamard Condition In Quantum Field Theory On Curved Space-Time. Commun. Math. Phys. 179, 529 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. SV08.
    Schlemmer J., Verch R.: Local Thermal Equilibrium States and Quantum Energy Inequalities. Ann. Henri Poincaré 9, 945–978 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  24. Ta89.
    Tadaki S.: Hadamard Regularization and Conformal Transformation. Prog. Theor. Phys. 81, 891 (1989)CrossRefADSMathSciNetGoogle Scholar
  25. Wa84.
    Wald R.M.: General Relativity. University of Chicago Press, Chicago (1984)zbMATHGoogle Scholar
  26. Wa94.
    Wald R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, Chicago (1994)zbMATHGoogle Scholar

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany

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