On quantum field theory in gravitational background


We discuss quantum fields on Riemannian space-time. A principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows us to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non-stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non-inertial motion are added.

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  1. 1.

    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys.43, 199 (1975)

    Google Scholar 

  2. 2.

    Unruh, W.G.: Notes on black-hole evaporation. Phys. Rev. D14, 870 (1976); compare also: Davies, P.C.W.: Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A8, 609 (1975)

    Google Scholar 

  3. 2a.

    Fulling, S.A.: Nonuniqueness of canonical field quantization in Riemannian space-time. Phys. Rev. D7, 2850 (1973)

    Google Scholar 

  4. 3.

    Haag, R., Kastler, D.: Algebraic approach to quantum field theory. J. Math. Phys.5, 848 (1964)

    Google Scholar 

  5. 4.

    Kastler, D.: Topics in the algebraic approach to quantum field theory, Cargèse lectures 1965, Lurçat, F. (ed.). N.Y.: Gordon and Breach 1967

    Google Scholar 

  6. 5.

    Rindler, W.: Kruskal space and the uniformly accelerated frame. Am. J. Phys.34, 1174 (1966)

    Google Scholar 

  7. 6.

    Bisognano, J.J., Wichmann, E.H.: On the duality condition for a Hermitian scalar field. J. Math. Phys.16, 985 (1975); On the duality condition for quantum fields. J. Math. Phys.17, 303 (1976)

    Google Scholar 

  8. 7.

    Reeh, H., Schlieder, S.: Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern. Nuovo Cimento22, 1051 (1961)

    Google Scholar 

  9. 8.

    Takesaki, M.: Tomita's theory of modular Hilbert-algebras. Berlin, Heidelberg, New York: Springer 1970

    Google Scholar 

  10. 9.

    Haag, R., Hugenholz, N.M., Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys.5, 215 (1967)

    Google Scholar 

  11. 10.

    Sewell, G.L.: Relativity of temperature and the Hawking effect. Phys. Lett.79A, 23 (1980)

    Google Scholar 

  12. 11.

    Sewell, G.L.: Quantum fields on manifolds. Ann. Phys. (USA)141, 201 (1982)

    Google Scholar 

  13. 12.

    Wald, R.M.: On particle creation by black holes. Commun. Math. Phys.45, 9 (1975)

    Google Scholar 

  14. 13.

    Dimock, D., Kay, B.S.: Classical and quantum scattering on stars and black holes. IAMP-Conference, Boulder 1983

  15. 14.

    Magnus, W., Oberhettinger, F.: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik. Berlin: Springer 1943

    Google Scholar 

  16. 15.

    H. Bateman Manuscript Project: Tables of Integral Transforms, Vol. II. California Institute of Technology. Erdélyi, A. (ed.). New York: McGraw-Hill 1954

    Google Scholar 

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Communicated by R. Haag

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Haag, R., Narnhofer, H. & Stein, U. On quantum field theory in gravitational background. Commun.Math. Phys. 94, 219–238 (1984). https://doi.org/10.1007/BF01209302

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