Abstract
As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon \({{\Im^-}}\) common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on M, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables \({{\mathcal{W}(\Im^-)}}\) constructed on the cosmological horizon. There is exactly one pure quasifree state λ on \({{\mathcal{W}(\Im^-)}}\) which fulfills a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore λ induces a preferred physically meaningful quantum state λ M for the quantum theory in the bulk. If M admits a timelike Killing generator preserving \({{\Im^-}}\) , then the associated self-adjoint generator in the GNS representation of λ M has positive spectrum (i.e., energy). Moreover λ M turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, λ M coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for λ M in more general spacetimes are presented.
Similar content being viewed by others
References
Aharony O., Gubser S.S., Maldacena J.M., Ooguri H., Oz Y.: Large n field theories, string theory and gravity. Phys. Rept. 323, 183 (2000)
Allen B.: Vacuum states in the Sitter space. Phys. Rew. D 32, 3136 (1985)
Araki, H.: Mathematical Theory of Quantum Fields. Oxford: Oxford University Press, 1999
Ashtekar A., Xanthopoulos B.C.: Isometries compatible with asymptotic flatness at null infinity: a complete description. J. Math. Phys. 19, 2216 (1978)
Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization, ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House, 2007
Birrel, N.D., Davies, P.C.W.: Quantum Field Theory in Curved Space. Cambridge: Cambridge University Press, 1982
Bros J., Moschella U., Gazeau J.P.: Quantum field theory in the de Sitter universe. Phys. Rev. Lett. 73, 1746 (1994)
Bros J., Moschella U.: Two-point Functions and Quantum Fields in de Sitter Universe. Rev. Math. Phys. 8, 327 (1996)
Bunch T.S., Davies P.C.W.: Quantum Fields theory in de Sitter space: renoramlization by point-splitting. Proc. R. Soc. Lond. A 360, 117 (1978)
Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. Vol. 1: C* and W* algebras, symmetry groups, decomposition of states. 2nd edition, Berlin-Heidelberg-New York: Springer-Verlag, 2002
Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. Vol. 2: Equilibrium states. Models in quantum statistical mechanics. 2nd edition, Berlin-Heidelberg-New York: Springer, 2002
Dappiaggi C., Moretti V., Pinamonti N.: Rigorous steps towards holography in asymptotically flat spacetimes. Rev. Math. Phys. 18, 349 (2006)
Dappiaggi C.: Projecting massive scalar fields to null infinity. Ann. Henri Poinc. 9, 35 (2008)
Dimock J.: Algebras of Local Observables on a Manifold. Commun. Math. Phys. 77, 219 (1980)
Duetsch M., Rehren K.H.: Generalized free fields and the AdS-CFT correspondence. Annales Henri Poincare 4, 613 (2003)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Fifth Edition, London-New York: Academic Press, 1995
Geroch, R.: In: Esposito, P., Witten, L. eds., Asymptotic Structure of Spacetime. London: Plenum, 1977
Geroch R.: Limits of Spacetimes. Commun. Math. Phys 13, 180–193 (1969)
Haag, R.: Local quantum physics: Fields, particles, algebras. Second Revised and Enlarged Edition, Berlin-Heidelberg-New York: Springer, 1992
Hall, G.S.: Symmetries and Curvature Structire in General Relativity. River Edge, NJ: World Scientific Publishing, 2004
Hollands, S.: Aspects of quantum field theory in curved spacetimes. Ph.D. Thesis, University of York (2000), advisor B.S. Kay, unpublished
Hollands S., Wald R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005)
Islam, J.N.: An introduction to mathematical cosmology. Cambridge: Cambridge Univ. Press, 2004
Junker W., Schrohe E.: Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties. Annales Poincare Phys. Theor. 3, 1113 (2002)
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)
Leray, J.: Hyperbolic Differential Equations, Unpublished. Lecture Notes, Princeton, 1953
Linde, A.: Particle Physics and Inflationary Cosmology. London: Harwood Academic Publishers, 1996
Lüders C., Roberts J.E.: Local Quasiequivalence and Adiabatic Vacuum States. Commun. Math. Phys. 134, 29 (1990)
Moretti V.: Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable. Commun. Math. Phys. 268, 726 (2006)
Moretti V.: Quantum out-states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property. Commun. Math. Phys. 279, 31 (2008)
Moretti V.: Comments on the stress energy tensor operator in curved space-time. Commun. Math. Phys. 232, 189 (2003)
Pinamonti, N.: De Sitter quantum scalar field and horizon holography. http://arXiv.org/list/hep-th/0505179, 2005
Olbermann H.: States of low energy on Robertson-Walker spacetimes. Class. Quantum. Grav. 24, 5011 (2007)
Parker L.: Quantized fields and particle creation in expanding universes. 1. Phys. Rev. 183, 1057 (1969)
Rindler, W.: Relativity. Special, General and Cosmological. Second Edition, Oxford: Oxford University Press, 2006
Schomblond C., Spindel P.: Conditions d’unicité pour le propagateur Δ(1)(x,y) du champ scalaire dans l’univers de de Sitter. Ann. Inst. Henri Poincaré 25, 67 (1976)
Wald, R.M.: General Relativity, Chicago: University of Chicago Press, 1984
Wald, R.M.: Quantum field theory in curved space-time and black hole thermodynamics. Chicago: The University of Chicago Press, 1994
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Dedicated to Professor Klaus Fredenhagen on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Dappiaggi, C., Moretti, V. & Pinamonti, N. Cosmological Horizons and Reconstruction of Quantum Field Theories. Commun. Math. Phys. 285, 1129–1163 (2009). https://doi.org/10.1007/s00220-008-0653-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0653-8