A distinguished vacuum state for a quantum field in a curved spacetime: formalism, features, and cosmology

  • Niayesh Afshordi
  • Siavash Aslanbeigi
  • Rafael D. Sorkin


We define a distinguished “ground state” or “vacuum” for a free scalar quantum field in a globally hyperbolic region of an arbitrarily curved spacetime. Our prescription is motivated by the recent construction [1, 2] of a quantum field theory on a background causal set using only knowledge of the retarded Green’s function. We generalize that construction to continuum spacetimes and find that it yields a distinguished vacuum or ground state for a non-interacting, massive or massless scalar field. This state is defined for all compact regions and for many noncompact ones. In a static spacetime we find that our vacuum coincides with the usual ground state. We determine it also for a radiation-filled, spatially homogeneous and isotropic cosmos, and show that the super-horizon correlations are approximately the same as those of a thermal state. Finally, we illustrate the inherent non-locality of our prescription with the example of a spacetime which sandwiches a region with curvature in-between flat initial and final regions.


Nonperturbative Effects Space-Time Symmetries Thermal Field Theory Cosmology of Theories beyond the SM 


  1. [1]
    S. Johnston, Feynman Propagator for a Free Scalar Field on a Causal Set, Phys. Rev. Lett. 103 (2009) 180401 [arXiv:0909.0944] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    R.D. Sorkin, Scalar Field Theory on a Causal Set in Histories Form, J. Phys. Conf. Ser. 306 (2011) 012017 [arXiv:1107.0698] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206-206] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J. Bisognano and E. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16 (1975) 985 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  5. [5]
    W. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].ADSGoogle Scholar
  6. [6]
    V.F. Mukhanov, H. Feldman and R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    B. Allen, Vacuum States in de Sitter Space, Phys. Rev. D 32 (1985) 3136 [INSPIRE].ADSGoogle Scholar
  8. [8]
    R. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, Chicago lectures in physics, University of Chicago Press, Chicago U.S.A. (1994).Google Scholar
  9. [9]
    R.M. Wald, The Formulation of Quantum Field Theory in Curved Spacetime, arXiv:0907.0416 [INSPIRE].
  10. [10]
    F. Dowker, S. Johnston and R.D. Sorkin, Hilbert Spaces from Path Integrals, J. Phys. A 43 (2010) 275302 [arXiv:1002.0589] [INSPIRE].MathSciNetADSGoogle Scholar
  11. [11]
    D. Sorkin, Rafael, Quantum mechanics as quantum measure theory, Mod. Phys. Lett. A 9 (1994) 3119 [gr-qc/9401003] [INSPIRE].MathSciNetADSGoogle Scholar
  12. [12]
    R.D. Sorkin, Toward afundamental theorem of quantal measure theory’, arXiv:1104.0997 [INSPIRE].
  13. [13]
    J.B. Hartle, Space-time quantum mechanics and the quantum mechanics of space-time, gr-qc/9304006 [INSPIRE].
  14. [14]
    A. Ashtekar and A. Magnon-Ashtekar, A curiosity concerning the role of coherent states in quantum field theory, Pramana 15 (1980) 107.ADSCrossRefGoogle Scholar
  15. [15]
    M. Reed and B. Simon, I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) (vol 1), Academic Press (1981).Google Scholar
  16. [16]
    C.J. Fewster and R. Verch, On a Recent Construction ofVacuum-likeQuantum Field States in Curved Spacetime, arXiv:1206.1562 [INSPIRE].
  17. [17]
    N. Afshordi et al., A Ground State for the Causal Diamond in 2 Dimensions, arXiv:1207.7101 [INSPIRE].
  18. [18]
    N. Afshordi, S. Aslanbeigi, M. Buck, F. Dowker and R. Sorkin, in preparation.Google Scholar
  19. [19]
    S. Fulling, Remarks on positive frequency and hamiltonians in expanding universes, Gen. Rel. Grav. 10 (1979) 807 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  20. [20]
    C. Dappiaggi, T.-P. Hack and N. Pinamonti, Approximate KMS states for scalar and spinor fields in Friedmann-Robertson-Walker spacetimes, Annales Henri Poincaré 12 (2011) 1449 [arXiv:1009.5179] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  21. [21]
    N. Birrell and P. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge U.K. (1982).MATHCrossRefGoogle Scholar
  22. [22]
    F.W. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, New York U.S.A. (2010).MATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Niayesh Afshordi
    • 1
    • 2
  • Siavash Aslanbeigi
    • 1
    • 2
  • Rafael D. Sorkin
    • 1
    • 3
  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of Physics and AstronomyUniversity of WaterlooWaterlooCanada
  3. 3.Department of PhysicsSyracuse UniversitySyracuseU.S.A.

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