Advertisement

Cosmological Applications of Algebraic Quantum Field Theory

  • Thomas-Paul Hack
  • Nicola Pinamonti
Chapter
Part of the Mathematical Physics Studies book series (MPST)

Abstract

Quantum field theory on curved spacetime is a generalisation of quantum field theory in flat spacetime which is expected to be the proper fundamental description of non–trivial physical phenomena in the presence of a spacetime curvature which is large but below Planck scale. Two prominent physical situations which fall under this characterisation are phenomena both in the vicinity of black holes and in the early universe. Focusing on the latter, we review several applications of algebraic quantum field theory on curved spacetimes to cosmology, as well as foundational results and constructions on which these applications are based. On the foundational side, we collect several proposals to construct Hadamard states on cosmological spacetimes, as this class of states is believed to encompass all physically meaningful states in quantum field theory on curved spacetimes. Afterwards we consider the solution theory of the semiclassical Einstein equation, quote a theorem of existence and uniqueness of solutions to this equation and indicate directions to go beyond the semiclassical Einstein equation. Then we highlight how the observed cosmological expansion may be understood qualitatively and quantitatively in this framework, before we discuss the quantization of perturbations in inflation in the context of algebraic quantum field theory. In the latter subject, the starting point is the assumption that the classical, rather than the semiclassical, Einstein equation is satisfied. We close this chapter briefly discussing how one may generalise the analysis of perturbations in inflation by allowing for spacetimes backgrounds which solve the semiclassical Einstein equations.

Keywords

Curve Spacetimes Local Observable Cosmological Time Spatial Infinity Hadamard State 
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.

References

  1. 1.
    Ade, P.A.R., et al.: [Planck Collaboration]: Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. 571, A16 (2014)Google Scholar
  2. 2.
    Ade, P.A.R., et al.: [Planck Collaboration]: Planck 2013 results. XXII. Constraints on inflation. Astron. Astrophys. 571, A22 (2014)Google Scholar
  3. 3.
    Ade, P.A.R., et al.: [BICEP2 Collaboration]: Detection of B-Mode polarization at degree angular scales by BICEP2. Phys. Rev. Lett. 112, 241101 (2014)CrossRefADSGoogle Scholar
  4. 4.
    Afshordi, N., Aslanbeigi, S., Sorkin, R.D.: A distinguished vacuum state for a quantum field in curved spacetime: formalism, features, and cosmology. JHEP 2012, 1–12 (2012)MathSciNetGoogle Scholar
  5. 5.
    Anderson, P.R.: Effects of quantum fields on singularities and particle horizons in the early universe. 4. initially empty universes. Phys. Rev. D 33, 1567 (1986)CrossRefADSGoogle Scholar
  6. 6.
    Brum, M., Fredenhagen, K.: ‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes. Class. Quantum Grav. 31, 025024 (2014)MathSciNetCrossRefADSzbMATHGoogle Scholar
  7. 7.
    Brunetti, R., Fredenhagen, K., Hollands, S.: A remark on alpha vacua for quantum field theories on de Sitter space. JHEP 0505, 063 (2005)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996)CrossRefADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003)MathSciNetCrossRefADSzbMATHGoogle Scholar
  10. 10.
    Dappiaggi, C., Hack, T.-P., Pinamonti, N.: Approximate KMS states for scalar and spinor fields in Friedmann-Robertson-Walker spacetimes. Ann. Henri Poincare 12, 1449 (2011)MathSciNetCrossRefADSzbMATHGoogle Scholar
  11. 11.
    Dappiaggi, C., Moretti, V., Pinamonti, N.: Cosmological horizons and reconstruction of quantum field theories. Commun. Math. Phys. 285, 1129–1163 (2009)MathSciNetCrossRefADSzbMATHGoogle Scholar
  12. 12.
    Dappiaggi, C., Moretti, V., Pinamonti, N.: Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50, 062304 (2009)MathSciNetCrossRefADSzbMATHGoogle Scholar
  13. 13.
    Dimock, J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4, 223–233 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Degner, A.: Properties of states of low energy on cosmological spacetimes. PhD thesis, University of Hamburg (2013)Google Scholar
  15. 15.
    Eltzner, B.: Quantization of perturbations in Inflation. arXiv:1302.5358 [gr-qc]
  16. 16.
    Eltzner, B., Gottschalk, H.: Dynamical backreaction in Robertson-Walker spacetime. Rev. Math. Phys. 23, 531–551 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fewster, C.J., Hunt, D.S.: Quantization of linearised gravity in cosmological vacuum spacetimes. Rev. Math. Phys. 25, 1330003 (2013)Google Scholar
  18. 18.
    Fewster, C.J., Verch, R.: On a recent construction of ‘Vacuum-like’ quantum field states in curved spacetime. Class. Quant. Grav. 29, 205017 (2012)MathSciNetCrossRefADSzbMATHGoogle Scholar
  19. 19.
    Fewster, C.J., Ford, L.H., Roman, T.A.: Probability distributions of smeared quantum stress tensors. Phys. Rev. D 81, 121901 (2010)CrossRefADSGoogle Scholar
  20. 20.
    Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudo-differential calculus. Comm. Math. Phys. 325, 713–755 (2014)MathSciNetCrossRefADSzbMATHGoogle Scholar
  21. 21.
    Hack, T.-P.: The Lambda CDM-model in quantum field theory on curved spacetime and dark radiation. arXiv:1306.3074 [gr-qc]
  22. 22.
    Hack, T.-P.: Quantization of the linearised Einstein-Klein-Gordon system on arbitrary backgrounds and the special case of perturbations in Inflation. Class. Quantum Grav. 31, 215004 (2014)MathSciNetCrossRefADSzbMATHGoogle Scholar
  23. 23.
    Hack, T.-P., Schenkel, A.: Linear bosonic and fermionic quantum gauge theories on curved spacetimes. Gen. Rel. Grav. 45, 877 (2013)MathSciNetCrossRefADSzbMATHGoogle Scholar
  24. 24.
    Hollands, S., Wald, R.M.: Local wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001)MathSciNetCrossRefADSzbMATHGoogle Scholar
  25. 25.
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 2309 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hu, B.L., Verdaguer, E.: Stochastic gravity: theory and applications. Living Rev. Rel. 11, 3 (2008); Living Rev. Rel. 7, 3 (2004)Google Scholar
  28. 28.
    Junker, W., Schrohe, E.: Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties. Ann. Henri Poincaré 3, 1113–1181 (2002)MathSciNetCrossRefADSzbMATHGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefADSzbMATHGoogle Scholar
  30. 30.
    Kolb, E.W., Turner, M.S.: The early universe. Front. Phys. 69, 1 (1990)Google Scholar
  31. 31.
    Küskü, M.: A class of almost equilibrium states in Robertson-Walker spacetimes. DESY-THESIS-2008-020Google Scholar
  32. 32.
    Lüders, C., Roberts, J.E.: Local quasiequivalence and adiabatic vacuum states. Commun. Math. Phys. 134, 29–63 (1990)CrossRefADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Moretti, V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189–221 (2003)MathSciNetCrossRefADSzbMATHGoogle Scholar
  34. 34.
    Olbermann, H.: States of low energy on Robertson-Walker spacetimes. Class. Quantum Grav. 24, 5011–5030 (2007)MathSciNetCrossRefADSzbMATHGoogle Scholar
  35. 35.
    Parker, L.: Quantized fields and particle creation in expanding universes I. Phys. Rev. 183, 1057–1068 (1969)CrossRefADSzbMATHGoogle Scholar
  36. 36.
    Pinamonti, N.: On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario. Commun. Math. Phys. 305, 563–604 (2011)MathSciNetCrossRefADSzbMATHGoogle Scholar
  37. 37.
    Pinamonti, N., Siemssen, D.: Scale-invariant curvature fluctuations from an extended semiclassical gravity. J. Math. Phys. 56, 022303 (2015)MathSciNetCrossRefADSzbMATHGoogle Scholar
  38. 38.
    Pinamonti, N., Siemssen, D.: Global existence of solutions of the semiclassical Einstein equation for cosmological spacetimes. Commun. Math. Phys. 334(1), 171–191 (2015)MathSciNetCrossRefADSzbMATHGoogle Scholar
  39. 39.
    Radzikowski, M.J.: Micro-local approach to the hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529–553 (1996)MathSciNetCrossRefADSzbMATHGoogle Scholar
  40. 40.
    Riegert, R.J.: A nonlocal action for the trace anomaly. Phys. Lett. B 134, 56 (1984)MathSciNetCrossRefADSzbMATHGoogle Scholar
  41. 41.
    Schlemmer, J.: Ph.D. thesis, Faculty of Physics, University of Leipzig (2010)Google Scholar
  42. 42.
    Sewell, G. L.: Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Phys. (NY) 141, 201 (1982)Google Scholar
  43. 43.
    Straumann, N.: From primordial quantum fluctuations to the anisotropies of the cosmic microwave background radiation. Annalen Phys. 15, 70 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99 (1980)CrossRefADSGoogle Scholar
  45. 45.
    Starobinsky, A.A.: The perturbation spectrum evolving from a nonsingular initially de-sitter cosmology and the microwave background anisotropy. Sov. Astron. Lett. 9, 302 (1983)ADSGoogle Scholar
  46. 46.
    Them, K., Brum, M.: States of low energy in homogeneous and inhomogeneous. Expanding Spacetimes. Class. Quant. Grav. 30, 235035 (2013)CrossRefADSzbMATHGoogle Scholar
  47. 47.
    Verch, R.: Local covariance, renormalization ambiguity, and local thermal equilibrium in cosmology. In: Finster, F., Müller, O., Nardmann, M., Tolksdorf, J., Zeidler, E. (eds.) Quantum field theory and gravity. Conceptual and mathematical advances in the search for a unified framework, Birkhäuser (2012)Google Scholar
  48. 48.
    Wald, R.M.: Axiomatic renormalization of stress tensor of a conformally invariant field in conformally flat spacetimes. Ann. Phys. 110, 472 (1978)CrossRefADSzbMATHGoogle Scholar
  49. 49.
    Wald, R.M.: Trace anomaly of a conformally invariant quantum field in curved space-time. Phys. Rev. D 17, 1477 (1978)MathSciNetCrossRefADSGoogle Scholar
  50. 50.
    Zschoche, J.: The Chaplygin gas equation of state for the quantized free scalar field on cosmological spacetimes. Ann. Henri Poincare 15, 1285 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GenovaGenova Italy
  2. 2.INFN—Sezione di GenovaGenovaItaly

Personalised recommendations