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Advances in Algebraic Quantum Field Theory pp 253–288Cite as

Cosmological Applications of Algebraic Quantum Field Theory

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Part of the book series: Mathematical Physics Studies ((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.

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Notes

  1. 1.

    Strictly speaking \({\mathcal {L}}_\varsigma EL({\mathfrak {G}})=0\) is satisfied even if \(\Box \varphi + \partial _\varphi V=c\) with c constant but non–zero. However, one may absorb c by redefining \(V(\varphi )\).

References

  1. Ade, P.A.R., et al.: [Planck Collaboration]: Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. 571, A16 (2014)

    Google Scholar 

  2. Ade, P.A.R., et al.: [Planck Collaboration]: Planck 2013 results. XXII. Constraints on inflation. Astron. Astrophys. 571, A22 (2014)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  6. Brum, M., Fredenhagen, K.: ‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes. Class. Quantum Grav. 31, 025024 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Brunetti, R., Fredenhagen, K., Hollands, S.: A remark on alpha vacua for quantum field theories on de Sitter space. JHEP 0505, 063 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Dappiaggi, C., Moretti, V., Pinamonti, N.: Cosmological horizons and reconstruction of quantum field theories. Commun. Math. Phys. 285, 1129–1163 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Dimock, J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4, 223–233 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Degner, A.: Properties of states of low energy on cosmological spacetimes. PhD thesis, University of Hamburg (2013)

    Google Scholar 

  15. Eltzner, B.: Quantization of perturbations in Inflation. arXiv:1302.5358 [gr-qc]

  16. Eltzner, B., Gottschalk, H.: Dynamical backreaction in Robertson-Walker spacetime. Rev. Math. Phys. 23, 531–551 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fewster, C.J., Hunt, D.S.: Quantization of linearised gravity in cosmological vacuum spacetimes. Rev. Math. Phys. 25, 1330003 (2013)

    Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Fewster, C.J., Ford, L.H., Roman, T.A.: Probability distributions of smeared quantum stress tensors. Phys. Rev. D 81, 121901 (2010)

    Article  ADS  Google Scholar 

  20. Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudo-differential calculus. Comm. Math. Phys. 325, 713–755 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Hack, T.-P.: The Lambda CDM-model in quantum field theory on curved spacetime and dark radiation. arXiv:1306.3074 [gr-qc]

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Hack, T.-P., Schenkel, A.: Linear bosonic and fermionic quantum gauge theories on curved spacetimes. Gen. Rel. Grav. 45, 877 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Junker, W., Schrohe, E.: Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties. Ann. Henri Poincaré 3, 1113–1181 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Kolb, E.W., Turner, M.S.: The early universe. Front. Phys. 69, 1 (1990)

    Google Scholar 

  31. Küskü, M.: A class of almost equilibrium states in Robertson-Walker spacetimes. DESY-THESIS-2008-020

    Google Scholar 

  32. Lüders, C., Roberts, J.E.: Local quasiequivalence and adiabatic vacuum states. Commun. Math. Phys. 134, 29–63 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Moretti, V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189–221 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Olbermann, H.: States of low energy on Robertson-Walker spacetimes. Class. Quantum Grav. 24, 5011–5030 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Parker, L.: Quantized fields and particle creation in expanding universes I. Phys. Rev. 183, 1057–1068 (1969)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Pinamonti, N., Siemssen, D.: Scale-invariant curvature fluctuations from an extended semiclassical gravity. J. Math. Phys. 56, 022303 (2015)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Riegert, R.J.: A nonlocal action for the trace anomaly. Phys. Lett. B 134, 56 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Schlemmer, J.: Ph.D. thesis, Faculty of Physics, University of Leipzig (2010)

    Google Scholar 

  42. Sewell, G. L.: Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Phys. (NY) 141, 201 (1982)

    Google Scholar 

  43. Straumann, N.: From primordial quantum fluctuations to the anisotropies of the cosmic microwave background radiation. Annalen Phys. 15, 70 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99 (1980)

    Article  ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  46. Them, K., Brum, M.: States of low energy in homogeneous and inhomogeneous. Expanding Spacetimes. Class. Quant. Grav. 30, 235035 (2013)

    Article  ADS  MATH  Google Scholar 

  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. Wald, R.M.: Axiomatic renormalization of stress tensor of a conformally invariant field in conformally flat spacetimes. Ann. Phys. 110, 472 (1978)

    Article  ADS  MATH  Google Scholar 

  49. Wald, R.M.: Trace anomaly of a conformally invariant quantum field in curved space-time. Phys. Rev. D 17, 1477 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  50. Zschoche, J.: The Chaplygin gas equation of state for the quantized free scalar field on cosmological spacetimes. Ann. Henri Poincare 15, 1285 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Genova, via Dodecaneso 35, I-16146, Genova, Italy

    Thomas-Paul Hack & Nicola Pinamonti

  2. INFN—Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy

    Nicola Pinamonti

Authors
  1. Thomas-Paul Hack
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  2. Nicola Pinamonti
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Correspondence to Thomas-Paul Hack .

Editor information

Editors and Affiliations

  1. University of Trento, Trento, Italy

    Romeo Brunetti

  2. University of Pavia, Pavia, Italy

    Claudio Dappiaggi

  3. University of Hamburg, Hamburg, Germany

    Klaus Fredenhagen

  4. University of Vienna, Vienna, Austria

    Jakob Yngvason

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Hack, TP., Pinamonti, N. (2015). Cosmological Applications of Algebraic Quantum Field Theory. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds) Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-21353-8_6

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