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Quantum Gravity, Information Theory and the CMB

  • Achim Kempf
Article
Part of the following topical collections:
  1. Black holes, Gravitational waves and Space Time Singularities

Abstract

We review connections between the metric of spacetime and the quantum fluctuations of fields. We start with the finding that the spacetime metric can be expressed entirely in terms of the 2-point correlator of the fluctuations of quantum fields. We then discuss the open question whether the knowledge of only the spectra of the quantum fluctuations of fields also suffices to determine the spacetime metric. This question is of interest because spectra are geometric invariants and their quantization would, therefore, have the benefit of not requiring the modding out of diffeomorphisms. Further, we discuss the fact that spacetime at the Planck scale need not necessarily be either discrete or continuous. Instead, results from information theory show that spacetime may be simultaneously discrete and continuous in the same way that information can. Finally, we review the recent finding that a covariant natural ultraviolet cutoff at the Planck scale implies a signature in the cosmic microwave background (CMB) that may become observable.

Keywords

Quantum gravity Planck scale Information theory CMB 

References

  1. 1.
    Kiefer, C.: Quantum Gravity. Clarendon Press, Oxford (2004)zbMATHGoogle Scholar
  2. 2.
    Seiberg, N., Witten, E.: String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lloyd, S.: Computational capacity of the universe. Phys. Rev. Lett. 88(23), 237901 (2002)ADSCrossRefGoogle Scholar
  4. 4.
    Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hawking, S.W.: The path-integral approach to quantum gravity. In: Hawking, S.W., Israel, K.W. (eds.) General Relativity: An Einstein Centenary Survey. Cambridge University Press, Cambridge (1979)Google Scholar
  6. 6.
    Oriti, D.: Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity. Rep. Prog. Phys. 64(12), 1703 (2001)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bojowald, M., Kempf, A.: Generalized uncertainty principles and localization of a particle in discrete space. Phys. Rev. D 86(8), 085017 (2012)ADSCrossRefGoogle Scholar
  8. 8.
    Sato, Y.: Space-Time Foliation in Quantum Gravity, pp. 37–56. Springer, Tokyo (2014)Google Scholar
  9. 9.
    Henson, J.: The causal set approach to quantum gravity. In: Oriti, D. (ed.) Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, pp. 393–413. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  10. 10.
    ’t Hooft, G.: The cellular automaton interpretation of quantum mechanics, vol. 185. Springer, New York (2016)zbMATHGoogle Scholar
  11. 11.
    ’t Hooft, G.: Classical cellular automata and quantum field theory. Int. J. Modern Phys. A 25(23), 4385–4396 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sorkin, R.D.: Causal sets: discrete gravity. In: Gomberoff, A., Marolf, D. (eds.) Lectures on Quantum Gravity, pp. 305–327. Springer, Boston (2005)CrossRefGoogle Scholar
  13. 13.
    Kempf, A.: Quantum gravity on a quantum computer? Found. Phys. 44(5), 472–482 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kempf, A.: Spacetime could be simultaneously continuous and discrete, in the same way that information can be. New J. Phys. 12(11), 115001 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    Kempf, A., Martin, R.: Information theory, spectral geometry, and quantum gravity. Phys. Rev. Lett. 100(2), 021304 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kempf, A.: Covariant information-density cutoff in curved space-time. Phys. Rev. Lett. 92(22), 221301 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kempf, A.: Fields over unsharp coordinates. Phys. Rev. Lett. 85(14), 2873 (2000)ADSCrossRefGoogle Scholar
  18. 18.
    Kempf, A.: Black holes, bandwidths and Beethoven. J. Math. Phys. 41(4), 2360–2374 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, Hoboken (2006)zbMATHGoogle Scholar
  21. 21.
    Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)zbMATHGoogle Scholar
  22. 22.
    Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press on Demand, Oxford (1996)zbMATHGoogle Scholar
  23. 23.
    Benedetto, J.J.: Ferreira, Paulo J.S.G. (ed.): Modern Sampling Theory: Mathematics and Applications. Springer Science & Business Media, New York (2012)Google Scholar
  24. 24.
    Pye, J., Donnelly, W., Kempf, A.: Locality and entanglement in bandlimited quantum field theory. Phys. Rev. D 92(10), 105022 (2015)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Witten, E.: Reflections on the fate of spacetime. In: Callender, C. (ed.) Physics Meets Philosophy at the Planck Scale, pp. 125–137. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  26. 26.
    Kempf, A.: In: Proceedings of the XXII DGM Conference on Sept.93 Ixtapa (Mexico), Adv. Appl. Cliff. Alg (Proc. Suppl.) (S1) (1994)Google Scholar
  27. 27.
    Kempf, A.: Uncertainty relation in quantum mechanics with quantum group symmetry. J. Math. Phys. 35(9), 4483–4496 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52(2), 1108 (1995)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Garay, L.J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10(02), 145–165 (1995)ADSCrossRefGoogle Scholar
  30. 30.
    Scardigli, F., Lambiase, G., Vagenas, E.C.: GUP parameter from quantum corrections to the Newtonian potential. Phys. Lett. B 767, 242–246 (2017)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Casadio, R., Garattini, R., Scardigli, F.: Point-like sources and the scale of quantum gravity. Phys. Lett. B 679(2), 156–159 (2009)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Scardigli, F.: Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment. Phys. Lett. B 452(1–2), 39–44 (1999)ADSCrossRefGoogle Scholar
  33. 33.
    Hossenfelder, S.: Minimal length scale scenarios for quantum gravity. Living Rev. Relativ. 16(1), 2 (2013)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Martin, R.T.W., Kempf, A.: Quantum uncertainty and the spectra of symmetric operators. Acta Appl. Math. 106(3), 349–358 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kempf, A.: Information-theoretic natural ultraviolet cutoff for spacetime. Phys. Rev. Lett. 103(23), 231301 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Gilkey, P.B.: The spectral geometry of a Riemannian manifold. J. Differ. Geom. 10(4), 601–618 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hawking, S.W.: Quantum gravity and path integrals. Phys. Rev. D 18(6), 1747 (1978)ADSCrossRefGoogle Scholar
  38. 38.
    Kempf, A.: On nonlocality, lattices and internal symmetries. EPL (Europhys. Lett.) 40(3), 257 (1997)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Kempf, A., Chatwin-Davies, A., Martin, R.T.W.: A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes. J. Math. Phys. 54(2), 022301 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Chatwin-Davies, A., Kempf, A., Martin, R.T.W.: Natural covariant Planck scale cutoffs and the cosmic microwave background spectrum. Phys. Rev. Lett. 119(3), 031301 (2017)ADSCrossRefGoogle Scholar
  41. 41.
    Kempf, A.: Mode generating mechanism in inflation with a cutoff. Phys. Rev. D 63(8), 083514 (2001)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Kempf, A., Niemeyer, J.C.: Perturbation spectrum in inflation with a cutoff. Phys. Rev. D 64(10), 103501 (2001)ADSCrossRefGoogle Scholar
  43. 43.
    Ashoorioon, A., Kempf, A., Mann, R.B.: Minimum length cutoff in inflation and uniqueness of the action. Phys. Rev. D 71(2), 023503 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    Kempf, A., Lorenz, L.: Exact solution of inflationary model with minimum length. Phys. Rev. D 74(10), 103517 (2006)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Martin, J., Martin, J., Brandenberger, R.H.: J. Martin and RH Brandenberger, Phys. Rev. D 63, 123501 (2001). Phys. Rev. D 63, 123501 (2001)ADSCrossRefGoogle Scholar
  46. 46.
    Shiu, G.: Inflation as a probe of trans-Planckian physics: a brief review and progress report. J. Phys. Conf. Ser. 18(1), 188–223 (2005)ADSCrossRefGoogle Scholar
  47. 47.
    Brandenberger, R.H., Martin, J.: The robustness of inflation to changes in super-Planck-scale physics. Mod. Phys. Lett. A 16(15), 999–1006 (2001)ADSCrossRefzbMATHGoogle Scholar
  48. 48.
    Brandenberger, R.H., Martin, J.: On signatures of short distance physics in the cosmic microwave background. Int. J. Mod. Phys. A 17, 3663 (2002)ADSCrossRefzbMATHGoogle Scholar
  49. 49.
    Easther, R., Greene, B.R., Kinney, W.H., Shiu, G.: Generic estimate of trans-Planckian modifications to the primordial power spectrum in inflation. Phys. Rev. D 66(2), 023518 (2002)ADSCrossRefGoogle Scholar
  50. 50.
    Greene, B.R., Schalm, K., Shiu, G., van der Schaar, J.P.: Decoupling in an expanding universe: backreaction barely constrains short distance effects in the cosmic microwave background. J. Cosmol. Astropart. Phys. 2005(02), 001 (2005)ADSCrossRefGoogle Scholar
  51. 51.
    Saravani, M., Aslanbeigi, S., Kempf, A.: Spacetime curvature in terms of scalar field propagators. Phys. Rev. D 93(4), 045026 (2016)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time, vol. 1. Cambridge University Press, Cambridge (1973)CrossRefzbMATHGoogle Scholar
  53. 53.
    Yazdi, Y.K., Kempf, A.: Towards spectral geometry for causal sets. Class. Quantum Gravity 34(9), 094001 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Datchev, K., Hezari, H.: Inverse problems in spectral geometry. Inverse Prob. Appl. 60, 455–486 (2011)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Aasen, D., Bhamre, T., Kempf, A.: Shape from sound: toward new tools for quantum gravity. Phys. Rev. Lett. 110(12), 121301 (2013)ADSCrossRefGoogle Scholar
  56. 56.
    Panine, M., Kempf, A.: Towards spectral geometric methods for Euclidean quantum gravity. Phys. Rev. D 93(8), 084033 (2016)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    Panine, M., Kempf, A.: A convexity result in the spectral geometry of conformally equivalent metrics on surfaces. Int. J. Geom. Methods Mod. Phys. 14(11), 1750157 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

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