Skip to main content

Quantum Gravity, Information Theory and the CMB

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.

This is a preview of subscription content, access via your institution.

Notes

  1. Technically, in Shannon’s theorem, a bandlimited signal is assumed to obey Dirichlet boundary conditions in the Fourier domain. A conventional lattice theory implements periodic boundary conditions in the Fourier domain.

References

  1. Kiefer, C.: Quantum Gravity. Clarendon Press, Oxford (2004)

    MATH  Google Scholar 

  2. Seiberg, N., Witten, E.: String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  3. Lloyd, S.: Computational capacity of the universe. Phys. Rev. Lett. 88(23), 237901 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  4. Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  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)

    MATH  Google Scholar 

  6. Oriti, D.: Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity. Rep. Prog. Phys. 64(12), 1703 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  7. Bojowald, M., Kempf, A.: Generalized uncertainty principles and localization of a particle in discrete space. Phys. Rev. D 86(8), 085017 (2012)

    ADS  Article  Google Scholar 

  8. Sato, Y.: Space-Time Foliation in Quantum Gravity, pp. 37–56. Springer, Tokyo (2014)

  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)

    Chapter  Google Scholar 

  10. ’t Hooft, G.: The cellular automaton interpretation of quantum mechanics, vol. 185. Springer, New York (2016)

    MATH  Google Scholar 

  11. ’t Hooft, G.: Classical cellular automata and quantum field theory. Int. J. Modern Phys. A 25(23), 4385–4396 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  12. Sorkin, R.D.: Causal sets: discrete gravity. In: Gomberoff, A., Marolf, D. (eds.) Lectures on Quantum Gravity, pp. 305–327. Springer, Boston (2005)

    Chapter  Google Scholar 

  13. Kempf, A.: Quantum gravity on a quantum computer? Found. Phys. 44(5), 472–482 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  15. Kempf, A., Martin, R.: Information theory, spectral geometry, and quantum gravity. Phys. Rev. Lett. 100(2), 021304 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  16. Kempf, A.: Covariant information-density cutoff in curved space-time. Phys. Rev. Lett. 92(22), 221301 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  17. Kempf, A.: Fields over unsharp coordinates. Phys. Rev. Lett. 85(14), 2873 (2000)

    ADS  Article  Google Scholar 

  18. Kempf, A.: Black holes, bandwidths and Beethoven. J. Math. Phys. 41(4), 2360–2374 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  19. Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)

    MathSciNet  Article  Google Scholar 

  20. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, Hoboken (2006)

    MATH  Google Scholar 

  21. Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  22. Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press on Demand, Oxford (1996)

    MATH  Google Scholar 

  23. Benedetto, J.J.: Ferreira, Paulo J.S.G. (ed.): Modern Sampling Theory: Mathematics and Applications. Springer Science & Business Media, New York (2012)

  24. Pye, J., Donnelly, W., Kempf, A.: Locality and entanglement in bandlimited quantum field theory. Phys. Rev. D 92(10), 105022 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  26. Kempf, A.: In: Proceedings of the XXII DGM Conference on Sept.93 Ixtapa (Mexico), Adv. Appl. Cliff. Alg (Proc. Suppl.) (S1) (1994)

  27. Kempf, A.: Uncertainty relation in quantum mechanics with quantum group symmetry. J. Math. Phys. 35(9), 4483–4496 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  28. Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52(2), 1108 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  29. Garay, L.J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10(02), 145–165 (1995)

    ADS  Article  Google Scholar 

  30. Scardigli, F., Lambiase, G., Vagenas, E.C.: GUP parameter from quantum corrections to the Newtonian potential. Phys. Lett. B 767, 242–246 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  31. Casadio, R., Garattini, R., Scardigli, F.: Point-like sources and the scale of quantum gravity. Phys. Lett. B 679(2), 156–159 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  32. Scardigli, F.: Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment. Phys. Lett. B 452(1–2), 39–44 (1999)

    ADS  Article  Google Scholar 

  33. Hossenfelder, S.: Minimal length scale scenarios for quantum gravity. Living Rev. Relativ. 16(1), 2 (2013)

    ADS  Article  Google Scholar 

  34. Martin, R.T.W., Kempf, A.: Quantum uncertainty and the spectra of symmetric operators. Acta Appl. Math. 106(3), 349–358 (2009)

    MathSciNet  Article  Google Scholar 

  35. Kempf, A.: Information-theoretic natural ultraviolet cutoff for spacetime. Phys. Rev. Lett. 103(23), 231301 (2009)

    ADS  Article  Google Scholar 

  36. Gilkey, P.B.: The spectral geometry of a Riemannian manifold. J. Differ. Geom. 10(4), 601–618 (1975)

    MathSciNet  Article  Google Scholar 

  37. Hawking, S.W.: Quantum gravity and path integrals. Phys. Rev. D 18(6), 1747 (1978)

    ADS  Article  Google Scholar 

  38. Kempf, A.: On nonlocality, lattices and internal symmetries. EPL (Europhys. Lett.) 40(3), 257 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  41. Kempf, A.: Mode generating mechanism in inflation with a cutoff. Phys. Rev. D 63(8), 083514 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  42. Kempf, A., Niemeyer, J.C.: Perturbation spectrum in inflation with a cutoff. Phys. Rev. D 64(10), 103501 (2001)

    ADS  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  44. Kempf, A., Lorenz, L.: Exact solution of inflationary model with minimum length. Phys. Rev. D 74(10), 103517 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  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)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  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)

    ADS  Article  Google Scholar 

  51. Saravani, M., Aslanbeigi, S., Kempf, A.: Spacetime curvature in terms of scalar field propagators. Phys. Rev. D 93(4), 045026 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  52. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time, vol. 1. Cambridge University Press, Cambridge (1973)

    Book  Google Scholar 

  53. Yazdi, Y.K., Kempf, A.: Towards spectral geometry for causal sets. Class. Quantum Gravity 34(9), 094001 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  54. Datchev, K., Hezari, H.: Inverse problems in spectral geometry. Inverse Prob. Appl. 60, 455–486 (2011)

    MathSciNet  MATH  Google Scholar 

  55. Aasen, D., Bhamre, T., Kempf, A.: Shape from sound: toward new tools for quantum gravity. Phys. Rev. Lett. 110(12), 121301 (2013)

    ADS  Article  Google Scholar 

  56. Panine, M., Kempf, A.: Towards spectral geometric methods for Euclidean quantum gravity. Phys. Rev. D 93(8), 084033 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Achim Kempf.

Additional information

This work has been supported by the Discovery Program of the National Science and Engineering Research Council of Canada (NSERC).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kempf, A. Quantum Gravity, Information Theory and the CMB. Found Phys 48, 1191–1203 (2018). https://doi.org/10.1007/s10701-018-0163-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-018-0163-2

Keywords

  • Quantum gravity
  • Planck scale
  • Information theory
  • CMB