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.
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Notes
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
Kiefer, C.: Quantum Gravity. Clarendon Press, Oxford (2004)
Seiberg, N., Witten, E.: String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999)
Lloyd, S.: Computational capacity of the universe. Phys. Rev. Lett. 88(23), 237901 (2002)
Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2007)
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)
Oriti, D.: Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity. Rep. Prog. Phys. 64(12), 1703 (2001)
Bojowald, M., Kempf, A.: Generalized uncertainty principles and localization of a particle in discrete space. Phys. Rev. D 86(8), 085017 (2012)
Sato, Y.: Space-Time Foliation in Quantum Gravity, pp. 37–56. Springer, Tokyo (2014)
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)
’t Hooft, G.: The cellular automaton interpretation of quantum mechanics, vol. 185. Springer, New York (2016)
’t Hooft, G.: Classical cellular automata and quantum field theory. Int. J. Modern Phys. A 25(23), 4385–4396 (2010)
Sorkin, R.D.: Causal sets: discrete gravity. In: Gomberoff, A., Marolf, D. (eds.) Lectures on Quantum Gravity, pp. 305–327. Springer, Boston (2005)
Kempf, A.: Quantum gravity on a quantum computer? Found. Phys. 44(5), 472–482 (2014)
Kempf, A.: Spacetime could be simultaneously continuous and discrete, in the same way that information can be. New J. Phys. 12(11), 115001 (2010)
Kempf, A., Martin, R.: Information theory, spectral geometry, and quantum gravity. Phys. Rev. Lett. 100(2), 021304 (2008)
Kempf, A.: Covariant information-density cutoff in curved space-time. Phys. Rev. Lett. 92(22), 221301 (2004)
Kempf, A.: Fields over unsharp coordinates. Phys. Rev. Lett. 85(14), 2873 (2000)
Kempf, A.: Black holes, bandwidths and Beethoven. J. Math. Phys. 41(4), 2360–2374 (2000)
Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)
Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, Hoboken (2006)
Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)
Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press on Demand, Oxford (1996)
Benedetto, J.J.: Ferreira, Paulo J.S.G. (ed.): Modern Sampling Theory: Mathematics and Applications. Springer Science & Business Media, New York (2012)
Pye, J., Donnelly, W., Kempf, A.: Locality and entanglement in bandlimited quantum field theory. Phys. Rev. D 92(10), 105022 (2015)
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)
Kempf, A.: In: Proceedings of the XXII DGM Conference on Sept.93 Ixtapa (Mexico), Adv. Appl. Cliff. Alg (Proc. Suppl.) (S1) (1994)
Kempf, A.: Uncertainty relation in quantum mechanics with quantum group symmetry. J. Math. Phys. 35(9), 4483–4496 (1994)
Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52(2), 1108 (1995)
Garay, L.J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10(02), 145–165 (1995)
Scardigli, F., Lambiase, G., Vagenas, E.C.: GUP parameter from quantum corrections to the Newtonian potential. Phys. Lett. B 767, 242–246 (2017)
Casadio, R., Garattini, R., Scardigli, F.: Point-like sources and the scale of quantum gravity. Phys. Lett. B 679(2), 156–159 (2009)
Scardigli, F.: Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment. Phys. Lett. B 452(1–2), 39–44 (1999)
Hossenfelder, S.: Minimal length scale scenarios for quantum gravity. Living Rev. Relativ. 16(1), 2 (2013)
Martin, R.T.W., Kempf, A.: Quantum uncertainty and the spectra of symmetric operators. Acta Appl. Math. 106(3), 349–358 (2009)
Kempf, A.: Information-theoretic natural ultraviolet cutoff for spacetime. Phys. Rev. Lett. 103(23), 231301 (2009)
Gilkey, P.B.: The spectral geometry of a Riemannian manifold. J. Differ. Geom. 10(4), 601–618 (1975)
Hawking, S.W.: Quantum gravity and path integrals. Phys. Rev. D 18(6), 1747 (1978)
Kempf, A.: On nonlocality, lattices and internal symmetries. EPL (Europhys. Lett.) 40(3), 257 (1997)
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)
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)
Kempf, A.: Mode generating mechanism in inflation with a cutoff. Phys. Rev. D 63(8), 083514 (2001)
Kempf, A., Niemeyer, J.C.: Perturbation spectrum in inflation with a cutoff. Phys. Rev. D 64(10), 103501 (2001)
Ashoorioon, A., Kempf, A., Mann, R.B.: Minimum length cutoff in inflation and uniqueness of the action. Phys. Rev. D 71(2), 023503 (2005)
Kempf, A., Lorenz, L.: Exact solution of inflationary model with minimum length. Phys. Rev. D 74(10), 103517 (2006)
Martin, J., Martin, J., Brandenberger, R.H.: J. Martin and RH Brandenberger, Phys. Rev. D 63, 123501 (2001). Phys. Rev. D 63, 123501 (2001)
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)
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)
Brandenberger, R.H., Martin, J.: On signatures of short distance physics in the cosmic microwave background. Int. J. Mod. Phys. A 17, 3663 (2002)
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)
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)
Saravani, M., Aslanbeigi, S., Kempf, A.: Spacetime curvature in terms of scalar field propagators. Phys. Rev. D 93(4), 045026 (2016)
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time, vol. 1. Cambridge University Press, Cambridge (1973)
Yazdi, Y.K., Kempf, A.: Towards spectral geometry for causal sets. Class. Quantum Gravity 34(9), 094001 (2017)
Datchev, K., Hezari, H.: Inverse problems in spectral geometry. Inverse Prob. Appl. 60, 455–486 (2011)
Aasen, D., Bhamre, T., Kempf, A.: Shape from sound: toward new tools for quantum gravity. Phys. Rev. Lett. 110(12), 121301 (2013)
Panine, M., Kempf, A.: Towards spectral geometric methods for Euclidean quantum gravity. Phys. Rev. D 93(8), 084033 (2016)
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)
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This work has been supported by the Discovery Program of the National Science and Engineering Research Council of Canada (NSERC).
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Kempf, A. Quantum Gravity, Information Theory and the CMB. Found Phys 48, 1191–1203 (2018). https://doi.org/10.1007/s10701-018-0163-2
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DOI: https://doi.org/10.1007/s10701-018-0163-2