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The Problem of Time pp 531–538Cite as

Tempus Post Quantum. ii. Semiclassical Machian Emergent Time

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 190))

Abstract

We now contemplate dealing with frozen quantum wave equations by finding an emergent time once already at the quantum level, within semiclassical quantum cosmology’s physical regime of interest. This could be the regime in which currently observed inhomogeneities in galaxy distribution and cosmic microwave background temperatures originated. This regime is characterized by heavy–light, adiabatic slow–fast, Born–Oppenheimer, and WKB approximations. In particular, inhomogeneities are light and fast to the scalefactor of the universe and homogeneous modes being heavy and slow. At the level of a Problem of Time strategy, the WKB approximation leads to an emergent semiclassical time. We argue furthermore that this admits a Machian interpretation, in close but not identical parallel with Part II’s emergent classical Machian time, and also cast the semiclassical approach in Temporal Relationalism compatible form. One is then left with an emergent time dependent Schrödinger equation for the light fast modes. We end by discussing how using a WKB approximation is crucial for this outcome but requires further justification.

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Notes

  1. 1.

    Some other earlier models—such as [93, 170, 228, 689, 690], which consider particles in absolute space as semiclassical model arenas—also fall within this ansatz.

  2. 2.

    These are not potentials per se, due to, firstly, overall factors of 2 and that some of the original equations having an \(\mbox{N}^{hh}(h)\) function prefactor being divided out. Secondly, due to terms going like \(\hbar^{2}\) originating from the conformal operator ordering being packaged into \(v\), by which (46.1) does not necessarily display all \(\hbar \) dependence.

  3. 3.

    For the case in which the velocities feature solely homogeneous quadratically in the kinetic term, these are ± the same expression. However, more generally, the 2 solutions are ± in the sense of being a complex conjugate pair.

  4. 4.

    A further argument involves constructive interference underlying classicality [897, 899]. However this amounts to imposing, rather than deducing (semi)classicality. This also applies to using (semi)classicality as a ‘final condition’ restriction on quantum-cosmological solutions. In this manner, LQC can also be argued to not address this point either, and with further issue due to the proportion of solutions rejected on such grounds being larger than is elsewise usual in Quantum Cosmology (see Sect. 43.5). Such a restriction is also akin to how Griffiths and Omnès remove by hand the superposition states which they term “grotesque universes” [390] due to their behaviour being very unlike that we experience today.

References

  1. Anderson, E.: On the semiclassical approach to quantum cosmology. Class. Quantum Gravity 28, 185008 (2011). arXiv:1101.4916

    Article  ADS  MATH  Google Scholar 

  2. Anderson, E.: Machian classical and semiclassical emergent time. Class. Quantum Gravity 31, 025006 (2014). arXiv:1305.4685

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Anderson, E.: Minisuperspace model of machian resolution of problem of time. I. Isotropic case. Gen. Relativ. Gravit. 46, 1708 (2014). arXiv:1307.1916

    Article  ADS  MATH  Google Scholar 

  4. Anderson, E.: Origin of structure in the universe: quantum cosmology reconsidered. Gen. Relativ. Gravit. 47, 101 (2015). arXiv:1501.02443

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Anderson, E.: The problem of time and quantum cosmology in the relational particle mechanics arena. arXiv:1111.1472

  6. Anderson, E., Kneller, S.A.R.: Relational quadrilateralland. II. The quantum theory. Int. J. Mod. Phys. D 23, 1450052 (2014). arXiv:1303.5645

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Banks, T.: TCP, quantum gravity, the cosmological constant and all that. Nucl. Phys. B 249, 322 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  8. Barbour, J.B.: Time and complex numbers in canonical quantum gravity. Phys. Rev. D 47, 5422 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  9. Barbour, J.B.: The timelessness of quantum gravity. II. The appearance of dynamics in static configurations. Class. Quantum Gravity 11, 2875 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  10. Barbour, J.B.: The End of Time. Oxford University Press, Oxford (1999)

    Google Scholar 

  11. Barvinsky, A.O., Kiefer, C.: Wheeler–DeWitt equation and Feynman diagrams. Nucl. Phys. B 526, 509 (1998). gr-qc/9711037

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Brack, M., Bhaduri, R.: Semiclassical Physics. Addison–Wesley, Reading (1997)

    MATH  Google Scholar 

  13. Briggs, J.S., Rost, J.M.: On the derivation of the time-dependent equation of Schrödinger. Found. Phys. 31, 693 (2001). quant-ph/9902035

    Article  MathSciNet  Google Scholar 

  14. Brout, R., Venturi, G.: Time in semiclassical gravity. Phys. Rev. D 39, 2436 (1989)

    Article  ADS  Google Scholar 

  15. Datta, D.P.: Notes on the Born-Oppenheimer approach in a closed dynamical system. Class. Quantum Gravity 14, 2825 (1997). gr-qc/9706077

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. DeWitt, B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113 (1967)

    Article  ADS  MATH  Google Scholar 

  17. Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O., Zeh, H.D.: Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  18. Goldstein, H.: Classical Mechanics. Addison–Wesley, Reading (1980)

    MATH  Google Scholar 

  19. Griffiths, R.B.: Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36, 219 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Halliwell, J.J., Hawking, S.W.: Origin of structure in the universe. Phys. Rev. D 31, 1777 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  21. Isham, C.J.: Canonical quantum gravity and the problem of time. In: Ibort, L.A., Rodríguez, M.A. (eds.) Integrable Systems, Quantum Groups and Quantum Field Theories. Kluwer Academic, Dordrecht (1993). gr-qc/9210011

    Google Scholar 

  22. Kiefer, C.: Quantum cosmology and the emergence of a classical world. In: Rudolph, E., Stamatescu, I-O. (eds.) FESt-Proceedings on the Concepts of Space and Time. Springer, Berlin (1993). gr-qc/9308025

    Google Scholar 

  23. Kiefer, C.: Conceptual issues in quantum cosmology. Lect. Notes Phys. 541, 158 (2000). gr-qc/9906100

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  25. Kiefer, C., Singh, T.P.: Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 44, 1067 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  26. Kuchař, K.V.: Time and interpretations of quantum gravity. In: Kunstatter, G., Vincent, D., Williams, J. (eds.) Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics. World Scientific, Singapore (1992); Reprinted as Int. J. Mod. Phys. Proc. Suppl. D 20, 3 (2011)

    Google Scholar 

  27. Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1949)

    MATH  Google Scholar 

  28. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon, New York (1965)

    MATH  Google Scholar 

  29. Landsman, N.P.: Mathematical Topics Between Classical and Quantum Mechanics. Springer, New York (1998)

    Book  MATH  Google Scholar 

  30. Landsman, N.P.: Between classical and quantum. In: Handbook of the Philosophy of Physics. Elsevier, Amsterdam (2005). quant-ph/0506082

    Google Scholar 

  31. Lapchinski, V.G., Rubakov, V.A.: Canonical quantization of gravity and quantum field theory in curved space-time. Acta Phys. Pol. B 10 (1979)

    Google Scholar 

  32. Messiah, A.: Quantum Mechanics, vols. 1 and 2. North-Holland, Amsterdam (1965)

    Google Scholar 

  33. Muga, G., Sala Mayato, R., Egusquiza, I. (eds.): Time in Quantum Mechanics, vol. 1. Springer, Berlin (2008)

    Google Scholar 

  34. Muga, G., Sala Mayato, R., Egusquiza, I. (eds.): Time in Quantum Mechanics, vol. 2. Springer, Berlin (2010)

    Google Scholar 

  35. Padmanabhan, T.: Semiclassical approximations for gravity and the issue of back-reaction. Class. Quantum Gravity 6, 533 (1989)

    Article  ADS  Google Scholar 

  36. Padmanabhan, T., Singh, T.P.: On the semiclassical limit of the Wheeler–DeWitt equation. Class. Quantum Gravity 7, 411 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  37. Parentani, R.: The background field approximation in (quantum) cosmology. Class. Quantum Gravity 17, 1527 (2000). gr-qc/9803045

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Wheeler, J.A.: Geometrodynamics and the issue of the final state. In: DeWitt, B.S., DeWitt, C.M. (eds.) Groups, Relativity and Topology. Gordon & Breach, New York (1964)

    Google Scholar 

  39. Wheeler, J.A.: Superspace and the nature of quantum geometrodynamics. In: DeWitt, C., Wheeler, J.A. (eds.) Battelle Rencontres: 1967 Lectures in Mathematics and Physics. Benjamin, New York (1968)

    Google Scholar 

  40. Zeh, H.D.: Emergence of classical time from a universal wavefunction. Phys. Lett. A 116, 9 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  41. Zeh, H.D.: Time in quantum gravity. Phys. Lett. A 126, 311 (1988)

    Article  ADS  Google Scholar 

  42. Zeh, H.D.: The Physical Basis of the Direction of Time. Springer, Berlin (1989)

    Book  Google Scholar 

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    Edward Anderson

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Anderson, E. (2017). Tempus Post Quantum. ii. Semiclassical Machian Emergent Time. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_46

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