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Quantum cosmology

  • Claus Kiefer EMAIL logo and Barbara Sandhöfer
Published/Copyright: February 11, 2022
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 77 Issue 6

Abstract

We give an introduction into quantum cosmology with emphasis on its conceptual parts. After a general motivation, we review the formalism of canonical quantum gravity on which discussions of quantum cosmology are usually based. We then present the minisuperspace Wheeler–DeWitt equation and elaborate on the problem of time, the imposition of boundary conditions, the semiclassical approximation, the origin of irreversibility, and singularity avoidance. Restriction is made to the framework of quantum geometrodynamics.

Keywords: cosmology; Hamiltonian formalism; quantum cosmology; quantum gravity
Corresponding author: Claus Kiefer, Faculty of Mathematics and Natural Sciences, Institute for Theoretical Physics, University of Cologne, Cologne, Germany, E-mail:

Acknowledgments

We thank Alexander Vilenkin and Hongbao Zhang for their comments on our manuscript.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: B.S. thanks the Friedrich-Ebert-Stiftung for financial support.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

A Derivation of the Wheeler–DeWitt equation for a concrete model

The starting point is the Einstein–Hilbert action of general relativity,

(18) S EH = 1 2 κ 2 ∫ d 4 x − g ( R − 2 Λ ) ,

where κ 2 = 8πG, R is the Ricci scalar, Λ denotes the cosmological constant, g stands for the determinant of the metric and integration is carried out over all spacetime. The general procedure of the canonical formulation and quantization was outlined in Section 2 and can be found in detail in, for example, [3, 20].

We want to apply the formalism for the case of a Friedmann universe, which is of central relevance for quantum cosmology. In the homogeneous and isotropic setting the line element is given by

(19) d s 2 = − N 2 ( t ) d t 2 + a 2 ( t ) d Ω 3 2 ,

where d Ω 3 2 is the line-element of an constant curvature space with curvature index k = 0, ±1.

Here, N is the lapse function, measuring the change of coordinate time with respect to proper time. Setting N = 1 yields the conventional Friedmann time. This is a preferred foliation given for the isotropic and homogeneous setting by observers comoving with the cosmological perfect fluid, cf. Figure 2.

The cosmological fluid shall here be mimicked by a scalar field ϕ with potential V(ϕ). The model is thus described by the action

(20) S = S grav + S matter = 3 V 0 κ 2 ∫ d t N − a a ̇ 2 N 2 + k a − Λ a 3 3 + V 0 2 ∫ d t N a 3 ϕ ̇ 2 N 2 − 2 V ( ϕ ) .

Here, V 0 denotes the (dimensionless) coordinate volume of a spatial slice. In the following we choose, for convenience, V 0 = 2π 2 (assuming the three-space is a three-sphere). We thus have a Lagrangian system with two degrees of freedom, a and ϕ. The canonical momenta read

(21) π a = ∂ L ∂ a ̇ = − 6 a a ̇ κ 2 N , π ϕ = ∂ L ∂ ϕ ̇ = a 3 ϕ ̇ N , π N = ∂ L ∂ N ̇ ≈ 0 .

Curly equal signs stand for weak equalities, that is, equalities which hold after the equations of motion are satisfied. The equation for π N is thus a primary constraint. The Hamiltonian reads

(22) H = π a a ̇ + π ϕ ϕ ̇ + π N N ̇ − L = − κ 2 12 a π a 2 + 1 2 π ϕ 2 a 3 + a 3 Λ κ 2 + a 3 V − 3 k a κ 2 .

As we assumed homogeneity and isotropy, the diffeomorphism constraints are satisfied trivially. From the preservation of the primary constraint π N ≈ 0 one finds that the Hamiltonian is constrained to vanish, H ≈ 0 . Expressed in terms of the ‘velocities’, a ̇ and ϕ ̇ , this constraint becomes identical to the Friedmann equation,

(23) a ̇ a 2 ≡ H 2 = κ 2 3 ϕ ̇ 2 2 + V ( ϕ ) + Λ 3 − k a 2 .

The space spanned by the canonical variable (the three-metric) in the full theory is called superspace, see the main text. In dependence on this denotation, the space spanned by (a, ϕ) is called minisuperspace. The metric on this space is named after DeWitt and is for this model given by

(24) G A B = − 6 a κ 2   0 0   a 3 .

Note the indefinite nature of the metric, resulting in an indefinite DeWitt metric for general relativity. The Hamiltonian constraint in this minisuperspace model thus reads

(25) H = N 1 2 G A B π A π B + V ( q ) ≈ 0 ,

where

(26) V ( q ) = 1 2 − 6 k a κ 2 + 2 Λ a 3 κ 2 + a 3 V ( ϕ ) ,

A, B = {a, ϕ}, that is, q 1 = a, q 2 = ϕ, and G AB is the inverse DeWitt metric. It is an artefact of the two-dimensionality of the model considered here that minisuperspace is conformally flat. In more complicated settings, minisuperspace can also be curved and additional curvature terms may occur depending on the choice of factor ordering [54].

Dirac’s constraint quantization requires an implementation of (22) as

(27) H ̂ Ψ = 0 .

The operator H ̂ is constructed from the conventional Schrödinger representation of canonical variables,

(28) q ̂ A Ψ ≡ q A Ψ , π ̂ A Ψ ≡ ℏ i ∂ ∂ q A Ψ ,

for A ∈ {a, ϕ}. The Hamiltonian operator is, of course, not uniquely defined in this way. On the contrary, factor-ordering ambiguities occur here as in ordinary quantum mechanics – with the important difference that here factor ordering cannot be justified by experiment. Usually one decides on the covariant ordering, the Laplace–Beltrami ordering,

(29) G A B π A π B → − ℏ 2 ▽ LB 2 = − ℏ 2 − G ∂ A − G G A B ∂ B ,

where G denotes the determinant of the DeWitt–metric. Choosing Laplace–Beltrami factor ordering, the Wheeler–DeWitt equation reads

(30) ℏ 2 κ 2 12 a ∂ ∂ a a ∂ ∂ a − ℏ 2 2 ∂ 2 ∂ ϕ 2 + a 6 V ( ϕ ) + Λ κ 2 − 3 k a 4 κ 2 × Ψ ( a , ϕ ) = 0 .

Introducing α ≡ ln a/a 0, with a 0 as reference scale, one obtains the following equation, which is (6) of the main text,

(31) ℏ 2 κ 2 12 ∂ 2 ∂ α 2 − ℏ 2 2 ∂ 2 ∂ ϕ 2 + a 0 6 e 6 α V ϕ + Λ κ 2   − 3 a 0 4 e 4 α k κ 2 Ψ ( α , ϕ ) = 0 .

This is an equation of the same form as the Klein–Gordon equation: the derivative with respect to α corresponds to a time derivative, the derivative with respect to ϕ to a spatial derivative, and the remaining terms constituting a ‘time and space dependent’ mass term, which is a nontrivial potential term. Equation (31) is the starting point for many discussions in quantum cosmology. The physical units are often chosen to be κ 2 = 6 for convenience.

References

[1] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed., Berlin, Springer, 2003.10.1007/978-3-662-05328-7Search in Google Scholar

[2] M. Schlosshauer, Decoherence and the Quantum-To-Classical Transition, Berlin, Springer, 2007.Search in Google Scholar

[3] C. Kiefer, Quantum Gravity, 3rd ed., Oxford, Oxford University Press, 2012.10.1093/oxfordhb/9780199298204.003.0024Search in Google Scholar

[4] M. Bojowald, “Quantum cosmology,” in Lecture Notes in Physics, vol. 835, New York, Springer, 2011. https://doi.org/10.1007/978-1-4419-8276-6.Search in Google Scholar

[5] C. Kiefer, “Conceptual problems in quantum gravity and quantum cosmology,” ISRN Math. Phys., vol. 2013, 2013, Art no. 509316. https://doi.org/10.1155/2013/509316.Search in Google Scholar

[6] D. H. Coule, “Quantum cosmological models,” Classical Quant. Grav., vol. 22, pp. R125–R166, 2005. https://doi.org/10.1088/0264-9381/22/12/r02.Search in Google Scholar

[7] D. L. Wiltshire, “An introduction to quantum cosmology,” in Cosmology: The physics of the Universe, B. Robson, N. Visvanathon, and W. S. Woolcock, Eds., Singapore, World Scientific, 1996, pp. 473–531.Search in Google Scholar

[8] J. J. Halliwell, “Introductory lectures on quantum cosmology,” in Quantum Cosmology and Baby Universes, S. Coleman, J. B. Hartle, T. Piran, and S. Weinberg, Eds., Singapore, World Scientific, 1991, pp. 159–243.10.1142/9789814503501_0003Search in Google Scholar

[9] B. S. DeWitt, “Quantum theory of Gravity. I. The canonical theory,” Phys. Rev., vol. 160, pp. 1113–1148, 1967. https://doi.org/10.1103/physrev.160.1113.Search in Google Scholar

[10] C. W. Misner, “Minisuperspace,” in Magic without magic, J. R. Klauder, Ed., San Francisco, Freeman, 1972, pp. 441–473.Search in Google Scholar

[11] M. P. Ryan, “Hamiltonian cosmology,” in Lecture Notes in Physics, vol. 13, Berlin, Springer, 1972.Search in Google Scholar

[12] J. B. Hartle and S. W. Hawking, “Wave function of the universe,” Phys. Rev. D, vol. 28, pp. 2960–2975, 1983. https://doi.org/10.1103/physrevd.28.2960.Search in Google Scholar

[13] A. Vilenkin, “Quantum cosmology and eternal inflation,” in The future of theoretical physics and cosmology, G. W. Gibbons, Ed., Cambridge, Cambridge University Press, 2003, pp. 649–666.Search in Google Scholar

[14] H. D. Conradi and H. D. Zeh, “Quantum cosmology as an initial value problem,” Phys. Lett. A, vol. 154, pp. 321–326, 1991. https://doi.org/10.1016/0375-9601(91)90026-5.Search in Google Scholar

[15] A. O. Barvinsky, Quantum Cosmology at the Turn of Millenium, 2001, arXiv:gr-qc/0101046.10.1142/9789812777386_0037Search in Google Scholar

[16] M. Gasperini and G. Veneziano, “The pre-big bang scenario in string cosmology,” Phys. Rep., vol. 373, pp. 1–212, 2003. https://doi.org/10.1016/s0370-1573(02)00389-7.(b) M. P. Dąbrowski and C. Kiefer, “Boundary conditions in quantum string cosmology,” Phys. Lett. B, vol. 397, pp. 185–192, 1997.Search in Google Scholar

[17] P. V. Moniz, “Quantum Cosmology: The Supersymmetric Perspective, Vols. 1 and 2,” in Lecture Notes in Physics, vol. 804, Berlin, Springer, 2010.10.1007/978-3-642-11570-7Search in Google Scholar

[18] C. Jonas, J.-L. Lehners, and V. Meyer, Revisiting the No-Boundary Proposal with a Scalar Field, 2021, arXiv:2112.07986 [hep-th].10.1103/PhysRevD.105.043529Search in Google Scholar

[19] M. Bouhmadi-López, C. Kiefer, and P. Martín-Moruno, “Phantom singularities and their quantum fate: general relativity and beyond – a CANTATA COST action topic,” Gen. Relat. Gravit., vol. 51, p. 135, 2019. https://doi.org/10.1007/s10714-019-2618-y.Search in Google Scholar

[20] D. Giulini and C. Kiefer, “The canonical approach to quantum gravity – general ideas and geometrodynamics,” in Approaches to fundamental physics – An assessment of current theoretical ideas, E. Seiler and I.-O. Stamatescu, Eds., Berlin, Springer, 2007, pp. 131–150.10.1007/978-3-540-71117-9_8Search in Google Scholar

[21] R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravitation: an introduction to current research, L. Witten, Ed., New York, Wiley, 1962, pp. 227–265.Search in Google Scholar

[22] C. Rovelli, Quantum Gravity, Cambridge, Cambridge University Press, 2004.10.1017/CBO9780511755804Search in Google Scholar

[23] D. Giulini, “The superspace of geometrodynamics. Gen. Rel. Grav. 41, 785–815,” Phys. Rev. D, vol. 81, p. 043530, 2009.10.1007/s10714-009-0771-4Search in Google Scholar

[24] J. C. Feng, “Volume average regularization for the Wheeler-DeWitt equation,” Phys. Rev. D, vol. 98, p. 026024, 2018. https://doi.org/10.1103/physrevd.98.026024.Search in Google Scholar

[25] J. Klauder, The Benefits of Affine Quantization, 2019, arXiv:1912.08047 [physics.gen-ph]. Higgs inflaton.Search in Google Scholar

[26] C. Kiefer and P. Peter, “Time in quantum cosmology,” Universe, vol. 8, p. 22, 2022. https://doi.org/10.3390/universe8010036.Search in Google Scholar

[27] C. J. Isham, Canonical Quantum Gravity and the Problem of Time, 1992, arXiv:gr-qc/9210011.10.1007/978-94-011-1980-1_6Search in Google Scholar

[28] K. Kuchař, “Time and interpretations of quantum gravity,” in Proc. 4th Canadian Conf. General Relativity and Relativistic Astrophysics, G. Kunstatter, Ed., Singapore, World Scientific, 1992, pp. 211–314.10.1142/S0218271811019347Search in Google Scholar

[29] E. Anderson, The Problem of Time, Cham, Springer, 2017.10.1007/978-3-319-58848-3Search in Google Scholar

[30] C. G. Torre, “Is general relativity an ‘already parametrized’ theory?” Phys. Rev. D, vol. 46, pp. 3231–3234, 1993. https://doi.org/10.1103/physrevd.46.r3231.Search in Google Scholar

[31] C. Kiefer, “Non-minimally coupled scalar fields and the initial value problem in quantum gravity,” Phys. Lett. B, vol. 225, pp. 227–232, 1989. https://doi.org/10.1016/0370-2693(89)90810-1.Search in Google Scholar

[32] M. P. Dąbrowski, C. Kiefer, and B. Sandhöfer, “Quantum phantom cosmology,” Phys. Rev. D, vol. 74, 2006, Art no. 044022.10.1103/PhysRevD.74.044022Search in Google Scholar

[33] A. Vilenkin, “Creation of universes from nothing,” Phys. Lett. B, vol. 117, pp. 25–28, 1982. https://doi.org/10.1016/0370-2693(82)90866-8.Search in Google Scholar

[34] C. Kiefer, “On the meaning of path integrals in quantum cosmology,” Ann. Phys. vol. 207, pp. 53–70, 1991. https://doi.org/10.1016/0003-4916(91)90178-b.Search in Google Scholar

[35] H. Matsui and T. Terada, “Swampland constraints on no-boundary quantum cosmology,” J. High Energy Phys., vol. 10, p. 162, 2020. https://doi.org/10.1007/jhep10(2020)162.Search in Google Scholar

[36] H. D. Zeh, “Time in quantum gravity,” Phys. Lett. A, vol. 126, pp. 311–317, 1988. https://doi.org/10.1016/0375-9601(88)90842-0.Search in Google Scholar

[37] H. D. Conradi, “Tunneling of macroscopic universes,” Int. J. Mod. Phys., vol. 7, pp. 189–200, 1998. https://doi.org/10.1142/s0218271898000152.Search in Google Scholar

[38] A. O. Barvinsky, A. Yu. Kamenshchik, C. Kiefer, and C. F. Steinwachs, “Tunneling cosmological state revisited: origin of inflation with a non-minimally coupled Standard Model Higgs inflaton,” Phys. Rev. D, vol. 81, p. 043530, 2010. https://doi.org/10.1103/physrevd.81.043530.Search in Google Scholar

[39] J. J. Halliwell and S. W. Hawking, “Origin of structure in the universe,” Phys. Rev. D, vol. 31, pp. 1777–1791, 1985. https://doi.org/10.1103/physrevd.31.1777.Search in Google Scholar

[40] C. Kiefer and T. P. Singh, “Quantum gravitational correction terms to the functional Schrödinger equation,” Phys. Rev. D, vol. 44, pp. 1067–1076, 1991. https://doi.org/10.1103/physrevd.44.1067.Search in Google Scholar

[41] A. O. Barvinsky and C. Kiefer, “Wheeler–DeWitt equation and Feynman diagrams,” Nucl. Phys. B, vol. 526, pp. 509–539, 1998. https://doi.org/10.1016/s0550-3213(98)00349-6.Search in Google Scholar

[42] L. Chataignier and M. Krämer, “Unitarity of quantum-gravitational corrections to primordial fluctuations in the Born-Oppenheimer approach,” Phys. Rev. D, vol. 103, 2021, Art no. 066005. https://doi.org/10.1103/physrevd.103.066005.Search in Google Scholar

[43] A. Vilenkin, “Interpretation of the wave function of the Universe,” Phys. Rev. D, vol. 39, pp. 1116–1122, 1989. https://doi.org/10.1103/physrevd.39.1116.Search in Google Scholar PubMed

[44] C. Kiefer, J. Marto, and P. V. Moniz, “Indefinite oscillators and black hole evaporation,” Ann. Phys., vol. 18, pp. 722–735, 2009. https://doi.org/10.1002/andp.200910366.Search in Google Scholar

[45] H. D. Zeh, The Physical Basis of the Direction of Time, 5th ed., Berlin, Springer, 2007.Search in Google Scholar

[46] C. Kiefer, On a Quantum Weyl Curvature Hypothesis, 2021, arXiv: 2111.02137 [gr-qc].Search in Google Scholar

[47] C. Kiefer and H. D. Zeh, “Arrow of time in a recollapsing quantum universe,” Phys. Rev. D, vol. 51, pp. 4145–4153, 1995. https://doi.org/10.1103/physrevd.51.4145.Search in Google Scholar PubMed

[48] C. Kiefer, D. Polarski, and A. A. Starobinsky, “Quantum-to-classical transition for fluctuations in the early universe,” Int. J. Mod. Phys., vol. 7, pp. 455–462, 1998. https://doi.org/10.1142/s0218271898000292.Search in Google Scholar

[49] B. Mashhoon, “Gravitation and non-locality,” in Proceedings of the 25th Johns Hopkins Workshop 2001: A Relativistic Spacetime Odyssey, Singapore, World Scientific, 2001, pp. 35–46.10.1142/9789812791368_0003Search in Google Scholar

[50] D. A. Konkowski, T. M. Helliwell, and V. Arndt, “Are classically singular spacetimes quantum mechanically singular as well?” in Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, Rio de Janeiro, Singapore, World Scientific, 2004, pp. 2169–2171.10.1142/9789812704030_0288Search in Google Scholar

[51] A. Y. Kamenshchik, C. Kiefer, and B. Sandhöfer, “Quantum cosmology with a big-brake singularity,” Phys. Rev. D, vol. 76, 2007, Art no. 064032. https://doi.org/10.1103/physrevd.76.064032.Search in Google Scholar

[52] M. Bojowald, “Singularities and quantum gravity,” AIP Conf. Proc., vol. 917, pp. 130–137, 2007. https://doi.org/10.1063/1.2752483.Search in Google Scholar

[53] H. D. Conradi, “Quantum cosmology of Kantowski-Sachs like models,” Classical Quant. Grav., vol. 12, pp. 2423–2439, 1995. https://doi.org/10.1088/0264-9381/12/10/005.Search in Google Scholar

[54] C. Kiefer, N. Kwidzinski, and D. Piontek, “Singularity avoidance in Bianchi I quantum cosmology,” Eur. Phys. J. C, vol. 79, p. 199, 2019. https://doi.org/10.1140/epjc/s10052-019-7193-6.Search in Google Scholar

[55] C. Kiefer, “Wave packets in minisuperspace,” Phys. Rev. D, vol. 38, pp. 1761–1772, 1988. https://doi.org/10.1103/physrevd.38.1761.Search in Google Scholar PubMed

Received: 2021-12-21
Revised: 2022-01-25
Accepted: 2022-01-25
Published Online: 2022-02-11
Published in Print: 2022-06-26

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/zna-2021-0384
Keywords for this article
cosmology; Hamiltonian formalism; quantum cosmology; quantum gravity

Articles in the same Issue

  1. Frontmatter
  2. General
  3. Anomalous skin effects and energy transfer of R-L waves in relativistic partially degenerate plasma
  4. Atomic, Molecular & Chemical Physics
  5. Optical fiber ammonia sensor based on porous Yb3+/Er3+ co-doped NaYF4/phenol red composites
  6. Dynamical Systems & Nonlinear Phenomena
  7. Relativistic electron dynamics in magnetic fields with low-degree of field nonlinearity
  8. Gravitation & Cosmology
  9. Quantum cosmology
  10. Hydrodynamics
  11. Translationally invariant exact steady flows of gas and fluid
  12. Quantum Theory
  13. Exact path integral solutions of Dirac wave equation for an exponentially decaying magnetic field
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