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Quantum cosmology

  • Claus Kiefer EMAIL logo und Barbara Sandhöfer
Veröffentlicht/Copyright: 11. Februar 2022
Zeitschrift für Naturforschung A
Aus der Zeitschrift Zeitschrift für Naturforschung A Band 77 Heft 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.

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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
Schlagwörter für diesen Artikel
cosmology; Hamiltonian formalism; quantum cosmology; quantum gravity

Artikel in diesem Heft

  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
  14. Solid State Physics & Materials Science
  15. Heat transfer analysis describing freezing of a eutectic system by a line heat sink with convection effect in cylindrical geometry
  16. Oscillating modes of thermomagnetic avalanches in superconductors
  17. Doped TiO2 slabs for water splitting: a DFT study
  18. The design of optical non-reciprocal abnormal transmission based on PT asymmetric system
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