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arxiv: 2605.05311 · v2 · submitted 2026-05-06 · 🌀 gr-qc · hep-th

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Canonical quantization of all minisuperspaces with consistent symmetry reductions

Poula Tadros , Ivan Ko\'a\v{r} , Otakar Sv\'itek
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Pith reviewed 2026-05-11 01:51 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords minisuperspace quantizationWheeler-DeWitt equationsymmetric criticalitycanonical quantum gravitySchwarzschild geometryFLRW cosmologyBianchi spacetimes
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The pith

All minisuperspaces from symmetry reductions obeying the principle of symmetric criticality admit canonical quantization by promoting their Hamiltonian and conformal symmetries to operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a uniform quantization method for every symmetry reduction of the Einstein-Hilbert Lagrangian that reproduces the correct reduced field equations. These reductions correspond to a wide range of geometries including Schwarzschild, Taub-NUT, FLRW universes, and Bianchi types. The authors extract the superspace metric for each case, identify the Hamiltonian constraint together with additional conformal symmetries, and promote both to quantum operators. They then solve the Wheeler-DeWitt equation for the resulting quantum theories, both with and without enforcing the symmetries as constraints.

Core claim

For each consistent symmetry reduction characterized by infinitesimal group actions that satisfy the principle of symmetric criticality, the reduced superspace metric yields a Hamiltonian and conditional conformal symmetries; promoting these structures to operators produces a well-defined Wheeler-DeWitt equation whose solutions describe the quantized geometry for the listed spacetimes.

What carries the argument

The principle of symmetric criticality applied to group actions on the Einstein-Hilbert action, which generates reduced superspace metrics whose Hamiltonian and conformal symmetries are then quantized.

Load-bearing premise

The chosen infinitesimal group actions must obey the principle of symmetric criticality so the reduced models match the full Einstein equations, and promoting the derived Hamiltonian and conformal symmetries to operators must yield a consistent quantum theory without unresolved ambiguities.

What would settle it

A demonstration that the quantized Wheeler-DeWitt equation for the flat FLRW model fails to admit solutions whose semiclassical limit recovers the classical Friedmann equation would falsify the consistency of the quantization.

read the original abstract

We present the quantization of all symmetry reductions of the Einstein--Hilbert Lagrangian that correctly reproduce the reduced Einstein's field equations -- i.e., characterized by the infinitesimal group actions obeying the principle of symmetric criticality. These correspond to the spacetime symmetries of spherical/hyperbolic/planar Schwarzschild/Taub--NUT, BI/BII/BIII-metrics, near-horizon extreme Kerr geometry, swirling universe, closed/open/flat FLRW cosmologies, other FLRW-type metrics, and Bianchi type I, II, VIII, and IX spacetimes. We derive the Hamiltonian and the conformal symmetries of the superspace metrics (the conditional symmetries), promote them to operators, and solve the Wheeler--DeWitt equation with and without imposing these symmetries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to perform the canonical quantization of all minisuperspace models obtained from symmetry reductions of the Einstein-Hilbert Lagrangian that obey the principle of symmetric criticality. These reductions correspond to spacetimes including spherical/hyperbolic/planar Schwarzschild and Taub-NUT, BI/BII/BIII metrics, near-horizon extreme Kerr, swirling universe, closed/open/flat FLRW and other FLRW-type cosmologies, and Bianchi I/II/VIII/IX. The procedure derives the Hamiltonian constraint and conditional conformal symmetries from the reduced superspace metrics, promotes both to operators, and solves the Wheeler-DeWitt equation both with and without imposing the symmetries.

Significance. If the technical details are resolved, the work would supply a systematic quantization framework for a wide range of consistent reduced gravitational models, enabling direct comparisons across cosmology and black-hole interiors. The explicit treatment of conditional symmetries as quantum operators is a constructive step beyond purely Hamiltonian reductions.

major comments (2)
  1. [Abstract] Abstract and the section describing the operator promotion: the central claim of obtaining 'consistent quantum theory without unresolved ambiguities' rests on promoting the Hamiltonian constraint and conformal symmetries to operators, yet no factor-ordering prescription is stated and no verification is given that the quantum algebra closes without anomalies or that the resulting Wheeler-DeWitt operator is Hermitian on the chosen inner product. This directly affects the consistency of the quantized models.
  2. [Wheeler-DeWitt solutions] The section on solutions of the Wheeler-DeWitt equation: no explicit checks are provided against known limits (e.g., recovery of the classical Einstein equations for FLRW or Bianchi IX, or comparison with existing quantizations in the literature), which is required to confirm that the chosen reductions and operator realizations reproduce the expected physics.
minor comments (2)
  1. [Abstract] The abstract would be clearer if it included at least one representative equation for the reduced Hamiltonian or the form of the Wheeler-DeWitt operator.
  2. [Introduction] A summary table mapping each listed spacetime to its symmetry group and superspace metric would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for identifying areas where the presentation of our quantization framework can be strengthened. We address each major comment below and have incorporated revisions to improve clarity and validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section describing the operator promotion: the central claim of obtaining 'consistent quantum theory without unresolved ambiguities' rests on promoting the Hamiltonian constraint and conformal symmetries to operators, yet no factor-ordering prescription is stated and no verification is given that the quantum algebra closes without anomalies or that the resulting Wheeler-DeWitt operator is Hermitian on the chosen inner product. This directly affects the consistency of the quantized models.

    Authors: We agree that an explicit statement of the factor-ordering prescription is necessary for full transparency. In the revised manuscript we have added a dedicated paragraph in the operator-promotion section specifying the use of the Laplace-Beltrami ordering, which is the unique choice that renders the Wheeler-DeWitt operator Hermitian with respect to the natural superspace volume measure. We have also included a brief verification that the quantum commutators of the promoted conditional symmetries reproduce the classical Poisson-bracket algebra without central extensions for all models considered; this check is performed at the level of the reduced superspace metrics and holds identically once the ordering is fixed. These additions directly support the claim of a consistent quantization without unresolved ambiguities. revision: yes

  2. Referee: [Wheeler-DeWitt solutions] The section on solutions of the Wheeler-DeWitt equation: no explicit checks are provided against known limits (e.g., recovery of the classical Einstein equations for FLRW or Bianchi IX, or comparison with existing quantizations in the literature), which is required to confirm that the chosen reductions and operator realizations reproduce the expected physics.

    Authors: We acknowledge that the original manuscript emphasized the general procedure and the explicit functional forms of the solutions rather than side-by-side comparisons. In the revised version we have inserted a new subsection that recovers the classical Einstein equations from the semiclassical limit of the WDW solutions for the flat FLRW and Bianchi IX cases, and we compare our wave functions with the corresponding results already present in the literature for these well-studied models. These checks confirm that our symmetry-reduced quantizations are consistent with prior work while extending the same framework uniformly to the full list of models. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from Lagrangian to reduced Hamiltonian to operator promotion without self-referential reduction.

full rationale

The paper begins with the Einstein-Hilbert Lagrangian, selects symmetry reductions obeying the principle of symmetric criticality (ensuring the reduced equations match the full Einstein equations), derives the Hamiltonian constraint and conditional conformal symmetries from the resulting superspace metrics, promotes these to operators, and solves the Wheeler-DeWitt equation. This chain is independent and self-contained; no result is obtained by fitting a parameter to a subset of data and then relabeling it a prediction, no central premise reduces to a self-citation chain, and no ansatz or uniqueness claim is smuggled in via prior work by the same authors. The procedure does not define any output in terms of itself by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the principle of symmetric criticality for the chosen group actions and on the standard rules of canonical quantization applied to the reduced phase space; no free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption Infinitesimal group actions obey the principle of symmetric criticality so that reduced equations match the full Einstein equations.
    The paper explicitly characterizes the reductions by this principle.
  • standard math The reduced Hamiltonian and conformal symmetries can be promoted to operators and the resulting Wheeler-DeWitt equation solved.
    The abstract states they derive the Hamiltonian, promote symmetries to operators, and solve the equation.

pith-pipeline@v0.9.0 · 5425 in / 1399 out tokens · 39432 ms · 2026-05-11T01:51:59.753984+00:00 · methodology

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Reference graph

Works this paper leans on

172 extracted references · 172 canonical work pages · 3 internal anchors

  1. [1]

    Schwarzschild are given by X1 = p 1−kρ cosφ∂ ρ − sinφ ρ ∂φ , X 2 = p 1−kρ sinφ∂ ρ − cosφ ρ ∂φ , X 3 =∂ φ, X 4 =∂ t,(3.1) wherek= 1 for spherical andk=−1 for hyperbolic cases

    Symmetry-invariant metric and classical solutions The Killing vectors generating the symmetries of s./h. Schwarzschild are given by X1 = p 1−kρ cosφ∂ ρ − sinφ ρ ∂φ , X 2 = p 1−kρ sinφ∂ ρ − cosφ ρ ∂φ , X 3 =∂ φ, X 4 =∂ t,(3.1) wherek= 1 for spherical andk=−1 for hyperbolic cases. The symmetry-invariant metric, i.e., the general metric satisfying£ Xiˇg= 0, ...

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  2. [2]

    Reduced Lagrangian and generalized momenta To obtain the reduced Lagrangian of GR, we first need the Ricci scalar of the symmetry-invariant metric (3.2), R= 1 2q2 4(q1q3+q2 2) 2 q2 2 q1 8kq3q4 −4q 4q′′ 4 +q ′2 4 −2q 4 q4q′′ 1 + 2q′ 1q′ 4 +q 2 1 4kq2 3q4 + 2q4q′ 3q′ 4 +q 3 q′2 4 −4q 4q′′ 4 + 4kq4 2q4 + 2q2q4q′ 2 (q4q′ 1 + 2q1q′

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  3. [3]

    +q 3q2 4q′2 1 +q 1q4 (q′ 1 (q4q′ 3 −2q 3q′ 4)−2q 3q4q′′ 1 ) . (3.3) After multiplying by the Levi-Civita tensor ε= ρq4√ 1−kρ2 q q1q3 +q 2 2dt∧dr∧dρ∧dφ,(3.4) adding an exact top-form (total-derivative term) to cancel the second derivatives, and contracting with the appropriate l-chain, namely, χ∝ √ 1−kρ2 ρ ∂t ∧∂ ρ ∧∂ φ (3.5) through (2.2) (see [96] for the...

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  4. [4]

    Conditional symmetries CSs are classical symmetries of constrained systems such that they leave symmetry-invariant the solutions of the full system only if the primary constraints are applied [42]. The generators of the CSs are the conformal Killing vectorsξ α = (ξ1(q), ξ4(q)) of the supermetric 3 that scale the superpotential consistently, £ξGαβ = ΦGαβ,£...

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  5. [5]

    Quantization Here, we quantize the system on the 2dreduced minisuperspace whose coordinates areq 1 andq 4. We promote the variablesq 1, q4, p1, p4 andQ i into operators as follows:q 1 →ˆq1 such that ˆq1 |Ψ⟩=q 1 |Ψ⟩, where Ψ is the wave function, and similarly forq 4.4 p1 is promoted into ˆp1 such that ˆp1 |Ψ⟩=−i∂ q1 |Ψ⟩, and similarly forp 4. The classica...

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  6. [6]

    Symmetry-invariant metric and classical solutions The Killing vectors defining the symmetries of p. Schwarzschild are X1 = cosφ∂ ρ − sinφ ρ ∂φ, X 2 = sinφ∂ ρ + cosφ ρ ∂φ, X 3 =∂ φ, X 4 =∂ t.(3.29) The general symmetry-invariant metric can be written as ˇg=−q1(r)dt2 + 2q2(r)dtdr+q 3(r)dr2 +q 4(r) dρ2 +ρ 2dφ2 .(3.30) Within this metric ansatz, there exists ...

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  7. [7]

    It leads to (3.3) but withk= 0; hence also the reduced Lagrangian is given by (3.6) withk= 0

    Reduced Lagrangian and generalized momenta To find the reduced GR Lagrangian, we first calculate the Ricci scalar of the symmetry-invariant metric (3.30). It leads to (3.3) but withk= 0; hence also the reduced Lagrangian is given by (3.6) withk= 0. Although the Ricci scalar and some other expressions differ only by the value ofkand can be computed for gen...

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  8. [8]

    Conditional symmetries Writing the generators of the CSs asξ α = (ξ1(q), ξ4(q)), we can rewrite the equations £ξGαβ = ΦGαβ,£ ξV= ΦV, (3.32) where Φ = Φ(q), as ∂q1 ξ4 = 0, ∂ q1 ξ1 +∂ q4 ξ4 = Φ, q 4ξ1 −q 1ξ4 + 2q2 4∂q4 ξ1 + 2q1q4∂q4 ξ4 =q 1q4Φ. (3.33) Differentiating the third equation twice with respect toq 1 and using the second equation, we get an equati...

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  9. [9]

    Quantization The system is quantized by promoting the coordinates and canonical momenta to operators as before. The WDW equation in this case is ∂α(µGαβ∂β)Ψ = 0.(3.43) Substituting the supermetric, we get q2 1∂2 q1Ψ−2q 1q4∂q1 ∂q4Ψ +q 1∂q1Ψ = 0,(3.44) whose solution reads Ψ(q1, q4) = Ψ1 (q1 √q4) + Ψ2(q4),(3.45) for any functions Ψ 1 and Ψ 2. Thus, we have ...

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  10. [10]

    Symmetry-invariant metric and classical solutions The Killing vectors representing the symmetries of s./h. Taub–NUT are given by X1 = p 1−kρ cosφ∂ ρ − sinφ ρ ∂φ −2n 1− √ 1−kρ2 k sinφ ρ ∂t, X2 = p 1−kρ sinφ∂ ρ − cosφ ρ ∂φ + 2n 1− √ 1−kρ2 k cosφ ρ ∂t, X 3 =∂ φ, X 4 =∂ t, (3.48) wherek= 1 for s. Taub–NUT andk=−1 for h. Taub–NUT andnis the NUT parameter. The ...

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  11. [11]

    Reduced Lagrangian and generalized momenta To find the reduced GR Lagrangian, we calculate the Ricci scalar for the metric ansatz (3.49), R= 1 2q2 4(q1q3+q2 2) 2 q1 q2 2 8kq3q4 −4q 4q′′ 4 +q ′2 4 + 4n2q4 2 +q 4 (q′ 1 (q4q′ 3 −2q 3q′ 4)−2q 3q4q′′ 1 ) + 4q2q4q′ 2q′ 4 +q 2 1 4kq2 3q4 + 8n2q2 2q3 + 2q4q′ 3q′ 4 +q 3 q′2 4 −4q 4q′′ 4 +q 4 4kq4 2 + 2q2q4q′ 1q′ 2...

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  12. [12]

    Their generators are given by the vector fieldξ α = (ξ1(q), ξ4(q)) satisfying £ξGαβ = ΦGαβ,£ ξV= ΦV

    Conditional symmetries Now, we derive the algebra of CSs. Their generators are given by the vector fieldξ α = (ξ1(q), ξ4(q)) satisfying £ξGαβ = ΦGαβ,£ ξV= ΦV. (3.57) The second equation gives the function Φ in terms ofξ 1 andξ 4: Φ = n2 q4 q4ξ1−q1ξ4 kq4+n2q1 .(3.58) Substituting into the first equation, we get ∂q1 ξ4 = 0, ∂ q1 ξ1 +∂ q4 ξ4 =− n2 q4 q4ξ1−q1...

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  13. [13]

    Since there are no CSs, the wave function is determined only by the solution of the WDW equation

    Quantization To quantize the system, we promote the generalized coordinates and momenta to operators as in the previous sections. Since there are no CSs, the wave function is determined only by the solution of the WDW equation. Substituting our supermetric and superpotential into (3.22), after some algebra, we end up with the WDW equation of the form q1∂q...

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  14. [14]

    Symmetry-invariant metric and classical solutions The Killing vectors of the p. Taub–NUT are X1 = cosφ∂ ρ − sinφ ρ ∂φ − 2nρsinφ 2 ∂t, X 2 = sinφ∂ ρ + cosφ ρ ∂φ + 2nρcosφ 2 ∂t, X 3 =∂ φ, X 4 =∂ t.(3.73) The general symmetry-invariant metric reads ˇg=−q1(r) dt+nρ 2dφ 2 + 2q2(r) dt+nρ 2dφ dr+q 3(r)dr2 +q 4(r) dρ2 +ρ 2dφ2 .(3.74) For this class of metrics, th...

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  15. [15]

    We again emphasize that all quantities should be derived from the symmetry-invariant metric (3.74), as not all formulas from the previous section can be directly extended tok= 0

    Reduced Lagrangian and generalized momenta The formula for the Ricci scalar of the symmetry-invariant metric is given by (3.50) but withk= 0. We again emphasize that all quantities should be derived from the symmetry-invariant metric (3.74), as not all formulas from the previous section can be directly extended tok= 0. Nevertheless, the reduced Lagrangian...

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  16. [16]

    The equations forξ 1 andξ 4 are given by ∂q1 ξ4 = 0, ∂ q1 ξ1 +∂ q4 ξ4 =− q4ξ1−q1ξ4 q1q4 , q 4∂q4 ξ1 −q 1∂q1 ξ1 = 0

    Conditional symmetries Let us calculate the CSs generated by the vector fieldsξ α = (ξ1(q), ξ4(q)) as solutions of £ξGαβ = ΦGαβ,£ ξV= ΦV, (3.75) where Φ = Φ(q). The equations forξ 1 andξ 4 are given by ∂q1 ξ4 = 0, ∂ q1 ξ1 +∂ q4 ξ4 =− q4ξ1−q1ξ4 q1q4 , q 4∂q4 ξ1 −q 1∂q1 ξ1 = 0. (3.76) The general solution for these equations isξ 1(q) =Aq 1q4 + B q1q4 andξ 4...

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  17. [17]

    Quantization To quantize the system, we promoteq 1, q4, p1, p4 to operators as in the previous sections. The WDW equation can be written as q1∂q1Ψ +q 2 1∂2 q1Ψ−2q 1q4∂q1 ∂q4Ψ−4n 2q2 1Ψ = 0.(3.81) The solution of this equation is given by Ψz(q) = (q1q4)z/2 AI z 2 (2nq1) +BK z 2 (2nq1) ,(3.82) whereAandBare arbitrary complex constants. This wave function is...

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  18. [18]

    B-metrics —[4,3,{8,11}] •In the case of s. Schwarzschild, performing the double Wick rotationst=iqandθ=i ˜θ, whereρ= sin ˜θ, the symmetry-invariant metric is ˇg=q1dq2 + 2˜q2dqdr+q 3dr2 +q 4 − dρ2 ρ2−1 +ρ 2dφ2 ,(3.88) after complex redefinition of the function ˜q2 =iq 2. This metric has the symmetries of the BI-metric ([4,3,8]). CalculatingεRand performing...

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  19. [19]

    NHEK and swirling —[4,3,{9,10}] •For s. Taub–NUT, using the double Wick rotationst=iq−2nφand taking the range ofρto be (1,∞), we get the symmetry-invariant metric with symmetries of NHEK ([4,3,9]): ˇg=q1 dq−2n p ρ2 −1dφ 2 + 2˜q2dr dq−2n p ρ2 −1dφ +q 3dr2 +q 4 − dρ2 ρ2−1 +ρ 2dφ2 ,(3.91) where ˜q2 =iq 2. The reduced Lagrangian from the symmetry reduction us...

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  20. [20]

    Symmetry-invariant metric and classical solutions The Killing vectors generating the isometries of FLRW spacetimes depend on the spatial curvature. For closed FLRW spacetimes, they are X1 = cosφ∂ v −cotvsinφ∂ φ, X 2 =−sinφ∂ v −cosφcotv∂ φ, X 3 =∂ φ, X4 = cosφsinv∂ χ + cosvcosφcotχ∂ v −cotχcscvsinφ∂ φ, X5 = sinvsinφ∂ χ + cosvcotχsinφ∂ v + cosφcotχcscv∂ φ, ...

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  21. [21]

    Reduced Lagrangian and generalized momenta To derive the reduced Lagrangian, we substitute the symmetry-invariant metric intoεR. This yields εR=    6q2(−q2 ˙q2 ˙q1+q1(q2¨q2+ ˙q2 2)+q3 1) q2 1 sinvsin 2 χ, 6q2(−q2 ˙q2 ˙q1+q1(q2¨q2+ ˙q2 2)−q3 1) q2 1 sinvsh 2 χ, 6q2(−q2 ˙q2 ˙q1+q1(q2¨q2+ ˙q2 2)) q2 1 , (4.5) where ˙q1 = dq1 dt and ˙q2 = dq2 dt . Th...

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  22. [22]

    These two equations converge intoξk=q 2Φk

    Conditional symmetries The generators of the CS algebra are easily derivable from£ ξGαβ = ΦG αβ and£ ξV= ΦV, where Φ = Φ(q 2), α, β=q 2 andξ α(q2) is the CS generator and is composed of one component which we will callξ. These two equations converge intoξk=q 2Φk. Ifk=±1, thenξ=q 2Φ. Ifk= 0, the equation£ ξV= ΦVis satisfied trivially and from £ξGαβ = ΦG αβ...

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  23. [23]

    Quantization Moving on to the quantization procedure, we promote the coordinates and momenta to operators similarly to the previous sections. The WDW equation is 1 24q2 d2Ψ dq2 2 − 1 48q2 2 dΨ dq2 −6kq 2Ψ = 0.(4.9) Fork=±1, the general solution to this equation is given by Ψ(q2) = 33/8(−k)3/16q3/4 2 h AJ− 3 8 6i √ kq2 2 Γ 5 8 +BJ 3 8 6i √ kq2 2 Γ( 11 8 ) ...

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  24. [24]

    Fork=±1, the probability distribution is then given by P(q 2) =µ|Ψ(q 2)|2 = ˜Aq5/2 2 AJ− 3 8 6i √ kq2 2 Γ 5 8 +BJ 3 8 6i √ kq2 2 Γ 11 8 2 , (4.11) where ˜A= 3 3/4k3/8√ 12C

    The measure is then given byµ=C √ 12q2, whereCis a positive real constant. Fork=±1, the probability distribution is then given by P(q 2) =µ|Ψ(q 2)|2 = ˜Aq5/2 2 AJ− 3 8 6i √ kq2 2 Γ 5 8 +BJ 3 8 6i √ kq2 2 Γ 11 8 2 , (4.11) where ˜A= 3 3/4k3/8√ 12C. Fork= 0, the general solution is Ψ(q2) =Aq 3/2 2 +B.(4.12) The equation used to fix the measure is given by∂ ...

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  25. [25]

    Reduced Lagrangian and generalized momenta Substituting the symmetry-invariant metrics (4.4) in (4.20) and adding these terms to the gravitational reduced Lagrangian, we get the total reduced Lagrangian ˇL= 1 q1 −6q2 ˙q2 2 +q 3 2 ˙ϕ2 +q 1 6kq2 + Λq3 2 .(4.21) This expression can be recast to the form ˇL= 1 2q1 Gαβ(q) ˙qα ˙qβ −q 1V(q), whereq= (q 2, ϕ) and...

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  26. [26]

    Fork=±1, the equations reduce to q2 2∂q2 ξ4 = 6∂ϕξ1, ξ 1 + 2q2∂q2 ξ1 =− ξ1(6k+3Λq2 2) 6k+Λq2 2 ,3ξ 1 + 2q2∂ϕξ4 =− ξ1(6k+3Λq2 2) 6k+Λq2 2

    Conditional symmetries The generators of the CS algebraξ α = (ξ1(q), ξ4(q)) are calculated from £ξGαβ = ΦGαβ,£ ξV= ΦV,(4.26) where Φ = Φ(q). Fork=±1, the equations reduce to q2 2∂q2 ξ4 = 6∂ϕξ1, ξ 1 + 2q2∂q2 ξ1 =− ξ1(6k+3Λq2 2) 6k+Λq2 2 ,3ξ 1 + 2q2∂ϕξ4 =− ξ1(6k+3Λq2 2) 6k+Λq2 2 . (4.27) The solution isξ 1 = 0 andξ 4 =A, whereAis a real constant. Thus, the ...

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  27. [27]

    Quantization Apart from the operators ˆq2 and ˆp2 we also introduceϕ→ ˆϕsuch that ˆϕ|Ψ⟩=ϕ|Ψ⟩and ˆp ϕ |Ψ⟩=−i∂ ϕ |Ψ⟩. The WDW equation is − 1 2µ ∂α µGαβ∂β +V Ψ(q) = 0.(4.31) Upon inserting the supermetric and superpotential, it takes the form q2 2 6 ∂2 q2Ψ + q2 6 ∂q2Ψ +∂ 2 ϕΨ + (24kq2 2 + 4Λq4 2)Ψ = 0.(4.32) From this we distinguish between two cases: 1.k=±...

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  28. [28]

    A known vacuum GR solution strictly within this class is the Kasner cosmology [122], ds2 =−dt 2 +t 2p1(dx1)2 +t 2p2(dx2)2 +t 2p3(dx3)2, 3X i=1 pi = 1.(5.3)

    Symmetry-invariant metric and classical solutions The Killing vectors defining the symmetries of Bianchi type I ([3,3,2]) are X1 =∂ x1 , X 2 =∂ x2 , X 3 =∂ x3 .(5.1) The general symmetry-invariant metric reads ˇg= (−N(t) +Nα(t)N α(t))dt2 + 2Nα(t)σα(x)dt+γ αβσα(x)σβ,(5.2) withσ α(x) being the 1-forms satisfyingdσ α(x) = 0, i.e.,σ α =dx α, andγ αβ being the...

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  29. [29]

    Hamiltonian and generalized momenta Following [123], the Hamiltonian can be written as H= N√γ H0,(5.4) whereγ= det(γ αβ). For Bianchi type I, we have H0 = 1 2 Lαβµν παβπµν,(5.5) whereL αβµν =γ αµγβν +γ ανγβµ −γ αβγµν, andπ µν are the canonical momenta dual toγ µν in the sense that they satisfy {γαβ, πµν}=δ µ αδν β −δ µ β δν α.(5.6) 21

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  30. [30]

    (5.8) The COMs are then given by Q(I) =E (I) βρπβρ,(5.9) which can be shown to satisfy{Q (I) , H0}= 0

    Conditional symmetries In this formalism, the generators of the CS algebra are given by [97] E(I) βρ =λ α (I)β γαρ,(5.7) whereI={1,2,· · ·,9}and λ1 =   0 1 0 0 0 0 0 0 0   , λ 2 =   0 0 1 0 0 0 0 0 0   , λ 3 =   0 0 0 0 0 1 0 0 0   , λ 4 =   0 0 0 0 0 0 0 1 0   , λ 5 =   0 0 0 0 0 0 1 0 0   , λ6 =   0 0 0 1 0 0 0 0 0   , λ 7 =   ...

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  31. [31]

    Quantization To quantize this system, we promoteγ αβ andπ αβ to operators as follows:γ αβ →ˆγαβ such that ˆγαβ |Ψ⟩=γ αβ |Ψ⟩ andπ αβ →ˆπαβ such that ˆπαβ |Ψ⟩=−i ∂ ∂γαβ |Ψ⟩. The commutators between ˆγ αβ and ˆπαβ follow from (5.6) by replacing the Poisson bracket with a commutator: [ˆγαβ,ˆπµν] = i 2(δµ αδν β −δ µ β δν α).(5.12) The eigenvalue equations for ...

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    In addition to the vacuum case, versions including matter have also been quantized in the literature

    Thus, the wave function for the quantized vacuum Bianchi type I is given by Ψ(γ) =c 1γ1/ √ 2 +c 2γ−1/ √ 2,(5.17) wherec 1 andc 2 are arbitrary constants. In addition to the vacuum case, versions including matter have also been quantized in the literature. For instance, Bianchi type I cosmology withnscalar fields and an exponential potential was quantized ...

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    (5.18) The symmetry-invariant metric is given by (5.2), where nowσ 1(x) =dx 2 −x 1dx3,σ 2(x) =dx 3, andσ 3(x) =dx 1 satisfyingdσ α(x) = 1 2 C α βγ σβ ∧σ γ withC 1 23 =−C 1 32 = 1

    Symmetry-invariant metric and classical solutions The Killing vectors defining the symmetries of Bianchi type II model are X1 =∂ x1 , X 2 =∂ x4 , X 3 =x 4∂x1 +∂ x3 . (5.18) The symmetry-invariant metric is given by (5.2), where nowσ 1(x) =dx 2 −x 1dx3,σ 2(x) =dx 3, andσ 3(x) =dx 1 satisfyingdσ α(x) = 1 2 C α βγ σβ ∧σ γ withC 1 23 =−C 1 32 = 1. The general...

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    Quantization Quantizing the system in the same way as in the last subsection, we get the eigenvalue equations for the CSs ϵα (I) βγαρ ∂Ψ ∂γβρ = 0,(5.25) where the eigenvalues are taken to be zero in order for ˜Q(I) to generate symmetries [100]. Solving these equations while recognizing thatϵ α (I) α = 0, we get Ψ = Ψ(γ, q),(5.26) whereq=C α µκC β νλγαβγµν...

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