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For sharply peaked initial states, the Krylov complexity of the quantum universe grows as (σ/2)²(φ−φ0)² for states and double that for operators—so it diverges at the WDW singularity but stays finite through the loop-quantum bounce.

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2026-08-03 20:55 UTC pith:MKFG7JNC

load-bearing objection Genuine first: Krylov complexity for a constrained quantum cosmological system with a relational clock, cleanly derived at leading order; the uncontrolled sharp-peaking approximation is the main soft spot but not disqualifying. the 3 major comments →

arxiv 2511.17711 v2 pith:MKFG7JNC submitted 2025-11-21 gr-qc quant-ph

Quantum Cosmology in Krylov Space: Complexity and Entropy

Meysam Motaharfar , Maxwell R. Siebersma , Parampreet Singh This is my paper
classification gr-qc quant-ph PACS 04.60.Kz04.60.Pp
keywords Krylov complexityquantum cosmologyWheeler-DeWittloop quantum cosmologyKrylov entropyLanczos algorithmspread complexitymassless scalar field clock
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to show that the notion of Krylov complexity—normally used to track how quantum states and operators spread under a Hamiltonian—can be transplanted into canonical quantum cosmology, where there is no external time, by using the massless scalar field as an internal clock. For sharply peaked Gaussian wave packets, the authors construct the Krylov basis exactly in both the Wheeler-DeWitt (WDW) quantization and the exactly solvable loop quantum cosmology (sLQC), and find that the Krylov state complexity grows quadratically with the clock while the operator complexity is exactly twice as large. The same formula describes both models; the physical difference shows up in global behavior. In WDW quantum cosmology the complexity and its associated Krylov entropy diverge as the clock approaches the big-bang or big-crunch singularity, while in sLQC they both remain finite at the quantum bounce. A sympathetic reader would care because this gives a quantitative, purely quantum-mechanical sense in which loop quantum gravity resolves the classical singularity, and opens a route to studying chaos in more realistic cosmological models.

Core claim

The central claim is that, for sharply peaked states (k0 >> σ), the Krylov state complexity of the universe in both WDW quantum cosmology and sLQC is C_K^(ψ) = (σ²/4)(φ−φ0)² and the Krylov operator complexity, computed for the pure-state density operator, is C_K^(O) = (σ²/2)(φ−φ0)², so C_K^(O) = 2 C_K^(ψ). The Lanczos coefficients are b_n = (√n/2)σ for states and b_n = √(n/2)σ for operators, giving Poisson probability distributions with means σ²(φ−φ0)²/4 and σ²(φ−φ0)²/2 respectively. Because the same Hamiltonian and inner product appear in both frameworks, the formulas are identical; the two theories differ only in which values of φ correspond to physical events. Consequently, Krylov complex

What carries the argument

The Lanczos algorithm applied to the physical Hamiltonian H = −√Θ, which in momentum space multiplies by −k. The key approximations are replacing the physical half-line inner product 2∫_0^∞ dk k χ̃₁*χ̃₂ by the full-line version 2∫_{-∞}^∞ dk k χ̃₁*χ̃₂ and extending the initial Gaussian to all k; this yields the exact Hermite-polynomial Krylov basis |K_n⟩ = (1/√(2^n n!)) H_n(−√2(k−k0)/σ) χ̃(k), with Lanczos coefficients a_n ≈ −k0 and b_n = (√n/2)σ for states. For operators, the Liouvillian L = [−√Θ, ·] gives b_n = √(n/2)σ. These b_n are the machine that produces the Poisson distributions and hence the quadratic growth and the factor-2 operator-to-state ratio.

Load-bearing premise

Everything hinges on the sharp-peaking approximation that replaces the physical half-line momentum inner product with a full-line inner product and extends the Gaussian initial state to all momenta; if it fails, the exact Hermite-polynomial basis, the Poisson distribution, the quadratic growth, and the factor-2 relation all become approximate or invalid.

What would settle it

Compute the exact Lanczos coefficients for the unapproximated physical inner product (χ1,χ2)_phys = 2∫_0^∞ dk k χ̃₁*χ̃₂ for finite k0/σ, either analytically or numerically. If b_n deviates from √n σ/2 by more than O(σ³/k0²), or if the resulting Krylov complexity is not C_K^(ψ) = σ²(φ−φ0)²/4 up to that error, the central claim is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In both WDW quantum cosmology and sLQC, the Krylov state and operator complexity grow as the square of the scalar-field clock, consistent with integrable (non-chaotic) dynamics.
  • The exact relation C_K^(O) = 2 C_K^(ψ) holds for these infinite-dimensional systems, not just for qubits.
  • Krylov entropy is minimized at the initial time φ0 and increases monotonically in each branch; in sLQC the bounce provides a canonical φ0, while in WDW the initial time is ad hoc.
  • Complexity and entropy diverge at the WDW big-bang/big-crunch singularities but remain finite throughout sLQC, including at the bounce.
  • The construction demonstrates a template for defining Krylov complexity in constrained systems by using a relational clock instead of external time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the factor-2 and quadratic-growth results rest on the sharp-peaking approximation, an exact (or numerical) treatment of finite σ/k0 is likely to show the relation holds only approximately; the paper itself flags corrections of order σ³/k0².
  • The use of the density operator as the probe makes the two frameworks mathematically identical; choosing a Dirac observable with different spectra, like the volume operator, would likely expose theory-specific complexity behavior.
  • The monotone growth of Krylov entropy away from the bounce is suggestive of an information-theoretic arrow of time, but the paper correctly notes that no rigorous link to thermodynamic entropy has been established.
  • Extending this analysis to anisotropic models is a natural test: if quantum chaos survives, Krylov complexity should switch from quadratic to exponential growth, providing a sharp signature of chaos near singularities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies Krylov (state and operator) complexity and Krylov entropy for a flat FLRW universe with a massless scalar field in two canonical quantum cosmology frameworks: Wheeler-DeWitt (WDW) and solvable loop quantum cosmology (sLQC). Using the physical Hilbert space and Hamiltonian of Ref. [11], the authors construct the Krylov basis analytically, starting from a Gaussian state sharply peaked in the scalar-field momentum (k0 >> sigma). They replace the physical half-line inner product with a full-line approximation and obtain Lanczos coefficients b_n ~ sqrt(n) sigma/2 for states and b_n ~ sqrt(n/2) sigma for operators. This leads to Poisson probability distributions, Krylov state complexity C_K^(psi) = sigma^2 (phi-phi0)^2 / 4, Krylov operator complexity C_K^(O) = sigma^2 (phi-phi0)^2 / 2, and hence C_K^(O) = 2 C_K^(psi). The paper interprets these results globally: in WDW the complexity and entropy diverge at the big bang/big crunch (phi -> +- infinity), whereas in sLQC they remain finite at the finite-time bounce.

Significance. If the central results hold, this is the first construction of Krylov complexity for a totally constrained quantum cosmological system with an internal clock, and the first direct bridge between Krylov complexity and canonical quantum gravity/LQC. The paper is explicit and analytic, uses the independently established physical Hilbert space of Ref. [11], introduces no fitted parameters, and gives falsifiable predictions (quadratic growth, factor-2 relation, finite-at-bounce behavior). These are genuine strengths. However, the entire quantitative structure rests on the sharp-peaking approximation introduced in Sec. IV.A, and the manuscript does not provide a controlled error analysis for the Lanczos coefficients or the resulting probabilities. Since the main physical conclusions (divergence vs. finiteness) are global in the internal time phi, the absence of such control is load-bearing.

major comments (3)
  1. [Sec. IV.A, Eqs. (4.3)-(4.17)] The Hermite basis (4.13) is not orthonormal with respect to the approximate inner product (4.3), because the factor k in the measure breaks the Gaussian-Hermite orthogonality. For the normalized Gaussian (4.4), a direct evaluation gives <K0|K1> = O(sigma^2/k0), not zero. The Lanczos algorithm with the exact a0 = -k0(1 + sigma^2/(4 k0^2)) would shift A1 in Eq. (4.7) by an O(sigma^2/k0) term. Thus the claims b_n = sqrt(n) sigma/2, the Poisson distribution (4.16), and the complexity (4.17) are leading-order statements; they are not exact consequences of the stated inner product. Moreover, retaining the k variation in the integral (4.14) gives corrections to the Krylov coefficients that are not uniformly small in phi - phi0; for late times the correction can become O(1). The citation to Ref. [64] concerns semiclassical expectation values, not orthonormality of a Krylov basis. The authors sho
  2. [Sec. IV.A and IV.B, Eq. (4.28) and the induction for b_n] The paper asserts, after constructing K0, K1, K2, K3, that 'continuing this iteration' yields a_n ~ -k0 and b_n ~ sqrt(n) sigma/2. No induction proof is given for either the state or the operator Lanczos coefficients. The quoted error term O(sigma^3 k0^-2) in Eq. (4.28) is not derived from a recurrence and is stated only for fixed n. The subsequent Poisson statistics and the exact factor-2 relation depend on all n, and the divergence at phi -> +- infinity probes arbitrarily large n. A general argument (or an explicit recurrence bound) is needed to justify that the approximation remains valid for the entire support of the probability distribution and for all relevant phi.
  3. [Sec. IV.A, Eqs. (4.2)-(4.4)] The replacement of the physical inner product 2 int_0^infinity dk k ... by 2 int_-infinity^infinity dk k ... is not a positive-definite inner product on L^2(R). For states with support at negative k the measure is negative. While this is harmless for the very-low-n overlap of a Gaussian peaked at k0 >> sigma, the Hermite polynomials in (4.13) grow at negative k as n grows; for n of order (k0/sigma)^2 the negative-k tail may no longer be negligible. Since the claimed divergence of Krylov complexity and entropy in WDW occurs at late internal times, where the relevant Krylov index grows without bound, the global interpretation is not established by the leading-order calculation. A numerical or analytic computation of the actual Lanczos coefficients for the half-line measure would settle whether the qualitative conclusions survive.
minor comments (4)
  1. [Sec. IV.A, Eq. (4.4)] The notation '.=' is used for an approximate equality but is never defined. Please define this notation explicitly (e.g., 'equality up to the sharp-peaking approximation').
  2. [Sec. IV.B, Eq. (4.31)] The solution is written with 'for some constant m' and then m is set to phi0 by the initial condition. This is fine mathematically, but the symbol m is unnecessary and can be eliminated for clarity.
  3. [Sec. IV.B, Eq. (4.24)] There appear to be typographical issues with subscripts/superscripts in the trace expression (e.g., 'brho2_0' and 'brho_2_0'). Please proofread the equation and define all terms.
  4. [Sec. IV.A, reference [64]] The statement that extending the Gaussian to all k introduces only 'negligible error' is cited to Ref. [64]. That reference studies coherent states and expectation values of Dirac observables; it does not address the orthonormality of the Krylov basis or the Lanczos coefficients. This justification should be supplemented with a direct estimate in the context of the present calculation.

Circularity Check

0 steps flagged

No significant circularity: the Krylov complexity results are derived from the independently established WDW/sLQC framework and the stated sharp-peaking approximation, not assumed as inputs.

full rationale

The derivation chain is not circular. The physical Hamiltonian, inner product, and singularity/bounce structure are imported from Ref. [11], which shares a co-author but is an independently established and widely used framework; the paper's new content is the Lanczos construction and the resulting Krylov coefficients, probabilities, and complexities. The initial Gaussian and parameters k0 and sigma are chosen initial-state data, not fitted to produce the claimed quadratic growth. The replacement of the half-line inner product (4.2) by the full-line approximation (4.3) is an explicit approximation: it makes the Hermite-basis results leading-order in sigma/k0, and the paper consistently states that all results hold only for k0 >> sigma, with Eq. (4.28) carrying O(sigma^3 k0^-2) corrections. This is an accuracy/justification limitation, not a circular reduction, because the complexity is computed honestly within the approximate model rather than imposed by construction. The relation C_K^(O) = 2 C_K^(psi) follows algebraically from the Lanczos coefficients b_n^(state) = sqrt(n) sigma/2 and b_n^(op) = sqrt(n/2) sigma, and the divergence/finiteness at singularities versus the bounce is read off from the known dependence of the physical volume on the internal clock. No fitted parameter is renamed as a prediction, and no load-bearing step reduces to a self-citation. The work is self-contained against the Krylov-complexity formalism and the standard WDW/sLQC quantization.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central derivation relies on the standard physical Hilbert space and Hamiltonian from the LQC literature (Ref. [11]), plus an analytic-control approximation (full-line inner product, k0 >> σ) that underlies the exact Lanczos coefficients. The Gaussian width σ, mean k0, and initial time φ0 are choices for the initial state, not fitted parameters. No new entities are introduced. The main non-standard input is the sharp-peaking approximation, which is flagged but not tightly error-controlled.

free parameters (3)
  • sigma (Gaussian width)
    Width of the initial Gaussian in scalar-field momentum; chosen by hand with k0 >> σ. It sets the coefficient of the quadratic Krylov complexity growth: C^(ψ)_K = σ²(φ−φ0)²/4. The functional form of the growth (quadratic) is independent of σ, but the magnitude is not.
  • k0 (mean momentum)
    Peak of the initial Gaussian; the sharp-peaking condition k0 >> σ underlies the approximation of the physical inner product. In the leading approximation, k0 drops out of the final complexity formulas except via the physical interpretation (contracting vs expanding branch).
  • phi0 (initial internal time)
    The time at which the initial state is defined and the Krylov complexity vanishes. It is an arbitrary origin; in sLQC it can be set to the bounce time φ_B, in WDW it is ad hoc.
axioms (5)
  • domain assumption The physical Hilbert space, physical inner product, and physical Hamiltonian for the flat FLRW massless-scalar universe in WDW and sLQC are those constructed in Ref. [11]; in the k-representation both reduce to a free-particle Hamiltonian H_phys = -√Θ ≈ -|k| on positive-frequency solutions.
    The entire Krylov construction is built on this framework, adopted from Ref. [11] which shares an author with the present paper but is independently established.
  • domain assumption The massless scalar field φ is a global monotonic internal clock for the entire non-singular history in sLQC and for each branch (expanding/contracting) in WDW.
    Needed to interpret φ as the time in the Schrödinger evolution (Eq. 3.8/3.23) and to give physical meaning to the φ-dependent Krylov complexity. Standard in LQC literature (Refs. [9-11]).
  • ad hoc to paper The physical inner product can be approximated by the full-line inner product (Eq. 4.3) and the initial state can be taken as a Gaussian on all of k (Eq. 4.4) with k0 >> σ.
    This approximation is introduced 'to gain analytical control over the construction of the Krylov basis' and is the basis for the exact Hermite-polynomial Krylov basis and the Poisson probability distribution. The error is stated as negligible (Ref. [64]) but not computed in detail.
  • standard math The Lanczos algorithm for the Gaussian state yields b_n = √n σ/2 for states and b_n = √(n/2) σ for operators for all n (Eqs. 4.13, 4.28).
    This recurrence follows from the oscillator-algebra structure of the Gaussian state acting with the operator -k, but the paper does not present an induction proof; it is inferred from the first few steps.
  • standard math The Hilbert-Schmidt inner product is well-defined and normalizable for the pure-state density matrix and all Liouvillian iterates used in the operator Lanczos algorithm.
    Needed for the operator Krylov basis. The authors assert normalizability ('we will only work with operators that are normalizable', Sec. II.B) and it holds for the Gaussian state.

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read the original abstract

We study the quantum dynamics in Krylov space of a spatially flat, homogeneous, and isotropic universe sourced with a massless scalar field within Wheeler-DeWitt (WDW) quantum cosmology and loop quantum cosmology (LQC) frameworks. The availability of a physical Hilbert space and physical Hamiltonian and the presence of an internal clock enable us to construct the Krylov basis analytically by applying the Lanczos algorithm. We then evaluate both the Krylov state and operator complexity for WDW quantum cosmology and LQC on this basis. In regimes where the wave function of the universe is sharply peaked, our results indicate that the Krylov complexity grows quadratically with the scalar field clock for the state and operator complexities in both the WDW quantum cosmology and LQC. We further show that the operator complexity is exactly twice the state complexity in these regimes. We discuss the interpretation of the global behavior of these systems by calculating the Krylov entropy for both quantum cosmological frameworks. We observe that in LQC, the Krylov complexity and entropy remain finite at the bounce, whereas in the WDW quantum cosmology, they diverge at the big bang/crunch singularity. Our work provides the first example of computing Krylov complexity for a system with a totally constrained Hamiltonian and no external time, a framework to calculate a purely quantum-mechanical entropy in quantum cosmology, and, to our knowledge, the first direct bridge between Krylov complexity and canonical quantum cosmology, as a first step toward understanding how polymerized quantum geometry modifies complexity and entropy.

Figures

Figures reproduced from arXiv: 2511.17711 by Maxwell R. Siebersma, Meysam Motaharfar, Parampreet Singh.

Figure 1
Figure 1. Figure 1: FIG. 1: Krylov entropy of the density operator ( [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗

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Reference graph

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