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REVIEW 2 major objections 5 minor 108 references

The holographic growth-rate/momentum relation is not generic: it holds only for symmetry-adapted coherent states.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 16:20 UTC pith:P4TOIQGF

load-bearing objection Worth a careful read: the central claim is right, the AdS part is honestly framed as a consistency check, and the flat/dS examples genuinely sharpen when a momentum-complexity relation can hold. the 2 major comments →

arxiv 2607.18024 v1 pith:P4TOIQGF submitted 2026-07-20 hep-th

Comments on holographic spread complexity

Zhehan Li , Jia Tian This is my paper
classification hep-th
keywords spread complexityholographic complexitymomentum–complexity correspondenceKrylov complexitygeneralized coherent statesprincipal-series representationde Sitter static patchAdS/CFT
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 revisits the holographic proposal that the growth rate of spread complexity equals the radial momentum of an infalling bulk particle. It argues that this relation is not a general semiclassical consequence: it holds only when the initial state is a generalized coherent state adapted to the spacetime's symmetry algebra and the Hamiltonian itself lies in that algebra. The authors give a bulk derivation in global AdS and then test the proposal in flat and de Sitter spacetimes. In flat spacetime the physical Hamiltonian disperses coherent packets, so spread complexity measures wave-packet broadening rather than momentum; the same happens for generic de Sitter energy packets, while a principal-series-adapted coherent state gives exponential growth with a momentum-like rate. The upshot is that a semiclassical limit alone is insufficient—the classical interpretation of spread complexity depends on a symmetry-adapted choice of state.

Core claim

On global AdS2, a boundary primary state is shown to be a one-particle generalized coherent state; its spread complexity C(t)=8Δ z0²/(1-z0²)² sin²(t/2) has growth rate equal to the radial momentum along the geodesic. In flat spacetime the physical Hamiltonian disperses coherent packets, so C(t)=M²t²/8 counts packet width, not momentum. In the de Sitter static patch, generic Gamma-distributed energy packets give quadratic growth, while a sharply localized principal-series coherent state gives b_n=n/2 and C(t)=sinh²(t/2), matching the classical radial momentum rate. The relation is therefore a property of symmetry-adapted coherent states, not a general semiclassical fact.

What carries the argument

The work is carried by the choice of a generalized coherent state adapted to the spacetime isometry algebra: the Krylov basis is then generated by ladder operators, the Hamiltonian stays inside the algebra, and spread complexity reduces to a classical charge on the particle phase space. The relevant spectral measures are negative-binomial (Meixner) in AdS, Gamma (Laguerre) in flat space, and sech (Meixner–Pollaczek) in the dS principal series; the last gives Lanczos coefficients b_n=n/2 and exponential growth.

Load-bearing premise

The strongest assumption is that the β→0 limit of the regulated boundary-localized de Sitter state yields a single static-patch particle with exactly the sech[π(E±μ)] spectral density and b_n=n/2; if the limit instead gives a different weight or mixes both patches, the exponential momentum-tracking example does not follow.

What would settle it

Evolve the de Sitter boundary-localized state |ψ_{β,φ}⟩ at finite β instead of taking β→0 and compute its Lanczos coefficients from the spectral density ρ_β(E)=|∫ dt e^{iEt} A_β(t)|^2. If ρ_β develops a second peak (the opposite static patch) or b_n deviates from n/2, the exponential momentum-tracking example fails. Alternatively, in flat spacetime, evolve the Gaussian packet with the exact relativistic Hamiltonian without the nonrelativistic approximation E_p≃M+p^2/2M; if C(t) departs from M^2t^2/8, the dispersive interpretation needs revision.

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

If this is right

  • The momentum–complexity proposal of the literature is not a generic statement: it is valid precisely when the initial state is a generalized coherent state and the Hamiltonian lies in the corresponding symmetry algebra.
  • Spread complexity can be computed directly in the bulk from the quantum probe state; the boundary theory serves only to suggest a convenient coherent initial state, so the AdS/CFT dictionary is not essential for the classical interpretation.
  • In flat spacetime, the physical Hamiltonian preserves momentum but not coherent states, so spread complexity measures the squared width of the dispersing wave packet (C(t)=M²t²/8) rather than any momentum.
  • In the de Sitter static patch, generic semiclassical Gamma wave packets give only quadratic growth, whereas a boundary-localized principal-series coherent state gives exponential growth C(t)=sinh²(t/2), with rate proportional to the classical radial momentum.
  • The semiclassical limit alone is insufficient: a sharply peaked energy packet does not automatically produce a momentum–complexity relation.

Where Pith is reading between the lines

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

  • Editorial inference: The mechanism is not tied to holography; any spacetime whose isometry group admits a coherent-state representation should exhibit a similar momentum–complexity relation, as the dS principal-series example suggests.
  • Editorial inference: The flat-space result offers a practical diagnostic: measuring whether spread complexity grows as t² (dispersion) or with a momentum-driven rate could reveal whether a given semiclassical state is symmetry-adapted.
  • Editorial inference: The dS exponential example depends on the sharp-localization limit; finite-β corrections may change the spectral density and the prefactor of sinh²(t/2), so the exact late-time growth rate may be regularization-dependent.
  • Editorial inference: The Groenewold–van Hove obstruction noted for dS charges implies that only some complexity=anything observables admit consistent quantization, which may limit which classical complexity observables have quantum counterparts.

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

2 major / 5 minor

Summary. The paper revisits the proposal dC_O/dt ∝ p_infalling and argues that the semiclassical limit alone is not sufficient: the relation requires an initial generalized coherent state adapted to the spacetime symmetry algebra and a Hamiltonian lying in that algebra. After a bulk-side derivation in global AdS2 (negative-binomial spectral weights and C(t)=8Δ z0^2/(1-z0^2)^2 sin^2(t/2)), the authors quantize a free particle in AdS2 and interpret spread complexity as a quantized complexity=anything observable. In flat spacetime, the physical Hamiltonian does not preserve the Heisenberg coherent-state structure and spread complexity measures wave-packet dispersion (C=M^2 t^2/8). In the dS static patch, Gamma-distributed energy packets give C ∝ T^2, while a β→0 boundary-localized principal-series state gives C=sinh^2(t/2), whose growth rate has the same time dependence as the classical radial momentum. The paper concludes that a semiclassical limit alone is insufficient for a momentum–complexity relation.

Significance. The paper provides a useful conceptual clarification: the momentum-complexity relation is not a generic semiclassical statement but requires symmetry-adapted coherent states and algebra-valued Hamiltonians. The bulk derivation in §2 is clean and the explicit connection to the complexity=anything framework is valuable. The negative examples in §3 and §4.2 are concrete and falsifiable: for a Gaussian packet under H=sqrt(p^2+M^2), spread complexity tracks dispersion, not momentum; for Gamma packets in the dS static patch, it grows quadratically. The positive dS example in §4.3, if fully derived, is a nice demonstration that symmetry-adapted states restore exponential momentum-tracking growth. The paper is honest that the AdS extrapolation dictionary acts as a change of representation, and it does not overclaim an independent derivation of the correspondence.

major comments (2)
  1. [§4.3, Eqs. (4.83)–(4.91)] The central dS positive example rests on the evaluation of the overlap integral (4.83) in the β→0 limit. The text states “The overlap integral can then be evaluated analytically. Keeping only the leading term in β…” but does not display the integration. Please include the full evaluation (or an appendix) leading to A_0^-(t)=e^{-iμt}/cosh(t/2). Without this, the spectral density (4.96), the Lanczos coefficients b_n=n/2 (4.100), and the final C(t)=sinh^2(t/2) are not independently checkable.
  2. [§4.3, Eq. (4.79)] The state |ψ_{β,φ}⟩=N_β e^{-β|L0|} O_Δ(φ)|Ω⟩ is introduced as “motivated by the extrapolation dictionary []” with an explicit empty citation. Since this construction is the basis of the single-patch result, the missing reference should be supplied. In addition, the paper should clarify whether this state is a generalized coherent state in the Perelomov sense discussed earlier (4.50), since the formal coherent state |ζ⟩ is not the one used for the exact result.
minor comments (5)
  1. [Abstract and §4.2] The abstract states that “generic semiclassical wave packets” in the static-patch energy representation show dispersive behavior, but the computation is carried out only for the Gamma-distribution family (4.29). Please qualify the claim or provide an argument that the Gamma family captures the generic case.
  2. [§2.5, after Eq. (2.93)] There is a sentence fragment: “the classical phase space is identified with the The resulting phase space is the Poincaré disk.” Please fix.
  3. [§4.3, Eq. (4.70)] For the |0⟩ state, Eq. (4.70) states C(t)∼sinh t, while the exact result for b_n=n/2 is C(t)=sinh^2(t/2). State the exponential growth rate consistently.
  4. [§3, Eq. (3.12)] The symbol M is used both for the particle mass and for the reference scale in the complex coordinates (3.12). Consider renaming the reference scale to avoid confusion.
  5. [§4.3, Eq. (4.94)] The spectral densities are written without normalization constants. Since the subsequent orthonormality relation (4.66) fixes normalization, a brief note would avoid ambiguity.

Circularity Check

1 steps flagged

No significant circularity: the AdS 'derivation' is an author-acknowledged change-of-representation consistency check; the [62] self-citation is non-load-bearing; the central classification claim rests on independent flat-space and de Sitter computations.

specific steps
  1. self definitional [§2.2, Eqs. (2.30)–(2.31) and the acknowledgment sentence after (2.31)]
    "This agreement is not surprising. Through the extrapolation dictionary, the bulk and boundary calculations involve the same quantum state and the same Hamiltonian; the dictionary therefore acts here simply as a change of representation."

    The bulk initial state (2.11) is defined as the extrapolation-dictionary image of the boundary state (2.10) under the same Hamiltonian, so the bulk spectral weights (2.18) equal the boundary weights by construction and the bulk result (2.31) identically reproduces the boundary result (1.16). The momentum–complexity link (1.20)–(1.22) likewise inherits the dictionary as an input, which the paper itself flags as 'not a derivation from first principles.' Thus the AdS 'derivation' is a change-of-representation consistency check. The paper acknowledges this explicitly, and the central claim (coherent state plus algebra-valued Hamiltonian required for the momentum–complexity relation) is carried by the independent flat-space (§3) and de Sitter (§4) computations, so the circularity is confined an

full rationale

The paper's central claim — that a semiclassical limit alone is insufficient for dC/dt ∝ p_infalling and that one additionally needs a generalized coherent state adapted to the symmetry algebra with the Hamiltonian in that algebra — is supported by independent, self-contained computations. The flat-space analysis (§3) derives C(t) = M²t²/8 and identifies it with packet-width growth σ(t)² (Eqs. (3.50)–(3.52)) from a Gamma/Laguerre computation; no data are fitted and no parameter is renamed as a prediction. The dS Gamma-packet example (§4.2) is likewise self-contained and is a negative result: quadratic complexity vs. exponentially growing classical p_ρ(T) (Eq. (4.36)). I checked the delicate positive dS example (§4.3): the β→0 limit of |ψ_{β,φ}⟩ giving A⁻₀(t) = e^{−iμt}/cosh(t/2) (Eq. (4.91)) is sound (the overlap integral with x = βy reduces to (8/π)∫dy/[(1+4y²)(1+4e^{−2t}y²)] = 2e^t/(1+e^t)); the Fourier transform to sech[π(E±μ)] and the Meixner–Pollaczek identification with b_n = n/2 then follow. The momentum match is only 'up to an overall factor' (Eq. (4.103)), so there is no dressed-up fitted coefficient. The only circular-adjacent element is the AdS 'derivation' in §2.2: the bulk state (2.11) is defined as the dictionary image of the boundary state (2.10) under the same Hamiltonian, so the bulk C(t) (2.31) coincides with the boundary result (1.16) by construction. The authors state this reduction openly and list the dictionary as an assumption, so it is an acknowledged consistency check, not a hidden circularity. The self-citation [62] supplies the boundary-side L̂₀ construction in §1.1, but Eqs. (1.12)–(1.17) are re-derived within the paper (§§2.2, 2.5), so [62] is not load-bearing. Missing support (flagged): §4.3 says the state is 'motivated by the extrapolation dictionary []' with an empty citation — a missing reference, not a circular step, since the boundary-operator expansion (4.75) is standard (cf. [92, 96]) and the limit calculation stands alone. The §5 sentence calling the bulk computation 'a proof of the proposal... without referring to the AdS/CFT correspondence' slightly oversells, since the state choice uses the dictionary, but the computation itself is bulk-internal. Verdict: no load-bearing circularity; score 2 reflects the minor non-load-bearing self-citation and the acknowledged representational tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The central claim rests mainly on standard Krylov/orthogonal-polynomial mathematics plus domain assumptions about the extrapolation dictionary, the discrete/principal-series identifications, and the chosen initial-state models. No numerical data fitting is involved; the only hand-chosen parameters are state-preparation and regulator parameters (Gamma β,λ and dS regulator β), which affect coefficients but not the qualitative conclusions.

free parameters (2)
  • Gamma-packet shape β and scale λ (dS §4.2) = β/λ = m(cosh χ0 − 1)
    State-preparation choices defining the 'generic semiclassical wave packet' in the static-patch energy representation. Their ratio is set to match a classical particle energy; the resulting spread complexity C(T)=β/λ² T² depends on the variance, but the qualitative conclusion (quadratic growth vs exponential momentum) does not.
  • dS boundary-localization regulator β (Eq. (4.79)) = β → 0 limit
    The regularization N_β e^{-β|L0|} is introduced solely to make the non-normalizable state |Δ,φ⟩ normalizable; the final results are taken in the β→0 limit. The paper does not check that a different regulator shape gives the same spectral density.
axioms (7)
  • domain assumption Extrapolation dictionary (2.7): near-boundary bulk field ~ (cosh ρ)^{-Δ} O(t), with the boundary Hamiltonian equal to the bulk Hamiltonian.
    Used in §2.1–2.2 to map the CFT local-quench state O(z0)|Ω⟩ to the one-particle bulk state (2.11). If this dictionary fails (e.g., with backreaction), the AdS part of the derivation no longer applies; the flat/dS sections do not use it.
  • standard math Krylov recursion is equivalent to orthogonal polynomials w.r.t. the spectral measure (Meixner/Laguerre/Hahn constructions).
    Standard result invoked in §2.2 (Eqs. (2.15)–(2.17)) and used throughout; no new proof supplied.
  • domain assumption The one-particle sector of the free massive scalar on AdS_2 realizes the lowest-weight discrete series of SL(2,R) with weight Δ = m in the large-Δ limit.
    Used in §2.5 to identify the quantized particle Hilbert space with the bulk Fock space; the exact scalar mass satisfies m² = Δ(Δ−1), so identifying m with Δ is only exact in the semiclassical limit (Eq. (2.34)).
  • domain assumption In flat space, the nonrelativistic truncation E_p ≃ M + p²/2M (Eq. (3.32)) is representative of the semiclassical regime.
    The flat-space conclusion (dispersion dominates) is derived in this limit; the paper does not analyze the fully relativistic spectral problem.
  • domain assumption The Gamma distribution (4.29) is a valid model of a generic semiclassical wave packet in the dS static-patch energy representation.
    Adopted in §4.2 to represent a sharply-peaked energy distribution; the quadratic spread complexity C(T)=β/λ² T² follows from this choice. The paper flags it as 'a convenient model'.
  • domain assumption The β→0 limit of the regularized boundary-localized states (4.79) gives a single-static-patch spectral density sech[π(E±μ)] with Lanczos coefficients b_n = n/2.
    Load-bearing for the dS positive example (§4.3, Eqs. (4.79)–(4.100)); the sharp-localization limit and the resulting eigenstate overlap are asserted rather than derived in detail. The same representation's |0⟩ state yielded a two-peak spectrum (4.72), so the limit is delicate.
  • domain assumption The principal-series representation L0|n⟩=n|n⟩, L±|n⟩=(n±Δ)|n±1⟩ with Δ=1/2+iμ describes static-patch one-particle states with H_s=(L_+ + L_-)/2 generating static time.
    Used in §4.1–4.3 to identify the dS particle Hilbert space; relies on [96] for the global-mode realization and on the static-patch quantization (4.24)–(4.26).

pith-pipeline@v1.3.0-alltime-deepseek · 31290 in / 26491 out tokens · 265498 ms · 2026-08-01T16:20:15.482577+00:00 · methodology

0 comments
read the original abstract

We revisit the holographic proposal relating the growth rate of spread complexity to the radial momentum of a bulk probe, aiming to identify its underlying assumptions and clarify how classical probe dynamics emerges from quantum dynamics. By quantizing AdS probes directly and using the extrapolation dictionary, we provide a more general derivation of the proposal. In particular, we interpret it as a concrete realization of a classical complexity observable and its quantization in the ``complexity=anything'' framework. Since probe dynamics is not intrinsically tied to the AdS/CFT correspondence, we also examine the proposal in flat and de Sitter spacetimes. In flat spacetime, the physical Hamiltonian does not preserve the coherent-state structure, and spread complexity measures the dispersive broadening of the wave packet rather than classical momentum. In de Sitter space, generic semiclassical wave packets in the static-patch energy representation show the same dispersive behavior. By contrast, coherent states adapted to the principal-series representation can exhibit exponential growth of spread complexity, with a growth rate that has the same time dependence as the classical radial momentum. These examples indicate that a semiclassical limit alone is not sufficient for a momentum--complexity relation.

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