Real-time dynamics from convex geometry

Scott Lawrence

REVIEW 3 major objections 4 minor 18 references

Real-time dynamics from convex geometry

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Finite, noisy Euclidean data plus one quantum axiom are enough to produce the tightest possible certified bounds on smeared real-time correlators, by solving a finite-dimensional convex dual.

desk verdict A clear, honest proceedings summary of a method whose full derivation and numerics live in a companion paper; treat it as a pointer, not a primary source. read the letter →

arxiv 2502.01880 v1 pith:TGTXO4G7 submitted 2025-02-03 hep-lat

Scott Lawrence This is my paper · ORCID

classification hep-lat

keywords real-timecorrelatorspectraldensityconvexoptimizationLagrangedualityanalyticcontinuationlatticefieldtheoryreflectionpositivityEuclidean

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that real-time response functions, normally unreachable from Euclidean lattice data because analytic continuation is ill-posed, can still be bounded rigorously. It casts the extraction of any linear functional of the spectral density—including a Gaussian-smoothed real-time correlator—as a convex program whose only physics input is that the spectral density is non-negative. Passing to the Lagrange dual reduces that infinite-dimensional problem to a finite one with one variable per Euclidean time-slice, so the bounds are computable in polynomial time. The solution is claimed to give the tightest bounds possible from the given Euclidean data, assuming nothing beyond quantum-mechanical axioms. Demonstrations on an anharmonic oscillator and a 2+1-dimensional scalar field theory show the bounds tracking the exact correlator for roughly one to two periods.

What carries the argument

The Lagrange dual of a convex program over spectral densities is the mechanism that carries the argument. Although the primal has infinitely many variables, the inner minimization in the dual forces the multiplier $\lambda(\omega)$ to lie in the finite span of the data kernels, $\lambda(\omega)=K(\omega)-\sum_i \ell_i K_i(\omega)$, collapsing the problem to an $N+1$-dimensional maximization over coefficients $\ell_i$ and a scalar $\mu\ge 0$, subject to the pointwise constraint $\lambda(\omega)\ge 0$. Strong duality makes that finite program reproduce the infinite primal optimum exactly, which is what turns an ill-posed inverse problem into polynomial-time certified bounds.

What would settle it

Take an exactly solvable model with a known spectral density, compute its Euclidean correlator at a finite set of separations, solve the dual program (13), and check whether the exact Gaussian-smeared real-time correlator stays inside the certified interval at every tested time; any excursion outside the interval would refute the claim that the bounds are the tightest possible.

Watch

Extended reading notes

Core claim

The central claim is that the convex program (8) over spectral densities $\rho(\omega)\ge 0$, constrained by the covariance ellipsoid $F[\rho]\le F_{\max}$, has a finite-dimensional Lagrange dual (13). The dual, with $N+1$ variables for $N$ Euclidean time slices, returns a certified lower bound on $\int K(\omega)\rho(\omega)\,d\omega$; solving again with $K\to -K$ gives the upper bound. Under standard constraint-qualification conditions the dual optimum equals the primal optimum, so the bounds are the tightest available from the supplied Euclidean data under the sole assumption $\rho(\omega)\ge 0$. Applied to the smeared real-time kernels $K_t^\sigma(\omega)=-e^{-\sigma^2\omega^2/2}\sin\omega t$, the dual yields validated intervals for $\tilde G_\sigma(t)$.

Load-bearing premise

The bounds are only as valid as the assumption that the Euclidean data could have come from a non-negative spectral density, which lattice data do not automatically satisfy.

Editorial extensions

If this is right

Real-time response functions become accessible from Euclidean lattice calculations without choosing a regulator, fitting an ansatz, or solving an ill-posed inverse problem: at each time $t$ and smearing width $\sigma$, the method returns certified upper and lower limits.
Under the stated assumptions, no method using the same Euclidean data and the same single physical axiom can produce a narrower interval for the same linear functional of the spectral density, because the dual optimum equals the primal optimum.
The computational cost is polynomial in the number of Euclidean time slices and independent of the spatial volume, so once Monte Carlo data exist, extracting real-time dynamics in a higher-dimensional theory is cheap relative to the lattice calculation itself.
Quantities that are not linear functionals of the spectral density, such as transport coefficients or particle lifetimes, are not bounded by this method; the paper notes that additional information would be needed.
If the input correlator violates $\rho(\omega)\ge 0$, the bounds are incorrectly tight, and the method can expose this as a lower bound that exceeds the upper bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One practical extension is a feasibility-first pipeline: solve a minimal variant of (8) to test whether any non-negative spectral density lies inside the data ellipsoid before quoting bounds, turning the known failure mode into a diagnostic.
Because the duality collapse works for any finite set of smooth data kernels, the same convex framework could absorb extra constraints from multiple correlation functions or from lattice equations of motion without giving up certified bounds.
The dependence on temperature and smearing width suggests that scanning $\sigma$ at fixed $t$ would map how much frequency resolution a given Euclidean dataset actually contains, effectively charting the horizon of analytic continuation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Candid thoughts on Lawrence's proceedings piece. What you should know first: this is a talk write-up, not a new result. The method, the derivation of the dual program, and both figures are explicitly credited to the companion paper [17]. As a standalone contribution it is incomplete: equation (13) is stated without derivation and the matrix M in that equation is never defined. The 'tightest possible' claim in the Introduction is conditional on the feasible set containing the true spectral density, which the author himself flags in Section 4.

That said, the paper does its job as a proceedings talk. The exposition is clear and the convex-geometry framing is genuinely useful for understanding why analytic continuation is ill-posed and what a principled bound can and cannot give. The author is transparent about the method's limitations: only linear functionals of the spectral density can be bounded, transport coefficients are out, and lattice QCD applications are restricted to exactly reflection-positive correlators. That last point is the load-bearing assumption, and the stress-test is right to highlight it. But it is not a hidden flaw: Section 4 says plainly that if a correlator not obeying rho >= 0 is fed in, one obtains incorrectly tight bounds. So the central claim is accurately scoped by the author, even if the abstract and Introduction are a bit punchy.

The soft spots are the gaps that force a reader to go to [17]: the dual program's derivation, the definition of M, and the validation of the Gaussian/ellipsoid confidence region. For a proceedings paper, deferring details is normal, but it does mean the claims here cannot be fully checked from this text. The numerical checks shown in the figures come from [17] and compare against direct diagonalization, which is the right benchmark, so there is no sign of circularity or curve-fitting.

Who gets value from this: lattice practitioners who want a quick, readable orientation to the approach and a clear statement of its known limits. It is not the primary reference; cite [17] for the substance and this paper only if you need the talk record. I see no fatal error, and the author's honesty about the reflection-positivity assumption deserves credit. If the venue runs refereed proceedings, send it to a referee who can check that the claims match the deferred derivations and that the confidence-region discussion is not overclaimed. I would accept it with minor revision, mainly asking for a pointer to where M is defined and one sentence clarifying that the bounds are conditional on the spectral density being non-negative.

Referee Report

3 major / 4 minor

Summary. The manuscript, a Lattice 2024 proceedings contribution, proposes a convex-optimization method for bounding real-time correlators from finite, noisy Euclidean data. The idea is to formulate the set of spectral densities consistent with positivity and with the data as a convex feasible set, then optimize linear functionals (smeared real-time correlators) over that set. To make the infinite-dimensional optimization tractable, the author passes to a finite-dimensional Lagrange dual program (Eq. (13)) and demonstrates the method on the anharmonic oscillator and on a 2+1-dimensional φ^4 theory, comparing against direct diagonalization. The paper claims these bounds are the tightest possible from the given Euclidean data, assuming no information beyond quantum-mechanical axioms, and it explicitly flags in Section 4 that the positivity assumption (reflection positivity) is essential and can fail for non-reflection-positive lattice correlators.

Significance. If the derivation is correct, the method offers a principled, regulator-free alternative to maximum entropy and explicit-fit approaches to analytic continuation, with polynomial cost in the number of Euclidean time-slices. The paper's strengths include the absence of free fitting parameters, the use of an external benchmark (direct diagonalization) in the numerical checks, and an unusually explicit statement of the method's limitations, particularly the reflection-positivity caveat. The main limitation for archival value is that the central technical step—the derivation of the finite-dimensional dual program (13)—is delegated to a companion paper [17], and the matrix M in (13) is left undefined, so the present manuscript cannot be verified in isolation.

major comments (3)

[§2, Eqs. (9)–(13)] The matrix M appearing in the dual objective of Eq. (13) is never defined, and the derivation of Eq. (13) is delegated to Ref. [17]. Because the central claim that the infinite-dimensional primal (8) can be solved as a finite-dimensional program rests entirely on this reduction, the manuscript should either define M explicitly in terms of the kernels K_i and the covariance matrix Σ, or clearly state that Eq. (13) is a result quoted from the companion paper and cannot be verified from the present text. As written, the reader cannot check the finiteness of the dual space or the correctness of the objective.
[§2, Eqs. (5)–(8)] The statistical ellipsoid defined by F(x)=x^T Σ^{-1} x ≤ F_max is introduced as the set of data-consistent points, and F_max is said to be chosen by resampling to give a desired confidence interval. However, the manuscript gives no validation that this procedure yields a valid confidence region when Σ is itself a bootstrap estimate. Treating a noisy estimate of Σ as fixed in the optimization (8) means the reported bounds are not certified at the claimed confidence level; this is load-bearing for the claim of 'tightest possible' bounds from the given Euclidean data.
[§4, 'Brief discussion'] The paper correctly states that if a correlator not obeying ρ(ω)≥0 is fed into the method, 'one obtains incorrectly tight bounds.' This admission makes the 'tightest possible' statement in the Introduction conditional on an assumption that is not guaranteed for generic lattice actions (e.g., Wilson or staggered fermions at finite lattice spacing). The limitation should be stated prominently in the abstract and Introduction, and the scope of the method should be restricted accordingly; as written, the claimed universality for lattice QCD is not supported.

minor comments (4)

[§3, Figure 1 and text] The text says the left panel uses β=30 and that lowering the temperature to β=30 gives the right panel, but the right panel caption reports β=100. This should be corrected to β=100.
[§2, Eq. (13)] The expression in Eq. (13) could be misread as (F_max/4) μ^{-1} ℓ^T M^{-1}ℓ; adding parentheses, e.g. (F_max/(4μ)) ℓ^T M^{-1}ℓ, would improve clarity.
[§1, Introduction] The phrase 'This space is of exponential (e.g. in volume) dimension' is vague; it would be clearer to state that the convex space of expectation values grows exponentially with system volume.
[§4, 'Brief discussion'] The sentence 'the method is worthless, and perhaps worthless, unless the correlator comes from an exactly reflection-positive lattice' contains a duplicated word or an intended emphasis that reads as a typo; please revise.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central bounds are exact convex optima under the stated positivity assumption, which the paper itself flags as a limitation rather than a hidden input.

full rationale

The paper's central claim is that the convex program (8) and its finite-dimensional Lagrange dual (13) yield the tightest possible bounds on linear functionals of the spectral density consistent with the given Euclidean data and ρ(ω) ≥ 0. This is a direct mathematical consequence of strong duality under Slater's conditions [18], not a fitted or renamed input: the bounds are extremal values over the feasible set, and the numerical checks compare against direct diagonalization as an external benchmark. The only load-bearing assumption is reflection positivity, equivalent to ρ(ω) ≥ 0, and the paper explicitly acknowledges in Section 4 that feeding in a correlator that does not obey ρ(ω) ≥ 0 produces 'incorrectly tight bounds.' That is a scope/correctness limitation, not circularity. The main self-citation is [17], to which the talk delegates the full derivation, the precise algorithm, and the figures; however, the talk itself sketches the derivation of (12)-(13) and reports an external comparison, so this delegation is not load-bearing in a circular sense. No fitted parameter is renamed as a prediction, no ansatz is smuggled in via citation, and no uniqueness theorem from the authors' prior work is invoked to force the choice. The minor self-citation burden is real but does not reduce the central claim to its own inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The method introduces no fitted parameters or new physical entities; its central claim rests on the mathematical machinery of convex duality plus the physical input of spectral positivity. The three axioms listed are the load-bearing premises: positivity, the Gaussian statistical model for the data, and strong duality, which itself depends on feasibility.

assumptions (3)

domain assumption The spectral density rho(omega) is non-negative for all frequencies, i.e., the Euclidean lattice is reflection-positive.
The convex program's feasible set and the claimed bounds rest entirely on this positivity constraint. The paper warns in Section 4 that a non-positive correlator produces incorrectly tight bounds.
domain assumption Monte Carlo estimates of the Euclidean integrals are jointly Gaussian, so the 99% confidence region is the ellipsoid x^T Sigma^{-1} x <= F_max defined in Eq. (5).
This defines which spectral densities count as consistent with the data. If the noise is non-Gaussian or Sigma is biased, the claimed tightness at 99% confidence is not exact.
standard math Slater's condition holds for the primal convex program (8), implying strong duality d*=p*.
The paper asserts that Slater's conditions are satisfied but gives no proof or explicit strictly feasible point. Strong duality is necessary for the finite-dimensional dual (13) to yield the exact primal optimum.

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Cite this review

Pith. "Pith review of Real-time dynamics from convex geometry." pith.science (2026). https://pith.science/paper/TGTXO4G7

@misc{pith2026250201880,
  author       = {Pith},
  title        = {Pith review of: Real-time dynamics from convex geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TGTXO4G7}},
  note         = {Machine review of arXiv:2502.01880}
}

read the original abstract

A quantum-mechanical system comes naturally equipped with a convex space: each (Hermitian) operator has a (real) expectation value, and the expectation value of the square any Hermitian operator must be non-negative. This space is of exponential (e.g.~in volume) dimension, but low-dimensional projections can be efficiently explored by standard algorithms. Such approaches have been used to precisely constrain critical exponents of conformal field theories ("conformal bootstrap") and, more recently, to constrain the ground state physics of various quantum-mechanical systems, including lattice field theories. In this talk we discuss related approaches to systematically constraining the real-time dynamics of quantum systems, which are otherwise obstructed from study by sign problems and the ill-posed nature of analytic continuation.

Figures

Figures reproduced from arXiv: 2502.01880 by the authors.

**Figure 1.** Extraction of the real-time correlator in the anharmonic oscillator (𝜔 = 10−4 and 𝜆 = 10−5 ) from noisy Euclidean data. At left is a reconstruction of the smeared correlator at an inverse temperature of 𝛽 = 30. The two regions show how the bounds get somewhat tighter as the number of samples is increased. The smearing is defined by 𝜎 = 1. At right is a similar calculation but at much colder temperature 𝛽 = 100, and … view at source ↗

**Figure 2.** Extraction of a real-time correlator in 2 + 1 𝜙 4 theory, on a 162 × 80 lattice. The lowest-lying mass, in lattice units, is 𝑀 ≈ 10−1 . A total of ∼ 1.9 × 105 samples were used to determine the Euclidean correlator. Figure from [17]. weakly reminiscent of a common story in the conformal bootstrap, where interesting CFTs appear to lie very close to edges and corners of permitted regions in the space of scaling dimens… view at source ↗

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Reference graph

Works this paper leans on

18 extracted references · 3 canonical work pages

[17]

Lawrence,Model-free spectral reconstruction via Lagrange duality, 2408.11766

S. Lawrence,Model-free spectral reconstruction via Lagrange duality, 2408.11766

arXiv
[1]

Osterwalder and R

K. Osterwalder and R. Schrader,Axioms for Euclidean Green’s Functions, Commun. Math. Phys. 31(1973) 83

work page 1973
[2]

F.Kos, D.Poland, D.Simmons-DuffinandA.Vichi, PrecisionIslandsintheIsingand 𝑂(𝑁) Models,JHEP 08 (2016) 036 [1603.04436]

arXiv 2016
[3]

Poland, S

D. Poland, S. Rychkov and A. Vichi,The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,Rev. Mod. Phys.91(2019) 015002 [1805.04405]

arXiv 2019
[4]

Han, S.A

X. Han, S.A. Hartnoll and J. Kruthoff,Bootstrapping Matrix Quantum Mechanics, Phys. Rev. Lett.125 (2020) 041601 [2004.10212]

arXiv 2020
[5]

Berenstein and G

D. Berenstein and G. Hulsey,Bootstrapping Simple QM Systems, 2108.08757

arXiv
[6]

Lawrence,Bootstrapping Lattice Vacua, 2111.13007

S. Lawrence,Bootstrapping Lattice Vacua, 2111.13007. 2Inpractice,themethodisoftenabletoprovethatnopositivespectraldensityfunctioncouldhaveproducedthegiven correlator. Numerically this manifests as a lower bound which exceeds the upper bound. 8 Real-time dynamics from convex geometry Scott Lawrence

arXiv
[7]

Lawrence,Semidefinite programs at finite fermion density,Phys

S. Lawrence,Semidefinite programs at finite fermion density,Phys. Rev. D107 (2023) 094511 [2211.08874]

arXiv 2023

Show all 18 references

[8]

Lawrence, B

S. Lawrence, B. McPeak and D. Neill,Bootstrapping time-evolution in quantum mechanics, 2412.08721

arXiv
[9]

Asakawa, T

M. Asakawa, T. Hatsuda and Y. Nakahara,Maximum entropy analysis of the spectral functions in lattice QCD,Prog. Part. Nucl. Phys.46 (2001) 459 [hep-lat/0011040]

2001 arXiv
[10]

Tripolt, P

R.-A. Tripolt, P. Gubler, M. Ulybyshev and L. Von Smekal,Numerical analytic continuation of Euclidean data,Comput. Phys. Commun.237 (2019) 129 [1801.10348]

2019 arXiv
[11]

Bailas, S

G. Bailas, S. Hashimoto and T. Ishikawa,Reconstruction of smeared spectral function from Euclidean correlation functions, PTEP 2020(2020) 043B07 [2001.11779]

2020 arXiv
[12]

Chen, H.T

S.Y. Chen, H.T. Ding, F.Y. Liu, G. Papp and C.B. Yang,Machine learning spectral functions in lattice QCD, 2110.13521

arXiv
[13]

L. Wang, S. Shi and K. Zhou,Reconstructing spectral functions via automatic differentiation,Phys. Rev. D106 (2022) L051502 [2111.14760]

2022 arXiv
[14]

Fournier, L

R. Fournier, L. Wang, O.V. Yazyev and Q. Wu,Artificial neural network approach to the analytic continuation problem, Phys. Rev. Lett.124 (2020) 056401

2020
[15]

Kades, J.M

L. Kades, J.M. Pawlowski, A. Rothkopf, M. Scherzer, J.M. Urban, S.J. Wetzel et al.,Spectral Reconstruction with Deep Neural Networks,Phys. Rev. D102(2020) 096001 [1905.04305]

2020 arXiv
[16]

Horak, J.M

J. Horak, J.M. Pawlowski, J. Rodríguez-Quintero, J. Turnwald, J.M. Urban, N. Wink et al., Reconstructing QCD spectral functions with Gaussian processes,Phys. Rev. D105(2022) 036014 [2107.13464]

2022 arXiv
[18]

Slater,Lagrange multipliers revisited, Cowles Foundation Discussion Papers 80, Cowles Foundation for Research in Economics, Yale University (1959)

M. Slater,Lagrange multipliers revisited, Cowles Foundation Discussion Papers 80, Cowles Foundation for Research in Economics, Yale University (1959). 9

1959

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [17]

Lawrence,Model-free spectral reconstruction via Lagrange duality, 2408.11766

S. Lawrence,Model-free spectral reconstruction via Lagrange duality, 2408.11766

arXiv

[2] [1]

Osterwalder and R

K. Osterwalder and R. Schrader,Axioms for Euclidean Green’s Functions, Commun. Math. Phys. 31(1973) 83

work page 1973

[3] [2]

F.Kos, D.Poland, D.Simmons-DuffinandA.Vichi, PrecisionIslandsintheIsingand 𝑂(𝑁) Models,JHEP 08 (2016) 036 [1603.04436]

arXiv 2016

[4] [3]

Poland, S

D. Poland, S. Rychkov and A. Vichi,The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,Rev. Mod. Phys.91(2019) 015002 [1805.04405]

arXiv 2019

[5] [4]

Han, S.A

X. Han, S.A. Hartnoll and J. Kruthoff,Bootstrapping Matrix Quantum Mechanics, Phys. Rev. Lett.125 (2020) 041601 [2004.10212]

arXiv 2020

[6] [5]

Berenstein and G

D. Berenstein and G. Hulsey,Bootstrapping Simple QM Systems, 2108.08757

arXiv

[7] [6]

Lawrence,Bootstrapping Lattice Vacua, 2111.13007

S. Lawrence,Bootstrapping Lattice Vacua, 2111.13007. 2Inpractice,themethodisoftenabletoprovethatnopositivespectraldensityfunctioncouldhaveproducedthegiven correlator. Numerically this manifests as a lower bound which exceeds the upper bound. 8 Real-time dynamics from convex geometry Scott Lawrence

arXiv

[8] [7]

Lawrence,Semidefinite programs at finite fermion density,Phys

S. Lawrence,Semidefinite programs at finite fermion density,Phys. Rev. D107 (2023) 094511 [2211.08874]

arXiv 2023