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arxiv: 2511.08560 · v2 · submitted 2025-11-11 · ✦ hep-th · cond-mat.str-el· math.OC· quant-ph

Bootstrapping Euclidean Two-point Correlators

Minjae Cho , Barak Gabai , Henry W. Lin , Jessica Yeh , Zechuan Zheng This is my paper

Pith reviewed 2026-05-17 23:32 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elmath.OCquant-ph
keywords bootstrapsemidefinite programmingtwo-point correlatorsreflection positivityHeisenberg equationsKMS conditionmatrix quantum mechanics
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0 comments X

The pith

A semidefinite program bounds Euclidean two-point correlators using reflection positivity, Heisenberg equations of motion, and KMS or ground-state positivity.

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

The paper casts the task of bounding Euclidean two-point correlators in the thermal or ground state of quantum mechanical systems as a semidefinite programming problem. The constraints imposed are reflection positivity, the Heisenberg equations of motion, and either the Kubo-Martin-Schwinger condition or ground-state positivity. In the dual formulation these equations of motion become inequalities of motion on the Lagrange multipliers. The resulting finite-dimensional program produces rigorous bounds on continuous-time correlators, which are demonstrated by extracting the low-lying spectrum and matrix elements in ungauged one-matrix quantum mechanics. A sympathetic reader would care because the method supplies a systematic, non-perturbative way to constrain dynamical correlators without first solving the full theory.

Core claim

We develop a bootstrap approach to Euclidean two-point correlators by formulating the bounding problem as a semidefinite program subject to reflection positivity, the Heisenberg equations of motion, and the Kubo-Martin-Schwinger condition or ground-state positivity. In the dual formulation the equations of motion appear as inequalities of motion on the Lagrange multipliers. This construction yields rigorous bounds on continuous-time two-point correlators from a finite-dimensional semidefinite or polynomial matrix program. The method is illustrated on the ungauged one-matrix quantum mechanics, from which the spectrum and matrix elements of low-lying adjoint states are extracted. Along the way

What carries the argument

Semidefinite programming formulation of correlator bounds in which the Heisenberg equations of motion are converted into inequalities of motion on the Lagrange multipliers that enforce positivity and thermal conditions.

If this is right

  • Rigorous upper and lower bounds on continuous-time two-point correlators follow directly from the finite-dimensional semidefinite program.
  • The spectrum and matrix elements of low-lying adjoint states can be read off from the optimizing correlator in the one-matrix quantum mechanics example.
  • A new derivation of the energy-entropy balance inequality is obtained as a byproduct of the positivity constraints.
  • A direct connection appears between the high-temperature limit of the two-point correlator bootstrap and the matrix integral bootstrap.

Where Pith is reading between the lines

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

  • The same dual inequalities of motion could be adapted to bound higher-point functions or correlators in higher-dimensional quantum field theories.
  • Truncation errors in the SDP could be controlled by adding more derivatives of the equations of motion or by incorporating additional positivity conditions.
  • The method may supply an independent check on results obtained from Monte Carlo simulations of matrix models at finite temperature.

Load-bearing premise

That the listed constraints of reflection positivity, Heisenberg equations of motion, and KMS or ground-state positivity remain sufficient to produce usefully tight bounds once the problem is truncated to a finite-dimensional semidefinite or polynomial matrix program.

What would settle it

Compute the exact two-point correlator in the ungauged one-matrix quantum mechanics at a fixed temperature or ground state, then check whether the SDP upper and lower bounds approach that exact function as the truncation level is increased.

Figures

Figures reproduced from arXiv: 2511.08560 by Barak Gabai, Henry W. Lin, Jessica Yeh, Minjae Cho, Zechuan Zheng.

Figure 1
Figure 1. Figure 1: Comparison of the allowed region (in pink) for the level 8 two-point correlator bootstrap (28) using the KMS condition with the level 7 one-point function bootstrap using the EEB inequality [3, 4] for the anharmonic oscillator V (x) = 1 2 x 2 + 1 4 x 4 . We also show the “exact” value from Hamiltonian truncation (in black). 8 If Gc is log-convex, then G = Gc + constant is log-convex for positive constant. … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the integral bootstrap (in dashed blue) with the two-point correlator bootstrap [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The ground-state Euclidean two-point correlator of the anharmonic oscillator with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The lower and upper bounds converge as the parameters of either the polynomial or spline [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bootstrap bounds on the Euclidean two-point correlator in the thermal state of anharmonic [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Euclidean two-point correlator ⟨tr X(τ )X(0)⟩ in the ground state of the 1-MQM with V = 1 2 Tr X2 + g Tr X4 at g = 1 (left) and g = −0.075026 ≈ gc (right). The shaded region indicates the allowed bootstrap region at level-{4, 6, 8}. The green dashed line indicates the universal lower bound in (44). Bounds are obtained from the PMP formulation using SDPB with d = 16 (Appendix C). 4.4 Bounds on the adjoint g… view at source ↗
Figure 7
Figure 7. Figure 7: Euclidean two-point correlator [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Adjoint gap near criticality. On the right, we see clear evidence of the non-analytic behavior [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Z2-even adjoint gap near criticality. As with the Z2-odd adjoint gap, we see clear evidence of the non-analytic behavior as µ → 0. We observe that the behavior of the level 6 and 8 bounds on the Z2-even gap are qualitatively similar to the level 4 and 6 bounds on the Z2-odd gap. lvl 16 bd lvl 6 bd gap 0.000 0.002 0.004 0.006 0.008 0.010 2.0 2.5 3.0 3.5 lvl 16 bd lvl 8 bd gap 0.000 0.002 0.004 0.006 0.008 0… view at source ↗
Figure 10
Figure 10. Figure 10: Derivative of the adjoint gap near criticality. Here we show the derivative of the level 16 bound [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Convergence of the estimates for the gap [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Euclidean two-point correlator in the ground state of the 1-MQM at [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Bootstrap bounds on the thermal two-point correlator [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: LMN bootstrap for the anharmonic oscillator with the rigorous spline approach [PITH_FULL_IMAGE:figures/full_fig_p044_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Bootstrap results for the ground state of the 1-MQM with [PITH_FULL_IMAGE:figures/full_fig_p048_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Monte Carlo for various values of N and L results with 500k burn in + 100k measurements. For each value of N, we show the continuum extrapolation to L → ∞. 0.0 0.2 0.4 0.6 0.8 1.0 0.96 0.97 0.98 0.99 1.00 1.01 G( )/ G b o o t s tra p( ) Bootstrap allowed region N=6 N=8 N=10 N=12 N extrapolation 0.000 0.005 0.010 0.015 0.020 0.025 0.030 1/N 2 0.24 0.26 0.28 0.30 0.32 0.34 G( ) =0.00 =0.25 =0.50 [PITH_FULL… view at source ↗
Figure 17
Figure 17. Figure 17: Extrapolation to infinite N. Here we take the large L extrapolation of the finite N data points and extrapolate to N → ∞ by fitting G(τ ) = G∞(τ ) + 1 N2 g(τ ). 50 [PITH_FULL_IMAGE:figures/full_fig_p050_17.png] view at source ↗
read the original abstract

We develop a bootstrap approach to Euclidean two-point correlators, in the thermal or ground state of quantum mechanical systems. We formulate the problem of bounding the two-point correlator as a semidefinite programming problem, subject to the constraints of reflection positivity, the Heisenberg equations of motion, and the Kubo-Martin-Schwinger condition or ground-state positivity. In the dual formulation, the Heisenberg equations of motion become "inequalities of motion" on the Lagrange multipliers that enforce the constraints. This enables us to derive rigorous bounds on continuous-time two-point correlators using a finite-dimensional semidefinite or polynomial matrix program. We illustrate this method by bootstrapping the two-point correlators of the ungauged one-matrix quantum mechanics, from which we extract the spectrum and matrix elements of the low-lying adjoint states. Along the way, we provide a new derivation of the energy-entropy balance inequality and establish a connection between the high-temperature two-point correlator bootstrap and the matrix integral bootstrap.

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 develops a bootstrap method for Euclidean two-point correlators in quantum mechanical systems. It formulates the bounding of these correlators as a semidefinite programming problem incorporating constraints from reflection positivity, the Heisenberg equations of motion, and the Kubo-Martin-Schwinger condition or ground-state positivity. In the dual formulation, the equations of motion are recast as inequalities on the Lagrange multipliers. The approach is illustrated by applying it to the ungauged one-matrix quantum mechanics, from which the spectrum and matrix elements of low-lying adjoint states are extracted. Additionally, a new derivation of the energy-entropy balance inequality is provided, and a connection is established between the high-temperature two-point correlator bootstrap and the matrix integral bootstrap.

Significance. If the central claims hold, this work introduces a promising new tool for obtaining rigorous bounds on continuous-time correlators in quantum systems using finite-dimensional programs. The ability to extract physical data such as energies and matrix elements from the bootstrap is a significant strength. The new derivation of the energy-entropy inequality and the link to matrix integral bootstrap further enhance the paper's contribution. These elements position the method as potentially useful for non-perturbative studies in quantum mechanics and related field theories.

major comments (2)
  1. [§4] §4 (one-matrix QM application): the extracted adjoint-state energies and matrix elements are presented as physical results, but no data or plots demonstrate monotonic tightening or stabilization of these quantities as the SDP dimension or polynomial degree is increased. This is load-bearing for the claim that finite truncation yields accurate low-lying spectrum rather than artifacts, directly addressing the concern that missing higher-order constraints could leave gaps in the bounds.
  2. [§3] §3 (SDP formulation and dual): the central assumption that reflection positivity, Heisenberg EOM (via dual inequalities of motion), and KMS/ground-state positivity suffice for usefully tight continuous-time bounds in finite truncation is stated but not tested via explicit convergence checks in the example; without this, the practical extraction of spectrum remains unverified.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'along the way' for the energy-entropy derivation and matrix-integral connection is understated; a brief parenthetical on their novelty would better highlight these contributions.
  2. [Throughout] Notation: ensure consistent use of 'EOM' and 'KMS' with definitions on first appearance in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address the major points below and will revise the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§4] §4 (one-matrix QM application): the extracted adjoint-state energies and matrix elements are presented as physical results, but no data or plots demonstrate monotonic tightening or stabilization of these quantities as the SDP dimension or polynomial degree is increased. This is load-bearing for the claim that finite truncation yields accurate low-lying spectrum rather than artifacts, directly addressing the concern that missing higher-order constraints could leave gaps in the bounds.

    Authors: We agree that explicit demonstration of convergence with increasing truncation is important to support the reliability of the extracted spectrum and matrix elements. In the revised manuscript we will add figures and tables displaying the dependence of the low-lying adjoint energies and matrix elements on the SDP dimension and polynomial degree, showing their stabilization. revision: yes

  2. Referee: [§3] §3 (SDP formulation and dual): the central assumption that reflection positivity, Heisenberg EOM (via dual inequalities of motion), and KMS/ground-state positivity suffice for usefully tight continuous-time bounds in finite truncation is stated but not tested via explicit convergence checks in the example; without this, the practical extraction of spectrum remains unverified.

    Authors: We acknowledge the value of explicit convergence checks to verify the practical performance of the finite truncation. While the manuscript already compares extracted quantities to known results in the one-matrix model, we will include additional convergence plots and analysis in the revision to directly demonstrate the tightening of bounds and stabilization of the spectrum as the truncation parameters increase. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent physical constraints

full rationale

The paper's central formulation casts bounds on Euclidean two-point correlators as an SDP whose inputs are the standard axioms of reflection positivity, Heisenberg equations of motion, and KMS/ground-state positivity. These are external physical requirements, not quantities defined in terms of the correlator bounds or extracted spectrum. The dual formulation simply rewrites the EOM as linear inequalities on the Lagrange multipliers; this is a mechanical change of variables, not a self-definition. The energy-entropy inequality is derived anew from the same constraints rather than assumed, and the connection to matrix-integral bootstrap is presented as an observation rather than a load-bearing premise. The one-matrix QM illustration extracts low-lying data from the resulting program, but the program itself is not fitted to those data. No self-citation chain, ansatz smuggling, or renaming of known results is required for the method. Finite truncation is an explicit approximation whose practical utility is demonstrated rather than asserted by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard domain assumptions of quantum mechanics and statistical mechanics rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Reflection positivity holds for the Euclidean two-point functions under consideration.
    Invoked as a core constraint in the SDP formulation; standard for ensuring unitarity in Euclidean QFT.
  • domain assumption The system obeys the Heisenberg equations of motion and the Kubo-Martin-Schwinger condition (or ground-state positivity).
    Used to generate the inequalities of motion in the dual problem; standard in thermal quantum mechanics.

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Forward citations

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  3. Bootstrapping Tensor Integrals

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  4. Finite-$N$ Bootstrap Constraints in Matrix and Tensor Models

    hep-th 2026-03 unverdicted novelty 7.0

    Finite-N bootstrap yields N-independent bounds for matrix models but N-dependent novel bounds on the two-point function versus quartic coupling for tensor models.

  5. Regge trajectories from the adjoint sector of Matrix Quantum Mechanics

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