Monte Carlo simulations on a space‑time lattice give accurate estimates of Euclidean correlation functions, but turning those numbers into real‑time physics is an ill‑posed inverse problem. This paper proposes a different goal: instead of trying to reconstruct the full spectral density ρ(ω) (a function that encodes particle masses and other dynamical data), the authors ask how large or small any linear average of ρ can be. Concretely, they bound quantities of the form ∫ G(ω)ρ(ω)dω, where G is a chosen test function such as a window or a Gaussian.

The key input from the lattice is reflection positivity, a mathematical property of many lattice actions that implies the spectral density is non‑negative. The Euclidean two‑point function is related to ρ by a known kernel K(ω,t)=e^{-tω}+e^{-(T−t)ω}, so any candidate ρ must reproduce the measured correlator within its statistical errors. The authors encode this requirement using a Mahalanobis distance built from the Monte Carlo covariance matrix, and choose a threshold Fmax from bootstrap resamples of the data to set a confidence level.

Formulated this way the problem is a convex but infinite‑dimensional optimization: minimize or maximize the linear functional over all non‑negative spectral densities consistent with the data. The technical advance of the paper is to take the dual of this problem and relax it into a hierarchy of finite semidefinite programs (SDP). SDPs are standard convex programs that can be solved by off‑the‑shelf solvers. The hierarchy gives increasingly sharp bounds and comes with rigorous convergence guarantees: the computed upper and lower bounds are mathematically valid even when the relaxation is not exact.

Two practical benefits follow. First, the bounds converge quickly in examples to the point where the remaining uncertainty is dominated by the original Monte Carlo statistical errors rather than by the inversion procedure. Second, the SDP framework provides a consistency test for the data: if no non‑negative ρ can fit the correlator within the stated errors, the program becomes infeasible and returns a certificate. That certificate can reveal underestimated error bars or other problems in the statistical analysis.