Two-Point Correlator Bootstrap

• Two-point correlator bootstrap is a framework that applies symmetry, analytic continuation, and positivity to extract operator spectra and OPE coefficients in various quantum systems.
• It uses techniques such as analytic inversion, spectral representations, and semidefinite programming to enforce convex constraints like reflection and spectral positivity.
• This approach enables precise determination of spectral gaps, matrix elements, and dynamical data across conformal, thermal, and cosmological models.

The two-point correlator bootstrap concerns the extraction, characterization, and constraint of the structure of two-point correlation functions using symmetry, analytic continuation, positivity, and global consistency (bootstrap) principles. Two-point correlators probe fundamental operator spectra, OPE coefficients, and dynamical data in conformal, finite-temperature, quantum mechanical, and cosmological systems. Bootstrap approaches encompass analytic inversion, spectral representations, resampling-based covariance estimation, and semidefinite-programming frameworks, each targeting rigorous or optimal determination of correlator structure given underlying physical and statistical axioms.

1. Two-Point Correlators: Definitions, Symmetry, and Structure

In quantum field theory and statistical mechanics, the two-point function for an operator $O$ in a given state (thermal, ground state, or vacuum) is defined as

$G(\tau, x) = \langle O(\tau, x)\,O(0,0)\rangle,$

where the expectation value is taken under the statistical or quantum state of interest. In thermal settings, the definition generalizes to

$$G_{ij}(\tau) = \frac{1}{Z(\beta)}\text{Tr}\left[e^{-\beta H}\,\bar O_i(\tau)\,O_j(0)\right],$$

with $\tau\sim\tau+\beta$ enforcing KMS periodicity. In boundary CFT and related contexts, the correlator takes the form

$$\langle\phi(x_1)\phi(x_2)\rangle = \frac{F(z)}{[4 x_{1\perp} x_{2\perp}]^{\Delta_\phi}},$$

where $z$ is the BCFT cross-ratio. Correlators are highly constrained by symmetry (conformal, translation, rotation, time-reflection), analyticity, and positivity. Their spectral representations, such as the Källén–Lehmann integral

$$G(p^2) = \int_0^\infty d\mu^2\, \frac{\rho(\mu^2)}{p^2+\mu^2-i\epsilon},$$

enforce physical requirements including unitarity ($\rho(\mu^2)\geq0$) and encode operator content by their pole and cut structure (Karateev, 2020).

2. Analytic Bootstrap and Boundary CFT Two-Point Functions

In boundary conformal field theories, the bootstrap equation for the two-point function utilizes the existence of independent expansions in the bulk operator product expansion (bulk OPE) and the boundary operator expansion (BOE): $$\sum_{\hat\Delta}\mu_{\hat\Delta}^2\,g_i(\hat\Delta, z) = (z-\frac12)^{-\Delta_\phi} \sum_{\Delta} \lambda_\Delta a_\Delta\, g_b(\Delta, z).$$ Here, $g_i$ and $g_b$ are explicit conformal blocks (hypergeometric functions), and crossing relates bulk and boundary spectra (Bissi et al., 2018).

The analytic bootstrap involves:

• Analytic continuation and identification of discontinuities ($\mathrm{Disc}\,F(z)$) in Lorentzian signature, which isolate OPE data by vanishing infinite sums for integer or even-integer-exchanged operator dimensions.
• Orthogonality of Jacobi polynomials to invert sums over block contributions, thereby extracting specific OPE or BOE data via integral representations.
• Use of image symmetry ($z\to-z$) to project out specific parity sectors in the operator expansions.
• Direct inversion formulas (e.g., contour integrals employing hypergeometric orthogonality) for the bulk or boundary towers.

Applications include the explicit determination of Wilson–Fisher BCFT two-point correlators to $\mathcal{O}(\epsilon^2)$ with closed-form expressions for anomalous dimensions and BOE coefficients, with all details realized via analytic inversion of the bootstrap equation (Bissi et al., 2018).

3. Bootstrap Constraints: Spectral Positivity, Reflection Positivity, and Semidefinite Programming

Bootstrap approaches to Euclidean two-point functions in quantum mechanical and field-theoretic systems employ convex constraints:

• Spectral positivity: The spectral density in the Källén–Lehmann representation is non-negative due to unitarity, and this underlies all rigorous spectral sum rules (Karateev, 2020).
• Reflection positivity: For any operator set $\{O_i\}$, matrices $$M_{ij}(\tau) = \langle\bar O_i(\tau/2) O_j(-\tau/2)\rangle$$ must be positive semidefinite for $\tau\geq0$ (Cho et al., 11 Nov 2025).
• KMS and ground-state relations: At finite $\beta$, KMS imposes linear periodicity between $M(\beta)$ and $M(0)$; at $\beta=\infty$, ground-state positivity requires positive-definiteness of $N_{ij}(\tau) = -\partial_\tau M_{ij}(\tau)$.
• Heisenberg equations (eom): Commutator structure and time-translation invariance enforce linear relations between matrix derivatives and commutators of the operators.

Casting the bounding of $G_{ij}(\tau)$ as a semidefinite program (SDP), all correlator constraints become convex LMIs and can be numerically optimized, supporting rigorous determination of correlator structure, spectral gaps, and matrix elements. In the dual SDP, the Heisenberg equations of motion become "inequalities of motion" on dual variables, enabling practical solution via SDPB or analogous solvers (Cho et al., 11 Nov 2025).

4. Thermal and Finite-Volume Analytic Bootstrap for Two-Point Functions

At finite temperature, two-point functions are holomorphic on the complexified time plane with periodicity $\tau\sim\tau+\beta$. Their only singularities are located at integer multiples of $\beta$, and the analytic structure is exploited via a dispersion relation: $$g(\tau) = \kappa + \sum_{m\in\mathbb Z}\int_{-i\infty}^{0} \frac{d\tau'}{2\pi i}\,\frac{\mathrm{Disc}\,g(\tau')}{\tau' + m\beta - \tau}.$$ Given the operator product expansion data for $g(\tau)$ at short distances, the discontinuity $\mathrm{Disc}$ can be computed, enabling direct construction of $g(\tau)$ everywhere on $S_\beta^1$ with explicit Hurwitz zeta function representations for each OPE channel: $$g(\tau) = \kappa + \sum_\Delta a_\Delta \big[ \zeta_H(2\Delta_\phi-\Delta, \tau/\beta) + \zeta_H(2\Delta_\phi-\Delta, 1 - \tau/\beta) \big].$$ For nonzero spatial separation, the "method of images" ensures full KMS invariance and OPE consistency, with the cluster property at large separation providing a nontrivial bootstrap constraint. This formalism has been rigorously validated via agreement with high-precision Monte Carlo data for the $3d$ Ising model energy two-point function (Barrat et al., 6 Jun 2025).

5. Internal Covariance Estimation of Two-Point Correlators via Jackknife and Bootstrap

Estimating the covariance of two-point correlation functions $\xi(r)$ in cosmological and statistical ensembles using internal resampling requires careful treatment of statistical biases:

• The measured two-point function is based on Landy–Szalay pair counts, with $n_b$ bins in separation.
• Resampling is performed by dividing the simulation (or data sample) into $n_{\rm sv}$ cubic subvolumes. Delete-one Jackknife and Bootstrap resamples are constructed by removing or resampling subvolumes and remeasuring $\xi(r)$.
• Key Bias: The naive resampling covariance estimate systematically mis-weights contributions from auto-pairs and cross-pairs—auto-pairs are scaled correctly, but cross-pairs are mis-scaled by a factor of 2.

The bias is corrected via a bin-wise rescaling: $$C_{ij}^{JK,\,corrected} = \gamma(r_i)\, C_{ij}^{JK,\,naive}, \quad \gamma(r) = f(r) + 2[1 - f(r)],$$ where $f(r)$ is the fraction of auto-pairs in bin $r$. With geometric weighting and this correction, the principal components of the corrected covariance reproduce the true covariance matrix to within $\sim 10\%$ precision, as validated on $1000$ QUIJOTE simulations (Mohammad et al., 2021). Bootstrap and Jackknife give identical results under mean weighting and rescaling, whereas mult-weighted Bootstrap significantly overestimates variances.

6. Concrete Bootstrap Problems and Numerical Implementation

Two-point correlator bootstraps can be formulated as optimization problems with explicit variables, constraints, and objectives:

• In massive QFT, the problem includes S-matrix partial waves $\{S_j(s)\}$, form factors $\{\mathcal F_{\Theta}(s)\}$, and spectral densities, subject to LMIs ensuring unitarity, reflection positivity, KMS (or ground-state) conditions, and analyticity.
• Objectives include minimizing CFT central charges, bounding spectral gaps, or extracting ratios such as central-charge integrals.
• Variables can be expanded in positive function bases (polynomials, B-splines) mapped onto discretized time or spectral intervals. The resulting SDPs are solved numerically to provide rigorous bounds.

These numerical bootstraps enable extraction of spectral gaps and matrix elements, e.g., for adjoint states in single-matrix quantum mechanics, with high-precision comparison to integral-equation solutions (e.g., the Marchesini–Onofri operator) (Cho et al., 11 Nov 2025). Finite-volume and temperature extensions are achieved by enforcing appropriate periodicity, reflection positivity, and clustering axioms (Barrat et al., 6 Jun 2025).

7. Applications, Accuracy, and Prospects

The two-point correlator bootstrap enables:

• Extraction of precise (often optimal) spectral, OPE, and BOE data in CFT, BCFT, and massive QFT, including finite-temperature regimes and models with boundaries.
• Internal estimation and validation of covariance matrices in cosmological and statistical applications, reducing reliance on massive numerical ensembles when corrected resampling is employed.
• Formulation of rigorous bounds and duality-based tests for integrable and non-integrable quantum systems, with convergence and precision limited only by operator basis truncation or finite sample size.

Current developments include extension to higher-point correlator bootstraps, sign-problem-free constraints in gauge theory, real-time (Lorentzian) dynamics, and nontrivial topological or defect sectors. The field continues to leverage analytic, computational, and convex-optimization techniques to exhaustively constrain and compute two-point structure from first principles (Bissi et al., 2018, Karateev, 2020, Mohammad et al., 2021, Barrat et al., 6 Jun 2025, Cho et al., 11 Nov 2025).