Bootstrapping Euclidean Two-point Correlators

This paper introduces a rigorous semidefinite programming framework to bound Euclidean two-point correlators in quantum mechanical systems by enforcing reflection positivity, equations of motion, and thermal or ground-state constraints, successfully applying the method to extract spectral data from one-matrix quantum mechanics while linking high-temperature correlator bounds to matrix integral bootstraps.

The Gist

Imagine you are trying to guess the shape of a mysterious, invisible object hidden inside a dark room. You can't see it, and you can't touch it directly. However, you know a few strict rules about how the object behaves:

1. It must be solid (it can't be made of "negative" matter).
2. It follows the laws of physics (like a ball rolling down a hill).
3. If you wait long enough, it settles into a predictable pattern.

This paper is about a new, super-smart way to guess the shape of that object using only those rules. The "object" is a quantum system (like a tiny particle or a collection of them), and the "shape" we are trying to guess is how two parts of the system talk to each other over time. This "talking" is called a two-point correlator.

Here is the breakdown of their method, using everyday analogies:

1. The Problem: The Infinite Maze

Usually, to understand these quantum systems, scientists try to solve complex equations. But for very large or strongly interacting systems, the equations are like a maze with infinite paths. You can't solve them all at once.

The authors realized that instead of trying to solve the maze, they could build a fence around the possible answers. They wanted to find the tightest possible fence (bounds) that the answer must stay inside, without ever needing to know the exact answer.

2. The Tool: The "Mathematical Squeeze" (Semidefinite Programming)

The authors use a mathematical technique called Semidefinite Programming (SDP). Think of this as a giant, high-tech vice grip.

• You feed the vice the rules of the universe (Reflection Positivity, Heisenberg's equations, etc.).
• You ask the vice: "What is the absolute lowest and highest value this quantum conversation could possibly have?"
• The vice tightens until it finds the absolute limits.

3. The Magic Trick: Turning Rules into "Inequalities of Motion"

Here is the clever part. In physics, things usually follow strict equations (like $F=ma$). But in this "dual" version of the problem, the authors turn those strict equations into inequalities.

Imagine you are trying to guess the speed of a car.

• Old way: You need to know the exact engine force and friction to calculate the speed.
• This paper's way: You don't need the engine details. You just need to know that the car cannot go faster than light and cannot go slower than zero. By testing thousands of "what-if" scenarios (using Lagrange multipliers, which are like invisible weights), they prove that the car's speed must be between 50 and 60 mph, even if they don't know the exact speed.

They call these "inequalities of motion." It's like saying, "No matter how you drive, you can't break these laws of physics."

4. The Test Drive: The One-Matrix Machine

To prove their method works, they tested it on a specific system called Matrix Quantum Mechanics.

• The Analogy: Imagine a giant drumhead made of a grid of springs. Each spring can vibrate. In this system, the "springs" are actually giant matrices (grids of numbers).
• The Challenge: This system is so complex that even supercomputers struggle to simulate it perfectly, especially when it gets hot (thermal state) or when the springs are very stiff (strong coupling).
• The Result: The authors used their "mathematical vice" to predict how these matrices vibrate. They found the "gap" (the energy needed to make the system jump to a new state) and the "matrix elements" (how strongly the parts talk to each other).

Their results were incredibly precise. In fact, for some temperatures, their math-based bounds were more accurate than the results from massive computer simulations (Monte Carlo), which often get messy due to "noise" and approximation errors.

5. Why This Matters

• No "Sign Problem": In quantum physics, some systems are impossible to simulate on computers because the math involves "negative probabilities" that cancel everything out (the sign problem). This new method doesn't care about that. It just looks at the rules.
• Universal Bounds: They proved that even without knowing the specific details of a system, you can derive universal limits on how it behaves. For example, they derived a new version of the "Energy-Entropy Balance," which is like a rule saying, "You can't have too much energy without paying a price in disorder."
• Future Applications: This method could help us understand black holes, the early universe, or materials that conduct electricity without resistance, where traditional math fails.

Summary

Think of this paper as inventing a new type of X-ray vision for math. Instead of trying to see the quantum object directly (which is impossible), they use the laws of physics to cast a shadow. By analyzing the shape of that shadow, they can tell you exactly how big and heavy the object is, even if they've never seen it. They turned the infinite complexity of quantum mechanics into a manageable puzzle of "what is possible" and "what is impossible."

Technical Summary

1. Problem Statement

The paper addresses the challenge of constraining Euclidean two-point correlators in quantum mechanical systems, specifically in thermal and ground states. While the "quantum bootstrap" has been successfully applied to time-independent quantities (like one-point functions or energy eigenvalues) and Lorentzian time evolution, extending it to continuous-time Euclidean correlators $\langle O_1(\tau) O_2(0) \rangle_\beta$ presents significant difficulties:

• Infinite Dimensionality: The correlator is a function of continuous Euclidean time $\tau$, making the space of bootstrap variables infinite-dimensional.
• Dynamical Constraints: The problem requires enforcing the Heisenberg equations of motion, reflection positivity, and thermal boundary conditions (KMS condition) simultaneously.
• Computational Complexity: Traditional methods like Monte Carlo simulations struggle with strong coupling and large $N$ limits, particularly in systems with sign problems or where analytic solutions are unavailable.

The authors aim to develop a rigorous, non-perturbative method to bound these correlators and extract physical properties (spectra, matrix elements) without relying on perturbation theory or lattice discretization.

2. Methodology

The authors formulate the problem as a Semidefinite Programming (SDP) problem using a Primal-Dual approach.

A. Primal Problem

The primal variables are the matrix of two-point correlators $M_{ij}(\tau) = \langle \bar{O}_i(\tau) O_j(0) \rangle$. The constraints are:

1. Reflection Positivity: $M(\tau) \succeq 0$ for all $\tau \geq 0$.
2. Heisenberg Equations of Motion: $\partial_\tau M(\tau) = -M(\tau)D$, where $D$ encodes the commutators $[H, O_i]$.
3. State-Specific Constraints:
   • Ground State: Imposes "Ground-state positivity" ($N(\tau) \succeq 0$) and time-reversal symmetry.
   • Thermal State: Imposes the Kubo-Martin-Schwinger (KMS) condition: $M(\beta) = T M(0) T^\dagger$, enforcing periodicity.

B. Dual Formulation

To handle the infinite-dimensional nature of the primal problem, the authors derive the dual problem.

• Key Insight: The Heisenberg equations of motion (equalities) in the primal problem become "inequalities of motion" in the dual problem involving Lagrange multipliers $\lambda(\tau)$.
• Finite-Dimensional Truncation: Instead of solving for functions $\lambda(\tau)$ directly, the authors expand them in a finite-dimensional basis of positive functions (e.g., Clamped B-splines or Polynomials).
• Polynomial Matrix Program (PMP): By mapping the time domain $\tau \in [0, T]$ to a semi-infinite domain and using polynomial ansatzes for the Lagrange multipliers, the problem is converted into a PMP. This allows the use of efficient SDP solvers like SDPB or MOSEK.
• Rigorous Bounds: By the weak duality theorem, any feasible solution to the dual problem provides a rigorous lower (or upper) bound on the primal objective (the correlator at time $T$).

C. Handling Zero-Time Inputs

The method accommodates scenarios where zero-time correlators $M(0)$ are:

• Fully Known: (e.g., from large $N$ analytic solutions).
• Partially Known: Using bounds derived from one-point function bootstraps (e.g., Energy-Entropy Balance inequalities).
• Unknown: Treating $M(0)$ as a variable subject to linear constraints derived from operator algebra and positivity.

3. Key Contributions

1. Formalism for Time-Dependent Bootstraps: The paper establishes a rigorous framework for bootstrapping continuous-time Euclidean correlators, extending previous work on time-independent observables and Lorentzian evolution (LMN bootstrap).
2. Derivation of Analytic Bounds:
   • Exponential Bound: Proves that connected correlators are log-convex, leading to a universal lower bound $G_c(\tau) \geq G_c(0)e^{-\mu \tau}$.
   • Energy-Entropy Balance (EEB): Provides a new derivation of the EEB inequality directly from the two-point bootstrap, linking thermal correlators to thermodynamic quantities.
   • High-Temperature Limit: Demonstrates that the thermal two-point bootstrap reduces to the matrix integral bootstrap (classical statistical mechanics) in the $\beta \to 0$ limit, recovering Schwinger-Dyson equations.
3. Extremal Functional Method: Introduces a technique to extract the spectrum and matrix elements of excited states by analyzing the "null eigenvectors" of the constraint matrices at the optimal bound. This allows the extraction of energy gaps and transition amplitudes without diagonalizing the Hamiltonian.
4. Application to Matrix Quantum Mechanics (MQM): Successfully applies the method to the ungauged one-matrix quantum mechanics (1-MQM), a system that is analytically solvable only at large $N$ via fermionization but difficult for standard Monte Carlo due to sign issues in certain regimes.

4. Key Results

• Anharmonic Oscillator: The method reproduces the exact ground-state correlator of the anharmonic oscillator with high precision. The bounds converge rapidly as the truncation level (operator weight) increases.
• Adjoint Gap in 1-MQM:
  • The authors compute rigorous upper bounds on the adjoint gap ($\Delta_{adj}$) for the 1-MQM.
  • They observe that the gap remains finite as the coupling approaches the critical point ($g \to g_c$), contradicting naive expectations of divergence.
  • The bootstrap results show excellent agreement with the Marchesini-Onofri equation (the exact large $N$ solution for the adjoint sector).
  • They extract the first few excited state energies ($\Delta_1, \Delta_2, \Delta_3$) and matrix elements, matching the exact solutions to high precision.
• Thermal Correlators:
  • For the thermal state, the bootstrap bounds are more precise than Monte Carlo (MC) results in certain regimes, particularly where MC suffers from systematic errors due to finite lattice size ($L$) and finite $N$ extrapolation.
  • The method successfully handles the ungauged model at negative coupling ($g < 0$), where the potential is unbounded and MC simulations fail due to instability, demonstrating the bootstrap's ability to probe intrinsically large-$N$ physics.
• Convergence: The numerical results show rapid convergence with increasing operator level ($L$) and polynomial degree ($d$), validating the efficiency of the PMP formulation.

5. Significance and Future Directions

• Non-Perturbative Tool: This work provides a powerful, non-perturbative tool for studying strongly coupled quantum systems and large $N$ limits where traditional lattice methods fail (e.g., sign problems).
• Spectral Extraction: It offers a novel pathway to extract spectral data (gaps, matrix elements) directly from correlator bounds, bypassing the need for explicit diagonalization.
• Connection to Holography: The ability to study thermal correlators and extract quasi-normal modes (QNMs) suggests potential applications in AdS/CFT, particularly for understanding black hole physics and thermalization in the planar limit.
• Generalizability: The framework is designed to be extended to multi-point correlators, out-of-time-ordered correlators (OTOCs), and lattice gauge theories.

In summary, this paper successfully bridges the gap between the mathematical rigor of semidefinite programming and the physical requirements of quantum dynamics, establishing a new standard for bounding and extracting physical observables in Euclidean quantum field theories and quantum mechanics.