Bootstrapping Euclidean Two-point Correlators

TL;DR

This paper introduces a bootstrap method for bounding Euclidean two-point correlators in quantum systems using semidefinite programming, applicable to thermal and ground states, and demonstrates its effectiveness on matrix quantum mechanics.

Contribution

It formulates the correlator bounding problem as a semidefinite program incorporating physical constraints, providing a new approach to analyze quantum correlators.

Findings

Derived rigorous bounds on two-point correlators.

Extracted spectrum and matrix elements of low-lying states.

Connected high-temperature bootstrap to matrix integral bootstrap.

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.