A Positivity Bootstrap for Quantum Mechanics and Fermionic Lattice Field Theories

Abstract

The quantum bootstrap aims to extract spectral data from quantum systems without ever constructing wavefunctions. By imposing consistency conditions on expectation values, including Hilbert-space positivity, the eigenstate equation, and symmetry constraints, the bootstrap asks what spectra and correlators could be consistent with the general rules of quantum mechanics that any physical state must obey. In this thesis, we examine how far this framework can be extended - from simple quantum mechanical systems to fermionic lattice field theories. We first reproduce and extend the anharmonic-oscillator bootstrap of Han, Hartnoll, and Kruthoff, demonstrating that mixed-correlator constraints involving both position and momentum tighten the feasible regions around the ground and first excited states. We then construct a Level 2 bootstrap for the staggered-fermion Thirring model, exploiting charge and momentum symmetries to decompose the moment matrix into ten symmetry blocks. At L = 4, m = g = a = 1, the bootstrap recovers the exact-diagonalization ground state energy to ten decimal places, the sharpest numerical benchmark in the thesis. We then extend the framework to L = 6, 8, tracking exact diagonalization across the gapless regime at m = 0, and compare the finite-size behaviour with expectations of conformal field theory (CFT). We observe that the free theory matches the expected Casimir-plus-winding scaling, while the interacting g = 1 bootstrap data agree closely with exact diagonalization but are not yet at large enough system size for a clean extraction of asymptotic CFT data.