The Mathematics of Quantum Propagation

Introduction

Strongly interacting and strongly correlated quantum many-body systems are at the forefront of modern quantum physics. Experimentalists have obtained unprecedented control on the interaction parameters and are able to reliably produce striking fundamental phenomena. These problems demand a rigorous mathematical treatment, but analytical methods are extremely scarce.

Lieb-Robinson Bounds

Outside of special scaling limits, the gold standard are Lieb-Robinson bounds (LRBs) which provide an a priori bound on the speed of information propagation with broad physical implications. However, for the important classes of:

1. Lattice bosons
2. Continuum fermions and continuum bosons

the standard derivations of Lieb-Robinson bounds break down because these systems have unbounded interactions.

Project Goals

The first goal of this project is to establish propagation bounds, including LRBs, for lattice bosons and to identify the true behavior of information propagation for these systems. This is the missing puzzle piece to develop a quantum information theory of lattice bosons that is on par with the revolutionary findings for quantum spin systems.

The second goal is to develop propagation bounds, including LRBs, for continuum fermions and bosons. These systems present even more fundamental challenges due to ultraviolet divergences.

Application

As an application, I aim to close a glaring gap in our understanding of continuum quantum many-body systems: the existence of the thermodynamic limit of the dynamics.

Methodology

I recently developed the ASTLO method which uses bootstrapped differential inequalities, microlocal-inspired resolvent expansions, and multiscale iteration to pioneer particle propagation bounds for the paradigmatic Bose-Hubbard Hamiltonian. This resolved longstanding problems in mathematical physics.

My new ASTLO method is a robust proof template. In combination with the technique of truncated dynamics, it enables me to now tackle even more challenging open problems about information propagation.