In-Depth Study of Lieb-Robinson Bound Proof

Abstract

The goal of this paper is to understand the statement and proof of the locality estimate given in a Lieb-Robinson bound. In order to do so, we must first take a careful look at the necessary background information. The first section defines the notions of continuity, differentiation, and integration of Banach space valued functions and proves a few basics theorems involving them, such as the fundamental theorem of calculus. The tools listed above allow us to solve differential equations in Banach spaces; we look at an example of this in section two. The third section is a review of tensor products of Hilbert spaces. Lastly, we make the necessary definitions to mathematically describe a quantum spin system, state and prove a Lieb-Robinson bound, and explore an application of the theorem.