Linear Response Theory: A Modern Analytic-Algebraic Approach (1612.01710v1)

Abstract: Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. This monograph presents a thoroughly modern framework to make linear response theory rigorous for a wide array of systems, that is suitable for novel applications such as periodic and random light conductors not yet covered in the literature. Our analytic-algebraic approach, based on von Neumann algebras and associated non-commutative $L^p$-spaces, can deal with discrete and continuous models alike, and include effects of disorder. First, we explain the mathematical setting, give a complete list of our hypotheses and state the main results, which include Kubo and Kubo-Streda formulas. To make our book accessible to a wide audience, we spend Chapters 3 and 4 explaining the mathematical underpinnings such as non-commutative $L^p$- and Sobolev spaces, and generalized commutators. Furthermore, we show how to construct a von Neumann algebras from a topological dynamical system and a 2-cocycle, a procedure which applies to discrete and continuous quantum systems. We dedicate Chapters 5 and 6 to the proofs of our main results. We close the book by sketching a novel application, linear response theory for periodic and random light conductors. This monograph is aimed at advanced students in mathematical physics and researchers.