Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory

Lieb-Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of infinite volume dynamics of non-relativistic quantum particles with short-range, possibly time-dependent interactions.In particular, the existence of fundamental solutions is shown for those (non-autonomous) C-dynamical systems for which the usual conditions found in standard theories of (parabolic or hyperbolic) non-autonomous evolution equations are not given. In mathematical physics, bounds on multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting particles to external perturbations. These bounds are derived for lattice fermions, in view of applications to microscopic quantum theory of electrical conduction discussed in this book. All results also apply to quantum spin systems, with obvious modifications. In order to make the results accessible to a wide audience, in particular to students in mathematicswith little Physics background, basics of Quantum Mechanics are presented, keeping in mind its algebraic formulation. The C-algebraic setting for lattice fermions, as well as the celebrated Lieb-Robinson bounds for commutators, are explained in detail, for completeness.

This volume deals with Lieb-Robinson bounds for multi-commutators of fermionic systems. This book is reasonably self-contained and has complete proofs in the parts describing more recent developments of the theory (Chapters 4 6). It is recommended for researchers wanting to learn about generalization of Lieb-Robinson bounds and some of its important applications in the description of quantum matter. (Fernando Lledó, zbMATH 1379.81047, 2018)

Inhalt
Introduction.- Algebraic Quantum Mechanics.- Algebraic Setting for Interacting Fermions on the Lattice.- LiebRobinson Bounds for MultiCommutators.- LiebRobinson Bounds for NonAutonomous Dynamics.- Applications to Conductivity Measures.