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

  • J.-B. Bru
  • W. de Siqueira Pedra

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 13)

Table of contents

  1. Front Matter
    Pages i-vii
  2. J.-B. Bru, W. de Siqueira Pedra
    Pages 1-4
  3. J.-B. Bru, W. de Siqueira Pedra
    Pages 5-15
  4. J.-B. Bru, W. de Siqueira Pedra
    Pages 17-30
  5. J.-B. Bru, W. de Siqueira Pedra
    Pages 31-61
  6. J.-B. Bru, W. de Siqueira Pedra
    Pages 63-87
  7. J.-B. Bru, W. de Siqueira Pedra
    Pages 89-101
  8. Back Matter
    Pages 103-109

About this book


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 mathematics with 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.


Constructive Quantum Field Theory Constructive QFT Minimally connected graph Interacting fermions Quantum spins on lattices Perturbed autonomous dynamics AC-conductivity measure Paramagnetic conductivity Representation theory Fermion Fock spaces Lattice Fermi Quantum spin systems

Authors and affiliations

  • J.-B. Bru
    • 1
  • W. de Siqueira Pedra
    • 2
  1. 1.Departamento de Matemáticas, Facultad de Ciencia y TecnologíaUniversidad del País VascoBilbaoSpain
  2. 2.Department of Mathematical Physics, Institute of PhysicsUniversity of São Paulo São PauloBrazil

Bibliographic information

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