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This book presents a modern and systematic approach to Linear
Response Theory (LRT) by combining analytic and algebraic ideas.
LRT is a tool to study systems that are driven out of equilibrium
by external perturbations. In particular the reader is provided
with a new and robust tool to implement LRT for a wide array of
systems. The proposed formalism in fact applies to periodic and
random systems in the discrete and the continuum. After a short
introduction describing the structure of the book, its aim and
motivation, the basic elements of the theory are presented in
chapter 2. The mathematical framework of the theory is outlined in
chapters 3-5: the relevant von Neumann algebras, noncommutative
$L^p$- and Sobolev spaces are introduced; their construction is
then made explicit for common physical systems; the notion of
isopectral perturbations and the associated dynamics are studied.
Chapter 6 is dedicated to the main results, proofs of the Kubo and
Kubo-Streda formulas. The book closes with a chapter about possible
future developments and applications of the theory to periodic
light conductors. The book addresses a wide audience of
mathematical physicists, focusing on the conceptual aspects rather
than technical details and making algebraic methods accessible to
analysts.
Mathematics is the language of physics, and over time physicists
have developed their own dialect. The main purpose of this book is
to bridge this language barrier, and introduce the readers to the
beauty of mathematical physics. It shows how to combine the
strengths of both approaches: physicists often arrive at
interesting conjectures based on good intuition, which can serve as
the starting point of interesting mathematics. Conversely,
mathematicians can more easily see commonalities between very
different fields (such as quantum mechanics and electromagnetism),
and employ more advanced tools.Rather than focusing on a particular
topic, the book showcases conceptual and mathematical commonalities
across different physical theories. It translates physical problems
to concrete mathematical questions, shows how to answer them and
explains how to interpret the answers physically. For example, if
two Hamiltonians are close, why are their dynamics similar?The book
alternates between mathematics- and physics-centric chapters, and
includes plenty of concrete examples from physics as well as 76
exercises with solutions. It exploits that readers from either end
are familiar with some of the material already. The
mathematics-centric chapters provide the necessary background to
make physical concepts mathematically precise and establish basic
facts. And each physics-centric chapter introduces physical
theories in a way that is more friendly to mathematicians.As the
book progresses, advanced material is sprinkled in to showcase how
mathematics and physics augment one another. Some of these examples
are based on recent publications and include material which has not
been covered in other textbooks. This is to keep it interesting for
the readers.
Mathematics is the language of physics, and over time physicists
have developed their own dialect. The main purpose of this book is
to bridge this language barrier, and introduce the readers to the
beauty of mathematical physics. It shows how to combine the
strengths of both approaches: physicists often arrive at
interesting conjectures based on good intuition, which can serve as
the starting point of interesting mathematics. Conversely,
mathematicians can more easily see commonalities between very
different fields (such as quantum mechanics and electromagnetism),
and employ more advanced tools.Rather than focusing on a particular
topic, the book showcases conceptual and mathematical commonalities
across different physical theories. It translates physical problems
to concrete mathematical questions, shows how to answer them and
explains how to interpret the answers physically. For example, if
two Hamiltonians are close, why are their dynamics similar?The book
alternates between mathematics- and physics-centric chapters, and
includes plenty of concrete examples from physics as well as 76
exercises with solutions. It exploits that readers from either end
are familiar with some of the material already. The
mathematics-centric chapters provide the necessary background to
make physical concepts mathematically precise and establish basic
facts. And each physics-centric chapter introduces physical
theories in a way that is more friendly to mathematicians.As the
book progresses, advanced material is sprinkled in to showcase how
mathematics and physics augment one another. Some of these examples
are based on recent publications and include material which has not
been covered in other textbooks. This is to keep it interesting for
the readers.
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