Fundamental Mathematical Structures of Quantum Theory
Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation
Authors: Moretti, Valter
Free Preview With many examples and solved exercises
 Highlights the interconnection between logic, lattice theory, general probability theory, and general spectral theory
 Nicely presents together standard material usually only found scattered in the literature
 Accessible to physicists as well as mathematicians
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 About this Textbook

This textbook presents in a concise and selfcontained way the advanced fundamental mathematical structures in quantum theory. It is based on lectures prepared for a 6 months course for MSc students. The reader is introduced to the beautiful interconnection between logic, lattice theory, general probability theory, and general spectral theory including the basic theory of von Neumann algebras and of the algebraic formulation, naturally arising in the study of the mathematical machinery of quantum theories. Some general results concerning hiddenvariable interpretations of QM such as Gleason's and the KochenSpecker theorems and the related notions of realism and noncontextuality are carefully discussed. This is done also in relation with the famous Bell (BCHSH) inequality concerning local causality.
Written in a didactic style, this book includes many examples and solved exercises.The work is organized as follows. Chapter 1 reviews some elementary facts and properties of quantum systems. Chapter 2 and 3 present the main results of spectral analysis in complex Hilbert spaces. Chapter 4 introduces the point of view of the orthomodular lattices' theory. Quantum theory form this perspective turns out to the probability measure theory on the nonBoolean lattice of elementary observables and Gleason's theorem characterizes all these measures. Chapter 5 deals with some philosophical and interpretative aspects of quantum theory like hiddenvariable formulations of QM. The KochenSpecker theorem and its implications are analyzed also in relation BCHSH inequality, entanglement, realism, locality, and noncontextuality. Chapter 6 focuses on the algebra of observables also in the presence of superselection rules introducing the notion of von Neumann algebra. Chapter 7 offers the idea of (groups of) quantum symmetry, in particular, illustrated in terms of Wigner and Kadison theorems. Chapter 8 deals with the elementary ideas and results of the so called algebraic formulation of quantum theories in terms of both *algebras and C*algebras.
This book should appeal to a dual readership: on one hand mathematicians that wish to acquire the tools that unlock the physical aspects of quantum theories; on the other physicists eager to solidify their understanding of the mathematical scaffolding of quantum theories.
 About the authors

Prof. Valter Moretti is Full Professor of Mathematical Physics at the University of Trento (Italy). His research interests include rigorous quantum field theory in curved spacetime, mathematical aspects of quantum mechanics and general relativity, the applications of operator algebras, functional analysis and global analysis to quantum field theory, and mathematical (analytic and geometric) methods for physics.
 Reviews

“The book can be read on the basis of an elementary knowledge of Hilbert spaces and measure theory. For mathematicians, it can be used as an introduction to Quantum Mechanics and a detailed guide to the associated mathematical problems; for the reader familiar with Quantum Mechanics, it provides an extensive insight into the mathematical discussion of its structural and foundational issues.” (Giovanni Morchio, Mathematical Reviews, June, 2020)
 Table of contents (8 chapters)


General Phenomenology of the Quantum World and Elementary Formalism
Pages 116

Hilbert Spaces and Classes of Operators
Pages 1746

Observables and States in General Hilbert Spaces: Spectral Theory
Pages 47129

Fundamental Quantum Structures on Hilbert Spaces
Pages 131186

Realism, NonContextuality, Local Causality, Entanglement
Pages 187212

Table of contents (8 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Fundamental Mathematical Structures of Quantum Theory
 Book Subtitle
 Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation
 Authors

 Valter Moretti
 Copyright
 2019
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer Nature Switzerland AG
 eBook ISBN
 9783030183462
 DOI
 10.1007/9783030183462
 Hardcover ISBN
 9783030183455
 Softcover ISBN
 9783030183486
 Edition Number
 1
 Number of Pages
 XIII, 337
 Number of Illustrations
 1 b/w illustrations
 Topics
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