Overview
- 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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Table of contents (8 chapters)
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Front Matter
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Back Matter
Keywords
- Lattice theory
- spectral theory
- von Neumann algebras
- quantum formalism
- quantization procedures
- Heisenberg inequalities
- quantum symmetries
- algebraic formulation
- Schur's lemma
- superselection rules
- fundamental quantum structures
- quantum mechanics textbook
- Complex Hilbert spaces
- hidden variables
- Bell inequality
- Local causality
- Gleason's theorem
- Entanglement
- non-contextuality
- Quantum symmetry
About this book
This textbook presents in a concise and self-contained 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 hidden-variable interpretations of QM such as Gleason's and the Kochen-Specker theorems and the related notions of realism and non-contextuality 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 factsand 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 non-Boolean 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 hidden-variable formulations of QM. The Kochen-Specker theorem and its implications are analyzed also in relation BCHSH inequality, entanglement, realism, locality, and non-contextuality. 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 socalled 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.
Reviews
“Moretti’s text is a fun and insightful tour through analytic aspects of quantum mechanics. In particular, it is an excellent resource for the spectral theory of unbounded operators. … This reviewer would find the text most useful as a supplement to a second course in functional analysis based on any of the more classic graduate texts. It is also well suited to self-study by graduate students/researchers transitioning to mathematical physics.” (Ryan Grady, MAA Reviews, January 6, 2023)
“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)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Fundamental Mathematical Structures of Quantum Theory
Book Subtitle: Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation
Authors: Valter Moretti
DOI: https://doi.org/10.1007/978-3-030-18346-2
Publisher: Springer Cham
eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-18345-5Published: 01 July 2019
Softcover ISBN: 978-3-030-18348-6Published: 14 August 2020
eBook ISBN: 978-3-030-18346-2Published: 20 June 2019
Edition Number: 1
Number of Pages: XIII, 337
Number of Illustrations: 1 b/w illustrations
Topics: Quantum Physics, Mathematical Methods in Physics, Mathematical Physics