Fundamental Mathematical Structures of Quantum Theory : Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation / by Valter Moretti.

Format:

Book

Author/Creator:

Moretti, Valter, author.

Contributor:

SpringerLink (Online service)

Series:

Physics and Astronomy (Springer-11651)

Language:

English

Subjects (All):

Quantum theory.

Physics.

Mathematical physics.

Quantum Physics.

Mathematical Methods in Physics.

Mathematical Physics.

Local Subjects:

Quantum Physics.

Mathematical Methods in Physics.

Mathematical Physics.

Physical Description:

1 online resource (XIII, 337 pages) : 1 illustrations

Edition:

First edition 2019.

Contained In:

Springer eBooks

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2019.

System Details:

text file PDF

Summary:

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 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 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 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.

Contents:

General Phenomenology of the Quantum World and Elementary Formalism

Hilbert Spaces and Classes of Operators

Observables and States in General Hilbert Spaces: Spectral Theory

Fundamental Quantum Structures on Hilbert Spaces

Realism, Non-Contextuality, Local Causality, Entanglement

von Neumann Algebras of Observables and Superselection Rules

Quantum Symmetries

The Algebraic Formulation.

Other Format:

Printed edition:

ISBN:

978-3-030-18346-2

9783030183462

Access Restriction:

Restricted for use by site license.