Valter Moretti's Books

Fundamental Mathematical Structures of Quantum Theory

Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation

By: Valter Moretti

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Fundamental Mathematical Structures of Quantum Theory

Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation

By: Valter Moretti

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.

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345

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Springer

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ISBN 10

3030183467

ISBN 13

9783030183462

Spectral Theory and Quantum Mechanics

With an Introduction to the Algebraic Formulation

By: Valter Moretti

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Spectral Theory and Quantum Mechanics

With an Introduction to the Algebraic Formulation

By: Valter Moretti

This book pursues the accurate study of the mathematical foundations of Quantum Theories. It may be considered an introductory text on linear functional analysis with a focus on Hilbert spaces. Specific attention is given to spectral theory features that are relevant in physics. Having left the physical phenomenology in the background, it is the formal and logical aspects of the theory that are privileged. Another not lesser purpose is to collect in one place a number of useful rigorous statements on the mathematical structure of Quantum Mechanics, including some elementary, yet fundamental, results on the Algebraic Formulation of Quantum Theories. In the attempt to reach out to Master's or PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book should benefit established researchers to organise and present the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly.

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742

Publisher

Springer Science & Business Media

Published Date

ISBN 10

8847028353

ISBN 13

9788847028357

Analytical Mechanics

Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity

By: Valter Moretti

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Analytical Mechanics

Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity

By: Valter Moretti

This textbook aims at introducing readers, primarily students enrolled in undergraduate Mathematics or Physics courses, to the topics and methods of classical Mathematical Physics, including Classical Mechanics, its Lagrangian and Hamiltonian formulations, Lyapunov stability, plus the Liouville theorem and the Poincaré recurrence theorem among others. The material also rigorously covers the theory of Special Relativity. The logical-mathematical structure of the physical theories of concern is introduced in an axiomatic way, starting from a limited number of physical assumptions. Special attention is paid to themes with a major impact on Theoretical and Mathematical Physics beyond Analytical Mechanics, such as the Galilean symmetry of classical Dynamics and the Poincaré symmetry of relativistic Dynamics, the far-fetching relationship between symmetries and constants of motion, the coordinate-free nature of the underpinning mathematical objects, or the possibility of describing Dynamics in a global way while still working in local coordinates. Based on the author’s established teaching experience, the text was conceived to be flexible and thus adapt to different curricula and to the needs of a wide range of students and instructors.

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848

Publisher

Springer Nature

Published Date

ISBN 10

3031276124

ISBN 13

9783031276125

Spectral Theory and Quantum Mechanics

Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

By: Valter Moretti

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Spectral Theory and Quantum Mechanics

Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

By: Valter Moretti

This book discusses the mathematical foundations of quantum theories. It offers an introductory text on linear functional analysis with a focus on Hilbert spaces, highlighting the spectral theory features that are relevant in physics. After exploring physical phenomenology, it then turns its attention to the formal and logical aspects of the theory. Further, this Second Edition collects in one volume a number of useful rigorous results on the mathematical structure of quantum mechanics focusing in particular on von Neumann algebras, Superselection rules, the various notions of Quantum Symmetry and Symmetry Groups, and including a number of fundamental results on the algebraic formulation of quantum theories. Intended for Master's and PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book also benefits established researchers by organizing and presenting the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly.

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962

Publisher

Springer

Published Date

ISBN 10

331970706X

ISBN 13

9783319707068

From Classical Mechanics To Quantum Field Theory, A Tutorial

By: Manuel Asorey,Elisa Ercolessi,Valter Moretti

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From Classical Mechanics To Quantum Field Theory, A Tutorial

By: Manuel Asorey,Elisa Ercolessi,Valter Moretti

This book collects an extended version of the lectures delivered by the authors at the Fall Workshop on Geometry and Physics in the years 2014, 2015, 2016.It aims at introducing advanced graduate and PhD students, as well as young researchers, to current research in mathematics and physics. In particular, it fills the gap between the more physical-oriented and the more mathematical-oriented literature on quantum theory. It introduces various approaches to methods of quantization, along with their impact on modern mathematical methods.

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255

Publisher

World Scientific

Published Date

ISBN 10

9811210500

ISBN 13

9789811210501

Hadamard States from Light-like Hypersurfaces

By: Claudio Dappiaggi,Valter Moretti,Nicola Pinamonti

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Hadamard States from Light-like Hypersurfaces

By: Claudio Dappiaggi,Valter Moretti,Nicola Pinamonti

This book provides a rather self-contained survey of the construction of Hadamard states for scalar field theories in a large class of notable spacetimes, possessing a (conformal) light-like boundary. The first two sections focus on explaining a few introductory aspects of this topic and on providing the relevant geometric background material. The notions of asymptotically flat spacetimes and of expanding universes with a cosmological horizon are analysed in detail, devoting special attention to the characterization of asymptotic symmetries. In the central part of the book, the quantization of a real scalar field theory on such class of backgrounds is discussed within the framework of algebraic quantum field theory. Subsequently it is explained how it is possible to encode the information of the observables of the theory in a second, ancillary counterpart, which is built directly on the conformal (null) boundary. This procedure, dubbed bulk-to-boundary correspondence, has the net advantage of allowing the identification of a distinguished state for the theory on the boundary, which admits a counterpart in the bulk spacetime which is automatically of Hadamard form. In the last part of the book, some applications of these states are discussed, in particular the construction of the algebra of Wick polynomials. This book is aimed mainly, but not exclusively, at a readership with interest in the mathematical formulation of quantum field theory on curved backgrounds.

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Page Count

113

Publisher

Springer

Published Date

ISBN 10

3319643436

ISBN 13

9783319643434

Analytic Aspects Of Quantum Fields

By: Andrei A Bytsenko,Guido Cognola,Emilio Elizalde,Valter Moretti,Sergio Zerbini

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Analytic Aspects Of Quantum Fields

By: Andrei A Bytsenko,Guido Cognola,Emilio Elizalde,Valter Moretti,Sergio Zerbini

One of the aims of this book is to explain in a basic manner the seemingly difficult issues of mathematical structure using some specific examples as a guide. In each of the cases considered, a comprehensible physical problem is approached, to which the corresponding mathematical scheme is applied, its usefulness being duly demonstrated. The authors try to fill the gap that always exists between the physics of quantum field theories and the mathematical methods best suited for its formulation, which are increasingly demanding on the mathematical ability of the physicist.

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Page Count

370

Publisher

World Scientific

Published Date

ISBN 10

9814486035

ISBN 13

9789814486033

Book's by Valter Moretti

Fundamental Mathematical Structures of Quantum Theory

Fundamental Mathematical Structures of Quantum Theory

Spectral Theory and Quantum Mechanics

Spectral Theory and Quantum Mechanics

Analytical Mechanics

Analytical Mechanics

Spectral Theory and Quantum Mechanics

Spectral Theory and Quantum Mechanics

From Classical Mechanics To Quantum Field Theory, A Tutorial

From Classical Mechanics To Quantum Field Theory, A Tutorial

Hadamard States from Light-like Hypersurfaces

Hadamard States from Light-like Hypersurfaces

Analytic Aspects Of Quantum Fields

Analytic Aspects Of Quantum Fields

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