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Valter Moretti - Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

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Valter Moretti Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation
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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 Masters 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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Springer International Publishing AG 2017
Valter Moretti Spectral Theory and Quantum Mechanics UNITEXT
1. Introduction and Mathematical Backgrounds
Valter Moretti 1
(1)
Department of Mathematics, University of Trento, Povo, Trento, Italy
Valter Moretti
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O frati, dissi che per cento milia perigli siete giunti a loccidente, a questa tanto picciola vigilia di nostri sensi ch del rimanente non vogliate negar lesperienza, di retro al sol, del mondo sanza gente.
Dante Alighieri, the Divine Comedy, Inferno, canto XXVI. Note (Brothers I said, who through a hundred thousand dangers have reached the channel to the west, to the short evening watch which your own senses still must keep, do not choose to deny the experience of what lies past the Sun and of the world yet uninhabited. Dante Alighieri, The Divine Comedy, translated by J. Finn Cotter, edited by C. Franco, Forum Italicum Publishing, New York, 2006.)
1.1 On the Book
1.1.1 Scope and Structure
One of the aims of the present book is to explain the mathematical foundations of Quantum Mechanics (QM), and Quantum Theories in general, in a mathematically rigorous way. This is a treatise on Mathematics (or Mathematical Physics) rather than a text on Quantum Mechanics. Except for a few cases, the physical phenomenology is left in the background in order to privilege the theorys formal and logical aspects. At any rate, several examples of the physical formalism are presented, lest one lose touch with the world of physics.
In alternative to, and irrespective of, the physical content, the book should be considered as an introductory text, albeit touching upon rather advanced topics, on functional analysis on Hilbert spaces, including a few elementary yet fundamental results on Picture 1 -algebras. Special attention is given to a series of results in spectral theory, such as the various formulations of the spectral theorem for bounded normal operators and not necessarily bounded, self-adjoint ones. This is, as a matter of fact, one further scope of the text. The mathematical formulation of Quantum Theories is confined to Chaps..
A third purpose is to collect in one place a number of rigorous and useful results on the mathematical structure of QM and Quantum Theories. These are more advanced than what is normally encountered in quantum physics manuals. Many of these aspects have been known for a long time but are scattered in the specialistic literature. We should mention Solrs theorem , Gleasons theorem , the theorem of Kochen and Specker , the theorems of Stonevon Neumann and Mackey , Stones theorem and von Neumanns theorem about one-parameter unitary groups, Kadisons theorem , besides the better known Wigner , Bargmann and GNS theorems ; or, more abstract results in operator theory such as Fugledes theorem , or the polar decomposition for closed unbounded operators (which is relevant in the TomitaTakesaki theory and statistical Quantum Mechanics in relationship to the KMS condition); furthermore, self-adjoint properties for symmetric operators, due to Nelson, that descend from the existence of dense sets of analytical vectors, and finally, Katos work (but not only his) on the essential self-adjointness of certain kinds of operators and their limits from the bottom of the spectrum (mostly based on the KatoRellich theorem ).
Some chapters suffice to cover a good part of the material suitable for advanced courses on Mathematical Methods in Physics; this is common for masters degrees in Physics or doctoral degrees, if we assume a certain familiarity with notions, results and elementary techniques of measure theory. The text may also be used for a higher-level course in Mathematical Physics that includes foundational material on QM. In the attempt to reach out to master or Ph.D. students, both in physics with an interest in mathematical methods or in mathematics with an inclination towards physical applications, the author has tried to prepare a self-contained text, as far as possible: hence a primer was included on general topology and abstract measure theory, together with an appendix on differential geometry. Most chapters are accompanied by exercises, many of which are solved explicitly.
The book could, finally, be useful to scholars to organise and present accurately the profusion of advanced material disseminated in the literature.
Results from topology and measure theory, much needed throughout the whole treatise, are recalled at the end of this introductory chapter. The rest of the book is ideally divided into three parts. The first part, up to Chap. , regards the general theory of operators on Hilbert spaces, and introduces several fairly general notions, like Banach spaces. Core results are proved, such as the theorems of Baire, HahnBanach and BanachSteinhaus, as well as the fixed-point theorem of BanachCaccioppoli, the Arzel-Ascoli theorem and Fredholms alternative, plus some elementary consequences. This part contains a summary of basic topological notions, in the belief that it might benefit physics students. The latters training on point-set topology is at times disparate and often presents gaps, because this subject is, alas, usually taught sporadically in physics curricula, and not learnt in an organic way like students in mathematics do.
Part two ends with Chap. spectral theory is applied to several practical and completely abstract contexts, both quantum and not.
Chapter in terms of direct decomposition of von Neumann factors of observables. In that chapter the notion of von Neumann algebra of observables is exploited to present the mathematical formulation of quantum theories in more general situations, where not all self-adjoint operators represent observables.
The third part of the book is devoted to the mathematical axioms of QM, and more advanced topics like quantum symmetries and the algebraic formulation of quantum theories . Quantum symmetries and symmetry groups (both according to Wigner and to Kadison) are studied in depth. Dynamical symmetries and the quantum version of Noethers theorem are covered as well. The Galilean group , together with its subgroups and central extensions, is employed repeatedly as reference symmetry group, to explain the theory of projective unitary representations. Bargmanns theorem on the existence of unitary representations of simply connected Lie groups whose Lie algebra obeys a certain cohomology constraint is proved, and Bargmanns rule of superselection of the mass is discussed in detail. Then the useful theorems of Grding and Nelson for projective unitary representations of Lie groups of symmetries are considered. Important topics are examined that are often neglected in manuals, like the uniqueness of unitary representations of the canonical commutation relations (theorems of Stonevon Neumann and Mackey), or the theoretical difficulties in defining time as the conjugate operator to energy (the Hamiltonian). The mathematical hurdles one must overcome in order to make the statement of Ehrenfests theorem precise are briefly treated. Chapter offers an introduction to the ideas and methods of the abstract formulation of observables and algebraic states via Picture 2 -algebras. Here one finds the proof of the GNS theorem and some consequences of purely mathematical flavour, like the general theorem of GelfandNajmark . This closing chapter also contains material on quantum symmetries in an algebraic setting. As an example the Weyl Picture 3
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