Summary and Info
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.Table of ContentsCoverSpectral Theory and Quantum Mechanics - With an Introduction to the Algebraic FormulationISBN 9788847028340 ISBN 9788847028357PrefaceContents1 Introduction and mathematical backgrounds 1.1 On the book 1.1.1 Scope and structure 1.1.2 Prerequisites 1.1.3 General conventions 1.2 On Quantum Mechanics 1.2.1 Quantum Mechanics as a mathematical theory 1.2.2 QM in the panorama of contemporary Physics 1.3 Backgrounds on general topology 1.3.1 Open/closed sets and basic point-set topology 1.3.2 Convergence and continuity 1.3.3 Compactness 1.3.4 Connectedness 1.4 Round-up on measure theory 1.4.1 Measure spaces 1.4.2 Positive s-additive measures 1.4.3 Integration of measurable functions 1.4.4 Riesz's theorem for positive Borel measures 1.4.5 Differentiating measures 1.4.6 Lebesgue's measure on Rn 1.4.7 The product measure 1.4.8 Complex (and signed) measures 1.4.9 Exchanging derivatives and integrals2 Normed and Banach spaces, examples and applications 2.1 Normed and Banach spaces and algebras 2.1.1 Normed spaces and essential topological properties 2.1.2 Banach spaces 2.1.3 Example: the Banach space C(K; Kn), the theorems of Dini and Arzel�-Ascoli 2.1.4 Normed algebras, Banach algebras and examples 2.2 Operators, spaces of operators, operator norms 2.3 The fundamental theorems of Banach spaces 2.3.1 The Hahn-Banach theorem and its immediate consequences 2.3.2 The Banach-Steinhaus theorem or uniform boundedness principle 2.3.3 Weak topologies. *-weak completeness of X 2.3.4 Excursus: the theorem of Krein-Milman, locally convex metrisable spaces and Fr�chet spaces 2.3.5 Baire's category theorem and its consequences: the open mapping theorem and the inverse operator theorem 2.3.6 The closed graph theorem 2.4 Projectors 2.5 Equivalent norms 2.6 The fixed-point theorem and applications 2.6.1 The xed-point theorem of Banach-Caccioppoli 2.6.2 Application of the xed-point theorem: local existence and uniqueness for systems of differential equations3 Hilbert spaces and bounded operators 3.1 Elementary notions, Riesz�s theorem and reflexivity 3.1.1 Inner product spaces and Hilbert spaces 3.1.2 Riesz's theorem and its consequences 3.2 Hilbert bases 3.3 Hermitian adjoints and applications 3.3.1 Hermitian conjugation, or adjunction 3.3.2 *-algebras and C*-algebras 3.3.3 Normal, self-adjoint, isometric, unitary and positive operators 3.4 Orthogonal projectors and partial isometries 3.5 Square roots of positive operators and polar decomposition of bounded operators 3.6 The Fourier-Plancherel transform4 Families of compact operators on Hilbert spaces and fundamental properties 4.1 Compact operators in normed and Banach spaces 4.1.1 Compact sets in (in nite-dimensional) normed spaces 4.1.2 Compact operators in normed spaces 4.2 Compact operators in Hilbert spaces 4.2.1 General properties and examples 4.2.2 Spectral decomposition of compact operators on Hilbert spaces 4.3 Hilbert�Schmidt operators 4.3.1 Main properties and examples 4.3.2 Integral kernels and Mercer's theorem 4.4 Trace-class (or nuclear) operators 4.4.1 General properties 4.4.2 The notion of trace 4.5 Introduction to the Fredholm theory of integral equations5 Densely-defined unbounded operators on Hilbert spaces 5.1 Unbounded operators with non-maximal domains 5.1.1 Unbounded operators with non-maximal domains in normed spaces 5.1.2 Closed and closable operators 5.1.3 The case of Hilbert spaces: the structure of H. H and the t operator 5.1.4 General properties of the Hermitian adjoint operator 5.2 Hermitian, symmetric, self-adjoint and essentially self-adjoint operators 5.3 Two major applications: the position operator and the momentum operator 5.3.1 The position operator 5.3.2 The momentum operator 5.4 Existence and uniqueness criteria for self-adjoint extensions 5.4.1 The Cayley transform and de ciency indices 5.4.2 Von Neumann's criterion 5.4.3 Nelson's criterion6 Phenomenology of quantum systems and Wave Mechanics: an overview 6.1 General principles of quantum systems 6.2 Particle aspects of electromagnetic waves 6.2.1 The photoelectric effect 6.2.2 The Compton effect 6.3 An overview of Wave Mechanics 6.3.1 De Broglie waves 6.3.2 Schr�dinger's wavefunction and Born's probabilistic interpretation 6.4 Heisenberg�s uncertainty principle 6.5 Compatible and incompatible quantities7 The first 4 axioms of QM: propositions, quantum states and observables 7.1 The pillars of the standard interpretation of quantum phenomenology 7.2 Classical systems: elementary propositions and states 7.2.1 States as probability measures 7.2.2 Propositions as sets, states as measures on them 7.2.3 Set-theoretical interpretation of the logical connectives 7.2.4 fIn nitef propositions and physical quantities 7.2.5 Intermezzo: basics on the theory of lattices 7.2.6 The distributive lattice of elementary propositions for classical systems 7.3 Propositions on quantum systems as orthogonal projectors 7.3.1 The non-distributive lattice of orthogonal projectors on a Hilbert space 7.3.2 Recovering the Hilbert space from the lattice 7.3.3 Von Neumann algebras and the classi cation of factors 7.4 Propositions and states on quantum systems 7.4.1 Axioms A1 and A2: propositions, states of a quantum system and Gleason's theorem 7.4.2 The Kochen-Specker theorem 7.4.3 Pure states, mixed states, transition amplitudes 7.4.4 Axiom A3: post-measurement states and preparation of states 7.4.5 Superselection rules and coherent sectors 7.4.6 Algebraic characterisation of a state as a noncommutative Riesz theorem 7.5 Observables as projector-valued measures on R 7.5.1 Axiom A4: the notion of observable 7.5.2 Self-adjoint operators associated to observables: physical motivation and basic examples 7.5.3 Probability measures associated to state/observable couples8 Spectral Theory I: generalities, abstract C*-algebras and operators in B(H) 8.1 Spectrum, resolvent set and resolvent operator 8.1.1 Basic notions in normed spaces 8.1.2 The spectrum of special classes of normal operators in Hilbert spaces 8.1.3 Abstract C*-algebras: Gelfand-Mazur theorem, spectral radius, Gelfand's formula, Gelfand-Najmark theorem 8.2 Functional calculus: representations of commutative 8.2.1 Abstract C*-algebras: functional calculus for continuous maps and self-adjoint elements 8.2.2 Key properties of *-homomorphisms of C*-algebras, spectra and positive elements 8.2.3 Commutative Banach algebras and the Gelfand transform 8.2.4 Abstract C*-algebras: functional calculus for continuous maps and normal elements 8.2.5 C*-algebras of operators in B(H): functional calculus for bounded measurable functions 8.3 Projector-valued measures (PVMs) 8.3.1 Spectral measures, or PVMs 8.3.2 Integrating bounded measurable functions in a PVM 8.3.3 Properties of operators obtained integrating bounded maps with respect to PVMs 8.4 Spectral theorem for normal operators in B(H) 8.4.1 Spectral decomposition of normal operators in B(H) 8.4.2 Spectral representation of normal operators in B(H) 8.5 Fuglede�s theorem and consequences 8.5.1 Fuglede's theorem 8.5.2 Consequences to Fuglede's theorem9 Spectral theory II: unbounded operators on Hilbert spaces 9.1 Spectral theorem for unbounded self-adjoint operators 9.1.1 Integrating unbounded functions with respect to spectral measures 9.1.2 Von Neumann algebra of a bounded normal operator 9.1.3 Spectral decomposition of unbounded self-adjoint operators 9.1.4 Example with pure point spectrum: the Hamiltonian of the harmonic oscillator 9.1.5 Examples with pure continuous spectrum: the operators position and momentum 9.1.6 Spectral representation of unbounded self-adjoint operators 9.1.7 Joint spectral measures 9.2 Exponential of unbounded operators: analytic vectors 9.3 Strongly continuous one-parameter unitary groups 9.3.1 Strongly continuous one-parameter unitary groups, von Neumann's theorem 9.3.2 One-parameter unitary groups generated by self-adjoint operators and Stone's theorem 9.3.3 Commuting operators and spectral measures10 Spectral Theory III: applications 10.1 Abstract differential equations in Hilbert spaces 10.1.1 The abstract Schr�dinger equation (with source) 10.1.2 The abstract Klein-Gordon/d'Alembert equation (with source and dissipative term) 10.1.3 The abstract heat equation 10.2 Hilbert tensor products 10.2.1 Tensor product of Hilbert spaces and spectral properties 10.2.2 Tensor product of operators (typically unbounded) and spectral properties 10.2.3 An example: the orbital angular momentum 10.3 Polar decomposition theorem for unbounded operators 10.3.1 Properties of operators A*A, square roots of unbounded positive self-adjoint operators 10.3.2 Polar decomposition theorem for closed and densely-de ned operators 10.4 The theorems of Kato-Rellich and Kato 10.4.1 The Kato-Rellich theorem 10.4.2 An example: the operator -.+V and Kato's theorem11 Mathematical formulation of non-relativistic Quantum Mechanics 11.1 Round-up and remarks on axioms A1, A2, A3, A4 and superselection rules 11.2 Axiom A5: non-relativistic elementary systems 11.2.1 The canonical commutation relations (CCRs) 11.2.2 Heisenberg's uncertainty principle as a theorem 11.3 Weyl�s relations, the theorems of Stone�von Neumann and Mackey 11.3.1 Families of operators acting irreducibly and Schur's lemma 11.3.2 Weyl's relations from the CCRs 11.3.3 The theorems of Stone-von Neumann and Mackey 11.3.4 The Weyl *-algebra 11.3.5 Proof of the theorems of Stone-von Neumann and Mackey 11.3.6 More on fHeisenberg's principlefl weakening the assumptions and extension to mixed states 11.3.7 The Stone-von Neumann theorem revisited, via the Heisenberg group 11.3.8 Dirac's correspondence principle and Weyl's calculus12 Introduction to Quantum Symmetries 12.1 Definition and characterisation of quantum symmetries 12.1.1 Examples 12.1.2 Symmetries in presence of superselection rules 12.1.3 Kadison symmetries 12.1.4 Wigner symmetries 12.1.5 The theorems of Wigner and Kadison 12.1.6 The dual action of symmetries on observables 12.2 Introduction to symmetry groups 12.2.1 Projective and projective unitary representations 12.2.2 Projective unitary representations are unitary or antiunitary 12.2.3 Central extensions and quantum group associated to a symmetry group 12.2.4 Topological symmetry groups 12.2.5 Strongly continuous projective unitary representations 12.2.6 A special case: the topological group R 12.2.7 Round-up on Lie groups and algebras 12.2.8 Symmetry Lie groups, theorems of Bargmann, G�rding, Nelson, FS3 12.2.9 The Peter-Weyl theorem 12.3 Examples 12.3.1 The symmetry group SO(3) and the spin 12.3.2 The superselection rule of the angular momentum 12.3.3 The Galilean group and its projective unitary representations 12.3.4 Bargmann's rule of superselection of the mass13 Selected advanced topics in Quantum Mechanics 13.1 Quantum dynamics and its symmetries 13.1.1 Axiom A6: time evolution 13.1.2 Dynamical symmetries 13.1.3 Schr�dinger's equation and stationary states 13.1.4 The action of the Galilean group in position representation 13.1.5 Basic notions of scattering processes 13.1.6 The evolution operator in absence of time homogeneity and Dyson's series 13.1.7 Antiunitary time reversal 13.2 The time observable and Pauli�s theorem. POVMs in brief 13.2.1 Pauli's theorem 13.2.2 Generalised observables as POVMs 13.3 Dynamical symmetries and constants of motion 13.3.1 Heisenberg's picture and constants of motion 13.3.2 A short detour on Ehrenfest's theorem and related mathematical issues 13.3.3 Constants of motion associated to symmetry Lie groups and the case of the Galilean group 13.4 Compound systems and their properties 13.4.1 Axiom A7: compound systems 13.4.2 Entangled states and the so-called fEPR paradoxf 13.4.3 Bell's inequalities and their experimental violation 13.4.4 EPR correlations cannot transfer information 13.4.5 The phenomenon of decoherence as a manifestation of the macroscopic world 13.4.6 Axiom A8: compounds of identical systems 13.4.7 Bosons and Fermions14 Introduction to the Algebraic Formulation of Quantum Theories 14.1 Introduction to the algebraic formulation of quantum theories 14.1.1 Algebraic formulation and the GNS theorem 14.1.2 Pure states and irreducible representations 14.1.3 Hilbert space formulation vs algebraic formulation 14.1.4 Superselection rules and Fell's theorem 14.1.5 Proof of the Gelfand-Najmark theorem, universal representations and quasi-equivalent representations 14.2 Example of a C*-algebra of observables: the Weyl C*-algebra 14.2.1 Further properties of Weyl *-algebras W (X,s) 14.2.2 The Weyl C*-algebra CW (X,s) 14.3 Introduction to Quantum Symmetries within the algebraic formulation 14.3.1 The algebraic formulation's viewpoint on quantum symmetries 14.3.2 (Topological) symmetry groups in the algebraic formalismAppendix A Order relations and groupsAppendix B Elements of differential geometryReferencesIndex
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