مشخصات کتاب

Spectral Theory and Quantum Mechanics: With an Introduction to the Algebraic Formulation

دسته بندی: مکانیک کوانتومی
ویرایش: 2013 
نویسندگان: Valter Moretti  
سری: UNITEXT / La Matematica per il 3+2 
ISBN (شابک) : 8847028345, 9788847028340 
ناشر: Springer 
سال نشر: 2013 
تعداد صفحات: 752 
زبان: English 
فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود) 
حجم فایل: 5 مگابایت 

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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 Contents Cover Spectral Theory and Quantum Mechanics - With an Introduction to the Algebraic Formulation ISBN 9788847028340 ISBN 9788847028357 Preface Contents 1 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 integrals 2 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 equations 3 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 transform 4 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 equations 5 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 criterion 6 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 quantities 7 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 couples 8 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 theorem 9 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 measures 10 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 theorem 11 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 calculus 12 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 mass 13 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 Fermions 14 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 formalism Appendix A Order relations and groups Appendix B Elements of differential geometry References Index

فهرست مطالب

1......Page 1
 Titlepage......Page 3
 Copyright......Page 4
 Preface......Page 5
 Contents......Page 8
  1.1 On the book 1.1.1 Scope and structure......Page 16
  1.1.3 General conventions......Page 19
  1.2 On Quantum Mechanics 1.2.1 Quantum Mechanics as a mathematical theory......Page 20
  1.2.2 QM in the panorama of contemporary Physics......Page 22
  1.3.1 Open/closed sets and basic point-set topology......Page 25
  1.3.2 Convergence and continuity......Page 27
  1.3.3 Compactness......Page 29
  1.3.4 Connectedness......Page 30
  1.4.1 Measure spaces......Page 31
  1.4.2 Positive s -additive measures......Page 34
  1.4.3 Integration of measurable functions......Page 37
  1.4.4 Riesz’s theorem for positive Borel measures......Page 40
  1.4.6 Lebesgue’s measure on Rn......Page 42
  1.4.7 The product measure......Page 46
  1.4.8 Complex (and signed) measures......Page 47
  1.4.9 Exchanging derivatives and integrals......Page 48
 2 Normed and Banach spaces, examples and applications......Page 50
  2.1.1 Normed spaces and essential topological properties......Page 51
  2.1.2 Banach spaces......Page 55
  Kn), the theorems of Dini and Arzelà–Ascoli......Page 57
  2.1.4 Normed algebras, Banach algebras and examples......Page 60
  2.2 Operators, spaces of operators, operator norms......Page 68
  2.3.1 The Hahn–Banach theorem and its immediate consequences......Page 75
  2.3.2 The Banach–Steinhaus theorem or uniform boundedness principle......Page 78
  weak completeness of X......Page 80
  2.3.4 Excursus: the theorem of Krein–Milman, locally convex metrisable spaces and Fréchet spaces......Page 84
  2.3.5 Baire’s category theorem and its consequences: the open mapping theorem and the inverse operator theorem......Page 88
  2.3.6 The closed graph theorem......Page 91
  2.4 Projectors......Page 93
  2.5 Equivalent norms......Page 95
  2.6.1 The fixed-point theorem of Banach-Caccioppoli......Page 97
  2.6.2 Application of the fixed-point theorem: local existence and uniqueness for systems of differential equations......Page 102
  Exercises......Page 105
  3.1 Elementary notions, Riesz’s theorem and reflexivity......Page 111
  3.1.1 Inner product spaces and Hilbert spaces......Page 112
  3.1.2 Riesz’s theorem and its consequences......Page 116
  3.2 Hilbert bases......Page 120
  3.3.1 Hermitian conjugation, or adjunction......Page 133
  algebras and C*-algebras......Page 136
  3.3.3 Normal, self-adjoint, isometric, unitary and positive operators......Page 141
  3.4 Orthogonal projectors and partial isometries......Page 144
  3.5 Square roots of positive operators and polar decomposition of bounded operators......Page 148
  3.6 The Fourier-Plancherel transform......Page 156
  Exercises......Page 167
 4 Families of compact operators on Hilbert spaces and fundamental properties......Page 175
  4.1.1 Compact sets in (infinite-dimensional) normed spaces......Page 176
  4.1.2 Compact operators in normed spaces......Page 178
  4.2 Compact operators in Hilbert spaces......Page 181
  4.2.1 General properties and examples......Page 182
  4.2.2 Spectral decomposition of compact operators on Hilbert spaces......Page 184
  4.3.1 Main properties and examples......Page 190
  4.3.2 Integral kernels and Mercer’s theorem......Page 198
   4.4.1 General properties......Page 201
   4.4.2 The notion of trace......Page 205
   4.5 Introduction to the Fredholm theory of integral equations......Page 209
   Exercises......Page 216
  5.1 Unbounded operators with non-maximal domains......Page 222
  Exercises......Page 248
  5.1.1 Unbounded operators with non-maximal domains in normed spaces......Page 223
  5.1.2 Closed and closable operators......Page 224
  and the t operator......Page 225
  5.1.4 General properties of the Hermitian adjoint operator......Page 226
  5.2 Hermitian, symmetric, self-adjoint and essentially self-adjoint operators......Page 228
  5.3.1 The position operator......Page 232
  5.3.2 The momentum operator......Page 233
  5.4.1 The Cayley transform and deficiency indices......Page 237
  5.4.2 Von Neumann’s criterion......Page 241
  5.4.3 Nelson’s criterion......Page 242
  6.1 General principles of quantum systems......Page 252
  6.2.1 The photoelectric effect......Page 254
  6.2.2 The Compton effect......Page 255
  6.3.1 De Broglie waves......Page 257
  6.3.2 Schrödinger’s wavefunction and Born’s probabilistic interpretation......Page 258
  6.4 Heisenberg’s uncertainty principle......Page 260
  6.5 Compatible and incompatible quantities......Page 261
 7 The first 4 axioms of QM: propositions, quantum states and observables......Page 263
  7.1 The pillars of the standard interpretation of quantum phenomenology......Page 264
  7.2.1 States as probability measures......Page 266
  7.2.2 Propositions as sets, states as measures on them......Page 268
  7.2.3 Set-theoretical interpretation of the logical connectives......Page 269
  7.2.4 “Infinite” propositions and physical quantities......Page 270
  7.2.5 Intermezzo: basics on the theory of lattices......Page 272
  7.2.6 The distributive lattice of elementary propositions for classical systems......Page 274
  7.3 Propositions on quantum systems as orthogonal projectors......Page 275
  7.3.1 The non-distributive lattice of orthogonal projectors on a Hilbert space......Page 276
  7.3.2 Recovering the Hilbert space from the lattice......Page 283
  7.4 Propositions and states on quantum systems......Page 285
  7.4.1 Axioms A1 and A2: propositions, states of a quantum system and Gleason’s theorem......Page 286
  7.4.2 The Kochen–Specker theorem......Page 293
  7.4.3 Pure states, mixed states, transition amplitudes......Page 294
  7.4.4 Axiom A3: post-measurement states and preparation of states......Page 299
  7.4.5 Superselection rules and coherent sectors......Page 301
  7.4.6 Algebraic characterisation of a state as a noncommutative Riesz theorem......Page 304
  7.5.1 Axiom A4: the notion of observable......Page 308
  7.5.2 Self-adjoint operators associated to observables: physical motivation and basic examples......Page 311
  7.5.3 Probability measures associated to state/observable couples......Page 316
  Exercises......Page 318
 8 Spectral Theory I: generalities, abstract C*-algebras and operators in B(......Page 320
  8.1.1 Basic notions in normed spaces......Page 322
  8.1 Spectrum, resolvent set and resolvent operator......Page 321
  8.1.2 The spectrum of special classes of normal operators in Hilbert spaces......Page 325
  8.1.3 Abstract C*-algebras: Gelfand-Mazur theorem, spectral radius, Gelfand’s formula, Gelfand–Najmark theorem......Page 327
  8.2.1 Abstract C*-algebras: functional calculus for continuous maps and self-adjoint elements......Page 333
  homomorphisms of C*-algebras, spectra and positive elements......Page 336
  8.2.3 Commutative Banach algebras and the Gelfand transform......Page 340
  8.2.4 Abstract C*-algebras: functional calculus for continuous maps and normal elements......Page 345
  functional calculus for bounded measurable functions......Page 347
  8.3.1 Spectral measures, or PVMs......Page 355
  8.3.2 Integrating bounded measurable functions in a PVM......Page 357
  8.3.3 Properties of operators obtained integrating bounded maps with respect to PVMs......Page 363
  8.4.1 Spectral decomposition of normal operators in B(......Page 370
  8.4.2 Spectral representation of normal operators in B(......Page 375
  8.5 Fuglede’s theorem and consequences......Page 382
  8.5.1 Fuglede’s theorem......Page 383
  8.5.2 Consequences to Fuglede’s theorem......Page 385
  Exercises......Page 386
  9.1 Spectral theorem for unbounded self-adjoint operators......Page 390
  9.1.1 Integrating unbounded functions with respect to spectral measures......Page 391
 9.1.2 Von Neumann algebra of a bounded normal operator......Page 403
 9.1.3 Spectral decomposition of unbounded self-adjoint operators......Page 404
 9.1.4 Example with pure point spectrum: the Hamiltonian of the harmonic oscillator......Page 412
 9.1.5 Examples with pure continuous spectrum: the operators position and momentum......Page 416
 9.1.6 Spectral representation of unbounded self-adjoint operators......Page 417
 9.1.7 Joint spectral measures......Page 418
 9.2 Exponential of unbounded operators: analytic vectors......Page 420
 9.3.1 Strongly continuous one-parameter unitary groups, von Neumann’s theorem......Page 424
 9.3.2 One-parameter unitary groups generated by self-adjoint operators and Stone’s theorem......Page 428
 9.3.3 Commuting operators and spectral measures......Page 435
  Exercises......Page 439
  10.1 Abstract differential equations in Hilbert spaces......Page 442
  10.1.1 The abstract Schrödinger equation (with source)......Page 444
  10.1.2 The abstract Klein–Gordon/d’Alembert equation (with source and dissipative term)......Page 450
  10.1.3 The abstract heat equation......Page 458
  10.2.1 Tensor product of Hilbert spaces and spectral properties......Page 461
  10.2.2 Tensor product of operators (typically unbounded) and spectral properties......Page 466
  10.2.3 An example: the orbital angular momentum......Page 469
  10.3 Polar decomposition theorem for unbounded operators......Page 472
  10.3.1 Properties of operators A*A, square roots of unbounded positive self-adjoint operators......Page 473
  10.3.2 Polar decomposition theorem for closed and densely-defined operators......Page 477
  10.4.1 The Kato-Rellich theorem......Page 479
  10.4.2 An example: the operator -. +V and Kato’s theorem......Page 481
  Exercises......Page 487
  11.1 Round-up and remarks on axioms A1, A2, A3, A4 and superselection rules......Page 489
 11.2 Axiom A5: non-relativistic elementary systems......Page 496
 11.2.1 The canonical commutation relations (CCRs)......Page 498
 11.2.2 Heisenberg’s uncertainty principle as a theorem......Page 499
 11.3.1 Families of operators acting irreducibly and Schur’s lemma......Page 500
 11.3.2 Weyl’s relations from the CCRs......Page 502
 11.3.3 The theorems of Stone–von Neumann and Mackey......Page 509
 11.3.4 The Weyl......Page 512
 11.3.5 Proof of the theorems of Stone–von Neumann and Mackey......Page 516
 11.3.6 More on “Heisenberg’s principle”: weakening the assumptions and extension to mixed states......Page 522
 11.3.7 The Stone–von Neumann theorem revisited, via the Heisenberg group......Page 523
 11.3.8 Dirac’s correspondence principle and Weyl’s calculus......Page 525
  Exercises......Page 528
  12.1 Definition and characterisation of quantum symmetries......Page 530
 12.1.1 Examples......Page 532
 12.1.2 Symmetries in presence of superselection rules......Page 533
 12.1.3 Kadison symmetries......Page 534
 12.1.4 Wigner symmetries......Page 536
 12.1.5 The theorems of Wigner and Kadison......Page 538
 12.1.6 The dual action of symmetries on observables......Page 548
 12.2.1 Projective and projective unitary representations......Page 553
 12.2.3 Central extensions and quantum group associated to a symmetry group......Page 558
 12.2.4 Topological symmetry groups......Page 561
 12.2.5 Strongly continuous projective unitary representations......Page 566
 12.2.6 A special case: the topological group R......Page 568
 12.2.7 Round-up on Lie groups and algebras......Page 573
 12.2.8 Symmetry Lie groups, theorems of Bargmann, Gårding, Nelson, FS3......Page 581
 12.2.9 The Peter–Weyl theorem......Page 592
  12.3.1 The symmetry group SO(3) and the spin......Page 597
  12.3.3 The Galilean group and its projective unitary representations......Page 601
  12.3.4 Bargmann’s rule of superselection of the mass......Page 608
  Exercises......Page 611
 13 Selected advanced topics in Quantum Mechanics......Page 616
  13.1.2 Dynamical symmetries......Page 619
  13.1.3 Schrödinger’s equation and stationary states......Page 622
  13.1.4 The action of the Galilean group in position representation......Page 630
  13.1.5 Basic notions of scattering processes......Page 632
  13.1.6 The evolution operator in absence of time homogeneity and Dyson’s series......Page 638
  13.1.1 Axiom A6: time evolution......Page 617
  13.1.7 Antiunitary time reversal......Page 642
  13.2.1 Pauli’s theorem......Page 644
  13.2.2 Generalised observables as POVMs......Page 645
  13.3.1 Heisenberg’s picture and constants of motion......Page 647
  13.3.2 A short detour on Ehrenfest’s theorem and related mathematical issues......Page 652
  13.3.3 Constants of motion associated to symmetry Lie groups and the case of the Galilean group......Page 654
  13.4.1 Axiom A7: compound systems......Page 659
  13.4.2 Entangled states and the so-called “EPR paradox”......Page 660
  13.4.3 Bell’s inequalities and their experimental violation......Page 662
  13.4.4 EPR correlations cannot transfer information......Page 665
  13.4.5 The phenomenon of decoherence as a manifestation of the macroscopic world......Page 668
  13.4.6 Axiom A8: compounds of identical systems......Page 669
  13.4.7 Bosons and Fermions......Page 671
  Exercises......Page 673
  14.1 Introduction to the algebraic formulation of quantum theories......Page 675
  14.1.1 Algebraic formulation and the GNS theorem......Page 676
  14.1.2 Pure states and irreducible representations......Page 682
  14.1.3 Hilbert space formulation vs algebraic formulation......Page 685
  14.1.4 Superselection rules and Fell’s theorem......Page 688
  14.1.5 Proof of the Gelfand–Najmark theorem, universal representations and quasi-equivalent representations......Page 691
 14.2.1 Further properties of Weyl......Page 694
 14.2.2 The Weyl C*-algebra CW (......Page 698
 14.3 Introduction to Quantum Symmetries within the algebraic formulation......Page 699
 14.3.1 The algebraic formulation’s viewpoint on quantum symmetries......Page 700
 14.3.2 (Topological) symmetry groups in the algebraic formalism......Page 702
 Appendix......Page 704
 References......Page 716
 Index......Page 721

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