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

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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 Arzelà–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 Fréchet 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, RieszÕs 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 HilbertÐSchmidt 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 Schrödinger’s wavefunction and Born’s probabilistic interpretation
6.4 HeisenbergÕs 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 FugledeÕs 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 Schrödinger 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 WeylÕs relations, the theorems of StoneÐvon 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, Gårding, 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 Schrödinger’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 PauliÕs 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