The Formalisms of Quantum Mechanics : An Introduction.

Intro -- Contents -- Chapter 1 Introduction -- 1.1 Motivation -- 1.2 Organization -- 1.3 What This Course is Not! -- Chapter 2 The Standard Formulations of Classical and Quantum Mechanics -- 2.1 Classical Mechanics -- 2.1.1 The Lagrangian Formulation -- 2.1.2 The Hamiltonian Formulation -- 2.1.2.1 The Phase Space and the Hamiltonian -- 2.1.2.2 Hamilton-Jacobi Equation -- 2.1.2.3 Symplectic Manifolds -- 2.1.2.4 Simple Example -- 2.1.2.5 Observables, Poisson Brackets -- 2.1.2.6 The Dynamics and Hamiltonian Flows -- 2.1.2.7 The Liouville Measure -- 2.1.2.8 A Less Trivial Example: The Classical Spin -- 2.1.2.9 Statistical States, Distribution Functions, the Liouville Equation -- 2.1.2.10 Canonical Transformations -- 2.1.2.11 Along the Hamiltonian Flows -- 2.1.3 The Algebra of Classical Observables -- 2.1.3.1 Functions as Elements of a Commutative C*-Algebra -- 2.1.3.2 Observable as Elements of a Commutative Poisson Algebra -- 2.2 Probabilities -- 2.2.1 The Frequentist Point of View -- 2.2.2 The Bayesian Point of View -- 2.2.3 Conditional Probabilities -- 2.3 Quantum Mechanics: The ``Canonical Formulation'' -- 2.3.1 Principles -- 2.3.1.1 Pure States and Hilbert Space -- 2.3.1.2 Observables and Operators -- 2.3.1.3 Measurements, Probabilities and the Born Rule -- 2.3.1.4 Unitary Dynamics -- 2.3.1.5 Multipartite Systems -- 2.3.1.6 Correspondence Principe, Canonical Quantization -- 2.3.2 Representations of Quantum Mechanics -- 2.3.2.1 The Schrödinger Picture -- 2.3.2.2 The Heisenberg Picture -- 2.3.3 Quantum Statistics and the Density Matrix -- 2.3.3.1 The Density Matrix -- 2.3.3.2 Quantum Ensembles Versus Classical Ensembles -- 2.3.3.3 The von Neumann Entropy -- 2.3.3.4 Example: Entanglement Entropy -- 2.3.3.5 Thermal States -- 2.3.3.6 Imaginary Time Formalism -- 2.4 Path and Functional Integrals Formulations -- 2.4.1 Path Integrals.

2.4.1.1 Path Integral in Configuration Space -- 2.4.1.2 Path Integral in Phase Space -- 2.4.2 Field Theories, Functional Integrals -- 2.5 Quantum Probabilities and Reversibility -- 2.5.1 Is Quantum Mechanics Reversible or Irreversible? -- 2.5.2 Reversibility of Quantum Probabilities -- 2.5.3 Causal Reversibility -- Chapter 3 The Algebraic Quantum Formalism -- 3.1 Introduction -- 3.1.1 Observables as Operators -- 3.1.2 Operator Algebras -- 3.1.3 The Algebraic Approach -- 3.2 The Algebra of Observables -- 3.2.1 The Mathematical Principles -- 3.2.1.1 Observables -- 3.2.1.2 The *-Conjugation -- 3.2.1.3 States -- 3.2.2 Physical Discussion -- 3.2.2.1 Observables and Causality -- 3.2.2.2 The *-Conjugation and Reversibility -- 3.2.2.3 States, Measurements and Probabilities -- 3.2.3 Physical Observables and Pure States -- 3.2.3.1 Physical (Symmetric) Observables -- 3.2.3.2 Pure States -- 3.2.3.3 Bounded Observables -- 3.3 The C*-Algebra of Observables -- 3.3.1 The Norm on Observables, A is a Banach Algebra -- 3.3.2 The Observables form a Real C*-Algebra -- 3.3.3 Spectrum of Observables and Results of Measurements -- 3.3.4 Complex C*-Algebras -- 3.4 The GNS Construction, Operators and Hilbert Spaces -- 3.4.1 Finite Dimensional Algebra of Observables -- 3.4.2 Infinite Dimensional Real Algebra of Observables -- 3.4.3 The Complex Case, the GNS Construction -- 3.5 Why Complex Algebras? -- 3.5.1 Dynamics -- 3.5.2 Locality and Separability -- 3.5.3 Quaternionic Hilbert Spaces -- 3.6 Superselection Sectors -- 3.6.1 Definition -- 3.6.2 A Simple Example: The Particle on a Circle -- 3.6.3 General Discussion -- 3.7 von Neumann Algebras -- 3.7.1 Definitions -- 3.7.2 Classification of Factors -- 3.7.3 The Tomita-Takesaki Theory -- 3.8 Locality and Algebraic Quantum Field Theory -- 3.8.1 Algebraic Quantum Field Theory in a Dash -- 3.8.2 Axiomatic QFT.

3.8.2.1 Wightman Axioms -- 3.8.2.2 CPT and Spin-Statistics Theorems -- 3.9 Discussion -- Chapter 4 The Quantum Logic Formalism -- 4.1 Introduction -- 4.1.1 Why an Algebraic Structure? -- 4.1.2 Measurements as ``Logical Propositions'' -- 4.1.3 The Quantum Logic Approach -- 4.2 A Presentation of the Principles -- 4.2.1 Projective Measurements as Propositions -- 4.2.2 Causality, POSET's and the Lattice of Propositions -- 4.2.2.1 Causal Order Relation -- 4.2.2.2 AND (Meet ) -- 4.2.2.3 Logical OR (Join ) -- 4.2.2.4 Trivial 1 and Vacuous Propositions -- 4.2.3 Reversibility and Orthocomplementation -- 4.2.3.1 Negations a and a -- 4.2.3.2 Causal Reversibility and Negation -- 4.2.3.3 Orthogonality -- 4.2.4 Subsystems of Propositions and Orthomodularity -- 4.2.4.1 What Must Replace Distributivity? -- 4.2.4.2 Sublattices and Weak-Modularity -- 4.2.4.3 Orthomodular Lattices -- 4.2.4.4 Weak-Modularity Versus Modularity -- 4.2.5 Pure States and AC Properties -- 4.2.5.1 Atoms -- 4.2.5.2 Atomic Lattices -- 4.2.5.3 Covering Property -- 4.3 The Geometry of Orthomodular AC Lattices -- 4.3.1 Prelude: The Fundamental Theorem of ProjectiveGeometry -- 4.3.2 The Projective Geometry of Orthomodular AC Lattices -- 4.3.2.1 The Coordinatization Theorem -- 4.3.2.2 Discussion: Which Division Ring K? -- 4.3.3 Towards Hilbert Spaces -- 4.4 Gleason's Theorem and the Born Rule -- 4.4.1 States and Probabilities -- 4.4.2 Gleason's Theorem -- 4.4.3 Principle of the Proof -- 4.4.4 The Born Rule -- 4.4.5 Physical Observables -- 4.5 Discussion -- Chapter 5 Information, Correlations, and More -- 5.1 Quantum Information Formulations -- 5.2 Quantum Correlations -- 5.2.1 Entropic Inequalities -- 5.2.2 Bipartite Correlations -- 5.2.2.1 The Tsirelson Bound -- 5.2.2.2 Popescu-Rohrlich Boxes -- 5.3 Hidden Variables, Contextuality and Local Realism.

5.3.1 Hidden Variables and ``Elements of Reality'' -- 5.3.2 Context-Free Hidden Variables -- 5.3.3 Gleason's Theorem and Contextuality -- 5.3.3.1 Gleason's Theorem Excludes General Context-Free Models -- 5.3.3.2 The Special Case of n=2 -- 5.3.3.3 Probabilistic Models=Quantum Mechanics -- 5.3.4 The Kochen-Specker Theorem -- 5.3.5 The Bell-CHSH Inequalities and Local Realism -- 5.3.5.1 The Local Realism Hypothesis -- 5.3.5.2 The Bell-CSHS Inequality -- 5.3.6 Contextual Models -- 5.3.6.1 Bell's Simple Model -- 5.3.6.2 de Broglie-Bohm Pilot-Wave Models -- 5.3.6.3 Stochastic Models and Nelson Stochastic Mechanics -- 5.3.6.4 An Adiabatic Argument -- 5.3.6.5 The Problems with Contextual Models -- 5.4 Summary Discussion on Quantum Correlations -- 5.4.1 Locality and Realism -- 5.4.2 Chance and Correlations -- 5.5 Measurements -- 5.5.1 What are the Questions? -- 5.5.2 The von Neumann Paradigm -- 5.5.3 Decoherence, Ergodicity and Mixing -- 5.5.4 Discussion -- 5.6 Formalisms, Interpretations and Alternatives to QuantumMechanics -- 5.6.1 What About Interpretations? -- 5.6.2 Formalisms -- 5.6.3 Interpretations -- 5.6.4 Alternatives -- 5.7 What About Gravity? -- Bibliography -- Index.

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