From Micro to Macro Quantum Systems : A Unified Formalism with Superselection Rules and Its Applications.
Başlık:
From Micro to Macro Quantum Systems : A Unified Formalism with Superselection Rules and Its Applications.
Yazar:
Wan, K. Kong.
ISBN:
9781860949081
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 online resource (709 pages)
İçerik:
Contents -- Preface -- I Aspects of Geometric and Operator Theories -- 1 Manifolds and Dynamical Systems -- 1.1 Topological Spaces and Topological Equivalence -- 1.1.1 Basic concepts and definitions -- 1.1.2 Topological equivalence -- 1.2 Euclidean Spaces -- 1.2.1 Basic concepts and definitions -- 1.2.2 Coordinate systems and coordinate transformations -- 1.2.3 Contravariant and covariant vectors in En -- 1.2.4 Contravariant covariant and mixed tensors -- 1.3 Differential Operators Vectors and Fields -- 1.3.1 Differential operators and derivations -- 1.3.2 Tangent vectors tangent vector fields and their integral curves -- 1.3.3 Transformation groups and complete vector fields -- 1.4 Cotangent Vectors and Differential Forms -- 1.4.1 Cotangent vectors differentials and one-forms -- 1.4.2 Tensor fields and two-forms -- 1.4.3 Exterior differentiation -- 1.4.4 Interior products closed and exact forms -- 1.5 Differentiable Manifolds -- 1.5.1 Definition and examples -- 1.5.2 Riemannian manifolds -- 1.5.3 Hamiltonian manifolds -- 1.6 Classical Dynamical Systems -- 1.6.1 Classical systems of finite order -- 1.6.2 First-order systems -- 1.6.3 Second-order Hamiltonian systems -- 1.6.4 Momentum observables vector fields and operators -- 1.6.5 Concluding remarks -- References -- 2 Operators and their Direct Integrals -- 2.1 Hilbert Spaces -- 2.2 Operators: Basic Definitions -- 2.2.1 Boundedness adjoints extensions and restrictions continuity and closure -- 2.2.2 Convergence of a family of bounded operators -- 2.2.3 Tensor products of Hilbert spaces and operators -- 2.3 Types of Operators and their Reductions -- 2.4 Unitary Operators and Unitary Transforms -- 2.5 Extensions of Symmetric Operators -- 2.5.1 Selfadjoint and maximal symmetric extensions.
2.5.2 Von Neumann's formula for selfadjoint extensions -- 2.6 Probability and Expectation Values -- 2.6.1 Borel sets measures and measurable functions -- 2.6.2 Probability measures and probability functions -- 2.6.3 Expectation values variances and uncertainties -- 2.7 Spectral Measures and Probability -- 2.8 Selfadjointness and Spectral Decomposition -- 2.8.1 Spectral theorem -- 2.8.2 Functions of a selfadjoint operator -- 2.8.3 Spectra of selfadjoint operators -- 2.8.4 Spectral representation spaces and spectral representations of selfadjoint operators -- 2.9 Generalized Spectral Measures and Probability -- 2.10 Spectral Functions of Symmetric Operators -- 2.10.1 Symmetric operators and their spectral functions -- 2.10.2 Strictly maximal symmetric operators and their spectral functions -- 2.10.3 The square of maximal symmetric operators -- 2.10.4 Spectra of symmetric operators -- 2.11 Probability and Operators -- 2.11.1 Probability measures spectral measures and selfadjoint operators -- 2.11.2 Probability measures generalized spectral measures and strictly maximal symmetric operators -- 2.12 Local Operators in Coordinate Space -- 2.12.1 Definitions -- 2.12.2 Localization of bounded operators -- 2.12.3 Local operator algebras -- 2.12.4 Localization of unbounded operators 1 -- 2.12.5 Localization of unbounded operators 2 -- 2.12.6 Local momentum and local Hamiltonian -- 2.13 Direct Integrals of Hilbert Spaces -- 2.13.1 Discrete composition of Hilbert spaces -- 2.13.2 Continuous composition of Hilbert spaces -- 2.14 Direct Integrals of Operators -- 2.14.1 Direct sums of operators -- 2.14.2 Direct integrals of operators -- 2.14.3 Density operators -- 2.14.4 Statistical operators -- 2.15 Direct Integrals of Tensor Products -- 2.15.1 Direct integrals of tensor product Hilbert spaces.
2.15.2 Direct integrals and tensor product of operators -- References -- II Orthodox and Generalized Quantum Mechanics -- 3 Orthodox Quantum Mechanics -- 3.1 Introduction -- 3.1.1 Structure of physical theories -- 3.1.2 Mathematical framework of quantum mechanics -- 3.2 Orthodox Quantum Statics -- 3.2.1 Postulate on orthodox quantum statics -- 3.2.2 Pure and mixed states -- 3.2.3 Correlation between states -- 3.2.4 Discretization of bounded and unbounded observables -- 3.2.5 Approximate nature of measurements -- 3.3 Quantization in En -- 3.3.1 Preliminaries on quantization -- 3.3.2 Failure of general schemes -- 3.3.3 Complete momentum observables -- 3.3.4 Observables linear in momenta -- 3.3.5 Incomplete momentum observables -- 3.3.6 Kinetic energy and the Hamiltonian -- 3.3.7 Constraint and quantization in circuit geometry -- 3.4 Orthodox Quantum Dynamics -- 3.4.1 Postulate on orthodox quantum dynamics -- 3.4.2 Asymptotic localization and separation: Free systems -- 3.4.3 Asymptotic localization and separation: Scattering systems -- 3.5 Quantum State Preparation -- 3.5.1 The problem -- 3.5.2 Mathematical preliminaries -- 3.5.3 Ideal particle source -- 3.5.4 Random particle source -- 3.5.5 Extension to spin-1/2 particles -- 3.6 Quantum Measurement -- 3.6.1 Local position observables and their measurability -- 3.6.2 Reduction to local position measurements -- 3.6.3 Spectral separation for spinless particles -- 3.6.4 Spectral separation for spin-1/2 particles -- 3.6.5 Local position measurement as an ionization process -- 3.6.6 A model ionization propagator -- 3.6.7 Projection postulate local position measurements and uncertainty relations -- 3.6.8 Concluding remarks -- References -- 4 Physical Theory in Hilbert Space -- 4.1 Introduction -- 4.2 Unified Statics in Direct Integral Space.
4.2.1 A unified postulate on quantum statics -- 4.2.2 Discrete and continuous direct integral decompositions -- 4.3 States and Superposition Principle -- 4.3.1 Regular and singular states pure and mixed states -- 4.3.2 Coherence and superposition principle -- 4.3.3 Superselection rules their origins and classical observables -- 4.4 Unified Dynamics in Direct Integral Space -- 4.4.1 Preliminaries -- 4.4.2 Preserving dynamics -- 4.4.3 Non-preserving dynamics 1: Motivation -- 4.4.4 Linear functionals for state description -- 4.4.5 Extensions and restrictions of linear functionals -- 4.4.6 Non-preserving dynamics 2: A general scheme -- 4.4.7 Non-preserving evolution and environments -- 4.5 Classical Systems of Finite Order -- 4.5.1 First-order systems in Hilbert space -- 4.5.2 Second-order Hamiltonian systems in Hilbert space -- 4.6 Mixed Quantum Systems -- 4.6.1 A model system -- 4.6.2 Classification of physical systems -- 4.6.3 Quantum/Classical divide 1 -- 4.6.4 Equilibrium and mixed quantum systems -- 4.7 Coupling of Systems of Different Types -- 4.7.1 Measuring devices -- 4.7.2 Coupling of orthodox quantum and classical systems -- 4.7.3 Coupling of orthodox and mixed quantum systems -- 4.7.4 Coupling of classical and mixed quantum systems -- 4.8 Concluding Remarks -- References -- 5 Generalized Quantum Mechanics -- 5.1 Introduction -- 5.2 Maximal Symmetric Operators and Observables -- 5.2.1 Observables: Concept and description -- 5.2.2 Measurement of intrinsically unsharp observables -- 5.3 Approximate and Related Observables -- 5.3.1 Approximate observables -- 5.3.2 Related family of observables -- 5.4 Implications on Quantization -- 5.5 Time Operators and Uncertainty Relation -- 5.6 Local Values in Coordinate and in Phase Spaces -- 5.6.1 Expectation values in terms of local values.
5.6.2 Local values and semi-local observables -- 5.6.3 Local values in generalized phase space -- 5.7 Appendix on Maximal Probability Families -- 5.8 Appendix on Time Operators -- 5.9 Concluding Remarks -- References -- III Point Interactions Macroscopic Quantum Systems and Superselection Rules -- 6 Point Interactions -- 6.1 Introduction -- 6.2 Extensions of Symmetric Operators -- 6.3 Extensions of Direct Sum Operators -- 6.3.1 Direct sums and their selfadjoint extensions -- 6.3.2 Selfadjoint extensions in terms of boundary conditions -- 6.4 Quantization by Parts and Point Interactions -- 6.5 Classification of Point Interactions in E -- 6.5.1 Type 1 (BC1): The step potential -- 6.5.2 Type 2 (BC2): 8-interaction as high-pass filters -- 6.5.3 Type 3 (BC3): 8'-interaction as low-pass filters -- 6.5.4 Type 4 (BC4): Perfect reflector -- 6.5.5 Type 5 (BC5): Elastic reflectors -- 6.5.6 Type 6 (BC6): Open end -- 6.5.7 Type 7 (BC7): Ideal n-phase shifters -- 6.5.8 Type 8 (BC8): High-pass n-phase shifters -- 6.5.9 Type 9 (BC9): Low-pass n-phase shifters -- 6.5.10 Type 10 (BC10): Ideal mid-pass 1/2n-phase shifters -- 6.5.11 Type 11 (BC11): Partial mid-pass filter -- 6.5.12 Type 12 (BC12): Ideal tunable phase shifters -- 6.6 Remarks on Quantization by Parts -- 6.7 Charged Particles in Circular Motion -- 6.7.1 Charged particles constrained to move in a circle -- 6.7.2 Charged particles in 3-dimensions -- 6.8 Point Interactions in a Circle -- 6.8.1 Momentum operators -- 6.8.2 Hamiltonians with reflection symmetry -- 6.9 Classification of Point Interactions in C -- 6.9.1 Type 1 (BCC1): Free motion -- 6.9.2 Type 2 (BCC2): 8-interaction -- 6.9.3 Type 3 (BCC3): 8'-interaction -- 6.9.4 Type 4 (BCC4): Perfect reflector -- 6.9.5 Type 5 (BCC5): Elastic reflector -- 6.9.6 Type 6 (BCC6): Open end.
6.9.7 Type 7 (BCC7): Ideal dynamic n-phase shifter.
Özet:
Traditional quantum theory has a very rigid structure, making it difficult to accommodate new properties emerging from novel systems. This book presents a flexible and unified theory for physical systems, from micro and macro quantum to classical. This is achieved by incorporating superselection rules and maximal symmetric operators into the theory. The resulting theory is applicable to classical, microscopic quantum and non-orthodox mixed quantum systems of which macroscopic quantum systems are examples. A unified formalism also greatly facilitates the discussion of interactions between these systems. A scheme of quantization by parts is introduced, based on the mathematics of selfadjoint and maximal symmetric extensions of symmetric operators, to describe point interactions. The results are applied to treat superconducting quantum circuits in various configurations. This book also discusses various topics of interest such as the asymptotic treatment of quantum state preparation and quantum measurement, local observables and local values, Schrödinger's cat states in superconducting systems, and a path space formulation of quantum mechanics. This self-contained book is complete with a review of relevant geometric and operator theories, for example, vector fields and operators, symmetric operators and their maximal symmetric extensions, direct integrals of Hilbert spaces and operators. Sample Chapter(s). Chapter 1: Manifolds and Dynamical Systems (494 KB). Contents: Aspects of Geometric and Operator Theories: Manifolds and Dynamical Systems; Operators and Their Direct Integrals; Orthodox and Generalized Quantum Mechanics: Orthodox Quantum Mechanics; Physical Theory in Hilbert Space; Generalized Quantum Mechanics; Point Interactions, Macroscopic Quantum Systems and Superselection Rules: Point Interactions; Macroscopic Quantum Systems; Asymptotic
Disjointness, Asymptotic Separability, Quantum Mechanics on Path Space and Superselection Rules: Separability and Decoherence; Quantum Mechanics on Path Space. Readership: Theoretical and mathematical physicists, applied and pure mathematicians, physicists and philosophers of science (with an interest in quantum theory).
Notlar:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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