From micro to macro quantum systems : a unified formalism with superselection rules and its applications / K. Kong Wan.

Format:

Book

Author/Creator:

Wan, K. Kong.

Language:

English

Subjects (All):

Quantum theory.

Physical Description:

xix, 688 pages ; 24 cm

Place of Publication:

London : Imperial College Press, [2006]

Summary:

Traditional quantum theory has a very rigid structure, making if 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, Schrodinger'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.

Contents:

I Aspects of Geometric and Operator Theories

1 Manifolds and Dynamical Systems 3

1.1 Topological Spaces and Topological Equivalence 4

1.1.1 Basic concepts and definitions 4

1.1.2 Topological equivalence 9

1.2 Euclidean Spaces 11

1.2.1 Basic concepts and definitions 11

1.2.2 Coordinate systems and coordinate transformations 13

1.2.3 Contravariant and covariant vectors in IE[superscript n] 15

1.2.4 Contravariant, covariant and mixed tensors 17

1.3 Differential Operators, Vectors and Fields 20

1.3.1 Differential operators and derivations 21

1.3.2 Tangent vectors, tangent vector fields and their integral curves 26

1.3.3 Transformation groups and complete vector fields 35

1.4 Cotangent Vectors and Differential Forms 40

1.4.1 Cotangent vectors, differentials and one-forms 41

1.4.2 Tensor fields and two-forms 47

1.4.3 Exterior differentiation 51

1.4.4 Interior products, closed and exact forms 53

1.5 Differentiable Manifolds 56

1.5.1 Definition and examples 56

1.5.2 Riemannian manifolds 60

1.5.3 Hamiltonian manifolds 63

1.6 Classical Dynamical Systems 67

1.6.1 Classical systems of finite order 67

1.6.2 First-order systems 68

1.6.3 Second-order Hamiltonian systems 69

1.6.4 Momentum observables, vector fields and operators 73

2 Operators and their Direct Integrals 81

2.1 Hilbert Spaces 81

2.2 Operators: Basic Definitions 87

2.2.1 Boundedness, adjoints, extensions and restrictions, continuity and closure 87

2.2.2 Convergence of a family of bounded operators 92

2.2.3 Tensor products of Hilbert spaces and operators 94

2.3 Types of Operators and their Reductions 97

2.4 Unitary Operators and Unitary Transforms 107

2.5 Extensions of Symmetric Operators 113

2.5.1 Selfadjoint and maximal symmetric extensions 113

2.5.2 Von Neumann's formula for selfadjoint extensions 128

2.6 Probability and Expectation Values 130

2.6.1 Borel sets, measures and measurable functions 132

2.6.2 Probability measures and probability functions 137

2.6.3 Expectation values, variances and uncertainties 140

2.7 Spectral Measures and Probability 142

2.8 Selfadjointness and Spectral Decomposition 148

2.8.1 Spectral theorem 148

2.8.2 Functions of a selfadjoint operator 154

2.8.3 Spectra of selfadjoint operators 157

2.8.4 Spectral representation spaces and spectral representations of selfadjoint operators 161

2.9 Generalized Spectral Measures and Probability 167

2.10 Spectral Functions of Symmetric Operators 170

2.10.1 Symmetric operators and their spectral functions 170

2.10.2 Strictly maximal symmetric operators and their spectral functions 173

2.10.3 The square of maximal symmetric operators 175

2.10.4 Spectra of symmetric operators 178

2.11 Probability and Operators 180

2.11.1 Probability measures, spectral measures and selfadjoint operators 180

2.11.2 Probability measures, generalized spectral measures and strictly maximal symmetric operators 183

2.12 Local Operators in Coordinate Space 185

2.12.2 Localization of bounded operators 187

2.12.3 Local operator algebras 188

2.12.4 Localization of unbounded operators 1 191

2.12.5 Localization of unbounded operators 2 192

2.12.6 Local momentum and local Hamiltonian 195

2.13 Direct Integrals of Hilbert Spaces 195

2.13.1 Discrete composition of Hilbert spaces 196

2.13.2 Continuous composition of Hilbert spaces 198

2.14 Direct Integrals of Operators 209

2.14.1 Direct sums of operators 209

2.14.2 Direct integrals of operators 213

2.14.3 Density operators 218

2.14.4 Statistical operators 221

2.15 Direct Integrals of Tensor Products 224

2.15.1 Direct integrals of tensor product Hilbert spaces 224

2.15.2 Direct integrals and tensor product of operators 225

II Orthodox and Generalized Quantum Mechanics

3 Orthodox Quantum Mechanics 231

3.1.1 Structure of physical theories 231

3.1.2 Mathematical framework of quantum mechanics 234

3.2 Orthodox Quantum Statics 236

3.2.1 Postulate on orthodox quantum statics 236

3.2.2 Pure and mixed states 239

3.2.3 Correlation between states 246

3.2.4 Discretization of bounded and unbounded observables 248

3.2.5 Approximate nature of measurements 250

3.3 Quantization in IE[superscript n] 252

3.3.1 Preliminaries on quantization 252

3.3.2 Failure of general schemes 256

3.3.3 Complete momentum observables 259

3.3.4 Observables linear in momenta 269

3.3.5 Incomplete momentum observables 272

3.3.6 Kinetic energy and the Hamiltonian 275

3.3.7 Constraint and quantization in circuit geometry 282

3.4 Orthodox Quantum Dynamics 286

3.4.1 Postulate on orthodox quantum dynamics 286

3.4.2 Asymptotic localization and separation: Free systems 290

3.4.3 Asymptotic localization and separation: Scattering systems 294

3.5 Quantum State Preparation 300

3.5.1 The problem 300

3.5.2 Mathematical preliminaries 302

3.5.3 Ideal particle source 303

3.5.4 Random particle source 305

3.5.5 Extension to spin-1/2 particles 307

3.6 Quantum Measurement 310

3.6.1 Local position observables and their measurability 310

3.6.2 Reduction to local position measurements 313

3.6.3 Spectral separation for spinless particles 314

3.6.4 Spectral separation for spin-1/2 particles 319

3.6.5 Local position measurement as an ionization process 320

3.6.6 A model ionization propagator 324

3.6.7 Projection postulate, local position measurements and uncertainty relations 328

4 Physical Theory in Hilbert Space 337

4.2 Unified Statics in Direct Integral Space 338

4.2.1 A unified postulate on quantum statics 339

4.2.2 Discrete and continuous direct integral decompositions 339

4.3 States and Superposition Principle 341

4.3.1 Regular and singular states, pure and mixed states 341

4.3.2 Coherence and superposition principle 344

4.3.3 Superselection rules, their origins and classical observables 345

4.4 Unified Dynamics in Direct Integral Space 351

4.4.2 Preserving dynamics 352

4.4.3 Non-preserving dynamics 1: Motivation 356

4.4.4 Linear functionals for state description 358

4.4.5 Extensions and restrictions of linear functionals 361

4.4.6 Non-preserving dynamics 2: A general scheme 364

4.4.7 Non-preserving evoluation and environments 367

4.5 Classical Systems of Finite Order 368

4.5.1 First-order systems in Hilbert space 368

4.5.2 Second-order Hamiltonian systems in Hilbert space 373

4.6 Mixed Quantum Systems 378

4.6.1 A model system 378

4.6.2 Classification of physical systems 379

4.6.3 Quantum/Classical divide 1 381

4.6.4 Equilibrium and mixed quantum systems 383

4.7 Coupling of Systems of Different Types 384

4.7.1 Measuring devices 384

4.7.2 Coupling of orthodox quantum and classical systems 385

4.7.3 Coupling of orthodox and mixed quantum systems 388

4.7.4 Coupling of classical and mixed quantum systems 390

5 Generalized Quantum Mechanics 395

5.2 Maximal Symmetric Operators and Observables 400

5.2.1 Observables: Concept and description 400

5.2.2 Measurement of intrinsically unsharp observables 406

5.3 Approximate and Related Observables 407

5.3.1 Approximate observables 407

5.3.2 Related family of observables 408

5.4 Implications on Quantization 409

5.5 Time Operators and Uncertainty Relation 409

5.6 Local Values in Coordinate and in Phase Spaces 413

5.6.1 Expectation values in terms of local values 413

5.6.2 Local values and semi-local observables 415

5.6.3 Local values in generalized phase space 418

5.7 Appendix on Maximal Probability Families 420

5.8 Appendix on Time Operators 423

III Point Interactions, Macroscopic Quantum Systems and Superselection Rules

6 Point Interactions 431

6.2 Extensions of Symmetric Operators 433

6.3 Extensions of Direct Sum Operators 435

6.3.1 Direct sums and their selfadjoint extensions 435

6.3.2 Selfadjoint extensions in terms of boundary conditions 439

6.4 Quantization by Parts and Point Interactions 443

6.5 Classification of Point Interactions in IE 446

6.5.1 Type 1 (BC1): The step potential 450

6.5.2 Type 2 (BC2): [delta]-interaction as high-pass filters 451

6.5.3 Type 3 (BC3): [delta]'-interaction as low-pass filters 455

6.5.4 Type 4 (BC4): Perfect reflector 463

6.5.5 Type 5 (BC5): Elastic reflectors 464

6.5.6 Type 6 (BC6): Open end 464

6.5.7 Type 7 (BC7): Ideal [pi]-phase shifters 465

6.5.8 Type 8 (BC8): High-pass [pi]-phase shifters 467

6.5.9 Type 9 (BC9): Low-pass [pi]-phase shifters 469

6.5.10 Type 10 (BC10): Ideal mid-pass 1/2[pi]-phase shifters 471

6.5.11 Type 11 (BC11): Partial mid-pass filter 473

6.5.12 Type 12 (BC12): Ideal tunable phase shifters 476

6.6 Remarks on Quantization by Parts 478

6.7 Charged Particles in Circular Motion 480

6.7.1 Charged particles constrained to move in a circle 480

6.7.2 Charged particles in 3-dimensions 486

6.8 Point Interactions in a Circle 489

6.8.1 Momentum operators 490

6.8.2 Hamiltonians with reflection symmetry 491

6.9 Classification of Point Interactions in C 495

6.9.1 Type 1 (BCC1): Free motion 495

6.9.2 Type 2 (BCC2): [delta]-interaction 495

6.9.3 Type 3 (BCC3): [delta]'-interaction 497

6.9.4 Type 4 (BCC4): Perfect reflector 498

6.9.5 Type 5 (BCC5): Elastic reflector 498

6.9.6 Type 6 (BCC6): Open end 499

6.9.7 Type 7 (BCC7): Ideal dynamic [pi]-phase shifter 500

6.9.8 Type 8 (BCC8): Static [pi]-phase shifter 500

6.9.9 Type 9 (BCC9): Gradient [pi]-phase shifter 501

6.9.10 Type 10 (BCC10): Ideal 1/2[pi]-phase shifter 502

6.9.11 Type 11 (BCC11): Static junction correlator 503

6.9.12 Type 12 (BCC12): Ideal tunable phase shifters 504

6.10 Current and Stationary States in a Circle 505

7 Macroscopic Quantum Systems 509

7.1 Single-Particle Representation 509

7.2 Macroscopic Wave Function Hypothesis 512

7.3 Uniformly Thick Superconducting Rings 513

7.3.1 Physical properties 513

7.3.2 Superconducting rings: Preliminaries 514

7.3.3 Superconducting rings as equilibrium mixed quantum systems 520

7.4 Superconducting Rings with a Junction 522

7.4.1 Josephson junction and dc Josephson effect 522

7.4.2 Supercurrent and magnetic flux operators 524

7.4.3 The Hamiltonian: Preliminary results 525

7.4.4 Superconducting ring with a Josephson junction as an equilibrium mixed quantum system 528

7.4.5 Superconducting ring with a [pi]-junction 530

7.4.6 Superconducting ring with a 1/2[pi]-junction 530

7.4.7 Superconducting ring with a Josephson junction in an external magnetic field 531

7.5 Feynman's Derivation of Josephson's Equation 533

7.6 Superconducting Wire with a Junction 535

7.6.1 Point interactions 535

7.6.2 Momentum and supercurrent operators 535

7.6.3 Hamiltonian operator 1: [pi]-junction 536

7.6.4 Hamiltonian operator 2: 1/2[pi]-junction 536

7.6.5 Hamiltonian operator 3: Josephson junction 537

7.6.6 Superconducting wire with a Josephson junction as a mixed equilibrium quantum system 539

7.7 Y-Shape Circuits 542

7.7.1 Momentum and supercurrent operators: Special cases 542

7.7.2 Hamiltonian operators: Special cases 544

7.7.3 Physics of strictly maximal symmetric operators 544

7.7.4 Momentum and supercurrent operators: General cases 545

7.7.5 Hamiltonian operators: General cases 547

7.7.6 Correlation 547

7.7.7 Superselection rules 548

7.7.8 Condensate in a pure or in a mixed state 549

7.8 Continuous Y-Shape Circuit 551

7.9 Superconducting Quantum Interference Devices 552

7.10 Non-Equilibrium Mixed Quantum System 554

7.11 BCS Theory and Superselection Rules 558

7.12 Conceptual Analyses 562

7.12.1 Non-uniqueness of quantization 562

7.12.2 Y-shape circuits, equilibrium mixed quantum systems and non-locality 562

7.12.3 Equilibrium states, globalization and non-locality 566

7.12.4 Quantum/Classical divide 2 568

7.13 Orthodox Quantum Systems 569

7.14 Prospects and Other Approaches 573

IV Asymptotic Disjointness, Asymptotic Separability, Quantum Mechanics on Path Space and Superselection Rules

8 Separability and Decoherence 581

8.2 Scattering Systems and de Broglie Paradox 586

8.2.1 Scattering systems 586

8.2.2 de Broglie paradox 587

8.3 Schrodinger's Cat States 589

8.3.1 Classical-like states 589

8.3.2 Classical cats and their states 592

8.3.3 Quantum cats and their states 593

8.3.4 Disjointness and Schrodinger's cat states 594

8.3.5 Scattering systems and Schrodinger's cat states 595

8.3.6 Quantized oscillator and Schrodinger's cat states 595

8.3.7 Weak Schrodinger's cat states 597

8.3.8 Periodic Schrodinger's cat states 599

8.3.9 Double-well potentials and chiral molecules 601

8.3.10 Dynamic and asymptotic decoherence 607

8.4 Superconducting Schrodinger's Cat States 608

8.4.1 Breakdown of superselection rules and capacitive junction 608

8.4.2 Schrodinger's cat states in superconducting systems 618

8.5 Asymptotically Separable Quantum Theory 620

8.5.1 Motivation 620

8.5.2 Asymptotically separable quantum mechanics 620

8.6 Entanglement and Decoherence 622

8.6.1 Distinguishable particles 623

8.6.2 Identical Fermions and Pauli exclusion principle 626

8.7 Chronological Disordering 629

8.7.1 The concept of chronological disordering 629

8.7.2 Two-particle correlation and conservation laws 630

9 Quantum Mechanics on Path Space 637

9.2 Physical Space and Path Space 638

9.3 Functions on Path Space 644

9.4 Quantum Mechanics on Path Space 649

9.4.1 Hilbert spaces H[subscript gamma]([Pi](C)) on path space [Pi](C) 649

9.4.2 Comparing H[subscript gamma]([Pi](C)) and L[superscript 2](C[subscript c]) 651

9.4.3 Position operators in H[subscript gamma]([Pi](C)) 654

9.4.4 Momentum operators in H[subscript gamma]([Pi](C)) 654

9.5 Josephson Effect and Superselection Rules 655.

Notes:

Includes bibliographical references (pages 659-673) and index.

ISBN:

1860946259

OCLC:

69243878