Local Quantum Measurement and Relativity

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Acknowledgements Contents 1 Introduction and Outline 2 Quantum Measurement the Emergence of POVMs and State Transformers 2.1 Groundwork 2.1.1 Motivation 2.1.2 Pointer States 2.2 Measurement(-like) Process 2.2.1 Efficient Measurements 2.2.2 Non-efficient Measurements 2.3 Intermezzo: How (not) to Understand Quantum Theory On the Results of Kochen and Specker and Bell 2.3.1 Against Naive Ignorance Interpretations 2.3.2 Nonlocality 2.4 Measurement Schemes 2.4.1 Projective Measurements I Ideal Measurements 2.4.2 Projective Measurements II Reproducibility and Pauli's Measurements of the First and Second Kind 2.4.3 Non-Projective Measurements I The Approximate Measurement Scheme 2.4.4 Non-projective Measurements II Indirect Measurement: The Ancilla Scheme 2.5 Modelling Experiments 2.5.1 Stern-Gerlach as Indirect Measurement 2.5.2 Von Neumann Measurements 2.6 Experimental Developments 2.6.1 Discrimination of Non-orthogonal States 2.6.2 Phase Space Observables 2.6.3 Weak Measurements 2.7 A General Operational Framework 2.7.1 Generic State Transitions: Superoperators and Complete Positivity 2.7.2 Quantum Operations: Channels and Instruments 2.7.3 Purification 2.7.4 Representations of CPMs and POVMs 2.7.5 Are all Selfadjoint Operators Observable? 2.7.6 Remark on Decoherence and the Measurement Problem 3 Local Quantum Measurement I Relativistic Consistency, No-Signalling and Local Commutativity 3.1 Quantum Measurements and Relativity 3.2 Relativistic Consistency and No-Signalling 3.3 The Meaning of Local Commutativity 3.3.1 Sufficiency 3.3.2 Necessity 3.3.3 Summary and Sketch of Further Results 3.4 Non-efficient Measurements 3.5 A Formal Counterexample 3.6 Conclusion 4 Local Quantum Measurement II Relativity and Particle Detectors 4.1 Introduction 4.2 Mathematical Groundwork 4.2.1 Some Analysis of Several Complex Variables 4.2.2 Hilbert Space, Space-Time Translations and the Spectrum Condition 4.2.3 The Basic Result 4.3 Causal Propagation, Positive Energy, Infinite Tails and Local Transformations 4.4 Malament Type Theorems 4.4.1 Prelude on Operators 4.4.2 Detecor Click Statistics 4.4.3 Projections: Malament & Detectors 4.4.4 N-States 4.4.5 Weakly Unsharp Effects: Busch's Result 4.4.6 General Effects: Halvorson–Clifton and Detectors 4.4.7 How To Understand? 4.5 Local Measurements Destroy the Vacuum: Reeh–Schlieder 4.5.1 AQFTs and the Present Framework 4.5.2 The Theorem and Implications 5 An Active Notion of Localization 5.1 Passive and Active Notions of Localization 5.2 Bohmian Mechanics in a Nutshell 5.3 Subsystems 5.3.1 Conditional Wave Function 5.3.2 Effective Collapse 5.4 Statistical Analysis 5.4.1 Measures of Typicality 5.4.2 Equivariance 5.4.3 Subsystems Again 5.4.4 Empirical Distributions 5.5 Measurement-Like Process Revisited 5.6 Relativistic Bohmian Mechanics 5.6.1 Lorentz Invariance 5.6.2 Quantum Field Theory 5.7 Getting Localized Appendix A Second Quantization in the Dirac Sea Picture Appendix B Hegerfeldt Theorem Faster Than Light Spreading Probability Distributions References Important Notions and Theorems