Foundations of Quantum Mechanics

Preface
Contents
Acronyms
1 Waves and Particles
1.1 Overview
1.2 The Schrödinger Equation
1.3 Unitary Operators in Hilbert Space
1.3.1 Existence and Uniqueness of Solutions of the Schrödinger Equation
1.3.2 The Time Evolution Operators
1.3.3 Unitary Matrices and Rotations
1.3.4 Inner Product
1.3.5 Abstract Hilbert Space
1.4 Classical Mechanics
1.4.1 Definition of Newtonian Mechanics
1.4.2 Properties of Newtonian Mechanics
1.4.3 Hamiltonian Systems
1.5 The Double-Slit Experiment
1.5.1 Classical Predictions for Particles and Waves
1.5.2 Actual Outcome of the Experiment
1.5.3 Feynman's Discussion
1.6 Bohmian Mechanics
1.6.1 Definition of Bohmian Mechanics
1.6.2 Properties of Bohmian Mechanics
1.6.3 Historical Overview
1.6.4 Equivariance
1.6.5 The Double-Slit Experiment in Bohmian Mechanics
1.6.6 Delayed-Choice Experiments
Afshar's Experiment
Exercises
References
2 Some Observables
2.1 Fourier Transform and Momentum
2.1.1 Fourier Transform
2.1.2 Momentum
2.1.3 Momentum Operator
2.1.4 Tunnel Effect
2.1.5 External Magnetic Field
2.2 Operators and Observables
2.2.1 Heisenberg's Uncertainty Relation
2.2.2 Limitation to Knowledge
2.2.3 Self-Adjoint Operators
2.2.4 The Spectral Theorem
2.2.5 Born's Rule
2.2.6 Conservation Laws in Quantum Mechanics
2.2.7 The Dirac Delta Function
2.3 Spin
2.3.1 Spinors and Pauli Matrices
2.3.2 The Pauli Equation
2.3.3 The Stern–Gerlach Experiment
2.3.4 Bohmian Mechanics with Spin
2.3.5 Is an Electron a Spinning Ball?
2.3.6 Are There Actual Spin Values?
2.3.7 Many-Particle Systems
2.3.8 Representations of SO(3)
2.3.9 Inverted Stern–Gerlach Magnet and Contextuality
Exercises
References
3 Collapse and Measurement
3.1 The Projection Postulate
3.1.1 Notation
3.1.2 The Projection Postulate
3.1.3 Projection and Eigenspace
3.1.4 Position Measurements
3.1.5 Consecutive Quantum Measurements
3.2 The Measurement Problem
3.2.1 What the Problem Is
3.2.2 How Bohmian Mechanics Solves the Problem
3.2.3 Decoherence
3.2.4 Schrödinger's Cat
3.2.5 Positivism and Realism
3.2.6 Experiments and Operators
3.3 The GRW Theory
3.3.1 The Poisson Process
3.3.2 Definition of the GRW Process
3.3.3 Definition of the GRW Process in Formulas
3.3.4 Primitive Ontology
3.3.5 How GRW Theory Solves the Measurement Problem
3.3.6 Empirical Tests
3.3.7 The Need for a Primitive Ontology
3.4 The Copenhagen Interpretation
3.4.1 Two Realms
3.4.2 Elements of the Copenhagen View
Positivism
Purported Impossibility of Non-paradoxical Theories
Completeness of the Wave Function
Language of Measurement
Narratives, But No Serious Ones
3.4.3 Complementarity
3.4.4 Reactions to the Measurement Problem
3.4.5 The Transactional Interpretation
3.5 Many Worlds
3.5.1 Schrödinger's Many-Worlds Theory
3.5.2 Everett's Many-Worlds Theory
3.5.3 Bell's First Many-Worlds Theory
3.5.4 Bell's Second Many-Worlds Theory
3.5.5 Probabilities in Many-Worlds Theories
3.6 Some Morals
3.7 Special Topics
3.7.1 Einstein's View
3.7.2 The Mach–Zehnder Interferometer
3.7.3 Path Integrals
3.7.4 Boundary Conditions
3.7.5 Point Interaction
3.7.6 No-Cloning Theorem
3.7.7 Aharonov–Bergmann–Lebowitz TimeReversal Symmetry
Exercises
References
4 Nonlocality
4.1 The Einstein–Podolsky–Rosen Argument
4.1.1 The EPR Argument
4.1.2 Square-Integrable Version
4.1.3 Further Conclusions
4.1.4 Bohm's Version of the EPR Argument Using Spin
4.1.5 Einstein's Boxes Argument
4.1.6 Too Good to Be True
4.2 Proof of Nonlocality
4.2.1 Bell's Experiment
4.2.2 Bell's 1964 Proof of Nonlocality
4.2.3 Bell's 1976 Proof of Nonlocality
4.3 Discussion of Nonlocality
4.3.1 Nonlocality in Bohmian Mechanics, GRW, Copenhagen, and Many-Worlds
4.3.2 Popular Myths About Bell's Theorem
4.3.3 Simultaneous Quantum Measurements
4.4 Special Topics
4.4.1 Bohr's Reply to EPR
4.4.2 The Frauchiger–Renner Paradox
Exercises
References
5 General Observables
5.1 POVMs: General Observables
5.1.1 Definition
5.1.2 The Main Theorem About POVMs
5.1.3 Limitations to Knowledge
5.1.4 Limitations to Knowledge as a General Fact
5.1.5 Limitations to Knowledge in Theories We Know
5.1.6 The Concept of Observable
5.2 Time of Detection
5.2.1 The Problem
5.2.2 The Quantum Zeno Effect
5.2.3 Allcock's Paradox
5.2.4 The Absorbing Boundary Rule
5.2.5 Time–Energy Uncertainty Relation
5.2.6 Historical Notes
5.3 Ontic Versus Epistemic
5.3.1 The Pusey–Barrett–Rudolph Theorem
5.4 Density Matrix and Mixed State
5.4.1 Trace
5.4.2 The Trace Formula in Quantum Mechanics
5.4.3 Pure and Mixed States
5.4.4 Empirically Equivalent Distributions
5.4.5 Density Matrix and Dynamics
5.5 Reduced Density Matrix and Partial Trace
5.5.1 Tensor Product
5.5.2 Definition of the Reduced Density Matrix
5.5.3 Partial Trace
5.5.4 The Trace Formula Again
5.5.5 The Measurement Problem and Density Matrices
5.5.6 POVM and Collapse
5.5.7 Completely Positive Superoperators
5.5.8 The Main Theorem About Superoperators
5.5.9 The No-Signaling Theorem
5.5.10 Canonical Typicality
5.5.11 The Possibility of a Fundamental Density Matrix
5.6 Quantum Logic
5.6.1 Boolean Algebras
5.6.2 Quantum Measures
5.7 No-Hidden-Variables Theorems
5.7.1 Bell's NHVT
5.7.2 Von Neumann's NHVT
5.7.3 Gleason's NHVT
5.7.4 Hidden Variables and Ontology
5.8 Special Topics
5.8.1 The Decoherent Histories Interpretation
5.8.2 The Hilbert–Schmidt Inner Product
Exercises
References
6 Particle Creation
6.1 Identical Particles
6.1.1 Symmetrization Postulate
6.1.2 Schrödinger Equation and Symmetry
6.1.3 The Space of Unordered Configurations
6.1.4 Identical Particles in Bohmian Mechanics
6.1.5 Identical Particles in GRW Theory
6.2 Hamiltonians of Particle Creation
6.2.1 Configuration Space of a Variable Number of Particles
6.2.2 Fock Space
The Fock Space of Spinless Bosons
The Fock Space of Spinless Fermions
General Fock Space
Two Species
6.2.3 Example: Emission–Absorption Model
6.2.4 Creation and Annihilation Operators
6.2.5 Ultraviolet Divergence
6.3 Particle Creation as Such
6.3.1 Jumps
6.3.2 Bell's Jump Process
6.3.3 Virtual Particles
6.3.4 GRW Theory and Many-Worlds in Fock Space
6.4 Interior-Boundary Conditions
6.4.1 What an IBC Is
6.4.2 Configuration Space with Two Sectors
Hilbert Space
Spherical Coordinates
Probability Transport
Hamiltonian and IBC
Delta Contribution
6.4.3 All Sectors
Hamiltonian
Jump Process
Ground State
6.5 A Brief Look at Quantum Field Theory
6.5.1 Problems of Quantum Field Theory
6.5.2 Field Ontology vs. Particle Ontology
Exercises
References
7 Relativity
7.1 Brief Introduction to Relativity
7.1.1 Galilean Relativity
7.1.2 Minkowski Space
7.1.3 Dual Space
7.1.4 Arc Length
7.1.5 Index Contraction
7.1.6 Classical Electrodynamics as a Paradigm of a Relativistic Theory
7.1.7 Cauchy Surfaces
7.1.8 Outlook on General Relativity
7.2 Relativistic Schrödinger Equations
7.2.1 The Klein-Gordon Equation
Fourier Transform
Dispersion Relation
The Klein-Gordon Equation
Positive Energy Solutions
7.2.2 Two-Spinors and Four-Vectors
Two-Spinors and Three-Vectors
Action of Lorentz Transformations
Conjugate Vector Space
Relation to 4-Vectors
Lorentz-Invariant Product
7.2.3 The Weyl Equation
Relation to the Klein-Gordon Equation
7.2.4 The Dirac Equation
Relation to the Klein-Gordon Equation
Lorentz Invariance
7.3 Probability
7.3.1 Current for the Weyl Equation
7.3.2 Current for the Dirac Equation
7.3.3 Probability Flow
Equation of Motion
Surface Equivariance
7.3.4 Evolution Between Cauchy Surfaces
7.3.5 Propagation Locality
7.3.6 External Fields
7.3.7 Non-Relativistic Limit
7.3.8 Probability and the Klein-Gordon Equation
Psi Squared
The Klein-Gordon Current
7.3.9 The Maxwell Equation as the Schrödinger Equation for Photons
Locally Plane Waves
The Poynting Vector
One Over Root omega
The Kappa Operator
Desiderata
7.4 Many Particles
7.4.1 Multi-Time Wave Functions
7.4.2 Surface Wave Functions
7.5 Which Theories Count as Relativistic?
7.5.1 Lorentz Invariance
7.5.2 Other Relativistic Properties
7.5.3 Relativistic Quantum Theories Without Observers
7.5.4 The Time Foliation
7.6 Bohmian Mechanics in Relativistic Space-Time
7.6.1 Law of Motion
7.6.2 Equivariance
Intersection Probability and Detection Probability
No Signaling
7.6.3 The Spin-0 Case
Definition of the Current Tensor
Trajectories
Time Travel
7.7 Predictions in Relativistic Space-Time
7.7.1 Is Collapse Compatible with Relativity?
The Aharonov-Albert Wave Function
Other Approaches to Relativistic Collapse
7.7.2 Tunneling Speed
7.8 GRW Theory in Relativistic Space-Time
7.8.1 1-Particle Case
Ingredients
Definition
POVM
7.8.2 The Case of N Non-Interacting Particles
Definition
Properties
Collapsed Wave Function
Criticisms
7.8.3 Interacting Particles
7.8.4 Matter Density
7.9 Other Approaches
7.9.1 Many-Worlds in Relativistic Space-Time
7.9.2 Wormholes as an Alternative?
Exercises
References
8 Further Morals
8.1 Controversy
8.2 Do We Need Ontology?
8.3 What If Two Theories Are Empirically Equivalent?
8.4 Positivism and Realism
8.5 If It Makes the Same Predictions, What Is It Good For?
8.6 Concluding Remarks
References
A Appendix
A.1 Topological View of the Symmetrization Postulate
A.2 Ultraviolet Divergence Problem in Classical Electrodynamics
A.3 Nelson's Stochastic Mechanics
A.4 Probability and Typicality in Bohmian Mechanics
A.4.1 Empirical Distributions in Bohmian Mechanics
The Law of Large Numbers
General Statement
A.4.2 Typicality
A.4.3 The Explanation of Quantum Equilibrium
A.4.4 Historical Notes
Quantum Potential
``And Then Throws It Away''
A.5 Philosophical Topics
A.5.1 Free Will
Free Will and Determinism
Free Will and Nonlocality
Conway and Kochen's ``Free Will Theorem''
A.5.2 Causation
A.5.3 The Mind-Body Problem
A.6 Differential Forms
References
Index