1. Waves and Particles
1.1 Overview
1.2 The Schrodinger Equation
1.3 Unitary Operators in Hilbert Space
1.3.1 Existence and Uniqueness of Solutions of the Schrodinger 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 Space1.4 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 Experiment1.5.1 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 Mechanics1.6.2 Historical Overview
1.6.3 Equivariance
1.6.4 The Double Slit Experiment in Bohmian Mechanics1.6.5 Delayed Choice Experiments
Summary
Exercises
References
2. Some Observables
2.1 Fourier Transform and Momentum
2.1.1 Fourier Transform
2.1.2 Momentum2.1.3 Momentum Operator
2.1.4 Tunnel Effect
2.2 Operators and Observables2.2.1 Heisenberg's Uncertainty Relation
2.2.2 Self-Adjoint Operators
2.2.3 The Spectral Theorem
2.2.4 Conservation Laws in Quantum Mechanics
2.3 Spin2.3.1 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 Is There an Actual Spin Vector?
2.3.7 Many-Particle Systems
2.3.8 Representations of SO(3)2.3.9 Inverted Stern-Gerlach Magnet and Contextuality
Summary
Exercises
References
3. Collapse and Measurement3.1 The Projection Postulate
3.1.1 Notation
3.1.2 The Projection Postulate
3.1.3 Projection and Eigenspace
3.1.4 Remarks3.2 The Measurement Problem
3.2.1 What the Problem Is
3.2.2 How Bohmian Mechanics Solves the Measurement Problem
3.2.3 Decoherence
3.2.4 Schrodinger's Cat3.2.5 Positivism and Realism
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 Tests3.3.7 The Need for a Primitive Ontology
3.4 The Copenhagen Interpretation
3.4.1 Two Realms3.4.2 Positivism
3.4.3 Purported Impossibility of Non-Paradoxical Theories
3.4.4 Completeness of the Wave Function3.4.5 Language of Measurement
3.4.6 Complementarity
3.4.7 Complementarity and Non-Commuting Operators3.4.8 Reactions to the Measurement Problem
3.5 Many Worlds
3.5.1 Schrodinger's Many-Worlds Theory3.5.2 Everett's Many-Worlds Theory
3.5.3 Bell's First Many-Worlds Theory
3.5.4 Bell's Second Many-Worlds Theory3.5.5 Probabilities in Many-World Theories
3.6 Special Topics
3.6.1 The Mach-Zehnder Interferometer3.6.2 Path Integrals
3.6.3 Point Interactions
3.6.4 No-Cloning Theorem3.6.5 Boundary Conditions
3.6.6 Aharonov-Bergmann-Lebowitz Symmetry and Two-State Vector Formalism
Summary
Exercises
References
4. Nonlocality
4.1 The Einstein-Podolsky-Rosen Argument
4.1.1 The EPR Argument
4.1.2 Further Conclusions
4.1.3 Bohm's Version of the EPR Argument Using Spin4.1.4 Einstein's Boxes Argument
4.1.5 Too Good to Be True
4.2 Proof of Nonlocality4.2.1 Bell's Experiment
4.2.2 Bell's 1964 Proof of Nonlocality
4.2.3 Bell's 1976 Proof of Nonlocality
4.2.4 Photons
4.3 Discussion of Nonlocality4.3.1 Nonlocality in Bohmian Mechanics, GRW, Copenhagen, Many-Worlds
4.3.2 Popular Myths About Bell's Proof
4.3.3 Bohr's Reply to EPR
Summary
Exercises
References
5. General Observables
5.1 POVMs: Generalized Observables
5.1.1 Definition
5.1.2 The Main Theorem About POVMs5.1.3 Limitations to Knowledge
5.1.4 The Concept of Observable
5.2 Time of Detection5.2.1 The Problem
5.2.2 The Quantum Zeno Effect
5.2.3 The Absorbing Boundary Rule
5.2.4 Historical Overview
5.3 Density Matrix5.3.1 Trace
5.3.2 The Trace Formula in Quantum Mechanics
5.3.3 Limitations to Knowledge
5.3.4 Density Matrix and Dynamics
5.4 Reduced Density Matrix and Partial Trace5.4.1 Partial Trace
5.4.2 The Trace Formula
5.4.3 Statistical Reduced Density Matrix
5.4.4 The Measurement Problem and Density Matrices
5.4.5 The No-Signaling Theorem5.4.6 Completely Positive Superoperators
5.4.7 Canonical Typicality
5.4.8 The Possibility of a Fundamental Density Matrix5.5 Quantum Logic
5.6 No-Hidden-Variables Theorems
5.6.1 Bell's NHVT
5.6.2 Von Neumann's NHVT
5.6.3 Gleason's NHVT
5.7 The Pusey-Barrett-Rudolph Theorem
5.8 The Decoherent Histories Interpretation
Summary
Exercises
References
6. Particle Creation
6.1 Identical Particles
6.1.1 Symmetrization Postulate
6.1.2 Schrodinger Equation and Symmetry
6.1.3 The Space of Unordered Configurations6.1.4 Identical Particles in Bohmian Mechanics
6.1.5 Identical Particles in GRW Theory
6.2 Particle Creation6.2.1 Configuration Space of a Variable Number of Particles
6.2.2 Fock Space
6.2.3 Example: Emission-Absorption Model
6.2.4 Creation and Annihilation Operators
6.2.5 Ultraviolet Divergence6.2.6 Bell's Jump Process
6.2.7 Determinism vs. Stochasticism
6.2.8 GRW Theory and Fock Space6.2.9 Many Worlds and Fock Space
6.2.10 Interior-Boundary Conditions
6.3 A Brief Look at Quantum Field Theory6.3.1 Historical Overview
6.3.2 Field Ontology vs. Particle Ontology
6.3.3 Scattering and the Dyson Series
6.3.4 Renormalization
SummaryExercises
References
7. Relativity
7.1 Brief Introduction to Relativity
7.1.1 Galilean Relativity
7.1.2 Minkowski Space
7.1.3 Arc Length
7.1.4 Classical Electrodynamics as a Paradigm of a Relativistic Theory
7.1.5 Cauchy Surfaces
7.1.6 Outlook on General Relativity7.2 Relativistic Schrodinger Equations
7.2.1 The Klein-Gordon Equation
7.2.2 Two-Spinors and Four-Vectors
7.2.3 The Weyl Equation
7.2.4 The Dirac Equation7.2.5 Bohmian Trajectories for the 1-Particle Weyl and Dirac Equations
7.2.6 Probability Conservation
7.2.7 Multi-Time Wave Functions7.2.8 Hypersurface Wave Functions
7.2.9 The Maxwell Equation as the Schrodinger Equation for Photons
7.3 Bohmian Mechanics in Relativistic Space-Time7.3.1 Law of Motion
7.3.2 Equivariance
7.3.3 Intersection Probability and Detection Probability
7.3.4 Possible Laws Governing the Time Foliation
7.3.5 Does This Count as Relativistic?7.4 Predictions in Relativistic Space-Time
7.4.1 Is Collapse Incompatible with Relativity?
7.4.2 Joint Distribution of Outcomes of Local Experiments
7.4.3 The Aharonov-Albert Wave Function
7.4.4 Tunneling Times
7.5 GRW Theory in Relativistic Space-Time
7.5.1 1-Particle Case
7.5.2 The Case of N Non-Interacting Particles7.5.3 Nonlocality in Relativistic GRW Theory
7.5.4 Interacting Particles
7.5.5 Primitive Ontology7.5.6 Which Theories Count as Relativistic?
7.6 Copenhagen Interpretation in Relativistic Space-Time
7.7 Many-Worlds in Relativistic Space-Time7.8 Special Topics
7.8.1 Multi-Time Equations of Particle Creation
7.8.2 The Tomonaga-Schwinger Equation
7.8.3 Born's Rule on Cauchy Surfaces
7.8.4 Negative Energy States and the Dirac SeaSummary
Exercises
References8. Some Morals Drawn
8.1 Positivism vs. Realism
8.2 Limitations to Knowledge
8.3 What if Two Theories Are Empirically Equivalent?8.4 Open Problems
References
Appendix
* Topological View of the Symmetrization Postulate
* Philosophical Topics
* Free Will
* Causation
* Nelson's Stochastic Mechanics
* Probability and Typicality in Bohmian Mechanics
- The Law of Large Numbers in Bohmian Mechanics
- The Explanation of Quantum Equilibrium
- Quantum Non-Equilibrium
* Vector Bundles- The Intuition Behind Vector Bundles
- Electromagnetic Vector Potential
- The Aharonov-Bohm Effect
- Using Bundles for the Symmetrization Postulate
Solutions
Index