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