Foundations of Quantum Mechanics

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

1.4 Classical Mechanics

1.4.1 Definition of Newtonian Mechanics

1.4.2 Properties of Newtonian Mechanics

1.4.3 Hamiltonian Systems

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

1.6.2 Historical Overview

1.6.3 Equivariance

1.6.5 Delayed Choice Experiments

Summary

Exercises

References

2. Some Observables

2.1 Fourier Transform and Momentum

2.1.1 Fourier Transform

2.1.3 Momentum Operator

2.1.4 Tunnel Effect

2.2.1 Heisenberg's Uncertainty Relation

2.2.2 Self-Adjoint Operators

2.2.3 The Spectral Theorem

2.2.4 Conservation Laws in Quantum Mechanics

2.3.1 Spinors and Pauli Matrices

2.3.2 The Pauli Equation

2.3.3 The Stern-Gerlach Experiment

2.3.4 Bohmian Mechanics with Spin

2.3.6 Is There an Actual Spin Vector?

2.3.7 Many-Particle Systems

2.3.9 Inverted Stern-Gerlach Magnet and Contextuality

Summary

Exercises

References

3.1 The Projection Postulate

3.1.1 Notation

3.1.2 The Projection Postulate

3.1.3 Projection and Eigenspace

3.2 The Measurement Problem

3.2.1 What the Problem Is

3.2.2 How Bohmian Mechanics Solves the Measurement Problem

3.2.3 Decoherence

3.2.5 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

3.3.7 The Need for a Primitive Ontology

3.4 The Copenhagen Interpretation

3.4.2 Positivism

3.4.3 Purported Impossibility of Non-Paradoxical Theories

3.4.5 Language of Measurement

3.4.6 Complementarity

3.4.8 Reactions to the Measurement Problem

3.5 Many Worlds

3.5.2 Everett's Many-Worlds Theory

3.5.3 Bell's First Many-Worlds Theory

3.5.5 Probabilities in Many-World Theories

3.6 Special Topics

3.6.2 Path Integrals

3.6.3 Point Interactions

3.6.5 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

4.1.4 Einstein's Boxes Argument

4.1.5 Too Good to Be True

4.2.1 Bell's Experiment

4.2.2 Bell's 1964 Proof of Nonlocality

4.2.3 Bell's 1976 Proof of Nonlocality

4.2.4 Photons

4.3.1 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

5.1.3 Limitations to Knowledge

5.1.4 The Concept of Observable

5.2.1 The Problem

5.2.2 The Quantum Zeno Effect

5.2.3 The Absorbing Boundary Rule

5.2.4 Historical Overview

5.3.1 Trace

5.3.2 The Trace Formula in Quantum Mechanics

5.3.3 Limitations to Knowledge

5.3.4 Density Matrix and Dynamics

5.4.1 Partial Trace

5.4.2 The Trace Formula

5.4.3 Statistical Reduced Density Matrix

5.4.4 The Measurement Problem and Density Matrices

5.4.6 Completely Positive Superoperators

5.4.7 Canonical Typicality

5.5 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

6.1.4 Identical Particles in Bohmian Mechanics

6.1.5 Identical Particles in GRW Theory

6.2.1 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

6.2.6 Bell's Jump Process

6.2.7 Determinism vs. Stochasticism

6.2.9 Many Worlds and Fock Space

6.2.10 Interior-Boundary Conditions

6.3.1 Historical Overview

6.3.2 Field Ontology vs. Particle Ontology

6.3.3 Scattering and the Dyson Series

6.3.4 Renormalization

Exercises

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

7.2 Relativistic Schrodinger Equations

7.2.1 The Klein-Gordon Equation

7.2.2 Two-Spinors and Four-Vectors

7.2.3 The Weyl Equation

7.2.5 Bohmian Trajectories for the 1-Particle Weyl and Dirac Equations

7.2.6 Probability Conservation

7.2.8 Hypersurface Wave Functions

7.2.9 The Maxwell Equation as the Schrodinger Equation for Photons

7.3.1 Law of Motion

7.3.2 Equivariance

7.3.3 Intersection Probability and Detection Probability

7.3.4 Possible Laws Governing the Time Foliation

7.4 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

7.5.3 Nonlocality in Relativistic GRW Theory

7.5.4 Interacting Particles

7.5.6 Which Theories Count as Relativistic?

7.6 Copenhagen Interpretation in Relativistic Space-Time

7.8 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

Summary

Exercises

8. Some Morals Drawn

8.1 Positivism vs. Realism

8.2 Limitations to Knowledge

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

- The Intuition Behind Vector Bundles

- Electromagnetic Vector Potential

- The Aharonov-Bohm Effect

- Using Bundles for the Symmetrization Postulate

Solutions

Index