Foundations of Quantum Mechanics

ISBN-13: 9783031095474 / Angielski / Miękka / 2022 / 468 str.

Roderich Tumulka

Opis

This book introduces and critically appraises the main proposals for how to understand quantum mechanics, namely the Copenhagen interpretation, spontaneous collapse, Bohmian mechanics, many-worlds, and others. The author makes clear what are the crucial problems, such as the measurement problem, related to the foundations of quantum mechanics and explains the key arguments like the Einstein-Podolsky-Rosen argument and Bell’s proof of nonlocality. He discusses and clarifies numerous topics that have puzzled the founding fathers of quantum mechanics and present-day students alike, such as the possibility of hidden variables, the collapse of the wave function, time-of-arrival measurements, explanations of the symmetrization postulate for identical particles, or the nature of spin. Several chapters are devoted to extending the different approaches to relativistic space-time and quantum field theory. The book is self-contained and is intended for graduate students and researchers who want to step into the fundamental aspects of quantum physics. Given its clarity, it is accessible also to advanced undergraduates and contains many exercises and examples to master the subject.

Spis treści

Preface

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 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        Historical Overview

1.6.3        Equivariance

1.6.4        The Double Slit Experiment in Bohmian Mechanics

1.6.5        Delayed Choice Experiments

Summary

Exercises

References

 

2.      Some Observables

2.1  Fourier Transform and Momentum

2.1.1        Fourier Transform

2.1.2        Momentum

2.1.3        Momentum Operator

2.1.4        Tunnel Effect

2.2  Operators and Observables

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  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        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 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        Remarks

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

3.2.4        Schrodinger's Cat

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

3.3.6        Empirical Tests

3.3.7        The Need for a Primitive Ontology

3.4  The Copenhagen Interpretation

3.4.1        Two Realms

3.4.2        Positivism

3.4.3        Purported Impossibility of Non-Paradoxical Theories

3.4.4        Completeness of the Wave Function

3.4.5        Language of Measurement

3.4.6        Complementarity

3.4.7        Complementarity and Non-Commuting Operators

3.4.8        Reactions to the Measurement Problem

3.5  Many Worlds

3.5.1        Schrodinger'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-World Theories

3.6  Special Topics

3.6.1        The Mach-Zehnder Interferometer

3.6.2        Path Integrals

3.6.3        Point Interactions

3.6.4        No-Cloning Theorem

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

4.1.3        Bohm's Version of the EPR Argument Using Spin

4.1.4        Einstein's Boxes Argument

4.1.5        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

4.2.4        Photons

4.3  Discussion of Nonlocality

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

5.1.2        The Main Theorem About POVMs

5.1.3        Limitations to Knowledge

5.1.4        The Concept of Observable

5.2  Time of Detection

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  Density Matrix

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  Reduced Density Matrix and Partial Trace

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

5.4.5        The No-Signaling Theorem

5.4.6        Completely Positive Superoperators

5.4.7        Canonical Typicality

5.4.8        The Possibility of a Fundamental Density Matrix

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

6.1.3        The Space of Unordered Configurations

6.1.4        Identical Particles in Bohmian Mechanics

6.1.5        Identical Particles in GRW Theory

6.2  Particle Creation

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

6.2.5        Ultraviolet Divergence

6.2.6        Bell's Jump Process

6.2.7        Determinism vs. Stochasticism

6.2.8        GRW Theory and Fock Space

6.2.9        Many Worlds and Fock Space

6.2.10    Interior-Boundary Conditions

6.3  A Brief Look at Quantum Field Theory

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

Summary

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

7.1.6        Outlook on General Relativity

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

7.2.4        The Dirac Equation

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

7.2.6        Probability Conservation

7.2.7        Multi-Time Wave Functions

7.2.8        Hypersurface Wave Functions

7.2.9        The Maxwell Equation as the Schrodinger Equation for Photons

7.3  Bohmian Mechanics in Relativistic Space-Time

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

7.3.5        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 Particles

7.5.3        Nonlocality in Relativistic GRW Theory

7.5.4        Interacting Particles

7.5.5        Primitive Ontology

7.5.6        Which Theories Count as Relativistic?

7.6  Copenhagen Interpretation in Relativistic Space-Time

7.7  Many-Worlds 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

7.8.4        Negative Energy States and the Dirac Sea

Summary

Exercises

References

8.      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

O autorze

Roderich Tumulka earned a Ph.D. in mathematics from Ludwig Maximilians University in Munich. During the period 2007-2016, he taught at Rutgers University (USA) and has since taught at Eberhard Karls University in Tübingen (Germany). His field of research is mathematical physics, focusing particularly on the foundations of quantum mechanics, quantum field theory, and quantum statistical mechanics.