Undergraduate Lecture Notes in Physics Jakob Schwichtenberg Physics from Symmetry www.com Undergraduate Lecture Notes in Physics www.com Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: • An exceptionally clear and concise treatment of a standard undergraduate subject. • A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject.
• A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career. Series editors Neil Ashby Professor Emeritus, University of Colorado, Boulder, CO, USA William Brantley Professor, Furman University, Greenville, SC, USA Michael Fowler Professor, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Professor, University of Oslo, Oslo, Norway Michael Inglis Professor, SUNY Suffolk County Community College, Long Island, NY, USA Heinz Klose Professor Emeritus, Humboldt University Berlin, Germany Helmy Sherif Professor, University of Alberta, Edmonton, AB, Canada More information about this series at http://www.com/series/8917 www.com Jakob Schwichtenberg Physics from Symmetry 123 www.com Jakob Schwichtenberg Karlsruhe Germany ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notes in Physics ISBN 978-3-319-19200-0 ISBN 978-3-319-19201-7 (eBook) DOI 10.1007/978-3-319-19201-7 Library of Congress Control Number: 2015941118 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright.
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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.com N AT U R E A L W AY S C R E AT E S T H E B E S T O F A L L O P T I O N S ARISTOTLE A S F A R A S I S E E , A L L A P R I O R I S TAT E M E N T S I N P H Y S I C S H A V E T H E I R O R I G I N I N S Y M M E T R Y. HERMANN WEYL T H E I M P O R TA N T T H I N G I N S C I E N C E I S N O T S O M U C H T O O B TA I N N E W F A C T S A S T O D I S C O V E R N E W W AY S O F T H I N K I N G A B O U T T H E M. W I L L I A M L AW R E N C E B R AG G www.com Dedicated to my parents www.com Preface The most incomprehensible thing about the world is that it is at all comprehensible.
- Albert Einstein1 1 As quoted in Jon Fripp, Deborah Fripp, and Michael Fripp. Speaking of Science. Newnes, 1st edition, 4 2000. ISBN 9781878707512 In the course of studying physics I became, like any student of physics, familiar with many fundamental equations and their solu- tions, but I wasn’t really able to see their connection.
I was thrilled when I understood that most of them have a com- mon origin: Symmetry. To me, the most beautiful thing in physics is when something incomprehensible, suddenly becomes comprehen- sible, because of a deep explanation. That’s why I fell in love with symmetries. For example, for quite some time I couldn’t really understand spin, which is some kind of curious internal angular momentum that almost all fundamental particles carry.
Then I learned that spin is a direct consequence of a symmetry, called Lorentz symmetry, and everything started to make sense. Experiences like this were the motivation for this book and in some sense, I wrote the book I wished had existed when I started my journey in physics. Symmetries are beautiful explanations for many otherwise incomprehensible physical phenomena and this book is based on the idea that we can derive the fundamental theories of physics from symmetry. One could say that this book’s approach to physics starts at the end: Before we even talk about classical mechanics or non-relativistic quantum mechanics, we will use the (as far as we know) exact sym- metries of nature to derive the fundamental equations of quantum field theory.
Despite its unconventional approach, this book is about standard physics. We will not talk about speculative, experimentally unverified theories. We are going to use standard assumptions and develop standard theories.com X PREFACE Depending on the readers experience in physics, the book can be used in two different ways: • It can be used as a quick primer for those who are relatively new to physics. The starting points for classical mechanics, electro- dynamics, quantum mechanics, special relativity and quantum field theory are explained and after reading, the reader can decide which topics are worth studying in more detail.
There are many good books that cover every topic mentioned here in greater depth and at the end of each chapter some further reading recommen- dations are listed. If you feel you fit into this category, you are encouraged to start with the mathematical appendices at the end 2 Starting with Chap. In addition, the of the book2 before going any further. corresponding appendix chapters are mentioned when a new mathematical • Alternatively, this book can be used to connect loose ends for more concept is used in the text.
Many things that may seem arbitrary or a little wild when learnt for the first time using the usual historical approach, can be seen as being inevitable and straightforward when studied from the symmetry point of view. In any case, you are encouraged to read this book from cover to cover, because the chapters build on one another. We start with a short chapter about special relativity, which is the foundation for everything that follows. We will see that one of the most powerful constraints is that our theories must respect special relativity.
The second part develops the mathematics required to utilize symmetry ideas in a physical context. Most of these mathe- matical tools come from a branch of mathematics called group theory. Afterwards, the Lagrangian formalism is introduced, which makes working with symmetries in a physical context straightforward. In the fifth and sixth chapters the basic equations of modern physics are derived using the two tools introduced earlier: The Lagrangian formalism and group theory.
In the final part of this book these equa- tions are put into action. Considering a particle theory we end up with quantum mechanics, considering a field theory we end up with quantum field theory. Then we look at the non-relativistic and classi- cal limits of these theories, which leads us to classical mechanics and electrodynamics. Every chapter begins with a brief summary of the chapter.
If you catch yourself thinking: "Why exactly are we doing this?", return to the summary at the beginning of the chapter and take a look at how this specific step fits into the bigger picture of the chapter. Every page has a big margin, so you can scribble down your own notes and 3 On many pages I included in the ideas while reading3. margin some further information or pictures.com PREFACE XI I hope you enjoy reading this book as much as I have enjoyed writing it. Karlsruhe, January 2015 Jakob Schwichtenberg www.com Acknowledgments I want to thank everyone who helped me create this book.
I am espe- cially grateful to Fritz Waitz, whose comments, ideas and corrections have made this book so much better. I am also very indebted to Arne Becker and Daniel Hilpert for their invaluable suggestions, comments and careful proofreading. Thanks to Robert Sadlier for his help with the English language and to Jakob Karalus for his comments. I want to thank Marcel Köpke for for many insightful discussions and Silvia Schwichtenberg and Christian Nawroth for their support.
Finally, my greatest debt is to my parents who always supported me and taught me to value education above all else. If you find an error in the text I would appreciate a short email to errors@jakobschwichtenberg. All known errors are listed at http://physicsfromsymmetry.com Contents Part I Foundations 1 Introduction 3 1.1 What we Cannot Derive .3 Elementary Particles and Fundamental Forces .1 The Invariant of Special Relativity .3 Upper Speed Limit .4 The Minkowski Notation .6 Invariance, Symmetry and Covariance. 20 Part II Symmetry Tools 3 Lie Group Theory 25 3.2 Rotations in two Dimensions .1 Rotations with Unit Complex Numbers .3 Rotations in three Dimensions .1 The Generators and Lie Algebra of SO(3) .2 The Abstract Definition of a Lie Algebra .3 The Generators and Lie Algebra of SU (2) .4 The Abstract Definition of a Lie Group .1 The Finite-dimensional Irreducible Representations of SU (2) .com XVI CONTENTS 3.2 The Casimir Operator of SU (2) .3 The Representation of SU (2) in one Dimension .4 The Representation of SU (2) in two Dimensions 57 3.5 The Representation of SU (2) in three Dimensions 58 3.7 The Lorentz Group O(1, 3) .1 One Representation of the Lorentz Group .2 Generators of the Other Components of the Lorentz Group .3 The Lie Algebra of the Proper Orthochronous Lorentz Group .7 Van der Waerden Notation .9 Spinors and Parity .10 Spinors and Charge Conjugation .11 Infinite-Dimensional Representations .8 The Poincare Group .10 Appendix: Rotations in a Complex Vector Space .2 Variational Calculus - the Basic Idea .3 Particle Theories vs.4 Euler-Lagrange Equation .1 Noether’s Theorem for Particle Theories .2 Noether’s Theorem for Field Theories - Spacetime Symmetries .3 Rotations and Boosts .5 Noether’s Theorem for Field Theories - Internal Symmetries .6 Appendix: Conserved Quantity from Boost Invariance for Particle Theories .7 Appendix: Conserved Quantity from Boost Invariance for Field Theories .com CONTENTS XVII Part III The Equations of Nature 5 Measuring Nature 113 5.1 The Operators of Quantum Mechanics .1 Spin and Angular Momentum .2 The Operators of Quantum Field Theory .1 Lorentz Covariance and Invariance .2 Klein-Gordon Equation .1 Complex Klein-Gordon Field .1 Internal Symmetry of Free Spin 12 Fields .2 Internal Symmetry of Free Spin 1 Fields .3 Putting the Puzzle Pieces Together .4 Inhomogeneous Maxwell Equations and Minimal Coupling .5 Charge Conjugation, Again .6 Noether’s Theorem for Internal U (1) Symmetry .7 Interaction of Massive Spin 0 Fields .8 Interaction of Massive Spin 1 Fields .3 Mass Terms and Unification of SU (2) and U (1) .5 Lepton Mass Terms .6 Quark Mass Terms .9 The Interplay Between Fermions and Bosons.
169 Part IV Applications 8 Quantum Mechanics 173 8.1 Particle Theory Identifications .2 Relativistic Energy-Momentum Relation .3 The Quantum Formalism .com XVIII CONTENTS 8.4 The Schrödinger Equation .