Physics From Symmetry.pdf

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Preface to the Second Edition

Preface to the First Edition

Acknowledgments

Contents

Part I Foundations

1 Introduction

1.1 What we Cannot Derive

1.2 Book Overview

1.3 Elementary Particles and Fundamental Forces

2 Special Relativity

2.1 The Invariant of Special Relativity

2.2 Proper Time

2.3 Upper Speed Limit

2.4 The Minkowski Notation

2.5 Lorentz Transformations

2.6 Invariance, Symmetry and Covariance

Part II Symmetry Tools

3 Lie Group Theory

3.1 Groups

3.2 Rotations in two Dimensions

3.2.1 Rotations with Unit Complex Numbers

3.3 Rotations in three Dimensions

3.3.1 Quaternions

3.4 Lie Algebras

3.4.1 The Generators and Lie Algebra of SO(3)

3.4.2 The Abstract Definition of a Lie Algebra

3.4.3 The Generators and Lie Algebra of SU(2)

3.4.4 The Abstract Definition of a Lie Group

3.5 Representation Theory

3.6 SU(2)

3.6.1 The Finite-dimensional Irreducible Representations of SU(2)

3.6.2 The Representation of SU(2) in one Dimension

3.6.3 The Representation of SU(2) in two Dimensions

3.6.4 The Representation of SU(2) in three Dimensions

3.7 The Lorentz Group O(1, 3)

3.7.1 One Representation of the Lorentz Group

3.7.2 Generators of the Other Components of the Lorentz Group

3.7.3 The Lie Algebra of the Proper Orthochronous Lorentz Group

3.7.5 The (1/2, 0) Representation

3.7.6 The (0, 1/2 ) Representation

3.7.7 Van der Waerden Notation

3.7.8 The (1/2, 1/2) Representation

3.7.9 Spinors and Parity

3.7.10 Spinors and Charge Conjugation

3.7.11 Infinite-Dimensional Representations

3.8 The Poincaré Group

3.9 Elementary Particles

3.10 Appendix: Rotations in a Complex Vector Space

3.11 Appendix: Manifolds

4 The Framework

4.1 Lagrangian Formalism

4.1.1 Fermat’s Principle

4.1.2 Variational Calculus - the Basic Idea

4.2 Restrictions

4.3 Particle Theories vs. Field Theories

4.4 Euler-Lagrange Equation

4.5 Noether’s Theorem

4.5.1 Noether’s Theorem for Particle Theories

4.5.2 Noether’s Theorem for Field Theories - Spacetime Symmetries

4.5.3 Rotations and Boosts

4.5.4 Spin

4.5.5 Noether’s Theorem for Field Theories - Internal Symmetries

4.6 Appendix: Conserved Quantity from Boost Invariance for Particle Theories

4.7 Appendix: Conserved Quantity from Boost Invariance for Field Theories

Part III The Equations of Nature

5 Measuring Nature

5.1 The Operators of Quantum Mechanics

5.1.1 Spin and Angular Momentum

5.2 The Operators of Quantum Field Theory

6 Free Theory

6.1 Lorentz Covariance and Invariance

6.2 Klein-Gordon Equation

6.2.1 Complex Klein-Gordon Field

6.3 Dirac Equation

6.4 Proca Equation

7 Interaction Theory

7.1 U(1) Interactions

7.1.1 Internal Symmetry of Free Spin 1/2 Fields

7.1.2 Internal Symmetry of Free Spin 1 Fields

7.1.3 Putting the Puzzle Pieces Together

7.1.4 Inhomogeneous Maxwell Equations and Minimal Coupling

7.1.5 Charge Conjugation, Again

7.1.6 Noether’s Theorem for Internal U(1) Symmetry

7.1.7 Interaction of Massive Spin 0 Fields

7.1.8 Interaction of Massive Spin 1 Fields

7.2 SU(2) Interactions

7.3 Mass Terms and "Unification" of SU(2) and U(1)

7.4 Parity Violation

7.5 Lepton Mass Terms

7.6 Quark Mass Terms

7.7 Isospin

7.7.1 Labelling States

7.8 SU(3) Interactions

7.8.1 Color

7.8.2 Quark Description

7.9 The Interplay Between Fermions and Bosons

Part IV Applications

8 Quantum Mechanics

8.1 Particle Theory Identifications

8.2 Relativistic Energy-Momentum Relation

8.3 The Quantum Formalism

8.3.1 Expectation Value

8.4 The Schrödinger Equation

8.4.1 Schrödinger Equation with an External Field

8.5 From Wave Equations to Particle Motion

8.5.1 Example: Free Particle

8.5.2 Example: Particle in a Box

8.5.3 Dirac Notation

8.5.4 Example: Particle in a Box, Again

8.5.5 Spin

8.6 Heisenberg’s Uncertainty Principle

8.7 Comments on Interpretations

8.8 Appendix: Interpretation of the Dirac Spinor Components

8.9 Appendix: Solving the Dirac Equation

8.10 Appendix: Dirac Spinors in Different Bases

8.10.1 Solutions of the Dirac Equation in the Mass Basis

9 Quantum Field Theory

9.1 Field Theory Identifications

9.2 Free Spin 0 Field Theory

9.3 Free Spin Field Theory

9.4 Free Spin 1 Field Theory

9.5 Interacting Field Theory

9.5.1 Scatter Amplitudes

9.5.2 Time Evolution of States

9.5.3 Dyson Series

9.5.4 Evaluating the Series

9.6 Appendix: Most General Solution of the Klein-Gordon Equation

10 Classical Mechanics

10.1 Relativistic Mechanics

10.2 The Lagrangian of Non-Relativistic Mechanics

11 Electrodynamics

11.1 The Homogeneous Maxwell Equations

11.2 The Lorentz Force

11.3 Coulomb Potential

12 Gravity

13 Closing Words

Part V Appendices

A Vector calculus

A.1 Basis Vectors

A.2 Change of Coordinate Systems

A.3 Matrix Multiplication

A.4 Scalars

A.5 Right-handed and Left-handed Coordinate Systems

B Calculus

B.1 Product Rule

B.2 Integration by Parts

B.3 The Taylor Series

B.4 Series

B.4.1 Important Series

B.4.2 Splitting Sums

B.4.3 Einstein’s Sum Convention

B.5 Index Notation

B.5.1 Dummy Indices

B.5.2 Objects with more than One Index

B.5.3 Symmetric and Antisymmetric Indices

B.5.4 Antisymmetric

B.5.5 Two Important Symbols

C Linear Algebra

C.1 Basic Transformations

C.2 Matrix Exponential Function

C.3 Determinants

C.4 Eigenvalues and Eigenvectors

C.5 Diagonalization

D Additional Mathematical Notions

D.1 Fourier Transform

D.2 Delta Distribution

Bibliography

Index

Undergraduate Lecture Notes in Physics Jakob Schwichtenberg Physics from Symmetry Second Edition

Undergraduate Lecture Notes in Physics

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 University of Colorado, Boulder, CO, USA William Brantley Department of Physics, Furman University, Greenville, SC, USA Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis SUNY Suffolk County Community College, Long Island, NY, USA More information about this series at http://www.springer.com/series/8917

Jakob Schwichtenberg Physics from Symmetry Second Edition 123

Jakob Schwichtenberg Karlsruhe Germany ISSN 2192-4791 Undergraduate Lecture Notes in Physics ISBN 978-3-319-66630-3 https://doi.org/10.1007/978-3-319-66631-0 ISBN 978-3-319-66631-0 (eBook) ISSN 2192-4805 (electronic) Library of Congress Control Number: 2017959156 1st edition: © Springer International Publishing Switzerland 2015 2nd edition: © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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 A R I S T O T L E A S FA 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. H E R M A N N W E Y L 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 FA 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 A W R E N C E B R A G G

Dedicated to my parents

Preface to the Second Edition In the two years since the ﬁrst edition of this book was published I’ve received numerous messages from readers all around the world. I was surprised by this large number of responses and how positive most of them were. Of course, I was cautiously conﬁdent that read- ers would like the book. Otherwise I wouldn’t have spent so many months writing it. However, there is certainly no shortage of books on group theory or on the role of symmetries in physics. To quote Predrag Cvitanovic1 Almost anybody whose research requires sustained use of group theory (and it is hard to think of a physical or mathematical problem that is wholly devoid of symmetry) writes a book about it. Moreover, I’m not a world renowned expert. Therefore, I knew no one would buy the book because my name is written on the cover. So the chances were high that "Physics from Symmetry" would simply drown in the ﬂood of new textbooks that are published every year. Therefore, it’s reasonable to wonder: Why and how did "Physics from Symmetry" avoid this fate? I think the main reason for the success of the ﬁrst edition is what I framed as something negative above: it wasn’t written by a world renowned expert. I wrote the book while I was still a student and, as I remarked in the preface to the ﬁrst edition, "I wrote the book I wished had existed when I started my journey in physics". So my motivation for writing the book wasn’t to create an authoritative ref- erence or a concise text that experts would love. Instead, my only fo- cus was to write a book that helps students understand. As a student myself I always had still fresh in memory what I found confusing and what ﬁnally helped me understand. This point of view is nicely summarized in the following quote by C.S. Lewis2 It often happens that two schoolboys can solve difﬁculties in their work for one another better than the master can. When you took the problem 1 Predrag Cvitanovi´c. Group Theory: Birdtracks, Lie’s, and Exceptional Groups. Princeton University Press, 7 2008. ISBN 9780691118369 2 C. S. Lewis. Reﬂections on the Psalms. HarperOne, reprint edition, 2 2017. ISBN 9780062565488

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