Student Friendly Quantum Field Theory [2nd. ed.] 978-0-9845139-5-6

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Student Friendb

Quantum Field Theory Second Edition

Basic Principles and Quantum Electrodynamics Robert D. Klauber

Student Friendly Quantum Field Theory,

2nd

Edition (with pedagogic improvements and conections)

Basic Principles and Quantum Electrodynamics

Copyright @ of Robert D. Klauber

WARNING! This book is in printed format only. ALL DIGITAL VERSIONS ARE UNAUTHORIZED. If you upload or download an ebook version of this work, you are causing hardship for the author and committing a crime. To obtain a hard copy, please go to the website shown on pg. xvi, which lists legitimate sources for purchasing this book.

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cite this book, or the book website, as the source. Published by

Sandtrove Press

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@

gmail.com

Cover by Aalto Design

Second Edition December 2013 (second revision with improvements and corrections July 2015)

Library of Congress Control Number: 2013920914

ISBN: Hard cover 978-0-9845139-4-9 Soft

cover 978-0-9845139-5-6

Printed in the United States of America

To the students May they find this the easiest, and thus the most efficient, physics text to learn from that they have ever used.

"Of all the communities available to us there is not one I would want to devote mryself to, except for the socieQ of the true searchers, which has very few living members at any time." Albert Einstein

Table of Contents

Charts Preface to Second Edition Table of Wholeness

....... ix ......... x

Acknowledgements for Second Edition ................ x

Preface .......... xi Prerequisites .............. xv Acknowledgements......... ........ xv Preparation

1. Bird's Bye View

Chapter............ ......... I 1.1 This Book's Approach to eFT ........ I 1.2 Why Quantum Field Theory? .......... 1.3 How Quantum Field Theory? .......... I 1.0 Purpose of the

1

1.4 From Whence Creation and

Destruction

Operators?.......

......... 3

1.5 Overview: The Structure of physics

Therein...

and QFT's Place .......... 3 1.6 Comparison of Three euantum Theories........ 5 1.7 Major Components of ...... g 1.8 Points to Keep in ..... g

eFT........ Mind 1.9 Big Picture of Our Goal ................... g .10 Summary of the Chapter ................ 9 1.11 Suggestions? ...................9 1.12 Problems....... .................. 9 1

2. Foundations

Overview............ .............. I I Dimensions ....... I I 2.2 Notation......... ................. 15 2.1 Natural Units and

2.3 Classical vs Quantum Plane Waves.............. 2.4 Review of Variational Methods............. ....... 2.5 Classical Mechanics: An Overview..............

l6

lj lg

2.6 Schrddinger vs Heisenberg pictures .......... ...25 2.7 Quantum Theory: An ....29

Overview 2.8 Chapter Summary ......... .................

3l

Problems

Components

0 Fields .....r..r....o.................. 40

3.0

Preliminaries

3. 1

Relativistic Quantum Mechanics:

....40

A History Lesson.... 3.2The Klein-Gordon Equation in

....41

Quantum Field Theory.... ...,.......47 3.3 Commutation Relations: The Crux of eFT...51 3.4 The Hamiltonian in eFI ................53 3.5 Expectation Values and the Hamiltonian......57 3.6 Creation and Destruction Operators.............. 5g 3.7 Probability, Four-Currents, and Charge ...........61

Density 3.8 More on Observables 3.9 Real Fields.....

......63 ..................65

3.10 Characteristics of Klein-Gordon States....... 65 3.1 I Odds and ........66

Ends.......

eFT

3.12 Harmonic Oscillarors and ....69 3.13 The Scalar Feynman propagator............. ....70 3.14 Chapter

Summary..........

...............7g

3.15 Appendix A: Klein-Gordon Equation

from H.P. Equation of Motion... ...............79 3.16 Appendix B: Vacuum euanta and Harmonic Oscillators............. ....90 3.17 Appendix C: Propagator Derivation Step 4 for ..........91 3.18 Appendix D: Enlarging the Inregration Path of Fig. ...........81

A-......

3-6 Problems........

................92

4. Spinors: Spin % Fietds 4.0 4.

Preliminaries

I Relativistic

.... g4

Quantum Mechanics

for Spinors............ 4.2The Dirac Equation in euantum Field

Theory

......95 ............. 103

4.3 Anti-commutation Relations

for Dirac

Fields

......... 104

4.4TheDirac Hamiltonian in

eF-I

.... 105

4.5 Expectation Values and the

2.9 Appendix: Understanding Contravariant 2.10

3. Scalars: Spin

3.19

2.0 Chapter

and Covariant

Part ong: Frgg Fields......................................39

Dirac

.......32 ........ 36

Hamiltonian.......... ......... 109

4.6 Creation and Destruction Operators............ 109 4.7

QIT Spinor Charge Operator and

Four-Curuent

...... l l1 4.8 Dirac Three Momentum Operator............... 113 4.9 Dirac Spin Operator in ...... 1 l3

QFT...

V1

Operator. Ends

.............. I l5

4.10 QF'f Helicity 4.1 1 Odds and

............

4.12 The Spinor Feynman Propagator ............. . 4.13 Appendix A. Dirac Matrices and u,,v,

Relations

1

15

ll7

....,.............. 122

Problems

......

131

5. Vectors: Spin I FieIds.............................. 134 5.0

Preliminaries

7

.

.. 134

5.1 Review of Classical Electromagnetism....... 135

Preliminaries

Theory

............ 147 Quantum Field 5.4 Commutation Relations for Photon Fields.. 148 5.5 The

QFf Hamiltonian for Photons .............

149

5.6 Other Photon Operators in QFT... ............... 149 5.7 The Photon Propagator............. ... 150 5.8 More on Quantization and Polarization ...... 150 5.9 Photon Spin Issues Similar to Spinors........ 154

Next?............ 5.l l Summary Chart

5.10 Where to

5.

1

.............. 155 ........... 155

2 Appendix: Completeness Relations .......... I 60

5.13

Problems.......

..............

161

6. Symmetry,

Invarianceo and Conservation for Free Fields ..........162

Preliminaries 6.1 Introduction to Symmetry..........

6.0

.. 162

..163 6.2 Symmetry in Classical Mechanics............. . 167 6.3 Transformations in Quantum Field Theory I7t 6.4 l,orentz Symmetry of the

Lagrangian

Density

.. 171

..187 Picture ..194 7.4The S Operator and the S Matrix 7.5 Finding the S Operator............. ....197 ..............200 T.6Expanding S,,pr, 7.7 Wick's Theorem Applied to

Theorem

... 172

6.6 Symmetry, Gauges, and Gauge Theory ......177

Summary 6.8 Problems........

6.7 Chapter

7.

Expansion.............

........201

8 Justifying Wick' s Theorem ........... .............. 204

7.9 Comment on Normal Ordering of the

Density............ ....249 7.10 Chapter Summary.......... .............210 Hamiltonian

7.1

I Appendix A: Justifying Wick's Theorem ..,210 via Induction.........

7.12 Appendix B: Operators in Exponentials

Ordering 7.13 Problems........

..212

and Time

2l3c

8. QED: Quantum Field Interaction Theory Applied to Blectromagnetism... 8.0

Preliminaries

21"4

..214

8.1 Dyson-Wick's Expansion for QED

Density............ ....215 .....217 8.2 s(0)Physically.......... (r) .....217 8.3 s Physically .......... .......22a 8.4 s(2) Physically........ Hamiltonian

8.5 The Shortcut Method: Feynman Rules...... ..235 8.6 Points to Be Aware of

...........

8.7 Including Other Charged

.......237

lrptons in QED ..24I

8.8 When to Add Amplitudes and

When to Add Probabilities ......... ............242

6.5 Other Symmetries of the Lagrangian

Density: Noether's

..182

7.3 The Interaction

Dyson .... 144

8L

7.1 Interactions in Relativistic Quantum Mechanics............. ... 183 T.2Interactions in Quantum Field Theory ........ 186

5.2 Relativistic Quantum Mechanics

for Photons........... 5.3 The Maxwell Equation in

L

Interactions: The Underlying Theory....l82 7.0

4.14 Appendix B. Relativistic Spin: Getting to the Real Bottom of It A11...... .............. 124 4.15

Part Two : Interacting FieIds.....................o..

8.9 Wave Packets and Complex Sinusoids .......243 8.10 Looking Closer at Attraction and

......... 178 ...............179

Repulsion

...........243

8.1 1 The Degree of the Propagator Contribution

to the Transition Amplitude............. ....... 246 8.12 Summary of Where We Have Been: ............247 Chaps. 7 and 8

............. 8.1 3 Problems........

9. Higher Order Correctiofls 9.0

.....................

Background

9.1 Higher Order Correction 9.2

..............252

Problems.........

Terms

.254 ....254 ...255

...,...........265

vii 10. The Vacuum Revisited......................... .. 267 10.0

Background....

.............267

Part Three: Renormalization - Taming Those Notorious Infinities . r......i...

10.1 Vacuum Fluctuations: The Theory ...........267 10.2 Vacuum Fluctuations and Experiment.... ..270 10.3 Further Considerations

12.0

...272 ..............274

12.3

Summary......... .............277 Addenda......... .............277

Bhabha

A: Theoretical Value for Vacuum Energy Density ..,......279 10.9 Appendix B: Symmetry Breaking, Mass Terms, and Vacuum pairs ....... 2g0 10.10 Appendix C: Comparison of eFT for Discrete vs Continuous Solutions .......... 2gl 10.11 Appendix D: Free Fields and "Pair Popping" Re-visited .......294

Bhabha

10.13

I

Problem.

............ ...31g Revisited ..........319 12.12 Where We Stand ........ 319 12.13 Chapter Summary ......320 12.14 Problems .....321 12.ll Regularization

..... 2g5

Preliminaries

...............2g6 1 1.1 A Helpful Modification to the Lagrangian}gT tl.2External Symmetry for Interacting Fields. 2g9

1

3. Renormalization Toolklt ....................... . 3Zz

Preliminaries ...............322 Integrals......... ....322 13.2 Relations We'll Need ..32s 13.0

13.1 The Three Key

I 1.3 Internal Symmetry and Conservation 1

.........290

1.4 Global vs Local Transformations

and

Symmetries

........292

11.5 Local Symmetry and Interaction Theory

Substitution tl.7 Chapter Summary ......... I 1.6 Minimal

11.8 Appendix: Showing [O,S] 1 1.9

Problems

.....310

12.10 Adiabatic Hypothesis

...2g5b

Interactions

Scattering

Other Symbol for Energy ....................... 313 12.9 Things You May Run Into ........ .317

Invarianceo and Conservation for Interacting Fields ...... 296

for

.....306

12.7 The Total Renormalization Scheme.......... 313 12.8 Express e (k) as e (p) or

l. Symmetry, 11.0

Scattering

12.5 Same Result for Any Interaction............. ..312 12.6We Also Need to Renormalize Mass ........312

10.12 Appendix E: Considerarions for Finite

Interactions...........

....305

A Renormalization Example:

l2.4Higher Order Contributions in

10.8 Appendix

Volume

...............304

Regularization.......

.............277

10.6 Chapter 10.7

Preliminaries

12.1 Whence the Term .,Renormalization,?..... 305 12.2 A Brief Mathematical Interlude:

...........

Considerations..

303

2. Overview of Renormalizatio[................ 304

of

Uncertainty Principle 10.4 Wave Packets 10.5 Further

I

i...

..293 .. Zgj

.............297

- 0.......... .........299 ...... 300

3.3 Ward ldentities, Renormalization, and Gauge Invariance........... ...329 13.4 Changes in the Theory with ms Instead of m .......330 I

.......

13.5 Showing the B in Fermion Loop Equals the L in Vertex Correction ...................... 33 I 13.6 Re-expressing 2nd Order Corrected Propagators, Vertex, and External Lines 332 13.7 Chapter

Summary..........

.............336

13.8 Appendix: Finding Ward Identities

Scattering 13.9 Problems........

for

Compton

..337

33ib

14. Renormalization:

Putting It All Together 14.0 1

Preliminaries

...............339

4. 1 Renormalization Example:

Compton's

Scattering..........

....340

14.2 Renormalizing 2nd Order

Divergent 14.3 The Total

Amplitudes..........

Amplitudeto

2nd

....342

Order.............351

vlll

Our

Approach........

... 351

14.5 Higher Order Renormalization Example:

Compton's

Scattering..........

....352

14.6 Renormalizing nth Order

.........

....354 ............364 nth Order to 14.7 The Total Amplitude ...364 14.8 Renormalization to All Orders Divergent Amplitudes

14.9 Chapter

Summary.........

............. 365

I{.I}Appendix: Showing ltkuB,t Term

Out......... l4.LlProblems

........372

Drops

....373

374

15. Regularization

Preliminaries 15.1 Relations We'll Need....

15.0

Preliminaries 17.1 The Cross Section 17.0

...............432

.......432

17.2 Review of Interaction Conservation Laws 445 17.3 Another Look at Macroscopic Charged

Particles

Interacting.............

....449

17.4 Scattering in QFT: An In Depth l-ook.......452 17.5 Scattering in QFT: Some Examples ..........463 17.6 Bremsstrahlung and

Infra-red

Divergences..........

....479

Closure......... Summary....,..... .............482 ..............485 17.9 Problems........ .........,.....482

17.7

17.8 Chapter

...............374 ............. 375

Addend8 .......................o.r1.ri.r.................. .......487

15.2 Finding Photon Self Energy Factor

Using the Cut-Off

.......o...... ..... 432

17. Scattering

14.4 Renormalization to Higher Orders:

Method

......379

1

5.3 Pauli-Villars Regulari2ation...................... 384

I

5.4 Dimensional Regulari2ation...................... 385

1

5.5 Comparing Various Regularization

Approaches............

... 388

15.6 Finding Photon Self Energy Factor

18. Path Integrals in Quantum Theories:...488 ...............488 Preliminaries ..488 l8.l Background Math...... ......489 Integral 18.2 Defining Functional

18.0

1

Using Dimensional Regularization ........ 388 15.7 Finding the Vertex Correction Factor

Using Dimensional Regularization ........ 393

8.3 The Transition Amplitude ......................... 490

18.4 Expressing the Wave Function Peak in Terms of the Lagrangian ............ .....,..492 18.5 Feynman's Path Integral Approach:

The Central

Idea

.....,.493

15.8 Finding Fermion Self Energy Factor Using Dimensional Regularization ........ 397 ....-........397 15.9 Chapter Summary .........

18.6 Superimposing a Finite Number of Paths..494 18.7 Summary of Approaches.......... ..497

15.10 Appendix: Additional Notes on Integrals 399

18.8 Finite Sums to Functional Integra1s...........498

15.11

Problems

""

400

I

8.9 An Example: Free Particle ........................ 502

18.10

QIT via Path Integrals

.............506

Summary 18.12 Appendix 18.13 Problem

......509

18.11 Chapter

Part Four: Application

to Experiment....-.. 401

16. Postdiction of Historical

Experimental ResuIts..........r..o............... . 402 16.0

Preliminaries

16.1 Coulomb Potential in 16.2 Coulomb Potential in

...............402 .......402

RQM QIIT.......

..404

16.3 Other Potentials and Boson Types............410 16.4 Anomalous Magnetic

Moment

16.6 A Note on QED Successes Over RQM..... 427

.........

.............428 16.8 Appendix: Deriving Feynman Rules for Static, External (Potential) Field ............ 430 16.9

Problems

......509

19. Looking Backward and Looking Forward:

Book Summary and Whatts Next..........5L0

..............: 19.1 Book Summary l9.2What'sNext

19.0 Preliminaries

...............

5 10

............ 51 1

.519

..411

16.5 The Lamb Shift

16.7 Chapter Summary

....509

.,...431a

Index......................................tt..t........""

tttttttt521

Table of Wholeness Charts Preparation 1-1 The Overall Structure of Physics.................... 5 1-2 Comparison of Three

Theories

........7

2-1 Conversions between Natural, Hybrid, and cgs Numeric Quantities ............. ........ I 4

.....20 ...28

2-5 Summary of Quantum Mechanics (Heisenberg Picture) .............

.... 30

Motion

..............65

3-3 Quantum Harmonic Oscillator Compared to QFT Free States.................. 69 3-4 Different Kinds of Operators in QFT ...........79 4-l Spin VzParticle Spin Summary................... 102 5-1 Summary of Classical Electromagnetism .......141 Potential

Theory

5-2 Comparing Spinor and Polarization Basis .............. 146

States

5-3 Gupta-Bleuler Weak Lorentz Condition

Overview

...154

QFI Overview, Part 1: From Field Equations to Propagators and Observables ............. 156

6-1 Symmetry

Summary..........

.......... 166

6-2 Galilean vs Lorentz Transformations ......... 168 6-3 Summary of Effect of Lorentz Transformation on Fields 6-4 Ways to Determine

is

........ 172

if a Quantity

Conserved.........

.... 177

Pictures

........... 188

7-2Examples from the Three Pictures.............. 193

8-l Keeping Four-momenta Signs Straight ......234 8-2 Comparing Typical Perturbation Theory

to

QED

......238

8-3 Summary of Virtual Photon Properties

for lD Attraction and Repulsion ............246 8-4

QFI Overview, Part 2: From Operators and Propagators to Feynman Rules ........248

9-1 Loop

Corrections...........

1

-1

QFT.28l

Types of Transformations ............. ............ 293

1-2 Summary of Global and Local Internal Symmetry for L and Lo .......................... 295

for

Interactions......

....298

l4-1 Two Routes to Renormalization ...............350 l4-2Types of Feynman Diagrams .....353 14-3 Comparing Certain Types of Feynman Diagrams ...354 l4-4 Renormalization Steps to 2nd Order in a.368 14-5 Renormalization Steps to nth Order..........368

15-l Wick Rotation

Summary

..............265

......,.....377

l5-2 Comparison of Four Regularization

Techniques............

....397

Part Four: Application to Experiment

l6-l

Boson Spin and Like

Charges

....411

16-2 Theoretical and Experimental Values .......412

17-l Summary of Definitions and Interpretations of o and d o I d Q.............. M4 l7-2 Scattering Off Stationary Target for Different Physical Theories .....M5 17-3 Two Particle Elastic Collisions

(Non-relativistic)

......450

l7-4 Two Particle Elastic Collisions

(Relativistic).........

....451

17-5 Fermion Spin Sum Relations ....................460 17-6 Differential Cross Section

Determination in

Part Two: Interacting Fields 7- I Comparing Schrcidinger, Heisenberg, and Interaction

1

Part Three: Renormalization

3-2 Physical, Hilbert, and F'ock Spaces............... 68

5-4

....278

I I -3 Summary of Symmetry Effects

Schrcidinger vs Heisenberg Picture Equations of

Part One: Free Fields 3-l Bosons vs Fermions.........

Scenarios

10-2 Discrete vs Continuous Versions of

1

2-2 Summary of Classical (Variational) Mechanics .............

24

10-1 Comparison of Vacuum Fluctuation

QFT

.............483

Addenda 18-1 From a Function of a Function to the

Functional

Integral.

...........489-490

18-2 Equivalent Approaches to Non-relativistic Quantum Mechanics ..... 498 l8-3 Adding Phasors at the Final Event for Three Discrete ..........499

Paths

l8-4 Comparing Particle Theory to Field Theory: Classical and Quantum....506 18-5 Comparing NRQM to QFI for the Many Paths Approach .............508

18-6 Super Simple

Summary

............508

Preface to the Second Edition In the eight months since publication of the frrst edition, I have received many excellent suggestions from readers for making cert-ain parts clearer and easier to understand. Though a new edition of a text typically comes out years after the prior one, I would be remiss to wait any longer to include those suggestions in a second edition. The changes made encompass the re-wording of several sections and the addition of a dozen new pages spread throughout th-e text, all of which should improve the leaming experience for many. These modifications are posted on the book-website (see URL on pg. xvi opposite pg. 1), for the benefit of those using the fint edition'

ThWs to note about newly aildeilmaterial To facilitate and simplify communication between usen of different editions, all equations and sections in the first edition have the same numbers in the second edition, Where new equations have been inserted into the second edition between two first edition equations, the new ones are numbered with that of the preceding first edition equation (first augmented with +1, +2, +3, etc. Where new pages have been inserted, they are numbered with the preceding edition) page number augmented with letters, a, b, etc. For example,

numbers New equations inserted Two pgs between p gs 321 and328 327a and3ZTb after (13-29), before (13-30) New pages inserted inside

chapter

N.ew page

New equation numbers (13-29)+1, (13-29)+2, etc

Where material has b€en added at the end of a chapter, equations are given new numbers that are simply incremented over the last numbered equation in tlat chapter of the first edition. For example' New appendix at end of

chapter

New page

nulnbers

Two pgs inserted before problems 43hand 43lb in Chap. 16, on pgs. 430-431

New equations inserted

after last equation in chapter,

(16-120)

New equation numbers (16-1

2t) (16-122), etc

Hence, at a glance, a reader can tell what has been inserted since the first edition, and confusion in communication between users of different editions should be minimized. justify insertion On page 203, in the middle of Chap. 7, new material has been added, but since it was not enough to by half a page which is off page numbering has page (from 203 to 212) chapter of additioial pages, the remainder of that of the frst mirroring that to page retums numbering edition second 8 onward, or so from thi fust edition. From Chap. giving by simply material reference be able to should one edition of ?, a reader edition. Other than this one area of Chip. edition. the relevant page number to a reader of the other In addition to new material being added, a significant number of typographical errors have been corrected. These are also listed on the book web site for the benefit of first edition users. Robert D. Klauber November 2013

Acknowleilgments

for the Sec ond Editinn

am simply incapable of adequately expressing my gratitude to Luc Longtin, Jimmy Snyder, and Holger Teutsch, each of whom I consider a candidate ?or tlre title- of world's greatest technical book editor. The three of them have offered exceptional insights into pedagogic improvement of various sections, pointed out typographical enors, and generally woiked tirelessly to help make this book better. I am indebted to, and thank them, deeply. I also thank Sebastian Allende, Sukruti Bansai, Ben Balmforth, John Davidson, Bill Foste(, Thomas Fowler, Michael Heiss, Kurt Huddleston, Michael Hyams, Dory Kodeih, Jason Koeller, Jeff Magill, Doug McKenzie, Pete Morcos, Marius paraschiv, Ramon Salazar, Andrew Solomon, and Brian Stephanik for other excellent suggestions and

I

corrections.

Additionally, several of those who reviewed and helped edit the first edition, Chris Locke, Christian Maennel, Mike Worsell, and David Scharf, helped once again, and I thank them sincerely for their continued support'

Preface "All of plrysics is either impossible or trivial. It is impossible until you undersnnd it, and then it beconxes trivinl,"

.

Emest Rutherford

This book is

l.

2. 3.

an attempt to make leaming quantum field theory (QFI) as easy, and thus as efficient, as is humanly possible, intended, first and foremost, for new students of QFT, and

an introduction to only the most fundamental and central concepts of the theory, panicularly as employed in quantum electrodynamics (QED).

It is not

1.

orthodox,

2. an exhaustive treatment of QFf, 3. concise (lacking extensive explanation), 4. written for seasoned practitioners in the field, or 5. a presentation of the latest, most modem approach to it. Students planning a career in field theory will obviously have to move on to more advanced texts, after they digest the more elementary material presented herein. This book is intended to provide a solid foundation in the most essential elements of the theory, nothing more.

In my own teaching experience, and in the course of researching pedagogy, I have come to see that "learning" has at its basis a fundamental three-in-one structure. The wholeness of leaming is composed of

i) ii) iii)

the knowledge to be leamed, the learneg and the process of learning itself.

It seems unfortunate that physics and physics textbooks have too often been almost solely concemed with the knowledge of physics and only rarely concemed with tlase who are learning it or how they couLd best go about learnlng. However, there are signs that this situation may be changing somewhat, and I hope that this book will be one stepping stone in that direction.

In writing this book, I have repeatedly tried to visualize the leaming process as a new leamer would. This viewpoint

is one we quickly lose when we, as teachers and researchers, gain familiarity with a given subject, and yet it is a perspective we must maintain if we are to be effe.tive educators. To this end, I have solicit€d guidance and suggestions from professional educators (those who make leaming and education, per se, lh€tr central focus in life), and more importantly, from those studying QFf for the first time. In addition, I have used my own notes, compiled when I was first studlng the theory myself, in which I carefully delineated ways the subject could be presented in a more studentfriendly manner. In this sense, the text incorporates "peer instruction", a pedagogic tool of recognized, and considerable, merit, wherein students help teach fellow students who arc learning the same subject. It is my sincere hope that the methodologies I have employed herein have helped me to remain sympatletic to, and in touch with, tie perspective of a new leamer. Of course, different students find different teaching techniques to have varying degrees of ffansparency, so there are no hard and fast rules. However, I do believe that most students would consider many of the following principles, which I have employed in the text, to be of pedagogic value.

l)

Brevity avoided

Conciseness is typically a honor for new students trying to fathom unfamiliar concepts. While it can be advantageous circles, be€ome a goal unto itself, extending even into pedagogy - an area for which it was never suited.

in some arenas, it is almost never so in education. Unfortunately, being succinct, has, in scientific/technical

Il

this book, I have gone to great lengths to avoid conciseness and to present extensive explanations. I often take a paragraph or more for what other authors cover in a single sentence. I do this because I learned a long time ago that the thinnest texts were the hardest. Thicker ones covering the same material actually took less time to get through, and I understood them better, because the authors took time and space to elaboraie, rather than leave significant gaps. Such gaps oflen contain ambiguities or possibilities for misunderstanding that the author has overlooked and left unresolved. Succinctness may impress peers, but can be terribly misleading and frustrating for students.

xlr

2)

Holistic previews

The entire book, each chapter, and many sections begin with simple, non-mathematical overviews of the material to be covered. These allow the student to gain a qualitative understanding of the "big picture" before he or she plunges into the rigors of the underlfng mathematics. Doing physics is a lot like doing ajig-saw puzzle. We assemble bits and pieces into small wholes and then gradually merge thoie small wholes into grealer ones, until ultimately we end up with the "big picture." Seeing the picture on the puzile box before w€ start has immense value in helping us put the whole thing together. We know the blue goes here, ihe green there, and the boundary of the two, somewhere in between. Without that picture preview to guide us, $e entire job lecornes considerably more difficult, more tedious, and less enjoyable. In this book, the holistic previews are much iike the picturrs on the puzzle boxes. The detail is not there, but the essence of the final goal is' These overviews should eliminate, or at least minimize, the 'lost in a maze of equations" syndrome by providing a "birds-eye road map" of where we.have corne from, and where we are going. By so doing we not only will keep sight of the forest in spite of the tlees, but will also have a feeling, from the beginning, for the relevance of each particular topic to the overriding structure of the wholeness of knowledge in which it is embedded.

3)

Schematic diagram summaries (wholeness charts)

Enhancing the "birds-eye road map" approach are block diagram summaries, whichl callwholeness charts, so named because they reveal in chart form the underlfng connections that unite various aspects of a given theory into a gleater whole. Unlike the chapter previews, these are often mathematical and contain considerable theoretical depth. I-earning a computer program line-byJine is immensely harder than leaming it with a block diagram of the program, showing major sections and sub-sections, and how they are all intenelated. There is a structure underlying the progranl which ii its issence and most important aspect, but which is not obvious by looking directly at the proglam code its€lf.

The same is true in physics, where line-byJine delineation of concepts and mathematics corresponds to program code, and in this text, wholeness charts play the role of block diagrams. In my own leaming experiences, in which I constructed such charts myself from my books and lecture notes, I found them to be invaluable aids. They coalesced a lot of different information into one central, compact, easy-to-see, easy-to-understand, and easy-to-reference framework. The specific advantages of wholeness charts are severalfold. First, in leaming any given material we are seeking, most importantly, an understanding of the kemel or conceptual essence, i.e., the main idea(s) underllng all the text. A picture is worth a thousand words, and a wholeness chart is a "snapshot" of those thousand words. Second, although the charts can summarize in-depth mathematics and concepts, they can be used to advantage even when reading through material for the first time. The holistic overview perspective can be more easily maintained by continual reference to the schematic as one leams the details. Third, comparison with similar diagrams in related areas can reveal parallel undedying threads running through seemingly diverse phenomena. (See, for example, Summary of Classical Mechanics Wholeness Chart 2-2 and Summary of Quaitum Mechanics Wholeness Chart 2-5 in Chap. 2, pgs, 2O-21 and 30-31.) This not only aids the learning process but also helps to reveal some of *re subtle workings and unified structure inherent in Mother Nature. Further, review of material for qualifying exams or any othet futur€ purpose is greatly facilitated. It is much easier to refresh one's memory, and even deepen understanding, from one or two summary sheets, rather than time consuming ventures through dozlns of pages of text. And by coplng all of the wholeness charts herein and stapling them together, you will have a pretty good summary of the entire book. Still further, the charts can be used as quick and easy-to-find references to key relations at future times, even years later.

4)

Reviews of background material

In situations where development of a given idea depends on material studied in previous courses (e.g., quantum mechanics) short reviews of thi relevant background subject matter are provided, usually in chapter introductory sections or later on, in special boxes separate from the main body ofthe text.

5)

Only basic concepts without peripheral subjects

believe it is of primary importance in the leaming process to focus on the fundamental concepts first, to the exclusion of all els". Th" ti*" to b.anch out into related (and usually more complex) areas is dJ?er the core knowledge is

I

assimilated, not drring the assimilation period.

xm All too often, students axe presented with a great deal of new material, some fundamental, other more peripheral or advanced. The peripheraVadvanced material not only consumes precious study time, but lends to confuse the student with regard to what precisely is essential (what he or she mrs, understand), and what is not (what it would be nice ifhe or she also understood at this point in their development). As one example, for those familiar with other approaches to QFT, this book does not inffoduce concepts appropriate to weak inieractions, such as pa theory, before students have first become grounded in the more elementary theory of quantum electrodynamics.

This book, by careful intention, restricts itself to only the most core principles of QFT. Once those principles are well in hand, the student should then be ready to glean maximum value from other, more extensive, texts.

6)

Optimal "retum on investment" exercises

All too often students get tied up, for what seem interminable periods, working through problems from which minimum actual learning is reaped. Study time is valuable, and spending it engulfed in great quantities of algebra and trigonometry is probably not its best use.

I

have tried, as best I could, to design the exercises in this book so that they consume minimum time but yield maximum return. Emphasis has been placed on gleaning an understanding of concepts without getting mired down. Later on, when students have become practicing researchers and time pressure is not so great, there will be ample oppomrnities to work through more involved problems down to every minute algebraic detail. If they are firmly in command of the corcepts and principles involved, the calculations, though often lengthy, become trivial. If, however, they never got grounded in the fundamentals because study time was not efficiently used, then research can go slowly indeed.

7)

Many small steps, rather than fewer large ones

Professional educators have known for some time now that learning progresses faster and more profoundly when new material is present€d in small bites. The longer, more moderately sloped hail can get one to the mountaintop much more readily than the agonizing climb up the nearly vertical face.

Unfortunately, from my personal experience as a student, it often seemed like my textbooks were trying to tak€ me up the steepest grade. I sincerely hope that those using this book do not have this experience. I have made every effort to include each and every relevant step in all derivations and examples.

In so doing, I have sought to avoid the common practice of letting students work out significant amounts of algebra that typically lies "between the lines". The thinking, as I understand it, is that students are perfectly capable of doing that themselves, so "why take up space with it in a text?" My answer is simply that including those missing steps makes the leaming process more efficient. If it takes the author ten minutes !o write out two or three more lines of algebra, then it probably takes the student twenty minutes to do so, provided h€,/she is not befuddled (which is not rare, and in which case, it can take a great deal longer). That ten minutes spent by the author saves hundreds, or even thousands, of student readers twenty minutes, or more, each, Multiply that by the number of times such things occur per chapter and the number of chapters per book, and we are talking enormous amounts of student tirne saved. Students leam very little, if anything, doing algebra. They recapture a lot of otherwise wast€d time that can be used for actual learning, if the author types out the missing lines.

8)

Liberal use of simple concrete examples

Professional educators have also known for quite some time that abstract concepts are best taught by leading into them with simple, physically visualizable examples. Further, understanding is deepened, broadened, and solidified with even more such concret€ examples. Some may argue that a more formal mathematical approach is preferable because it is important to have a profound, not superficial, understanding. While I completely agree that a profound understanding is essential, it is my experience that the mathematically rigorous introduction, more often than not, has quite the opposite result. (Ask any student about this,) Further, to know any field profoundly we must know it from all angles. We must know the underlying mathematics in detail plns we must have a grasp on what it all means in the real world, i.e., how the relevant systems behave, how they parallel other types of systems with which we are already familiar, etc. Since we have to cover the whole range of knowledge from abstract to physical anyway, it seems best to start with the end of the specffum most readily apprehensible (i.e., the visualizable, concrete, and analogous) and move on from there.

xlv This methodology is empl,oyed liberally in this book. It is hoped that so doing will ameliorate the "what is going on?" frustration common among students who are introduced to conceptually new ideas almost solely via routes heavily oriented toward abstraction and pure mathematics. In this context it is relevant that Richard Feynman, in his autobiography, notes,

"I can't understand. anything in general unless I'm carrying along in my mind a specific emmple and watching it go....(Others think) I'n following the steps mathematically but that's not v,hat I'm doinT. have the specifrc, physbal exanpk of what (is being analyzed) and I know from instinct and expeience the properties of the thing."

I

same way, and I have a suspicion that almost everyone else does as well. Yet few teac& that way. This book is an attempt to teach in that way.

I know from my own exp€rience that I learn in the

9)

Margin overview notes

Within a given section of any tertbook, one group of paragraphs can refer to one subjert, another group to another subject. When reading material for the first time, not knowing exactly where one train of the author's thought ends and a different one begins can oflentimes prove confusing. In this book, each new idea not set off with its own section heading is highlighted, along with its central message, by notations in tie margins. In this way, emphasis is once again placed on the overview, the "big picture" of each topic, even on the subordinate levels within sections and subsections. Additionally, the extra space in the margins can be used by students to make their own notes and comments. In my

own experience as a student I found this practice to be invaluable. My own remarks written in a book are, almost invariably, more comprehensible to me when reviewing later for exams or other purposes than are those of the author. 10) Definitions and key equations emphasized

As a student, I often found myself encountering a term that had been introduced earlier in the text, but not being clear on its exact meaning, I had to search back through pages clumsily trying to find the first use of tle word. In this boolq new terminology is underlined when it is introduced or defined, so that it "jumps out" at the reader later when trying to

find it again. In addition, key equations

-

the ones students really need to know

-

have borders around them.

11) Non-use of terms like "obvious", 'trivial", etc. The text avoids use of emotionally debilitating terms such as "obvious", "trivial", "simple", "easy", and the like to describe things that may, after years of familiarity, be easy or obvious to the author, but can be anything but that to the new student. (See "A Nontrivial Manifesto" by Matl Landreman, Physics Today'March2005 ' 52-53.) have undertaken here has been a challenging one. I have sought to produce a physics textbook which is relatively lucid and transparent to those studying quantum field theory for the first time. In so doing, I have employed some decidedly non-naditional tactics, and so anticipated resistance from main stream publishers, who typically have 'least motivations foi wanting to do things the way they have been done before. Their respective missions do not seerL at to me, to be focused primarily on optimizing the process of conveying knowledge. The job

I

As an example, a good friend of mine submitted a graduate level physics text manuscript, with student friendly notes in the margins, to oni of the world's top academic publishers. He was ordered to remove the margin notes before they would pubiish the book. Not wanting to fight (and lose) this kind of battle over methodologies I employ, and consider essential in making students' work easier, I have chosen a different route. I also anticipate resistance ftom some physics professors who may consider the book too verbose and too simple. I only ask them to try it and let their students be the judges. The proof will be in the pudding. If comprehension comes more quickly and more deeply, then the approach taken here will be vindicated. If you are a student now, appreciate the pedagogic methodologies used in this book, and end up one day writing a text your of own, I hope you will not forget what advantage you once gained from tlose methodologies. I hope you will use them in your own book. Above all, I hope your presentation will be profuse with elucidation and not terse. Good luck to the new students of quantum field theory! May their studies be personally rewarding ald professionally

fruitful'

Robert D. Klauber February 2013

xv

Prerequisiles Quantum field theory takes off where the following subjects end. Those beginning this book should be reasonably at the levels described below.

well versed in them,

Quantum Mechanics An absolute minimum of two undergraduate quarters, but far more preferably, an additional two graduale level quarters. Some exposure to relativistic quantum mechanics would be advantageous, but is not necessary. Optimal level proficiency: Eugen Merzbacher's Quantum Mechanics (John Wiley) or a similar book.

of

Classlcal Mechanics semester at the graduate level. Topics covered should include the lagrangian formulation (for particles, and importantly, also for fields), the legendre ffansformation, sperial relativity, and classical scattering. A familiarity with Poisson brackets would be helpful. Optimal level of proficiency: Herbert Goldstein's Classical Mechanics (Addison-

A

Wesley) or similar.

Electromagnetism

Two quarters at the undergraduate level plus two graduate quarters. Areas studied should comprise Maxwell's equalions, conservation laws, e,/m wave propagation, relativistic treatment, Maxwell's equations in terms of the four potential. Optimal level of proficiency: John David Jackson's Classical Electrodynamics (Iohn Wiley) or similar. Math/Relativity Advantageous but not essential, as it is covered in the appendix of Chap. 2: Exposure to covariant and contravariant coordinates, and metric tensors, for orthogonal4D syslems, at the level found in Jackson's chapters on special relativity.

Acknowledgements You cannot live a perfea day without doing something for sotneone who will never be able to repay you,"

John Wooden Hall of Fame UCI,A basketball coach. The people who reviewed, edited, made suggestions for, and corrected draft portions of perfect days. There is no way I can repay them.

tlis book had many

candidate

I am most indebted to three, Chris l-ocke, Christian Maennel and Mike Worsell, who read every word and made innumerable great contributions. Close behind on my gratitude list are Carlo Marino, David Scha{ Jean-Louis Sicaud, and Jon Tynell, each of whom read most of the text and provided a substantial number of valuable suggestions and corrections. David, Jon, and Morgan Orcutt deserve further heartfelt tlanks for working most of the problerns (and frnding errors in several of them). Others making significant, much appreciated contributions include Martin Biiker, Jim Bogan, Ben Brenneman, Brad Bill Cohwig, Trevor Daniels, Saurya Das, I-orenzo Del Re, Tony D'Esopo, Paul Drechsel, Michael Gildner, Esteban Herrera, Phil Jones, Rutl Kastner, Lorek Krzysztof, Claude Liechti, Rattan Mann, Lorenzo Massimi, Enda McGlynn, Gopi Rajagopal, Javier Rubio, Girish Sharma, and Dennis Smoot.

Carlile,

Many years before I started writing this text, I fell in debt to my teachers, Robin Ticciati and John Hagelin, who guided me through my earliest sojoums into the quantum theory of fields, and earned both my respect and deep gratitude. Robin, in particular, wBs generous well beyond rhe call of duty, in granting me numerous one-on-one sessions to discuss various aspects of the theory. Non-technical, but nonetheless vital support came from my wonderful wife Susan. I cannot thank her enough for her patience, understanding, love, and unswerving devotion throughout the days, weeks, months, and years I spent writing ... and re-writing. Last mentioned, yet anything but least, are my amazing and caring parents, without whose support and many, many sacrifices, I would never have gained the education I did, and thus, never have written this book. Thank you, mom and dad. This book, whatever it is, would be substantially less without these people. Regardless, any enors or insufficiencies that may still remain are my responsibility, and mine alone.

xvl

The website for this book is

www. quantumfieldtheorY.info It contains presentations of advanced topics

as

well as a list of

corrections and improvements to this printing. Please use the site to report any effors you might find and to suggest ways to make future versions of this book easier for students to understand.

Chapter

I Bird's Eye View Well begun is half done.

Old Proverb

7.0 Parpose of the Chapter Before starting on any journey, thoughtful people study a map of where they will be going. This

allows them to maintain their bearings as they progress, and not get lost en rbute. This chalter is like such a map, a schematic overview of ttri ienain of quantuni fietd theory eru) without the complication of details. You, the student, can get a feeling for the theory, and be somewhat at home with it, even before delving into the "nitty-gritty" mathematics. Hopefully, this will allow you to keep sight of the "big picture'', and minim\ze confusion, as you make your way, step-by-step, through this book.

1.7 This Book's Approach to QFT There are two main branches to (ways to do) quantum field theory called r the canonical quantization approach, and

o the path integral (maqy patns, sum over histories, or functional quantization) approach. The first of these is considered by many, and certainly by me, as the easiest way to be introduced to the subject, since it treats particles as objects that one can visualize as evolving along a particular path in spacetime, much as we commonly think of them doing. The path integral appioach (which goes by several names), on the other hand, treats particles and fields as if they *.r.-ri-ultaneously traveling all possible paths, a difficult concept with even more difficult mathematics behind it. This book is primarily devoted to the canonical quantization approach, though I have provided a simplified, brief introduction to the path integral approach in Chap. 18 near the end. Students wishing to make a career in field theory will eventually need to become well versed in both.

1.2 Why Qaantum Field Theory? The quantum mechanics (QM) courses students take prior to QFT generally treat a single particle such as an electron in a potential (e.g., square well, harmonic oscillatoi, etc.), and ihe

particle retains its integrity (e.g., an electron remains an electron throughout the interaction.) There is no general way to treat transmutations of particles, such as that of a particle and its antiparticle annihilating one another to yield neutral particles such as photons (e.g., e + e* - 2T) Nor is there any way to describe the decay of an elementary particle such as a muon into other particles (e.g. p+ e + v + /, where the latter two symbols represent neutrino and antineutrino, respectively).

Here is where QI]T comes to the rescue. It provides a means whereby partictes can be annihilated, created, and transmigrated from one type to another. In so doing, its utility su{passes that provided by ordinary QM. There are other reasons why QFT supersedes ordinary QM. For one, it is a relativistic theory, and thus more all encompassing. Further, as we will discuss more fully later on, the straightforwaid extrapolation of non-relativistic quantum mechanics (NROM) to relativistic quantum mechanics (ROM) results in states with negative energies, and in the early days of quantum theory, these were quite problematic. V/e will see in subsequent chapters how QF.f resolved this issue quit-e nicely.

7.3 How Quantum Field Theory? As an example of the type of problem QFf handles well, consider the interaction between an electron and a positron that produces a muon and anti-muon, i.e., e + e* -+ /t- + F*, zs shown in

Limitation of original QM: no transmutation of particles

QFT: transmutation included Energies