## Contents

Preface xiii

Part I QUANTUM THEORY OF RADIATION 1

1. Quantization of the Nonrelativistic String 3

1.1 The one-dimensional classical string 3

1.2 Normal modes of the string 7

1.3 Quantization of the string 10

1.4 Canonical commutation relations 12

1.5 The number operator and phonon states 13

1.6 The quanta as particles 15

1.7 The classical limit: Field-particle duality 18

1.8 Time translation 22 Problems 25

2. Quantization of the Electromagnetic Field 28

2.1 Lorentz transformations 28

2.2 Relativistic form of Maxwell's theory 31

2.3 Interactions between particles and fields 39

2.4 Plane wave expansions 42

2.5 Massive vector fields 44

2.6 Field quantization 46

2.7 Spin of the photon 50 Problems 56

3. Interaction of Radiation with Matter 57

3.1 Time evolution and the S-matrix 57

3.2 Decay rates and cross sections 65

3.3 Atomic decay

3.4 The Lamb shift

3.5 Deuteron photodisintegration Problems

Part II RELATIVISTIC EQUATIONS

4. The Klein-Gordon Equation

4.1 The equation

4.2 Conserved norm

4.4 Pair creation from a high Coulomb barrier

4.5 Two-component form

4.6 Nonrelativistic limit

4.7 Coulomb scattering

4.8 Negative energy states Problems

5. The Dirac Equation

5.1 The equation

5.2 Conserved norm

5.4 Charge conjugation

5.5 Coulomb scattering

5.6 Negative energy states

5.7 Nonrelativistic limit

5.8 The Lorentz group

5.9 Covariance of the Dirac equation

5.10 Bilinear covariants

5.11 Chirality and massless fermions Problems

6. Application of the Dirac Equation

6.1 Spherically symmetric potentials

6.3 Hydrogen-like atoms Problems

Part III ELEMENTS OF QUANTUM FIELD THEORY

7. Second Quantization

7.1 Schrôdinger theory

7.2 Identical particles

7.3 Charged Klein-Gordon theory

7.4 Dirac theory

7.5 Interactions: An introduction Problems

8. Symmetries I

8.1 Noether's theorem

8.2 Translations

8.3 Transformations of states and operators

8.4 Parity

8.5 Charge conjugation

8.6 Time reversal

8.7 The PCT theorem Problems

9. Interacting Field Theories

9.2 Relativistic decays

9.3 Relativistic scattering

9.4 Introduction to the Feynman rules

9.5 Calculation of the cross section

9.6 Effective nonrelativistic potential

9.7 Identical particles

9.8 Pion-nucleon interactions and isospin

9.9 One-pion exchange

9.10 Electroweak decays Problems

10. Quantum Electrodynamics

10.1 The Hamiltonian

10.2 Photon propagator: ep scattering

10.4 e+e- annihilation

10.5 Fermion propagator: Compton scattering Problems

11. Loops and Introduction to Renormalization

11.1 Wick's theorem

11.2 QED to second order

11.3 Electron self-energy

11.4 Vacuum bubbles

11.5 Vacuum polarization

11.6 Loop integrals and dimensional regularization

11.7 Dispersion relations

11.8 Vertex corrections

11.9 Charge renormalization

11.10 Bremsstrahlung and radiative corrections Problems

12. Bound States and Unitarity

12.2 The role of crossed ladders

12.3 Relativistic two-body equations

12.4 Normalization of bound states

12.5 The Bethe-Salpeter equation

12.6 The spectator equation

12.7 Equivalence of two-body equations

12.8 Unitarity

12.9 The Blankenbecler-Sugar equation

12.10 Dispersion relations and anomalous thresholds Problems

Part IV SYMMETRIES AND GAUGE THEORIES

13. Symmetries II

13.1 Abelian gauge invariance

13.2 Non-Abelian gauge invariance

13.3 Yang-Mills theories

13.4 Chiral symmetry

13.5 The linear sigma model

13.6 Spontaneous symmetry breaking

13.7 The non-linear sigma model

13.8 Chiral symmetry breaking and PCAC Problems

Path Integrals

14.1 The wave function and the propagator

14.2 The S-Matrix

14.3 Time-ordered products

14.4 Path integrals for scalar field theories

14.5 Loop diagrams in cf)3 theory

14.6 Fermions Problems

Quantum Chromodynamics and the Standard Model

15.1 Quantization of gauge theories

15.2 Ghosts and the Feynman rules for QCD

15.3 Ghosts and unitarity

15.4 The standard electro weak model

15.5 Unitarity in the Standard Model Problems

Renormalization

16.1 Power counting and regularization

16.2 cp3 theory: An example

16.3 Proving renormalizability

16.4 The renormalization of QED

16.5 Fourth order vacuum polarization

16.6 The renormalization of QCD Problems

The Renormalization Group and Asymptotic Freedom

17.1 The renormalization group equations

17.2 Scattering at large momenta

17.3 Behavior of the running coupling constant

17.4 Demonstration that QCD is asymptotically free

### 17.5 QCD corrections to the ratio R Problem

Relativistic Quantum Mechanics and Field Theory are among the most challenging and beautiful subjects in Physics. From their study we explain how states decay, can predict the existence of antimatter, learn about the origin of forces, and make the connection between spin and statistics. All of these are great developments which all physicists should know but it is a real challenge to learn them for the first time.

This book grew out of my struggle to understand these topics and to teach them to second year graduate students. It began with notes I prepared for my personal use and later shared with my students. About two years ago I decided to have these notes typed in TgX, little realizing that by so doing I had committed myself to eventually producing this book. My objectives in preparing this text > ¿fleet the original reasons I prepared my own notes: to write a book which (i) can be understood by students learning the subject for the first time, (ii) carries the development far enough so that a student is prepared to begin research, and (iii) gives meaning to the study through examples drawn from the fields of atomic, nuclear, and particle physics. In short, the goal was to produce a book which begins at the beginning, goes to the end, and is easy to read along the way.

The first two parts of this book (Part I: Quantum Theory of Radiation, and Part II: Relativistic Equations) assume no previous experience with advanced quantum mechanics. The subjects included here are quantization of the electromagnetic field, relativistic one-body wave equations, and the theoretical explanation for atomic decay, all fundamental subjects which can be regarded as necessary to a well rounded education in physics (even for classical physicists). The presentation is modeled after the first third of a year-long course which I have taught at various times over the past 15 years and these topics are given in the beginning so that those students who must leave the course at the end of the first semester will have some knowledge of these important areas.

To prepare a student for advanced work, the last two parts of this book include an introduction to many of the unique insights which relativistic field theory has contributed to modern physics, including gauge symmetry, functional methods (path integrals), spontaneous symmetry breaking, and an introduction to QCD, chiral symmetry, and the Standard Model. Part III also contains a chapter (Chapter 12) on relativistic bound state wave equations, an important topic frequently overlooked in studies at this level. I have tried to present even these more advanced topics from an elementary point of view and to discuss the subjects in sufficient detail so that the questions asked by beginning students are addressed. The entire book includes a little more material than can comfortably fit into a year long course, so that some selection must be made when used as a text.

To make the book easier to read, most proofs and demonstrations are worked out completely, with no important steps missing. Some topics, such as the quantization of fields, symmetries, and the study of the Lorentz group, are introduced briefly first, and returned to later as the reader gains more experience, and when a greater understanding is needed. This "spiral" structure (as it is sometimes referred to by the educators) is good for beginning students but may be frustrating for more advanced students who might prefer to find all the discussion of one topic in one place. I hope such readers will be satisfied by the table of contents and the index (which I have tried to make fairly complete). Considerable emphasis is placed on applications and some effort is made to show the reader how to carry out practical calculations. Problems can be found at the end of each chapter and four appendices include important material in a convenient place for ready reference.

There are many good texts on this subject and some are listed in the Reference section. Most of these books are either classics, written before the advent of modern gauge theories, or new books which treat gauge theories but omit some of the detail and elementary material found in older books. I believe that most of this elementary material is still very helpful (maybe even necessary) for students, rfid have tried to cover both modern gauge theories and these elementary topics in a single book. As a result the book is somewhat longer than many, and omits some advanced topics I would very much like to have included. Among these omissions is a discussion of anomalies in field theories.

Many people have helped me in this effort. I am grateful to Michael Frank, Joe Milana, and Michael Musolf for important suggestions and help with individual chapters. I also thank my colleagues Carl Carlson, Nathan Isgur, Anatoly Radyushkin, and Marc Sher. S. Bethke and C. Wohl kindly gave permission to use figures 17.4 and 10.9 (respectively). Many students suffered through earlier drafts, found numerous mistakes, and made many helpful suggestions. Among these are: S. Ananyan, A. Colman, K. Doty, D. Gaetano, C. Hoff, R. Kahler, Z. Li, R. Martin, D. Meekins, C. Nichols, J. Oh, X. Ou,, M. Sasinowski, P. Spickler, Y. Surya, X. Tang, A. B. Wakley, and C. Wang. Roger Gilson did an excellent job transforming my original notes into TgX. And no effort like this would be possible or meaningful without the support of my family. I am especially grateful to my wife, Chris, who assumed many of my responsibilities so I could complete the work on this book in a timely fashion. I could not have done it without her.

Franz Gross

PART I