Table Of Content

Preface to the second edition There has not been any drastic changes in the present edition. Of course some of the typographical and other smaH mistakes, posted on the internet page for the book, have been corrected. In addition, we have added OCC8S sional comments and clarifications at some places, hoping that they would make the discussions more accessible to a beginner. The only serious departure from the first edition has been in the no tation for spinor solutions of the Dirac equation. In the previous edition, we put a subscript on the spinors which corresponded to their helicities. In this edition, we change the notation'towards what is more conventionaL This notation has been introduced in §4.3 and explained in detail in §4.6. We have received many requests for posting the answers to the exercises. In this edition, we have added answers to some exercises which actually require an answer. The web page for the errata of the book is now at http://tnp.saha.ernet.in/~pbpal/books/qft/errata.html Ifyou find any mistake in the presen~edition, please inform us through this web page. It is a pleasure to thank everyone who has contributed to this book by pointing out errors or asking for clarifications. On the errata page we have acknowledged all correspondence which have affected some change. In particular, comments by C. S. Aulakh, Martin Einhorn and Scan Murray were most helpful. Amitabha Lahiri Palash B. Pal July 2004 Preface to the first edition It has been known for more than fifty years that Quantum Field Theory is necessary for describing precision experiments involving electromagnetic interactions. Within the last few decades ofthe twentieth century it has also become clear that the weak and the strong interactions are well described by interacting quantum fields. Although it is quite possible that at even smaller length scales some other kind of theory may be operative, it is clear today that quantum fields provides the appropriate framework to describe a wide class of phenomena in the energy range covered by all experiments to date. In this book, we wanted to introduce the subject as a beautiful but essentially simple piece of machinery with a wide range of applications. This book is meant as a textbook for advanced undergraduate or beginning post-graduate students. For this reason, we employ canonical quantization throughout the book. The name of the book is an echo of various chil dren's texts that were popular a long time ago, not a claim to primacy or originality. Our approach differs from many otherwise excellent textbooks at the introductory level which set up the description of electrons and photons as their goal. For example, decays are rarely discussed, since the electron and the photon are both stable particles. However decay processes are in l some sense simpler than scattering processes, since the former has only one particle in the initial state whereas the latter has two. We felt that the basic machinery of Feynman diagrams could be intro duced through decay processes even before talking about the quantization of spin-l fields. With that in mind, we start with some introductory ma terial in Chapters 1 and 2 and discuss the quantization of scalar fields in Chapter 3 and ofspin-4 fields in Chapter 4. Unlike many other texts at this level, we use a fermion normalization that should he applicable to massless as well as massive fermions. After that we discuss the generalities of the S-matrix theory in Chapter 5, and the methods of calculating Feynman diagrams, decay rates, scattering cross sections etc. with spin-O and spin-! fields, in Chapters 6 and 7. The quantization of the spin-l fields, with special reference to the pho- vii Vlll Preface ton, is taken up next in Chapter 8. After this, we have a detailed intro duction to quantum electrodynamics in Chapter 9, where we introduce the crucial concept of gauge invariance and give detailed derivations of impor tant scattering processes at their lowest orders in perturbation theory. Discrete symmetries can serve as a good guide in calculating higher or der corrections. With this in mind, we discuss parity, time reversal, charge conjugation, and their combinations, in Chapter ]O. Unlike most other textbooks on the subject which describe the P, T, C transformations only in the Pauli-Dirac representation of the Dirac matrices, we present them in a completely representation-independent way. Since other representations such as the Majorana or the chiral representations are very useful in some contexts, we hope that the general formulation will be of use to students and researchers. We calculate loop diagrams in Chapter 11, showing how to use symme tries of a problem to parametrize quantum corrections. In this chapter, we restrict ourselves to finite contributions only. Some basic concepts of renor malization are then described, with detailed calculations, in Chapter 12. Only the electromagnetic gauge symmetry is used in these two chapters for illustrative purpose. But this leads to a more general discussion of symme tries, which is done in Chapter 13. This chapter also discusses the general ideas of symmetry breaking, and the related physics of Nambu-Goldstone theorem as well as its evasion through the Higgs mechanism. This is fol lowed by an introduction to the Yang-Mills (or non-Abelian) gauge theories in Chapter 14. Finally, a basic introduction to standard electroweak the ory and electroweak processes is given in Chapter 15. In keeping with our overall viewpoint, we discuss decays as well as scattering processes in this chapter. We have tried to keep the book at the elementary level. In other words, this is a book for someone with no prior knowledge of the subject, and only .a reasonable familiarity with special relativity and quantum mechanics. To set the stage as well as to help the reader, we visit briefly the relevant parts of classical field theory in Chapter 2. A short but comprehensive introduction to group theory also appears in Chapter 13, since we did not want to assume any background in group theory for the reader. The most important tools to help the reader are the exercise problems in the book. These problems are not collected at the ends of chapters. Instead, any problem appears in the place of the text where we felt it would be most beneficial for the reader to have it worked out. Working it out at that stage should also prepare the reader for the ensuing parts of the chapter. Even if for some reason the reader does not want to work out a problem at that stage, we suggest strongly to at least read carefully the statement of the problem before proceeding further. Some of the problems come with notes or hints, some come with a relation that might be useful later. A few of the problems are marked with a * sign, implying that they Preface IX might be a little hard at that stage of the book, and the reader can leave it at that point to visit it later. A few have actually been worked out later in the book, but we have not marked them. The book was submitted in a camera-ready form to the publishers. This means that we are responsible for all the mistakes in the book, including typographical ones. We have spared no effort to avoid errors, but if any has crept in, we would like to hear about it frum the reader. We have set up a web site at http://tnp. saha.ernet. in/rvpbpal/qftbk.html containing errata for the hook, and a way of contacting us. The book grew out of courses that both of us have taught at several universities and research institutes in India ami ahroad. We have ben· efited from the enquiries and criticisms of students and colleagues. OUl' Indrajit Mitra went through the entire manw;cript carefully and offered numerous suggestions. Many other friends and colleagues also read parts of the manuscript and made useful comments, in particular Kaushik Bhat· tacharya, Ed Copeland, A. Harindranath, H. S. Mani, Jose Nieves, Saurabh Rindani. Those who have taught and influenced through their lectures, liS books and papers are too numerous to name sepa.rately. We thank them all. We thank our respective institutes for extending various facilities while the book was being written. Amitabha Lahiri Palash B. Pal April 2000 Notations JJ.,II, Space-time indices of a vector or tensor. i,i, Spatial indices of a vector or tensor. gpol.J (Components of) metric tensor, diag(l, -1, -1, -1). pJJ contravariant 4-vector. P,.. covariant 4-vectof. p 3-vector. Ipl. p Magnitude of the 3-vector p, i.e., We have used p2 and p2 interchangeably. The magnitude of the co-ordinate 3-vector has been denoted by r. a· b Scalar product of 3-vectors a and b, a .b = aibi == I:aibi. =- L: a· b Scalar product of 4-vectors a and b, a· b = al-'b,.. aP-b,.. = o aOb - a· b. ;. "'(/Jaw !L' Lagrangian density, frequently called Lagrangian. f L Total Lagrangian (; d3x Z). f f PI Action (; dt L ; d4~Z). .Ye Hamiltonian density, frequently called Hamiltonian. f H Total Hamiltonian (; d3x£'). lA, Blp Poisson bracket of A and B. lA,BI_ Commutator AB - BA. lA,BI+ Anticommutator AB + BA. a(p),at(p) Annihilation and creation operators for antiparticles ofparticles created by at(p) and annihilated by a(p), first encountered in §3.6. 8(x) Unit step function, defined in Eq. (3.13). :[000]: Normal ordered product, first defined in §3.4. 5' I···j Time ordered product, first defined in §3.7. xi Notations Xli AT Transpose of t.he matrix A. At For any matrix A, At =::. !'oAfl'D . Pauli matrices, given in Eq. (4.55). (Ti ri Same as (Ti, hut thought of as generators ofsome internal SU(2) symmetry. le-(p, s)) Electron state of3-mornentum p and spin s, first defined in §6.2. 6.F(p) Feynman propagator for scalar field in momentum space. SF(p) Feynman propagator for fermion field in momentum space. DJw{p) Feynman propagator for vector hoson field in momentum space. e Electric charge of proton. Electron carries charge -e. a Fine structure constant, first defined in §1.5. e- Electron. "'Y Photon. S/i S-matrix element between initial state Ii) and final state If}. .4lfi Feynman amplitude between initial sLate Ii} and final state If). 1_4f12 Magnitude squared of Feynman amplitude after making spin and polarization sums and averages. li,t Polarization vector for a vector boson. P Parity transformation operator, defined in Ch. 10. C Charge conjugation operator. T Time reversal operator. P Matrix of parity transformation acting on a fermion. C Matrix of charge conjugation acting on a fermion. T Matrix of time reversal acting on a fermion. x Parity transformed system of co-ordinate::;, i = (t, -x). rIt General electromagnetic vertex, first. defined in §11.1. E For dimen::;ional regularization in N dimensions, E = 2 - !N. dN Rank of I'-rnatrices in N-dimensional space-time, tr hpl'v) = dN9lw ' T Generators of a Lie group. a fabc Structure constants of a Lie group, IT", Tbl- = ifnbcTc. DI• Gauge covariant derivative D,! = a/. + i9I:,A~. Contents Preface to the second edition v Preface to the first edition vii Notations xi 1 Preliminaries 1 1.1 Why Quantum Field Theory 1 1.2 Creation and annihilation operators 3 1.3 Special relativity . . . . . . . 5 1.4 Space and time in relativistic quantum theory. 8 1.5 Natural units . . . . 9 2 Classical Field Theory 12 2.1 A quick review of particle mechanics 12 2.1.1 Action principle and Euler-Lagrange equations 12 2.1.2 Hamiltonian formalism and Poisson brackets 14 2.2 Euler-Lagrange equations in field theory . 15 2.2.1 Action functional and Lagrangian 15 2.2.2 Euler-Lagrange equations 17 2.3 Hamiltonian formalism. 19 2.4 Noether's theorem . . . . 21 3 Quantization of scalar field3 28 3.1 Equation of motion . . . . . . 28 3.2 The field and its canonical quantization 29 3.3 Fourier decomposition of the field. 30 3.4 Ground state of the Hamiltonian and Ilormal ordering 34 3.5 Fock space. .. 36 3.6 Complex scalar field . 37 3.6.1 Creation and annihilation operators 37 3.6.2 Particles and antiparticles . . . 39 3.6.3 Ground state and Hamiltonian 40 3.7 Propagator .. ..... 41 xiii Contents XIV -4 Quantization oj Dirac fields 47 4.1 Dirac Hamiltonian . . . . . 47 4.2 Dirac equation .. . . . . . 51 4.3 Plane wave solutions of Dirac equation. 54 4.3.1 Positive and negative energy spinor::; 54 4.3.2 Explicit solutions in Dirac-Pauli representation 56 4.4 Projection operators . .. 59 4.4.1 Projection operators for positive and negative energy states . . . . . . . 59 4.4.2 Helicity projection operators . 60 4.4.3 Chirality projection operators. 61 4.4.4 Spin projection operators . 62 4.5 Lagrangian for a Dirac field . . 63 4.6 Fourier decomposition of the field. 65 4.7 Propagator . 69 5 The 5-matrix expansion 72 5.1 Examples of interactions 73 5.2 Evolution operator 75 5.3 5-matrix. 80 5.4 Wick\; theorem . . 82 6 Prom Wick expansion to Feynman diagrams 87 6.1 Yukawa interaction: decay of a scalar . 87 6.2 Normalized states. . . . . . . . . 94 6.3 Sample calculation of a matrix element. 97 6.4 Another example: fermion scattering 101 6.5 Feynman amplitude 105 6.6 Feynman rules 106 6.7 Virtual particle~ 110 6.8 Amplitudes which are not S-matrix elements 112 7 Cross sections and decay rates 115 7.1 Decay rate. . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Examples of decay rate calculation. . . . . . . . . . . . 117 7.2.1 Decay of a scalar into a fermion-antifermion pair 117 7.2.2 Muon decay with 4-fermion interaction. 122 7.3 Scattering cross section 130 7.4 Generalities of 2-to-2 scattering 133 7.4.1 eM frame. . . 135 7.4.2 Lab frame. . . . . . . . 137 7.5 Inelastic scattering with 4-fermion interaction 140 7.5.1 Cross-section in CM frame 142 7.5.2 CroSl:i-section in Lab frame 143 7.6 Mandelstam variahles 144 Contents xv 8 Quantization of the electromagnetic field 146 8.1 Classical theory ofelectromagnetic fields. 146 8.2 Problems with quantization . . . . 149 8.3 Modifying the classical Lagrangian 150 8.4 Propagator .... . . . . . . . .. 153 8.5 Fourier decomposition of the field ". 156 8.6 Physical states ..... . ... 158 8.7 Another look at the propagator 162 8.8 Feynman rules for photons. 164 9 Quantum electrodynamics 166 9.1 Local gauge invariance . 166 9.2 Interaction Hamiltonian 170 9.3 Lowest order processes . 172 9.4 Electron-electron scattering 174 9.5 Electron-positron scattering 180 9.6 e-e+ --+ ~-~+ . 182 9.7 Consequence of gauge invariance 184 9.8 Compton scattering . 185 9.9 Scattering by an external field . . 194 9.10 Bremsstrahlung . 197 lOP, T, C and their combinations 200 10.1 Motivations from classical physics . 200 10.2 Parity . 201 10.2.1 Free scalar fields 201 10.2.2 Free Dirac field . 202 10.2.3 Free photon field 204 10.2.4 Interacting fields 205 10.3 Charge conjugation. 207 10.3.1 Free fields . 207 10.3.2 Interactions. 211 10.4 Time reversal . . . . 212 10.4.1 Antilinearity 212 10.4.2 Free fields . 213 10.4.3 Interactions 216 10.5 CP . 217 10.6 CPT . 218 11 Electromagnetic form factors 222 11.1 General electromagnetic vertex 222 11.2 Physical interpretation of form factors 224 11.2.1 Charge form factor F 224 1 ..... 11.2.2 Anomalous magnetic mOE1ent F 228 2 11.2.3 Electric dipole moment F 228 2 ....