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Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education)

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Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education) Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education) Roberto Soldati February 23, 2019 Contents 0.1 Prologue ............................ 4 1 Basics in Group Theory 8 1.1 Groups and Group Representations . 8 1.1.1 Definitions ....................... 8 1.1.2 Theorems ........................ 12 1.1.3 Direct Products of Group Representations . 14 1.2 Continuous Groups and Lie Groups . 16 1.2.1 The Continuous Groups . 16 1.2.2 The Lie Groups .................... 18 1.2.3 An Example : the Rotations Group . 19 1.2.4 The Infinitesimal Operators . 21 1.2.5 Lie Algebra ....................... 23 1.2.6 The Exponential Representation . 24 1.2.7 The Special Unitary Groups . 27 1.3 The Non-Homogeneous Lorentz Group . 34 1.3.1 The Lorentz Group . 34 1.3.2 Semi-Simple Groups . 43 1.3.3 The Poincar´eGroup . 49 2 The Action Functional 57 2.1 The Classical Relativistic Wave Fields . 57 2.1.1 Field Variations .................... 58 2.1.2 The Scalar and Vector Fields . 62 2.1.3 The Spinor Fields ................... 65 2.2 The Action Principle ................... 76 2.3 The N¨otherTheorem ................... 79 2.3.1 Problems ........................ 87 1 3 The Scalar Field 94 3.1 General Features ...................... 94 3.2 Normal Modes Expansion . 101 3.3 Klein-Gordon Quantum Field . 106 3.4 The Fock Space .......................112 3.4.1 The Many-Particle States . 112 3.4.2 The Lorentz Covariance Properties . 116 3.5 Special Distributions . 129 3.5.1 Euclidean Formulation . 133 3.6 Problems ............................137 4 The Spinor Field 151 4.1 The Dirac Equation . 151 4.1.1 Normal Modes Expansion . 152 4.1.2 Spin States .......................157 4.2 N¨otherCurrents . 160 4.3 Dirac Quantum Field . 165 4.4 Covariance of the Quantum Dirac Field . 172 4.5 Special Distributions . 179 4.6 Euclidean Fermions . 182 4.7 The C, P and T Transformations . 186 4.7.1 The Charge Conjugation . 186 4.7.2 The Parity Transformation . 188 4.7.3 The Time Reversal . 189 4.8 Problems ............................195 4.8.1 Belifante Energy-Momentum Tensor . 195 4.8.2 Mass-less Neutrino and Anti-Neutrino Fields . 197 4.8.3 Majorana Spinor Field . 204 4.8.4 Dirac and Majorana Mass Terms . 215 5 The Vector Field 219 5.1 General Covariant Gauges . 220 5.1.1 Conserved Quantities . 222 5.2 Normal Modes Decomposition . 226 5.2.1 Normal Modes of the Massive Vector Field . 227 5.2.2 Normal Modes of the Gauge Potential . 230 5.3 Covariant Canonical Quantum Theory . 234 2 5.3.1 The Massive Vector Field . 234 5.3.2 The St¨uckelberg Ghost Scalar . 238 5.3.3 The St¨uckelberg Vector Propagator . 240 5.3.4 The Gauge Vector Potential . 244 5.3.5 Fock Space with Indefinite Metric . 246 5.3.6 Photon Helicity . 252 5.4 Problems ............................258 3 0.1 Prologue The content of this manuscript is the first semester course in quantum field theory that I have delivered at the Physics Department of the Faculty of Sciences of the University of Bologna since the academic year 2007/2008. It is a pleasure for me to warmly thank all the students and in particular Pietro Longhi, Lorenzo Rossi, Fabrizio Sgrignuoli and Demetrio Vilardi for their invaluable help in finding mistakes and misprints. In addition it is mandatory to me to express my deep and warm gratitude to Dr. Paola Giacconi for her concrete help in reading my notes and her continuous encouragement. To all of them I express my profound thankfulness. Roberto Soldati Distanze Questo `eil tempo in cui tutto `ea portata di mano e niente a portata di cuore Sono sempre di pi´uquelli che aspettano uno sguardo. I nostri occhi non sanno pi´uvedere. Per questo, un giorno, qualcosa cambier´a. Maurizio Bacchilega A revolution is not a dinner party, or writing an essay, or painting a picture, or doing embroidery; it cannot be so refined, so leisurely and gentle, so temperate, kind, courteous, restrained and magnanimous. A revolution is an insurrection, an act of violence... Mao Tse Tung 4 Introduction : Some Notations Here I like to say a few words with respect to the notation adopted. The components of all the tetra-vectors or 4-vectors have been chosen to be real. The metric is defined by means of the Minkowski constant symmetric tensor of rank two 8 < 0 for µ 6= ν gµν = 1 for µ = ν = 0 µ, ν = 0; 1; 2; 3 : −1 for µ = ν = 1; 2; 3 i.e. the invariant product of two tetra-vectors a and b with components a0; a1; a2; a3 and b0; b1; b2; b3 is defined in the following manner µ ν 0 0 1 1 2 2 3 3 0 0 0 0 k k a · b ≡ gµν a b = a b − a b − a b − a b = a b − a · b = a b − a b Thus the Minkowski space will be denoted by M = R1;3. Summation over repeated indexes is understood { Einstein's convention { unless differently stated, while bold type a; b notation is used to denote ordinary vectors in the 3-dimensional Euclidean space. Indexes representing all four components 0,1,2,3 are usually denoted by Greek letters, while all the indexes representing the three space components 1,2,3 are denoted by Latin letters. The upper indexes are, as usual, contravariant while the lower indexes are covariant. Raising and lowering indexes are accomplished with the aid of the Minkowski metric tensor , e.g. ν µ µν µν aµ = gµν a a = g aν gµν = g in such a manner that the invariant product can also be written in the form µ ν µ 0 k 0 1 2 3 a · b = gµν a b = a bµ = a b0 + a bk = a b0 + a b1 + a b2 + a b3 Throughout the notes the natural system of units is used c = ~ = 1 for the speed of light and the reduced Planck's constant, unless explicitly stated. In turn, for physical units the Heaviside{Lorentz C. G. S. system of electromagnetic units will be employed. In the natural system of units it appears that energy, momentum and mass have the dimensions of a reciprocal length or a wave number, while the time x0 has the dimensions of a length. The Coulomb's potential created by a point charge q is q q α '(x) = = 4π jxj e2 r 5 and the fine structure constant is e2 e2 1 α = = = 7:297 352 568(24) × 10−3 ≈ 4π 4π~c 137 The symbol −e (e > 0) stands for the negative electron charge with [ e ] = eV G−1 cm−1 : We generally work with the four dimensional Minkowski form of the Maxwell equations 1 " µνρσ@ F = 0 @ F µν = J ν ν ρσ µ c where µ µ A = ('; A) F µν = @µAν − @νAµ J = (cρ, J) 1 B = r × AE = −∇' − A_ c in which 1 2 3 k k0 E = (E ;E ;E ) ;E = F = F0k ; ( k = 1; 2; 3 ) | 1 |k` 123 1230 B = (F32;F13;F21) B = 2 " Fk` ( " = " = −1 ) are the electric and magnetic fields respectively while @ 1 @ @ = = ; r = (c−1@ ; r ) = (c−1@ ;@ ;@ ;@ ) µ @x µ c @t t t x y z so that @E r · E = ρ c r × B = J + @t Notice that in the Heaviside{Lorentz C. G. S. system of electromagnetic units 1 − 3 µ −2 −1 −1 we have [ E ] = [ B ] = G = eV 2 cm 2 while [ J ] = esu cm s = G s : We will often work with the relativistic generalizations of the Schr¨odinger wave functions of 1-particle quantum mechanical states. We represent the energy-momentum operators acting on such wave functions following the convention: µ µ P = (P0 ; P) = i@ = ( i@t ; −ir ) µ In so doing the plane wave exp{−ip · xg has momentum p = ( p0 ; p) and the Lorentz covariant coupling between a charged particle of charge q and the electromagnetic field is provided by the so called minimal substitution i@ − q '(x) for µ = 0 P − q A (x) = t µ µ ir − q A(x) for µ = 1; 2; 3 6 The Pauli spin matrices are the three Hermitean 2 × 2 matrices 8 0 1 9 8 0 −i 9 8 1 0 9 σ = > > σ = > > σ = > > 1 :> 1 0 ;> 2 :> i 0 ;> 3 :> 0 −1 ;> which satisfy σi σj = δ ij + i " ijk σk 123 where " 123 = 1 = − " so that [ σi ; σj ] = 2i " ijk σk fσi ; σjg = 2δ ij The Dirac matrices in the Weyl, or spinorial, or even chiral representation are given by 8 9 8 9 0 1 0 σk γ0 = > > γ k = > > ( k = 1; 2; 3 ) : 1 0 ; : − σk 0 ; with the Hermitean conjugation property µy 0 µ 0 µ ν µν γ = γ γ γ fγ ; γ g = 2g I so that 8 9 8 9 0 1 − σk 0 β = γ0 = > > α k = > > ( k = 1; 2; 3 ) : 1 0 ; : 0 σk ; It follows that the Dirac operator in the Weyl representation of the Clifford algebra takes the form 8 9 imc=} 0 @0 + @z @x − i@y > > > 0 imc=} @x + i@y @0 − @z > i@= − m = i} > > > −@0 − @z −@x + i@y imc=} 0 > :> ;> −@x − i@y −@0 + @z 0 imc=} Concerning the Levi-Civita symbol " µνρσ ; i.e. the totally anti-symmetric unit tensors, I follow the conventions in Lev D. Landau and Evgenij M. Lifˇsits, Teoria dei campi, Editori Riuniti, Roma (1976) x6., pp. 30-40: namely, 0123 123 " = +1 = − " 0123 " 123 = +1 = − " 1 { {|k | k C = A × B C { = 2 " {|k A| Bk = − C = " A B 1 { 1 k B = r × A B { = 2 " {|k @| Ak = − B = − 2 " {|k @| A 7 Chapter 1 Basics in Group Theory The aim of this Chapter is to briefly summarize the main definitions and key results of group theory, a huge and tough mathematical subject, which turns out to be absolutely necessary in order to understand and appreciate the crucial role of the symmetries in the development of field theory models to describe High Energy and Particle Physics.
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