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Lectures on Quantum Field Theory Aleksandar R. Bogojevi´c1 Institute of Physics P. O. Box 57 11000 Belgrade, Yugoslavia March, 1998 1Email: [email protected] ii Contents 1 Linearity and Combinatorics 1 1.1 Schwinger–DysonEquations. 1 1.2 GeneratingFunctionals. 3 1.3 FreeFieldTheory ........................... 6 2 Further Combinatoric Structure 7 2.1 ClassicalFieldTheory ......................... 7 2.2 TheEffectiveAction .......................... 8 2.3 PathIntegrals.............................. 10 3 Using the Path Integral 15 3.1 Semi-ClassicalExpansion . 15 3.2 WardIdentities ............................. 18 4 Fermions 21 4.1 GrassmannNumbers.......................... 21 5 Euclidean Field Theory 27 5.1 Thermodynamics ............................ 30 5.2 WickRotation ............................. 31 6 Ferromagnets and Phase Transitions 33 6.1 ModelsofFerromagnets . 33 6.2 TheMeanFieldApproximation. 34 6.3 TransferMatrices............................ 35 6.4 Landau–Ginsburg Theory . 36 6.5 TowardsLoopExpansion . 38 7 The Propagator 41 7.1 ScalarPropagator ........................... 41 7.2 RandomWalk.............................. 43 iii iv CONTENTS 8 The Propagator Continued 47 8.1 The Yukawa Potential . 47 8.2 VirtualParticles ............................ 49 9 From Operators to Path Integrals 53 9.1 HamiltonianPathIntegral. 53 9.2 LagrangianPathIntegral . 55 9.3 QuantumFieldTheory. 57 10 Path Integral Surprises 59 10.1 Paths that don’t Contribute . 59 10.2 Lagrangian Measure from SD Equations . 62 11 Classical Symmetry 67 11.1 NoetherTechnique . 67 11.2 Energy-Momentum Tensors Galore . 70 12 Symmetry Breaking 73 12.1 GoldstoneBosons............................ 73 12.2 TheHiggsMechanism . 77 13 Effective Action to One Loop 81 13.1 TheEffectivePotential. 81 13.2 The O(N)Model............................ 86 14 Solitons 89 14.1 Perturbative vs. Semi-Classical . 89 14.2 ClassicalSolitons ............................ 90 14.3 The φ4 Kink .............................. 94 14.4 TheSine-GordonKink. 95 15 Solitons Continued 99 15.1 BogomolyniDecomposition . 99 15.2 Derrick’sTheorem . 102 16 Quantization of Solitons 105 16.1Stability................................. 105 16.2 PathIntegralFormalism . 107 17 Instanton Preliminaries 111 17.1 ClassicalSolutions . 111 17.2 TheDeterminant . 113 CONTENTS v 18 Instantons 117 18.1 DoubleWellPotential . 117 18.2 PeriodicPotential. 123 18.3 DecayoftheFalseVacuum . 124 19 Gauge Theories 127 19.1 GaugeTheoriesonaLattice. 127 19.2 TheContinuumLimit . 130 19.3 Electrodynamics . 132 20 Differential Geometry and Gauge Fields 137 20.1 DifferentialForms . 137 20.2 GaugeFieldsasForms . 140 21 Euclidian Yang-Mills and Topology 145 21.1 ThePontryaginIndex . 145 21.2 The Chern-Simons Action . 147 21.3 The Wess-Zumino Functional . 148 22 The Axial Anomaly 151 22.1 SchwingerModel ............................ 151 22.2 Current Correlators in d =2...................... 156 22.3 Axial Anomaly via Point-Splitting . 156 23 Gauge Anomalies 159 23.1 Cochains, Cocycles and the Coboundary Operator . 159 23.2 Chern Forms and the Descent Equations . 162 23.3 Atiyah-SingerIndexTheorem . 164 23.4 The Wess-Zumino-Witten Model . 168 24 Vacuum Polarization 171 24.1 Schwinger’sSolution . 171 24.2 PerturbativeSolution . 175 25 Perturbative vs. Exact 177 25.1 BorelSummation. 177 25.2 Theories that are not Borel Summable . 179 25.3 Getting Around Instantons . 180 26 Quantizing Gauge Theories 183 26.1 Faddeev-Popov Quantization . 183 26.2 BRSTSymmetry ............................ 186 26.3Examples ................................ 189 26.4 The U(1) Antisymmetric Tensor Model . 191 vi CONTENTS 27 Background Field Method 195 27.1 OneLoopCounterterms . 195 27.2 AnAuxilliaryGaugeSymmetry. 199 Preface These lecture notes form the material for the graduate course QFT 2, the second of three quantum field theory courses in the graduate program in High Energy Theory at the Institute of Physics in Belgrade. The course presents the functional for- malism of Schwinger-Dyson equations, generating functionals and Feynman path integrals. The topics covered include: perturbation theory, loop expansion, Eu- clidean field theory, solitons, instantons, vacuum polarization, geometry of gauge theories, quantization of gauge theories, anomalies, the background field method, renormalization, the renorm group, Borel summation, large N expansion and quan- tum field theory at non zero temperature. These general tools are presented on a host of models including: The Schwinger model, ’t Hooft model, sine-Gordon model, Liouville field theory, CPN model, Ising model, Landau-Ginsburg model, Heisenberg model, Wilson model, and the Kogut-Susskind model. vii Lecture 1 Linearity and Combinatorics 1.1 Schwinger–Dyson Equations We look at a given quantum field theory (QFT) with one-particle states φi. The theory is completely known if we are given all the n-particle transition amplitudes (Green’s functions) G ,G ,G We shall diagramaticaly depict these as the i ij ijk ··· following set of blobs: Gi = i Gi j = i j k G = i j k i j Instead of knowing an infinity of Green’s functons we seek a smaller set of basic amplitudes out of which we may build all the Green’s functions. For the moment let us suppose that we are given these basic amplitudes — also known as Feynman rules. k γ i J = ii j k = i i i jj i∆ = i j i j l k γ ii j k l = i jj The basic one-point amplitude is often called a source or external field. The 1 2 LECTURE 1. LINEARITY AND COMBINATORICS corresponding two-point amplitude is the propagator, while all the higher am- plitudes designate interaction vertices. The reasons for these names will become evident later. We are usualy going to be interested in dynamics without external fields, however, as we shall soon see, many of our calculations will be simplified if we keep the sources till the very end. Green’s functions are total amplitudes for a given process and they represent a sum of the amplitudes for all distinct ways in which this process can happen. This additive property is the central property of all quantum theories, and is mathematicaly stated in terms of the Schwinger–Dyson (SD) equations. To write down the SD equation for a given Green’s function we follow a simple rule: Pick any leg, follow it into the blob and list all possible outcomes consistent with the Feynman rules. For example, let us write the appropriate SD equation for Gijk for a model with only cubic and quartic interactions and no sources. k G = = i j k i + + j 1 1 + 2! + 3! If we were keeping sources we would have an additional piece On the other hand, if we also had an n-particle vertex we would in addition have 1 (!n-1) The numerical factors multiplying a given blob are just symmetry factors. The n-vertex has (n 1) internal legs which can be permuted (n 1)! times. To avoid − 1 − over-counting we multiply by (n 1)! as above. The SD equations are easily− seen to represent an infinite set of coupled equa- tions, and are extremely difficult to solve. We shall first present a systematic approximation scheme called perturbation theory. If the interaction aplitudes γ are small, then one can perform an expansion in powers of γ, i.e. in numbers 1.2. GENERATING FUNCTIONALS 3 of vertices. We, therefore, iterate the SD equations and disregard diagrams with more than a certain number of vertices. For example, we will calculate the two- point Green’s function to two vertices in a model with a purely cubic interaction and no source terms. We have 1 = + 2 2 1 Now the 3-point Green’s function obeys = + + 1 2 1 1 1 0 To proceed further we need the tadpole to one vertex 1 = 1 = 2 2 1 0 as well as the 4-point function to no vertices = + + = 0 0 0 0 = + + Putting this back into the expression for the two-point amplitude we get = ( + 1 + 2 4 + 1 + 1 2 2 ) 1.2 Generating Functionals Now we will present a compact way to write all the SD equations at once. To do this we introduce the generating functional ∞ im Z[J]= Gi1i2 im Ji1 Ji2 Jim = (1.1) m! ··· ··· m=0 X 4 LECTURE 1. LINEARITY AND COMBINATORICS = + + 1 +1 + ... = 2! 3! J Note that this is just the vacuum diagram in the presence of sources. Through its derivatives Z[J] generates all the Green’s functions, for example 1 ∂ ∂ ∂ G = Z[J] . (1.2) ijk i3 ∂J ∂J ∂J i j k J=0 If we do not set J = 0 at the end we get the corresponding Green’s function in the presence of an external field. Now we return to the SD equations. For simplicity let us look at a model with a purely cubic interaction. The tadpole SD equation with sources is just = + 1 J 2 J J In terms of Z[J] this is simply the differential equation 1 ∂ 1 1 ∂ 1 ∂ Z[J]= i iJ + iγ Z[J] . (1.3) i ∂J 4ij i 2 jkl i ∂J i ∂J i k l 1 We multiply both sides with − and the SD equation may then be compactly 4ij written as 1 ∂S + Ji Z[J] = 0 , (1.4) ∂φi φ= 1 ∂ i ∂J where we have introduced the quantum action functional 1 1 1 S[φ]= φ − φ + γ φ φ φ . (1.5) 2 i 4ij j 3! ijk i j k For a general theory the SD equation retains the same form, while the appropriate quantum action becomes 1 1 1 1 S[φ]= φ − φ + γ φ φ φ + γ φ φ φ φ + ... (1.6) 2 i 4ij j 3! ijk i j k 4! ijkl i j k l As we see S[φ] is the generating functional for the Feynman rules. Later we will see that it is strongly related to the classical action I[φ], hence its name. Note that 1.4 is just the tadpole SD equation. It is a homogenous linear differential equation. In the next lecture we shall find its formal solution. What is important for us now is that equation 1.4 completely determines Z[J], and hence our whole QFT.
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