Quantum Field Theory I Babis Anastasiou Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland E-mail: [email protected] December 16, 2020 Contents 1 Quantum Field Theory. Why? 7 2 Review of principles of classical and quantum mechanics 8 2.1 Time evolution in classical mechanics . .8 2.1.1 Properties of Poisson brackets . 10 2.1.2 A way to think of classical time evolution . 11 2.2 Time evolution in quantum mechanics . 12 2.3 Conservation and symmetries in classical mechanics . 13 2.3.1 A classical example . 15 2.4 Symmetries in Quantum mechanics . 15 3 Theory of Classical Fields 18 3.1 Fields from a discretised space (lattice) . 18 3.2 Euler-Lagrange equations for a classical field from a Lagrangian density . 21 3.3 Noether's theorem . 22 3.3.1 Internal field symmetry transformations . 22 3.3.2 Space-Time symmetry transformations . 23 3.3.3 Energy-momentum tensor . 26 3.3.4 Lorentz symmetry transformations and conserved currents . 28 3.4 Field Hamiltonian Density from discretization . 31 3.4.1 Hamilton equations for fields . 32 3.5 An example: acoustic waves . 33 4 Quantisation of the Schr¨odingerfield 34 4.1 The Schr¨odingerequation from a Lagrangian density . 34 4.2 Symmetries of the Schroedinger field . 35 4.3 Quantisation of Fields . 37 4.4 Quantised Schr¨odingerfield . 38 4.5 Particle states from quantised fields . 39 4.6 What is the wave-function in the field quantisation formalism? . 42 5 The Klein-Gordon Field 45 5.1 Real Klein-Gordon field . 45 5.1.1 Real solution of the Klein-Gordon equation . 45 5.1.2 Quantitation of the real Klein-Gordon field . 47 5.1.3 Particle states for the real Klein-Gordon field . 47 5.1.4 Energy of particles and \normal ordering" . 48 5.1.5 Field momentum conservation . 50 5.1.6 Labels of particle states? . 51 1 5.2 Casimir effect: the energy of the vacuum . 51 5.3 Two real Klein-Gordon fields . 54 5.3.1 Two equal-mass real Klein-Gordon fields . 55 5.3.2 Two real Klein-Gordon fields = One complex Klein-Gordon field . 58 5.4 Conserved Charges as generators of symmetry transformations . 59 5.5 Can the Klein-Gordon field be an one-particle wave-function? . 60 6 Quantisation of the free electromagnetic field 62 6.1 Maxwell Equations and Lagrangian formulation . 62 6.1.1 Classical gauge invariance and gauge-fixing . 64 6.1.2 Lagrangian of the electromagnetic field . 65 6.2 Quantisation of the Electromagnetic Field . 65 6.3 Massive photons: The Higgs mechanism∗ .................. 68 7 The Dirac Equation 69 7.1 Mathematical interlude . 70 7.1.1 Pauli matrices and their properties . 70 7.1.2 Kronecker product of 2 × 2 matrices . 71 7.2 Dirac representation of γ-matrices . 71 7.3 Traces of γ− matrices . 73 7.4 γ−matrices as a basis of 4 × 4 matrices . 73 7.5 Lagrangian for the Dirac field . 74 8 Lorentz symmetry and free Fields 76 8.1 Field transformations and representations of the Lorentz group . 77 8.1.1 Scalar representation M(Λ) = 1 . 78 8.1.2 Vector representation M(Λ) = Λ . 79 8.2 Generators of field representations of Lorentz symmetry transformations 79 8.2.1 Generators of the scalar representation . 80 8.2.2 Generators of the vector representation . 81 8.2.3 Lie algebra of continuous groups . 82 8.3 Spinor representation . 84 8.4 Lorentz Invariance of the Dirac Lagrangian . 85 8.5 General representations of the Lorentz group . 86 8.6 Weyl spinors . 87 8.7 Majorana equation . 90 8.7.1 Majorana Lagrangian and Majorana equation in a four-dimensional spinor notation* . 91 9 Classical solutions of the Dirac equation 92 9.1 Solution in the rest frame . 93 9.2 Lorentz boost of rest frame Dirac spinor along the z-axis . 94 9.3 Solution for an arbitrary vector . 96 9.4 A general solution . 96 10 Quantization of the Dirac Field 98 10.1 One-particle states . 100 10.1.1 Particles and anti-particles . 100 1 10.1.2 Particles and anti-particles of spin- 2 ................ 101 2 10.2 Fermions . 103 10.3 Quantum symmetries . 104 10.4 Lorentz transformation of the quantized spinor field . 104 10.4.1 Transformation of the quantized Dirac field . 106 10.5 Parity . 107 10.6 Other discrete symmetries . 109 11 Propagation of free particles 110 11.1 Transition amplitude for the Schr¨odingerfield . 110 11.2 Transition amplitude for the real Klein-Gordon field . 111 11.3 Time Ordering and the Feynman-St¨uckelberg propagator for the real Klein- Gordon field . 114 11.4 Feynman propagator for the complex Klein-Grodon field . 115 11.5 Feynman propagator for the Dirac field . 116 11.6 Feynman propagator for the photon field . 117 11.7 Wick's theorem: time-ordering, normal-ordering and propagation . 117 11.7.1 Wick's theorem for Dirac fermion fields∗ .............. 120 11.7.2 Wick's theorem for Majorana fermions* . 123 12 Scattering Theory (S-matrix) 124 12.1 Propagation in a general field theory . 124 12.1.1 A special case: free scalar field theory . 128 12.1.2 \Typical" interacting scalar field theory . 129 12.2 Spectral assumptions in scattering theory . 130 12.3 \In" and \Out" states . 130 12.4 Scattering Matrix-Elements . 132 12.5 S-matrix and Green's functions . 133 12.6 The LSZ reduction formula . 134 12.7 Truncated Green's functions . 136 12.8 Cross-sections∗ ................................ 137 13 Perturbation Theory and Feynman Diagrams 138 13.1 Time evolution operator in the interaction picture . 139 13.2 Field operators in the interacting and free theory . 141 13.3 The ground state of the interacting and the free theory . 142 13.4 Feynman Diagrams for φ4 theory . 144 13.5 Feynman rules in momentum space . 147 13.6 Truncated Green's functions in perturbation theory . 149 14 Loop Integrals 151 14.1 The simplest loop integral. Wick rotation . 151 14.2 Dimensional Regularization . 153 14.2.1 Angular Integrations . 154 14.2.2 Properties of the Gamma function . 156 14.2.3 Radial Integrations . 157 14.3 Feynman Parameters . 157 3 15 Quantum Electrodynamics 160 15.1 Gauge invariance . 160 15.2 Perturbative QED . 162 15.3 Dimensional regularization for QED . 164 15.3.1 Gamma-matrices in dimensional regularization . 166 15.3.2 Tensor loop-integrals . 166 15.4 The electron propagator at one-loop . 169 15.5 Electron propagator at all orders . 170 15.5.1 The electron mass . 172 15.6 The photon propagator at one-loop . 173 15.7 Ward identity∗ ................................ 174 15.8 Photon propagator at all orders . 177 16 Renormalisation of QED 178 16.1 Running of the QED coupling constant and the electron mass∗ ...... 181 A Special Relativity 182 A.1 Proper time . 182 A.2 Subgroups of Lorentz transformations . 184 A.3 Time dilation . 185 A.4 Doppler effect . 186 A.5 Particle dynamics . 186 A.6 Energy and momentum . 188 A.7 The inverse of a Lorentz transformation . 189 A.8 Vectors and Tensors . 190 A.9 Currents and densities . 191 A.10 Energy-Momentum tensor . 193 A.11 Relativistic formulation of Electrodynamics . 195 A.11.1 Energy-Momentum Tensor in the presence of an electromagnetic field198 4 Bibliography [1] The Quantum Theory of Fields, Volume I Foundations, Steven Weinberg, Cambridge University Press. [2] An introduction to Quantum Field Theory, M. Peskin and D. Schroeder, Addison- Wesley [3] Quantum Field Theory in a nutshell, A. Zee, Princeton University Press. [4] Quantum Field Theory, Mark Srednicki, Cambridge University Press. [5] An introduction to Quantum Field Theory, George Sterman, Cambridge University Press. [6] Classical Mechanics, Goldstein, Poole and Safko, Addison-Wesley [7] Lectures On Qed And Qcd: Practical Calculation And Renormalisation Of One- And Multi-loop Feynman Diagrams, Andrea Grozin, World Scientific 5 Conventions for Special Relativity Our metric convention is gµν = diag (1; −1; −1; −1) : (1) A contravariant position four-vector is xµ = (x0; x1; x2; x3) ≡ (ct; x; y; z) = (ct; ~x): (2) A covariant position four-vector is ν xµ = gµνx ; (3) which gives xµ = (x0; x1; x2; x3) ≡ (ct; −x; −y; −z) =.