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Quantum Field Theory I + II

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Quantum Field Theory I + II Quantum Field Theory I + II Institute for Theoretical Physics, Heidelberg University Timo Weigand Literature This is a writeup of my Master programme course on Quantum Field Theory I (Chapters 1-6) and Quantum Field Theory II. The primary source for this course has been • Peskin, Schröder: An introduction to Quantum Field Theory, ABP 1995, • Itzykson, Zuber: Quantum Field Theory, Dover 1980, • Kugo: Eichtheorie, Springer 1997, which I urgently recommend for more details and for the many topics which time constraints have forced me to abbreviate or even to omit. Among the many other excellent textbooks on Quantum Field Theory I particularly recommend • Weinberg: Quantum Field Theory I + II, Cambridge 1995, • Srednicki: Quantum Field Theory, Cambridge 2007, • Banks: Modern Quantum Field Theory, Cambridge 2008 as further reading. All three of them oftentimes take an approach different to the one of this course. Excellent lecture notes available online include • A. Hebecker: Quantum Field Theory, • D. Tong: Quantum Field Theory. Special thanks to Robert Reischke1 for his fantastic work in typing these notes. 1For corrections and improvement suggestions please send a mail to [email protected]. 4 Contents 1 The free scalar field9 1.1 Why Quantum Field Theory?..............................9 1.2 Classical scalar field: Lagrangian formulation..................... 11 1.3 Noether’s Theorem................................... 14 1.4 Quantisation in the Schrödinger Picture........................ 17 1.5 Mode expansion..................................... 18 1.6 The Fock space..................................... 21 1.7 Some important technicalities.............................. 23 1.7.1 Normalisation.................................. 23 1.7.2 The identity................................... 24 1.7.3 Position-space representation.......................... 24 1.8 On the vacuum energy................................. 25 1.9 The complex scalar field................................ 28 1.10 Quantisation in the Heisenberg picture......................... 30 1.11 Causality and Propagators............................... 33 1.11.1 Commutators.................................. 33 1.11.2 Propagators................................... 35 1.11.3 The Feynman-propagator............................ 36 1.11.4 Propagators as Green’s functions........................ 39 2 Interacting scalar theory 41 2.1 Introduction....................................... 41 2.2 Källén-Lehmann spectral representation........................ 42 2.3 S-matrix and asymptotic in/out-states......................... 46 2.4 The LSZ reduction formula............................... 48 2.5 Correlators in the interaction picture.......................... 53 2.5.1 Time evolution................................. 55 2.5.2 From the interacting to the free vacuum.................... 56 2.6 Wick’s theorem..................................... 59 2.7 Feynman diagrams................................... 61 5 6 CONTENTS 2.7.1 Position space Feynman-rules......................... 63 2.7.2 Momentum space Feynman-rules....................... 63 2.8 Disconnected diagrams................................. 65 2.8.1 Vacuum bubbles................................ 67 2.9 1-particle-irreducible diagrams............................. 67 2.10 Scattering amplitudes.................................. 70 2.10.1 Feynman-rules for the S -matrix........................ 72 2.11 Cross-sections...................................... 73 1 3 Quantising spin 2 -fields 77 3.1 The Lorentz algebra so(1, 3) .............................. 77 3.2 The Dirac spinor representation............................. 81 3.3 The Dirac action.................................... 84 3.4 Chirality and Weyl spinors............................... 86 3.5 Classical plane-wave solutions............................. 89 3.6 Quantisation of the Dirac field............................. 91 3.6.1 Using the commutator............................. 91 3.6.2 Using the anti-commutator........................... 94 3.7 Propagators....................................... 97 3.8 Wick’s theorem and Feynman diagrams........................ 99 3.9 LSZ and Feynman rules................................. 100 4 Quantising spin 1-fields 103 4.1 Classical Maxwell-theory................................ 103 4.2 Canonical quantisation of the free field......................... 105 4.3 Gupta-Bleuler quantisation............................... 108 4.4 Massive vector fields.................................. 112 4.5 Coupling vector fields to matter............................ 113 4.5.1 Coupling to Dirac fermions.......................... 114 4.5.2 Coupling to scalars............................... 116 4.6 Feynman rules for QED................................. 117 4.7 Recovering Coulomb’s potential............................ 121 4.7.1 Massless and massive vector fields....................... 124 5 Quantum Electrodynamics 127 5.1 QED process at tree-level................................ 127 5.1.1 Feynman rules for in/out-states of definite polarisation............ 127 5.1.2 Sum over all spin and polarisation states.................... 128 5.1.3 Trace identities................................. 129 5.1.4 Centre-of-mass frame.............................. 130 CONTENTS 7 5.1.5 Cross-section.................................. 130 5.2 The Ward-Takahashi identity.............................. 132 5.2.1 Relation between current conservation and gauge invariance......... 135 5.2.2 Photon polarisation sums in QED....................... 136 5.2.3 Decoupling of potential ghosts......................... 136 5.3 Radiative corrections in QED - Overview....................... 137 5.4 Self-energy of the electron at 1-loop.......................... 138 5.4.1 Feynman parameters.............................. 139 5.4.2 Wick rotation.................................. 140 5.4.3 Regularisation of the integral.......................... 141 5.5 Bare mass m0 versus physical mass m ......................... 144 5.5.1 Mass renormalisation.............................. 146 5.6 The photon propagator................................. 146 5.7 The running coupling.................................. 149 5.8 The resummed QED vertex............................... 150 5.8.1 Physical charge revisited............................ 153 5.8.2 Anomalous magnetic moment......................... 153 5.9 Renormalised perturbation theory of QED....................... 154 5.9.1 Bare perturbation theory............................ 155 5.9.2 Renormalised Perturbation theory....................... 158 5.10 Infrared divergences................................... 164 6 Classical non-abelian gauge theory 165 6.1 Geometric perspective on abelian gauge theory.................... 165 6.2 Non-abelian gauge symmetry.............................. 167 6.3 The Standard Model.................................. 171 7 Path integral quantisation 173 7.1 Path integral in Quantum Mechanics.......................... 173 7.1.1 Transition amplitudes.............................. 173 7.1.2 Correlation functions.............................. 178 7.2 The path integral for scalar fields............................ 179 7.3 Generating functional for correlation functions.................... 183 7.3.1 Functional calculus............................... 183 7.4 Free scalar field theory................................. 185 7.5 Perturbative expansion in interacting theory...................... 188 7.6 The Schwinger-Dyson equation............................ 191 7.7 Connected diagrams................................... 195 7.8 The 1PI effective action................................. 196 7.9 G(') as a quantum effective action and background field method........... 200 8 CONTENTS 7.10 Euclidean QFT and statistical field theory....................... 203 7.11 Grassman algebra calculus............................... 206 7.12 The fermionic path integral............................... 211 8 Renormalisation of Quantum Field Theory 217 8.1 Superficial divergence and power counting....................... 217 8.2 Renormalisability and BPHZ theorem......................... 220 8.3 Renormalisation of φ4 theory up to 2-loops...................... 222 8.3.1 1-loop renormalisation............................. 224 8.3.2 Renormalisation at 2-loop........................... 226 8.4 Renormalisation of QED revisited........................... 231 8.5 The renormalisation scale................................ 233 8.6 The Callan-Symanzyk (CS) equation.......................... 234 8.7 Computation of β-functions in massless theories.................... 237 8.8 The running coupling.................................. 239 8.9 RG flow of dimensionful operators........................... 243 8.10 Wilsonian effective action & Renormalisation Semi-Group.............. 245 9 Quantisation of Yang-Mills-Theory 253 9.1 Recap of classical YM-Theory............................. 253 9.2 Gauge fixing the path integral.............................. 255 9.3 Faddeev-Popov ghosts................................. 259 9.4 Canonical quantisation and asymptotic Fock space.................. 262 9.5 BRST symmetry and the physical Hilbert space.................... 264 Chapter 1 The free scalar field 1.1 Why Quantum Field Theory? In (non-relativistic) Quantum Mechanics, the dynamics of a particle is described by the time-evolution of its associated wave-function (t, ~x) with respect to the non-relativistic Schrödinger equation @ ih¯ (t, ~x) = H (t, ~x), (1.1) @t ~pˆ2 with the Hamilitonian given by H = 2m + V(xˆ). In order
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