Quantum field theory: Lecture Notes Rodolfo Alexander Diaz Sanchez Universidad Nacional de Colombia Departamento de F´ısica Bogot´a, Colombia August 23, 2015 Contents 1 Relativistic quantum mechanics 7 1.1 Surveyonquantummechanics . .............. 7 1.1.1 Vector subspaces generated by eigenvalues . ................... 8 1.2 Symmetriesinquantummechanics . ................ 9 1.3 Irreducible inequivalent representations of groups . .......................... 12 1.4 ConnectedLiegroups .............................. ............. 14 1.5 Lorentztransformations . ................ 17 1.6 The inhomogeneous Lorentz Group (or Poincar´egroup) . ...................... 19 1.6.1 Four-vectorsandtensors. .............. 20 1.7 SomesubgroupsofthePoincar´egroup . .................. 22 1.7.1 Proper orthochronous Lorentz group . ................ 23 1.7.2 Discrete transformations in the Lorentz group . .................... 24 1.7.3 Infinitesimal transformations within the proper orthochronus Lorentz group . 25 1.8 QuantumLorentzTransformations . ................. 25 1.8.1 Four-vector and tensor operators . ................ 26 1.8.2 Infinitesimal quantum Lorentz transformations . .................... 27 1.8.3 Lorentz transformations of the generators . ................... 28 1.8.4 Lie algebra of the Poincar´egenerators . .................. 29 1.8.5 Physical interpretation of Poincare’s generators . ....................... 30 1.9 One-particlestates .............................. ............... 32 1.9.1 One-particle states under pure translations . .................... 32 1.9.2 One-particle states under homogeneous Lorentz transformations ............... 32 1.9.3 Physicallittlegroups. .............. 35 1.9.4 Normalization of one-particle states . .................. 36 1.10 One-particle states with non-null mass . ..................... 38 1.10.1 Wigner rotation and standard boost . ................ 40 1.11 One-particle states with null mass . ................... 44 1.11.1 Determination of the little group . ................. 44 1.11.2 Lie algebra of the little group ISO (2) .............................. 48 1.11.3 Massless states in terms of eigenvalues of the generators of ISO (2).............. 50 1.11.4 Lorentz transformations of massless states . ..................... 51 1.12 Space inversion and time-reversal . .................... 53 1.13 Parity and time-reversal for one-particle states with M > 0....................... 56 1.13.1 Parity for M > 0.......................................... 56 1.13.2 Time reversal for M > 0...................................... 58 1.13.3 Parity for null mass particles . ................ 59 1.13.4 Time-reversal for null mass particles . ................... 62 1.14 Action of T 2 andKramer’sdegeneracy. 63 2 CONTENTS 3 2 Scattering theory 65 2.1 Construction of “in” and “out” states . .................. 65 2.2 The S matrix ............................................. 70 − 2.3 Symmetries of the S matrix ....................................... 75 − 2.3.1 Lorentzinvariance . .. .. .. .. .. .. .. .. ............ 75 2.3.2 Internalsymmetries . .. .. .. .. .. .. .. .. ............ 81 2.3.3 Parity ........................................ ........ 83 2.3.4 Time-reversal................................. ........... 86 2.3.5 PTsymmetry.................................... ........ 89 2.3.6 Charge-conjugationC,CPandCPT . ............. 90 2.4 Ratesandcross-sections . ................ 91 2.4.1 One-particle initial states . ................ 94 2.4.2 Two-particles initial states . ................. 95 2.4.3 Multi-particle initial states . ................. 95 2.4.4 Lorentz transformations of rates and cross-sections ....................... 95 2.5 Physical interpretation of the Dirac’s phase space factor δ4 (p p ) dβ ............... 98 β − α 2.5.1 The case of Nβ =2......................................... 98 2.5.2 The case with Nβ =3andDalitzplots.............................. 101 2.6 Perturbationtheory .............................. .............. 102 2.6.1 Distorted-wave Born approximation . ................ 107 2.7 Implicationsofunitarity . ................. 108 2.7.1 Generalized optical theorem and CPT invariance . ................... 111 2.7.2 Unitarity condition and Boltzmann H-theorem . .................. 112 3 The cluster decomposition principle 115 3.1 Physicalstates .................................. ............. 115 3.1.1 Interchange of identical particles . .................. 116 3.1.2 Interchange of non-identical particles . .................... 117 3.1.3 Normalization of multi-particle states . ................... 117 3.2 Creation and annihilation operators . ................... 118 3.2.1 Commutation and anti-commutation relations of a (q) and a† (q) ............... 119 3.3 Arbitrary operators in terms of creation and annihilationoperators ................. 120 3.4 Transformation properties of the creation and annihilation operators . 121 3.5 Cluster decomposition principle and connected amplitudes....................... 122 3.5.1 Someexamplesofpartitions. .............. 124 3.6 Structureoftheinteraction . ................. 127 3.6.1 Asimpleexample ................................ ......... 129 3.6.2 Connected and disconnected parts of the interaction . ..................... 131 3.6.3 Some examples of the diagrammatic properties . .................. 133 3.6.4 Implications of the theorem . .............. 136 4 Relativistic quantum field theory 137 4.1 Freefields....................................... ........... 137 4.2 Lorentz transformations for massive fields . ..................... 139 4.2.1 Translations.................................. ........... 142 4.2.2 Boosts........................................ ........ 144 4.2.3 Rotations ..................................... ......... 144 4.3 Implementation of the cluster decomposition principle ......................... 145 4.4 Lorentz invariance of the S matrix.................................... 146 − 4 CONTENTS 4.5 Internal symmetries and antiparticles . ..................... 147 4.6 Lorentz irreducible fields and Klein-Gordon equation . ........................ 149 5 Causal scalar fields for massive particles 151 5.1 Scalar fields without internal symmetries . ..................... 151 5.2 Scalar fields with internal symmetries . ................... 155 5.3 Scalar fields and discrete symmetries . ................... 158 6 Causal vector fields for massive particles 162 6.1 Vector fields without internal symmetries . .................... 162 6.2 Spinzerovectorfields ............................. .............. 165 6.3 Spinonevectorfields.............................. .............. 166 6.4 Spin one vector fields with internal symmetries . ..................... 174 6.4.1 Field equations for spin one particles . .................. 175 6.5 Inversion symmetries for spin-one fields . .................... 176 7 Causal Dirac fields for massive particles 178 7.1 Spinor representations of the Lorentz group . ..................... 178 7.2 Some additional properties of the Dirac matrices . ...................... 182 7.3 The chiral representation for the Dirac matrices . ....................... 185 7.4 CausalDiracfields ................................ ............. 190 7.5 Dirac coefficients and parity conservation . .................... 193 7.6 Charge-conjugation properties of Dirac fields . ...................... 201 7.7 Time-reversal properties of Dirac fields . ..................... 204 7.8 Majoranafermionsandfields . ............... 207 7.9 Scalar interaction densities from Dirac fields . ....................... 207 7.10TheCPTtheorem .................................. ........... 209 8 Massless particle fields 211 9 The Feynman rules 223 9.1 Generalframework ................................ ............. 223 9.2 Rules for the calculation of the S matrix ................................ 226 − 9.3 Diagrammatic rules for the S matrix .................................. 227 − 9.4 Calculation of the S matrixfromthefactorsanddiagrams . 229 − 9.5 Afermion-bosontheory . .. .. .. .. .. .. .. .. .............. 234 9.5.1 Fermion-boson scattering . .............. 234 9.5.2 Fermion fermion scattering . .............. 238 9.5.3 Boson-bosonscattering . ............. 238 9.6 Aboson-bosontheory .............................. ............. 238 9.7 Calculation of the propagator . ................. 240 9.7.1 Other definitions of the propagator . ................ 248 9.8 Feynman rules as integrations over momenta . ................... 249 9.9 Examples of application for the Feynman rules with integration over four-momenta variables . 252 9.9.1 Fermion-boson scattering . .............. 252 9.9.2 Fermion-fermion scattering . ............... 255 9.9.3 Boson-bosonscattering . ............. 255 9.10 Examples of Feynman rules as integrations over momenta ....................... 258 9.10.1 Fermion-boson scattering . ............... 259 9.10.2 Fermion-fermion and Boson-boson scattering . .................... 260 CONTENTS 5 9.11 Topological structure of the lines . .................... 260 9.12 Off-shell and on-shell four-momenta . .................. 262 9.12.1 The r thderivativetheorem ................................ 264 − 10 Canonical quantization 268 10.1 Canonicalvariables. ................ 268 10.1.1 Canonical variables for scalar fields . ................... 269 10.1.2 Canonical variables for vector fields . .................. 271 10.1.3 Canonical variables for Dirac fields . .................. 274 10.2 Functional derivatives for canonical variables . ......................... 275 10.3 FreeHamiltonians ..............................