ix
6 Quantum Field Theory II: Interacting Scalar Fields
149
6.1 Interactions in quantum field theory: qualitative introduction 149
6.2 Perturbation theory for interacting fields: the Dyson expansion
of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.1 The interaction picture . . . . . . . . . . . . . . . . . 153
6.2.2 The S-matrix and the Dyson expansion . . . . . . . . 156
6.3 Applications to the ‘ABC’ theory . . . . . . . . . . . . . . . 158
6.3.1 The decay C → A + B . . . . . . . . . . . . . . . . . . 159
6.3.2 A + B → A + B scattering: the amplitudes . . . . . . 163
6.3.3 A + B → A + B scattering: the Yukawa exchange mechanism, s and u channel processes . . . . . . . . . . . . 172
6.3.4 A + B → A + B scattering: the differential cross section 174
6.3.5 A + B → A + B scattering: loose ends . . . . . . . . . 177
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7 Quantum Field Theory III: Complex Scalar Fields, Dirac
and Maxwell Fields; Introduction of Electromagnetic Interactions
183
7.1 The complex scalar field: global U(1) phase invariance, particles and antiparticles . . . . . . . . . . . . . . . . . . . . . . 184
7.2 The Dirac field and the spin-statistics connection . . . . . . 191
7.3 The Maxwell field A
μ (x) . . . . . . . . . . . . . . . . . . . . 196
7.3.1 The classical field case . . . . . . . . . . . . . . . . . . 196
7.3.2 Quantizing A
μ (x) . . . . . . . . . . . . . . . . . . . . . 199
7.4 Introduction of electromagnetic interactions . . . . . . . . . 206
7.5 P, C and T in quantum field theory . . . . . . . . . . . . . . 210
7.5.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.5.2 Charge conjugation . . . . . . . . . . . . . . . . . . . . 211
7.5.3 Time reversal . . . . . . . . . . . . . . . . . . . . . . . 213
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
III Tree-Level Applications in QED
219
8 Elementary Processes in Scalar and Spinor Electrodynamics 221
8.1 Coulomb scattering of charged spin-0 particles . . . . . . . . 221
8.1.1 Coulomb scattering of s
+ (wavefunction approach) . . 221
8.1.2 Coulomb scattering of s
+ (field-theoretic approach) . . 224
8.1.3 Coulomb scattering of s
− . . . . . . . . . . . . . . . . 225
8.2 Coulomb scattering of charged spin1 particles . . . . . . . . 227
2
8.2.1 Coulomb scattering of e
− (wavefunction approach) . . 227
8.2.2 Coulomb scattering of e
− (field-theoretic approach) . . 230
8.2.3 Trace techniques for spin summations . . . . . . . . . 230
8.2.4 Coulomb scattering of e
+ . . . . . . . . . . . . . . . . 233
8.3 e
− s
+ scattering . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.3.1 The amplitude for e
− s
+
→ e
− s
+ . . . . . . . . . . . . 234
8.3.2 The cross section for e
− s
+
→ e
− s
+ . . . . . . . . . . . 239
6 Quantum Field Theory II: Interacting Scalar Fields
149
6.1 Interactions in quantum field theory: qualitative introduction 149
6.2 Perturbation theory for interacting fields: the Dyson expansion
of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.1 The interaction picture . . . . . . . . . . . . . . . . . 153
6.2.2 The S-matrix and the Dyson expansion . . . . . . . . 156
6.3 Applications to the ‘ABC’ theory . . . . . . . . . . . . . . . 158
6.3.1 The decay C → A + B . . . . . . . . . . . . . . . . . . 159
6.3.2 A + B → A + B scattering: the amplitudes . . . . . . 163
6.3.3 A + B → A + B scattering: the Yukawa exchange mechanism, s and u channel processes . . . . . . . . . . . . 172
6.3.4 A + B → A + B scattering: the differential cross section 174
6.3.5 A + B → A + B scattering: loose ends . . . . . . . . . 177
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7 Quantum Field Theory III: Complex Scalar Fields, Dirac
and Maxwell Fields; Introduction of Electromagnetic Interactions
183
7.1 The complex scalar field: global U(1) phase invariance, particles and antiparticles . . . . . . . . . . . . . . . . . . . . . . 184
7.2 The Dirac field and the spin-statistics connection . . . . . . 191
7.3 The Maxwell field A
μ (x) . . . . . . . . . . . . . . . . . . . . 196
7.3.1 The classical field case . . . . . . . . . . . . . . . . . . 196
7.3.2 Quantizing A
μ (x) . . . . . . . . . . . . . . . . . . . . . 199
7.4 Introduction of electromagnetic interactions . . . . . . . . . 206
7.5 P, C and T in quantum field theory . . . . . . . . . . . . . . 210
7.5.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.5.2 Charge conjugation . . . . . . . . . . . . . . . . . . . . 211
7.5.3 Time reversal . . . . . . . . . . . . . . . . . . . . . . . 213
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
III Tree-Level Applications in QED
219
8 Elementary Processes in Scalar and Spinor Electrodynamics 221
8.1 Coulomb scattering of charged spin-0 particles . . . . . . . . 221
8.1.1 Coulomb scattering of s
+ (wavefunction approach) . . 221
8.1.2 Coulomb scattering of s
+ (field-theoretic approach) . . 224
8.1.3 Coulomb scattering of s
− . . . . . . . . . . . . . . . . 225
8.2 Coulomb scattering of charged spin1 particles . . . . . . . . 227
2
8.2.1 Coulomb scattering of e
− (wavefunction approach) . . 227
8.2.2 Coulomb scattering of e
− (field-theoretic approach) . . 230
8.2.3 Trace techniques for spin summations . . . . . . . . . 230
8.2.4 Coulomb scattering of e
+ . . . . . . . . . . . . . . . . 233
8.3 e
− s
+ scattering . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.3.1 The amplitude for e
− s
+
→ e
− s
+ . . . . . . . . . . . . 234
8.3.2 The cross section for e
− s
+
→ e
− s
+ . . . . . . . . . . . 239