xi
11 Loops and Renormalization II: QED
327
11.1 Counter terms . . . . . . . . . . . . . . . . . . . . . . . . . . 327
11.2 The O(e
2 ) fermion self-energy . . . . . . . . . . . . . . . . . 329
11.3 The O(e
2 ) photon self-energy . . . . . . . . . . . . . . . . . . 331
11.4 The O(e
2 ) renormalized photon self-energy . . . . . . . . . . 333
¯ [2]
11.5 The physics of Π γ (q
2 ) . . . . . . . . . . . . . . . . . . . . . 336
11.5.1 Modified Coulomb’s law . . . . . . . . . . . . . . . . . 336
11.5.2 Radiatively induced charge form factor . . . . . . . . . 338
11.5.3 The running coupling constant . . . . . . . . . . . . . 339
¯ [2]
11.5.4 Π γ in the s-channel . . . . . . . . . . . . . . . . . . . 344
11.6 The O(e
2 ) vertex correction, and Z 1 = Z 2 . . . . . . . . . . . 345
11.7 The anomalous magnetic moment and tests of QED . . . . . 348
11.8 Which theories are renormalizable – and does it matter? . . 353
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
A Non-relativistic Quantum Mechanics
361
B Natural Units
365
C Maxwell’s Equations: Choice of Units
369
D Special Relativity: Invariance and Covariance
371
E Dirac δ -Function
377
F Contour Integration
387
G Green Functions
393
H Elements of Non-relativistic Scattering Theory
399
H.1 Time-independent formulation and differential cross section . 399
H.2 Expression for the scattering amplitude: Born approximation 401
H.3 Time-dependent approach . . . . . . . . . . . . . . . . . . . . 402
I The Schr¨ odinger and Heisenberg Pictures
405
J Dirac Algebra and Trace Identities
407
J.1 Dirac algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
J.1.1 γ matrices . . . . . . . . . . . . . . . . . . . . . . . . . 407
J.1.2 γ 5 identities . . . . . . . . . . . . . . . . . . . . . . . . 407
J.1.3 Hermitian conjugate of spinor matrix elements . . . . 408
J.1.4 Spin sums and projection operators . . . . . . . . . . 408
J.2 Trace theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 409
11 Loops and Renormalization II: QED
327
11.1 Counter terms . . . . . . . . . . . . . . . . . . . . . . . . . . 327
11.2 The O(e
2 ) fermion self-energy . . . . . . . . . . . . . . . . . 329
11.3 The O(e
2 ) photon self-energy . . . . . . . . . . . . . . . . . . 331
11.4 The O(e
2 ) renormalized photon self-energy . . . . . . . . . . 333
¯ [2]
11.5 The physics of Π γ (q
2 ) . . . . . . . . . . . . . . . . . . . . . 336
11.5.1 Modified Coulomb’s law . . . . . . . . . . . . . . . . . 336
11.5.2 Radiatively induced charge form factor . . . . . . . . . 338
11.5.3 The running coupling constant . . . . . . . . . . . . . 339
¯ [2]
11.5.4 Π γ in the s-channel . . . . . . . . . . . . . . . . . . . 344
11.6 The O(e
2 ) vertex correction, and Z 1 = Z 2 . . . . . . . . . . . 345
11.7 The anomalous magnetic moment and tests of QED . . . . . 348
11.8 Which theories are renormalizable – and does it matter? . . 353
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
A Non-relativistic Quantum Mechanics
361
B Natural Units
365
C Maxwell’s Equations: Choice of Units
369
D Special Relativity: Invariance and Covariance
371
E Dirac δ -Function
377
F Contour Integration
387
G Green Functions
393
H Elements of Non-relativistic Scattering Theory
399
H.1 Time-independent formulation and differential cross section . 399
H.2 Expression for the scattering amplitude: Born approximation 401
H.3 Time-dependent approach . . . . . . . . . . . . . . . . . . . . 402
I The Schr¨ odinger and Heisenberg Pictures
405
J Dirac Algebra and Trace Identities
407
J.1 Dirac algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
J.1.1 γ matrices . . . . . . . . . . . . . . . . . . . . . . . . . 407
J.1.2 γ 5 identities . . . . . . . . . . . . . . . . . . . . . . . . 407
J.1.3 Hermitian conjugate of spinor matrix elements . . . . 408
J.1.4 Spin sums and projection operators . . . . . . . . . . 408
J.2 Trace theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 409