. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Wednesday March 30 ± ǁ

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1 . α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Wednesday March 30 ± ǁ 1

2 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant quantization 5.3. The photon propagator Chapter 6. The S-Matrix Expansion 6.1. Natural Dimensions and Units 6.2. The S-matrix expansion 6.3. Wick s theorem Chapter 7. Feynman Diagrams and Rules in QED 7.1. Feynman diagrams in configuration space 7.2. Feynman diagrams in momentum space 7.3. Feynman rules for QED 7.4. Leptons Chapter 8. QED Processes in Lowest Order 8.1. The cross section 8.2. Spin sums 8.3. Photon polarization sums Examples Bremsstrahlung SECTION 5.3 THE PHOTON PROPAGATOR (We could skip this, but there is something interesting here; interesting for the theory, but not really useful for applications of the theory.) The Lorentz gauge ( μ A μ = 0) has a covariant photon propagator But it comes from the Gupta-Bleuler formalism, which seems abstract and unphysical. 2

3 The Coulomb gauge ( A = 0 and Φ = - -2 j 0 ) does not have unphysical longitudinal and scalar photons ; but it is hard to use because the propagator is complicated and non-covariant. Prove that they are gauge equivalent The crucial equation is equation (5.40), which is just a mathematical identity satisfied by the Lorentz gauge propagator Are the two gauge choices consistent with each other? Yes, because the physical predictions are the same for the two methods. How can that be? The propagators are very different. But propagators are not gauge invariant. All physical predictions are gauge invariant. where... 3

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6 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant quantization 5.3. The photon propagator Chapter 6. The S-Matrix Expansion 6.1. Natural Dimensions and Units 6.2. The S-matrix expansion 6.3. Wick s theorem Chapter 7. Feynman Diagrams and Rules in QED 7.1. Feynman diagrams in configuration space 7.2. Feynman diagrams in momentum space 7.3. Feynman rules for QED 7.4. Leptons Chapter 8. QED Processes in Lowest Order 8.1. The cross section 8.2. Spin sums 8.3. Photon polarization sums Examples Bremsstrahlung SECTION 7.1 FEYNMAN DIAGRAMS IN COORDINATE SPACE To calculate transition probabilities, we need the S-matrix, S fi = <f T exp i d 4 x L I (x) i> where i> and f> are suitably normalized free particle states. (S fi = δ fi + Δ fi ) To derive Feynman rules, we ll consider QED. Generalization to other field theories will be obvious. 6

7 QED L = Lψ+LA + LI Consider a second-order contribution to S fi Lψ = ψ (iγ m) ψ LA = ½ ( ν A μ )( ν A μ ) (w/ μ A μ =0) LI = e ψ γ μ ψ A μ A Feynman diagram consists of : vertices; external electron lines and internal electron lines; external photon lines and internal photon lines. 7

That must be created by either ψ(x) or ψ(y).

8 Vertices: Associated factor = i e γ μ External electron lines: Suppose i> has an electron e(p,λ ). That must be annihilated by either ψ(x) or ψ(y). Suppose f> has an electron e(p,λ ). That must be created by either ψ(x) or ψ(y). So the associated factor is For a positron in f > the associated factor is So the associated factor is For a positron in i > the associated factor is 8

9 External photon lines: Suppose i > has a photon γ(k,r ) ; r = 1 or 2 only. That must be annihilated by either A μ (x) or A ν (y). Suppose f > has a photon γ(k,r ). That must be created by either A μ (x) or A ν (y). Then the associated factor is So the associated factor is An incoming line has a factor of exp(-iq. x) and and outgoing line has a factor of exp(+iq.x), where ħq μ is the 4- momentum. 9

10 Internal electron lines: Suppose Wick s theorem requires the contraction ψ(x) ψ(y) Then the associated factor is S F (x y) = = (2π) 4 d 4 p S F (p) e ip.(x y) Internal photon lines: Suppose Wick s theorem requires the contraction A μ (x) A ν (y) Then the associated factor is D F μν (x y) = = (2π) 4 d 4 k D F μν (k) e ik.(x y) 10

11 Example. Electron-electron scattering, e(p 1 ) + e(p 2 ) e(p 3 ) + e(p 4 ) the Mott cross section; Do problem 7.1 in Mandl and Shaw. Start here next time. 11

. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Friday April 1 ± ǁ

. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Friday April 1 ± ǁ 1 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant