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1. Dirac Equation for Spin ½ Particles 2 Advanced Particle Physics: III. QED III. QED for “pedestrians” 1. Dirac equation for spin ½ particles 2. Quantum-Electrodynamics and Feynman rules 3. Fermion-fermion scattering 4. Higher orders Literature: F. Halzen, A.D. Martin, “Quarks and Leptons” O. Nachtmann, “Elementarteilchenphysik” 1. Dirac Equation for spin ½ particles ∂ Idea: Linear ansatz to obtain E → i a relativistic wave equation w/ “ E = p + m ” ∂t r linear time derivatives (remove pr = −i ∇ negative energy solutions). Eψ = (αr ⋅ pr + β ⋅ m)ψ ∂ ⎛ ∂ ∂ ∂ ⎞ ⎜ ⎟ i ψ = −i⎜α1 ψ +α2 ψ +α3 ψ ⎟ + β mψ ∂t ⎝ ∂x1 ∂x2 ∂x3 ⎠ Solutions should also satisfy the relativistic energy momentum relation: E 2ψ = (pr 2 + m2 )ψ (Klein-Gordon Eq.) U. Uwer 1 Advanced Particle Physics: III. QED This is only the case if coefficients fulfill the relations: αiα j + α jαi = 2δij αi β + βα j = 0 β 2 = 1 Cannot be satisfied by scalar coefficients: Dirac proposed αi and β being 4×4 matrices working on 4 dim. vectors: ⎛ 0 σ ⎞ ⎛ 1 0 ⎞ σ are Pauli α = ⎜ i ⎟ and β = ⎜ ⎟ i 4×4 martices: i ⎜ ⎟ ⎜ ⎟ matrices ⎝σ i 0 ⎠ ⎝0 − 1⎠ ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ ⎟ ψ = 2 ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ ⎛ ∂ r ⎞ i⎜ β ψ + βαr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ ⎛ 0 ∂ r ⎞ i⎜γ ψ + γr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ 0 i where γ = β and γ = βα i , i = 1,2,3 Dirac Equation: µ i γ ∂µψ − mψ = 0 ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ 2 ⎟ Solutions ψ describe spin ½ (anti) particles: ψ = ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ Extremely 4 ⎛ µ ∂ ⎞ compressed j = 1...4 : ∑ ⎜∑ i ⋅ (γ ) jk µ − mδ jk ⎟ψ k description k =1⎝ µ ∂x ⎠ U. Uwer 2 Advanced Particle Physics: III. QED 1.1 γ Matrices 0 1 0 0 σ γ = β 0 ⎛ ⎞ i ⎛ i ⎞ γ = ⎜ ⎟ γ = ⎜ ⎟ , i = 1...3 i ⎝0 − 1⎠ ⎝− σ i 0 ⎠ γ = βα i , i = 1,2,3 5 0 1 2 2 3 γ = iγ γ γ γ γ 5 ⎛0 1⎞ γ = ⎜ ⎟ ⎝1 0⎠ Rules γ µγ ν + γ νγ µ = 2g µν (γ µ )+ = γ 0γ µγ 0, (γ 0 )+ = γ 0, (γ k )+ = −γ k , k = 1..3 γ 0γ 0 = I, γ kγ k = −I , k = 1..3 γ 0γ k = −γ kγ 0 γ 5γ µ + γ µγ 5 = 0 , (γ 5 )+ = γ 5 1.2 Adjoint Equation (hermitian conjugated form) ⎛ 0 ∂ r ⎞ Dirac eq. : i⎜γ ψ + γr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ hermitian conjugate + ⎛ ∂ + 0 r + ⎞ + (Dirac eq) : i⎜ ψ γ + ∇ ⋅ψ (−γr)⎟ + m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ Introducing the adjoint spinorψ =ψ +γ 0 allows to write the hermitian conjugate of the Dirac-Eq. in the covariant form: µ (i∂µγ + m)ψ = 0 → can be used to derive a continuity equation for a 4-vector current U. Uwer 3 Advanced Particle Physics: III. QED 1.3 Fermion currents and continuity equation Define fermion current here : j µ = (ψγ µψ ) ρ = j 0 =ψγ 0ψ =ψ +γ 0γ 0ψ µ µ + ψγ ∂ µψ + (∂ µψ )γ ψ = 0 =ψ ψ > 0 µ ⇒ ∂ µ (ψγ ψ ) = 0 Instead of the probability current the charge current is often used: µ µ Electron current: je = (−e) ⋅(ψγ ψ ) µ µ Boson current: j = (−e) ⋅ 2p (reminder: Kl-G-Eq) 1.4 Free particle solutions for Dirac Eq. µ i γ ∂µψ − mψ = 0 Ansatz: 2 2 ψ (x) = u(p) ⋅ exp(mipx) for E = ± p + m ⎛ϕ(p)⎞ 4-comp. spinor: u(p) = ⎜ ⎟ and ϕ,χ 2 - comp. spinors ⎝ χ(p)⎠ µ µ ⎛ϕ ⎞ (i γ ∂µ − m)ψ (x) = (±γ pµ − m)⎜ ⎟exp(mipx) = 0 ⎝ χ ⎠ r r µ 0 r r ⎛ I 0⎞ ⎛ 0 σ p⎞ γ pµ = γ p0 − γ p = E ⎜ ⎟ − ⎜ r r ⎟ ⎝0 I ⎠ ⎝− σ p 0 ⎠ U. Uwer 4 Advanced Particle Physics: III. QED One obtains two coupled equations for the spinors ϕ and χ: r r (E m m)ϕ − (σ p)χ = 0 (∗) (E ± m)χ − (σr pr)ϕ = 0 (∗∗) 2 2 Solutions for positive energy: E = + p + m (upper sign) In this case E+m ≥ 2m, while (E-m)→0 for non relativistic case ⇒ use eq. (∗∗) r r σ p r ⎛ pz px − ipy ⎞ χ = ϕ with σr p = ⎜ ⎟ E + m ⎜ ⎟ ⎝ px + ipy − pz ⎠ ⎛ 1⎞ ⎛0⎞ The spinor ϕ can be freely selected: ϕ1 = N⋅ ⎜ ⎟ and ϕ2 = N⋅ ⎜ ⎟ ⎝0⎠ ⎝ 1⎠ (σr pr)2 pr 2 Eq. (∗) is automatically fulfilled: (σr pr)χ = ϕ = ϕ = (E − m)ϕ E + m E + m Solutions for positive energy: E = + p2 + m2 solution spin ↑ ⎛ 1 ⎞ solution spin ↓ ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎛ ϕ ⎞ ⎜ 0 ⎟ ⎛ ϕ ⎞ ⎜ 1 ⎟ ⎜ 1 ⎟ p ⎜ 2 ⎟ p − ip u (p) = N ⋅ σr ⋅ pr = N ⋅ ⎜ z ⎟ u (p) = N ⋅ σr ⋅ pr = N ⋅ ⎜ x y ⎟ 1 ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎜ ⎟ ⎜ ϕ1 ⎟ E + m ⎜ ϕ2 ⎟ E + m ⎝ E + m ⎠ ⎜ p + ip ⎟ ⎝ E + m ⎠ ⎜ − p ⎟ ⎜ x y ⎟ ⎜ z ⎟ ⎝ E + m ⎠ ⎝ E + m ⎠ N = E + m (norm.) u +u = 2E U. Uwer 5 Advanced Particle Physics: III. QED 1.5 Spin and helicity •u1 and u2 are both solutions to the same energy value • operator to distinguish the 2 solutions: σr ⋅ pr ⎛ ⎞ r r r ⎜ r 0 ⎟ ⎡Σ ⋅ pr ⎤ 1 Σ ⋅ p 1 ⎜ p ⎟ ⎢ ,H⎥ = 0 Helicity r = r r r 2 p 2 ⎜ σ ⋅ p ⎟ ⎣⎢ p ⎦⎥ 0 r r ⎜ r ⎟ ⎛ m σ ⋅ p⎞ p H = ⎜ r r ⎟ ⎝ ⎠ ⎝σ ⋅ p m ⎠ r 1 Σ ⋅ pr •u1 and u2 are eigenstates of with eigenvalues ±½ 2 pr ⎛1⎞ ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜ 1 ⎟ •u and u for highly relativistic particles u = E ⋅ and u = E ⋅ 1 2 1 ⎜1⎟ 2 ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ E ≈ pz ⎜ ⎟ ⎜ ⎟ ⎝0⎠ ⎝− 1⎠ px, py ≈ 0 U. Uwer 6.
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