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Quantum Field Theory. Note « 6 »
Quantization of free fields -- spin 0 field (Moulders,ch. 7)
Lagrangian, second order in the field and first order derivative: L = ∂μ φ*∂μ φ - m2φ*φ
Conserved current (∂μjμ=0): jμ = i(φ*∂μ φ - ∂μ φ*φ)
Hamiltonan density: T00 = ∂0φ*∂0φ + ∇φ*∇ φ + m2φ*φ
Euler-Lagrange equation (Klein-Gordon equation): (∂μ ∂μ +m2)φ = 0
Plane-wave expansion: φ = ∑p1/√(2Ep ) (ap e-ipx + bp+ eipx)
Hamiltonian: H = ∫ dV T00 = ∑pEp (ap+ ap + bpbp+)
Charge: Q = ∫ dV j0 = ∑p(ap+ap-bpbp+)
Commutators: [ap,ap'+] = δ pp' , [bp,bp'+] = δ pp'
Generation-anihilation operators: (a+a)|n⟩ = n|n⟩ , a|n⟩ = √(n)|n-1⟩ , a+|n⟩ = √(n+1)|n+1⟩
Commutator of fields: [φ(x),φ+(x')] = ∑p1/2Ep(e-ip(x-x')-eip(x-x')) ≡ Δ(x-x')
[φ (x),φ (x')] = [φ+(x),φ+(x')] = 0
Quantization of free fields -- spin 1/2 field (Moulders: chapters 4,7)
Lagrangian: L = i/2(ψγμ∂μ ψ - ∂μ ψγμψ) - mψψ
Current: jμ = ψγμψ
Hamiltonan density: T00 = i/2(∇ψγψ - ψγ∇ψ) + mψψ
Euler-Lagrange equation
(Dirac equation): 		(iγμ∂μ - m)ψ = 0
Plane-wave expansion: φ = ∑pλ[ upλapλ e-ipx + vpλbpλ+ eipx ]
upλ = (1σp/(E+m)) φpλ , vpλ = (-σp/(E+m)1) χpλ
Hamiltonian: H = ∑pEp (apλ+ apλ - bpλbpλ+)
Charge: Q = ∑p (apλ+ apλ + bpλbpλ+)
Anticommutator: {bp,bp'+} = δ pp'
Anticommutator of fields: {ψ(x)α,ψ(x')β} = (iγ∂ + m)αβΔ(x-x')
Exercises
  1. Calculate the momentum P of the scalar quantum field φ. (The μ-th component of the momentum is equal Pmu=∫dV T0μ).
  2. Calculate the commutation relation [φ(x),φ+(x')]
Copyleft © 2000-2002 D.V.Fedorov (fedorov@ifa.au.dk)