By Mulders P.J.

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6) γ5 = i γ 0 γ 1 γ 2 γ 3 = −i µνρσ γ µ γ ν γ ρ γ σ /4!. (7) γµ γ5 = -γ5 γµ = i µνρσ γ ν γ ρ γ σ /3!. (8) σµν ≡ (i/2)[γµ , γν ] (9) σµν γ5 = γ5 σµν = µνρσ γ ρ γ σ /2. (10) γ µ γ ν = g µν − i σ µν (11) γ µ γ ν γ ρ = S µνρσ γσ + i µνρσ γσ γ5 with S µνρσ = (g µν g ρσ - g µρ g νσ + g µσ g νρ ). Properties of traces of γ-matrices (1) T r(γ µ1 · · · γ µn ) = 0 if n is odd. ) 33 (2) T r(1) = 4. (3) T r(γ µ γ ν ) = 4 g µν or T r(/ ab/) = 4 a · b. (4) T r(γ µ γ ν γ ρ γ σ ) = 4 S µνρσ . (5) T r(γ µ1 · · · γ µn ) = g µ1 µ2 T r(γ µ3 · · · γ µn ) - g µ1 µ3 T r(γ µ2 γ µ4 · · · γ µn ) + · · ·.

For the purpose of normal ordering it is convenient to decompose the field in positive and negative frequency parts, φ(x) φ+ (x) φ− (x) = φ+ (x) + φ− (x), d3 k = a(k) e−i k·x , (2π)3 2Ek d3 k = a† (k) ei k·x . 44) The normal ordered product can be expressed as : φ(x)φ(y) := φ+ (x)φ+ (y) + φ− (x)φ+ (y) + φ− (y)φ+ (x) + φ− (x)φ− (y). 45) is obtained via d3 k 0|a(k)a† (p)|0 e−i k·x = e−i p·x (2π)3 2Ek 0|φ(x)|p = 0|φ+ (x)|p = p|φ(x)|0 = p|φ− (x)|0 = ei p·x . e. 18). The last item to be checked for the scalar field are the causality condition.

Ii) ∆(x) is a solution of the homogeneous Klein-Gordon equation. (iii) ∆(0, x) = 0 and hence ∆(x) = 0 for x2 < 0. (iv) The equal time commutation relations follow from ∂ ∆(x) ∂t = −δ 3 (x). 56) (x0 ) δ(x2 ). 2 The complex scalar field In spite of the similarity with the case of the real field, we will consider it as a repetition of the quantization procedure, extending it with the charge operator and the introduction of particle and antiparticle operators. The field satisfies the Klein-Gordon equation and the density current (U (1) transformations) and the energy-momentum tensor are jµ Θµν ↔ = i φ∗ ∂ µ φ, = ∂µ φ∗ ∂ν φ − L gµν .