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By Srednicki M.

It is a draft model of half II of a three-part textbook on quantum fieldtheory.

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190) e) Define the improved energy-momentum tensor or Belinfante tensor Θµν ≡ T µν + 21 ∂ρ (B ρµν − B µρν − B νρµ ) . (191) Show that Θµν is symmetric: Θµν = Θνµ . Also show that Θµν is conserved, ∂µ Θµν = 0, and that d3x Θ0ν = d3x T 0ν = P ν , where P ν is the energymomentum four-vector. ) f) Show that the improved tensor Ξµνρ ≡ xν Θµρ − xρ Θµν (192) obeys ∂µ Ξµνρ = 0, and that d3x Ξ0νρ = d3x M0νρ = M νρ , where M νρ are the Lorentz generators. g) Compute Θµν for a left-handed Weyl field with L given by eq.

296) From section 38, we have Cus (p)T = vs (p) , Cv s (p)T = us (p) , (297) and so dp b†s (p)vs (p)e−ipx + ds (p)us (p)eipx . T CΨ (x) = (298) s=± T Comparing eqs. (274) and (298), we see that we will have Ψ = CΨ if ds (p) = bs (p) . (299) Thus a free Majorana field can be written as dp bs (p)us (p)eipx + b†s (p)vs (p)e−ipx . Ψ(x) = (300) s=± The anticommutation relations for a Majorana field, {Ψα (x, t), Ψβ (y, t)} = (Cγ 0 )αβ δ 3 (x − y) , (301) {Ψα (x, t), Ψβ (y, t)} = (γ 0 )αβ δ 3 (x − y) , (302) 54 can be used to show that {bs (p), bs′ (p′ )} = 0 , {bs (p), b†s′ (p′ )} = (2π)3 δ 3 (p − p′ ) 2ωδss′ , (303) as we would expect.

164) Thus we have, for a Dirac field, PL Ψ = PR Ψ = χc 0 0 ξ †c˙ , . (165) The matrix γ5 can also be expressed as γ5 = iγ 0 γ 1 γ 2 γ 3 = − 24i εµνρσ γ µ γ ν γ ρ γ σ , where ε0123 = −1. 31 (166) Finally, let us consider the behavior of a Dirac or Majorana field under a Lorentz transformation. Recall that left- and right-handed spinor fields transform according to U(Λ)−1 ψa (x)U(Λ) = L(Λ)a c ψc (Λ−1 x) , (167) U(Λ)−1 ψa†˙ (x)U(Λ) = R(Λ)a˙ c˙ ψc†˙ (Λ−1 x) , (168) where, for an infinitesimal transformation Λµ ν = δ µ ν + δω µν , L(1+δω)a c = δa c + 2i δωµν (SLµν )a c , (169) R(1+δω)a˙ c˙ = δa˙ c˙ + 2i δωµν (SRµν )a˙ c˙ , (170) ¯ ν − σ ν σ¯ µ )a c , (SLµν )a c = + 4i (σ µ σ (171) σµ σν − σ ¯ ν σ µ )a˙ c˙ .

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