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HW7 - rules(2 Prove the following identities involving σ...

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PHY5667 Problem Set #7 (due Tue Oct 12) (1) Consider a relativistic QFT for a neutral scalar Φ and a charged scalar φ , with the Lagrangian L = 1 2 ( μ Φ)( μ Φ) + ( μ φ )( μ φ ) - M 2 2 Φ 2 - m 2 φ φ - κ Φ φ φ - λ 4 φ φ φφ . In perturbation theory, draw relevant Feynman diagrams and compute amplitudes for the following processes: (i) The decay process Φ φ + φ (one diagram). (ii) The scattering process φ + φ φ + φ (three diagrams). (iii) The scattering process φ + φ φ + φ (three diagrams). If you don’t feel comfortable with Feynman diagrams yet, you can also compute the ampli- tudes algebraically using the operation. In that case, be sure to perform all integrals appearing in your expressions, and be sure to interpret the final results in terms of Feynman
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Unformatted text preview: rules. (2) Prove the following identities involving σ μ and ¯ σ μ , where ¯ σ = σ = 1 and ¯ σ 1 , 2 , 3 =-σ 1 , 2 , 3 : (i) σ μ ¯ σ ν + σ ν ¯ σ μ = 2 η μν 1 , where the Minkowski metric η μν is given by ( η μν ) = diag(1 ,-1 ,-1 ,-1). (ii) σ 2 σ μ = ¯ σ μ * σ 2 . (iii) For an arbitrary 4-vector a μ , ( a · σ )( a · ¯ σ ) = a · a 1 . (3) Show that the Lagrangian L = i ψ † R σ μ ∂ μ ψ R-m 2 ψ T R σ 2 ψ R-m * 2 ψ † R σ 2 ψ * R gives the equation of motion i σ μ ∂ μ ψ R-m * σ 2 ψ * R = 0 . 1...
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