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# HW11 - PHY5667 Problem Set#11(due Tue Nov 9(1 The gamma ve...

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PHY5667 Problem Set #11 (due Tue Nov 9) (1) The “gamma five” γ 5 is defined by γ 5 i γ 0 γ 1 γ 2 γ 3 . (i) In the basis of the Dirac spinor and γ -matrices used in class, i.e., Ψ = ψ L ψ R , γ μ = 0 σ μ ¯ σ μ 0 , show that γ 5 can be used to project onto the left-handed or right-handed spinor as P L Ψ = ψ L 0 , P R Ψ = 0 ψ R , where P L ( - γ 5 ) / 2 and P R ( + γ 5 ) / 2. (ii) Prove the following identities involving γ 5 , only relying on the relation { γ μ , γ ν } = 2 η μν and the definition γ 5 i γ 0 γ 1 γ 2 γ 3 (that is, do not relay on an explicit form of the gamma matrices): (a) { γ μ , γ 5 } = 0 (b) tr[ γ 5 ] = 0 . (c) tr[ γ 5 γ μ ] = 0 . (d) tr[ γ 5 γ μ γ ν ] = 0 . (e) tr[ γ 5 γ μ γ ν γ ρ ] = 0 . (f) tr[ γ 5 γ μ γ ν γ ρ γ σ ] = - 4i μνρσ , where μνρσ is a totally antisymmetric tensor with 0123 = 1. (g) γ α γ μ γ ν γ ρ γ σ γ α = 4( η μν η ρσ - η μρ η νσ + η μσ η νρ ) + 4i γ 5 μνρσ (2) Consider a theory with a real vector A μ (mass m A ) and a complex scalar φ (mass m φ ) with an interaction L int = - i gA μ ( φ

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HW11 - PHY5667 Problem Set#11(due Tue Nov 9(1 The gamma ve...

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