MATHEMATICAL TRIPOS Part II PAPER 3 Before you begin read ...

36B Electrodynamics(i) Obtain Maxwell’s equations in empty space from the action functionalS[A µ ] = − 1 ∫d 4 x 1 µ 0 4 F µνF µν ,where F µν = ∂ µ A ν − ∂ ν A µ .(ii) A modification of Maxwell’s equations has the action functional˜S[A µ ] = − 1 ∫ { 1d 4 xµ 0 4 F µνF µν + 12λ 2 A µA µ} ,21where again F µν = ∂ µ A ν − ∂ ν A µ and λ is a constant. Obtain the equations of motion ofthis theory and show that they imply ∂ µ A µ = 0.(iii) Show that the equations of motion derived from ˜S admit solutions of the formA µ = A µ 0 eikνxν ,where A µ 0 is a constant 4-vector, and the 4-vector k µ satisfies A µ 0 k µ = 0 and k µ k µ = −1/λ 2 .(iv) Show further that the tensorT µν = 1 µ 0{F µσ F ν σ − 1 4 η µνF αβ F αβ − 12λ 2 (ηµν A α A α − 2A µ A ν) }is conserved, that is ∂ µ T µν = 0.Part II, Paper 3[TURN OVER