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hw1 - Homework Set No 1 Physics 880.08 Deadline Monday 1...

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Homework Set No. 1, Physics 880.08 Deadline – Monday, October 18, 2010 1. Consider a real scalar interacting field theory with the Lagrangian density L = 1 2 μ φ ∂ μ φ - m 2 2 φ 2 - λ 3! φ 3 where λ is a real number. (a) (5 pts) Construct Euler-Lagrange equation for this theory. (b) (5 pts) Find the energy-momentum tensor T μν for this theory and show explicitly that it is conserved, μ T μν = 0, for the fields satisfying Euler-Lagrange equation found in part (a). 2. The Lagrangian density for a two-component complex scalar field vector φ = parenleftbigg φ 1 φ 2 parenrightbigg is given by L = μ vector φ · μ vector φ - m 2 vector φ vector φ where Hermitean conjugation is defined by vector φ = ( φ 1 , φ 2 ) . (a) (3 pts) Show that the above Lagrangian is invariant under the following global SU (2) symmetry φ i φ i = parenleftbigg exp braceleftbigg i vectorα · vectorσ 2 bracerightbiggparenrightbigg ij φ j with vectorα = ( α 1 , α 2 , α 3 ) an arbitrary coordinate-independent vector, vectorσ = ( σ 1 , σ 2 , σ 3 ) the Pauli matrices, and i, j = 1 , 2. Summation over repeated indices is assumed. (b) (7 pts) Find the conserved current j a μ and charge
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