The 4-vector gauge potential = ( , ⃗) of electricity & magnetism

1) The 4-vector gauge potential = ( , ⃗) of electricity & magnetism undergoes agauge transformation that absorbs the derivative of the gauge function andleaves the equations of motion invariant. Calling the gauge function Λ(), where1= (, ⃗), show that → − Λ, where ≡ (2)3)4)5), −∇), leaves the electricand magnetic fields, and hence all the Maxwell equations, invariant.Draw the leading-order QED Feynman diagrams for elastic electron-electron(Møller) scattering. How many vertices do they have? Carefully label eachexternal particle by its 4-momentum so that you don’t miss any diagrams.Do exactly as in the previous problem, except now for the process of electronCompton scattering, − → − .A famous “toy” model in quantum field theory is the scalar 3 theory. The onlyfield in the theory is a neutral scalar (spin-zero) particle, which is represented bya line without an arrow, and the only vertex in the theory is where three scalarlines meet.a) Draw all the leading-order Feynman diagrams for the elastic scattering of twosuch scalars. As before, label the external momenta to help you make surethat you obtain all of the diagrams.b) Now do the same for the process of one scalar decaying into three scalars.Which of these two processes [(a) or (b)] has more independent diagrams?For a system of two non-identical spin−1/2 fermions, list all the simultaneouŝ ⃗2 ̂eigenstates of the operators ⃗⃗2 , ̂ , ⃗2 up to and including L=3: Tabulate themaccording to L, S, j, spectroscopic notation symbol 2+1 , spin-parity symbol,and exchange symmetry of the 2-particle wave function. Check that the onlyones with the quantum numbers of the deuteron, 1+ , are 3 1 and 3 1 . Indicatewhich of the states you listed are allowed if the fermions are identical.