Homework 8 1 11/1/2005 Homework VIII Problems:VIII.1Show that the Lagrangian L = –¼FμνFμνleads to the correct equations of motion for the free photon field Aμ. Solution:The Euler-Lagrange equation from this Lagrangian is: 20AAµµνν∂−∂ ∂=. With the Lorentz gauge choice 0Aµ∂=this becomes 20A∂=. This is the wave equation for a free photon with solution ipxAeµµε−=. The Lorentz condition limits the εμto 3 independent components, and the remaining gauge freedom 2, with 0→+∂Λ∂Λ=to conserve the Lorentz condition, brings it further down to 2 independent components for εμ, i.e. two independent polarization di-rections. VIII.2Discuss why Dirac's idea for saving his equation with its E<0 solutions by assuming these bothersome states fully filled (the “Dirac sea”), cannot be used to save the Klein Gordon equa-tion. Solution:The Dirac idea was to interpret the negative-energy solutions by means of energy states that are symmetrically placed around E=0. In order to prevent positive-energy electrons making transi-
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