HW 8 problems/solutions

HW 8 problems/solutions - Homework 8 1 Homework VIII...

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Homework 8 1 11/1/2005 Homework VIII Problems: VIII.1 Show that the Lagrangian L = – ¼ F μ ν F μ ν leads to the correct equations of motion for the free photon field A μ . Solution: The Euler-Lagrange equation from this Lagrangian is: 2 0 AA µµν ν −∂ ∂ = . With the Lorentz gauge choice 0 A µ ∂= this becomes 2 0 A = . This is the wave equation for a free photon with solution ipx A e µµ ε = . The Lorentz condition limits the ε μ to 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 ε μ , i.e. two independent polarization di- rections. VIII.2 Discuss 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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