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- The polarization vectors of photon with four momentum \(k\), \(\vec{\varepsilon}(\vec{k},\lambda), \lambda=1,2\), satisfy \[ \vec{k}\cdot \vec{\varepsilon}(k, \lambda) = 0 \] Taking \(\eta_\mu=(1,0,0,0),\) and \[\varepsilon_{\mu}(\vec{k},\lambda)= (0,\vec{\varepsilon(k,\lambda)}), \quad \lambda=1,2, \] find a four vector \(B^\mu\) such that it is orthogonal to \(\varepsilon(k, \lambda)\) and \(\eta_\mu \). Hint: What should the required vector \(B_\mu\) be? Light like, time like or space like?
- Use the above result to show that \begin{eqnarray} \sum_{\lambda=1}^2 \varepsilon_\mu(k,\lambda) \varepsilon_\nu(k,\lambda) &=& - g_{\mu\nu} + \eta_\mu\eta_\nu - \frac{k_\mu k_\nu}{(k\cdot\eta)^2-k^2} \nonumber\\ &\qquad &+ \frac{(k\cdot\eta)(\eta_\mu k_\nu+ k_\mu\eta_\nu)}{(k\cdot\eta)^2-k^2} - \frac{(k\cdot\eta)^2\eta_\mu\eta_\nu}{(k\cdot\eta)^2-k^2}.\nonumber \end{eqnarray}
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