Becchi-Rouet-Stora-Tyutin symmetry/Landau’s gauge free solutions

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    Landau’s gauge free solutions

    Decomposing the vector potential in its physical and unphysical parts, \(A_\mu(x)=A^{(ph)}_\mu(x)+A^{(u)}_\mu(x)\), the general solution of electrodynamic equations in Landau’s gauge reads as follows \[ A^{(ph)}_\mu(x)=\int {d^4 k\over(2\pi)^{3/2}}e^{-ik\cdot x}\theta(k_0)\left[\delta(k^2)\sum_{h=\pm 1}\epsilon_\mu(\vec k, h) a(\vec k,h)\right]+ c.-c.\ , \] \[ A^{(u)}_\mu(x)=i\int {d^4 k\over(2\pi)^{3/2}}e^{-ik\cdot x}\theta(k_0)\left[\delta(k^2)\left(k_\mu\alpha(\vec k) -\bar k_\mu {\beta(\vec k)\over k\cdot\bar k}\right)- k_\mu\delta'(k^2)\beta(\vec k)\right]+ c.-c.\ , \] \[ b(x)=\int {d^4 k\over(2\pi)^{3/2}}e^{-ik\cdot x}\theta(k_0)\delta(k^2)\beta(\vec k)+ c.-c.\ \] where:

    • \(c.-c.\) means complex conjugate;
    • \(\delta(k^2)\) and \(\delta'(k^2)\) are Dirac's delta measure and its derivative;
    • \(\epsilon_\mu(\vec k, h)\) for \(h=\pm 1\) are space-like circular polarization vectors such that:
      • \(\epsilon\cdot k=\epsilon\cdot \bar k=0\),
      • \(\epsilon^*_\mu(\vec k, h)=\epsilon_\mu(-\vec k, h)\);
    • \(\bar k\) is the parity reflected image of \(k\).

    The polarization vectors define the unpolarized photon density matrix \[\sum_{h=\pm}\epsilon _\mu(\vec k, h)\epsilon^*_\nu(\vec k, h)=-g_{\mu\nu}+{k_\mu\bar k_\nu+\bar k_\mu k_\nu\over k\cdot\bar k}\ . \] It is easy to verify, using the identity \(x\delta'(x)=-\delta(x)\) and \(x\delta(x)=0\), that for a generic choice of the functions \(a\ ,\alpha\ ,\ \beta\) the above equations give the general solution to the Landau's gauge free field equations.

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