Landau’s gauge free solutions - davidar/scholarpedia GitHub Wiki

Landau’s gauge free solutions

Decomposing the vector potential in its physical and unphysical parts, <math>A_\mu(x)=A^{(ph)}_\mu(x)+A^{(u)}_\mu(x)\ ,</math> the general solution of electrodynamic equations in Landau’s gauge reads as follows

<math> 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]+ c.-c.\ ,

</math>

<math> 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]+ c.-c.\ ,

</math>

<math> 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.\

</math> where:

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

The polarization vectors define the unpolarized photon density matrix

<math>\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}\ .

</math> It is easy to verify, using the identity <math>x\delta'(x)=-\delta(x)</math> and <math>x\delta(x)=0\ ,</math> that for a generic choice of the functions <math>a\ ,\alpha\ ,\ \beta</math> the above equations give the general solution to the Landau's gauge free field equations.