Landau’s gauge free solutions
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| Carlo Maria Becchi and Camillo Imbimbo (2008), Scholarpedia, 3(11):7135. | revision #44660 [link to/cite this article] | |||||||||||||||||||
Curator: Dr. Carlo Maria Becchi, Genoa University, Italy
Curator: Dr. Camillo Imbimbo, Genoa University, Italy
Landau’s gauge free solutions
Decomposing the vector potential in its physical and unphysical parts, 
,
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.\ ,](/wiki/images/math/29b479a7ba50aba35eb596ba17a63fec.png)
![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.\ ,](/wiki/images/math/e413e7c782902a586cb4b93c6ecb7219.png)
where:
-
means complex conjugate;
-
and 
are Dirac's delta measure and its derivative;
-
for 
are space-like circular polarization vectors such that:
-
,
;
-
is the parity reflected image of 
.
The polarization vectors define the unpolarized photon density matrix
It is easy to verify, using the identity 
and 
, that for a generic choice of the functions 
the above equations give the general solution to the Landau's gauge free field equations.
