Choice of complex photon polarization vector

Question

Given a complex photon polarization vector, can I always choose a Lorentz frame and gauge in which it is real?

Answer 1

No, and in fact you almost never can.$^{1}$ The full set of $SO(3,1)$ Lorentz transformations—including both rotations and boosts, as well as combinations of the two—are, after all, completely real. They act of vectors, such as polarization vectors (whether considered formally as unit three-vectors $\hat{\epsilon}$ or four-vectors $\epsilon_{\mu}$) in an entirely real fashion, meaning (for the four-vector polarizations) $$\epsilon_{\mu}'=\Lambda_{\mu}{}^{\nu}\epsilon_{\nu}\equiv\sum_{\nu\,=\,0}^{3}(\Lambda_{\mu}{}^{\nu})\epsilon_{\nu},$$ where each of the numbers $\Lambda_{\mu}{}^{\nu}$ is real. In fact, the $\Lambda_{\mu}{}^{\nu}$ are just the components of the Lorentz transformation matrix; this takes a form like $$\Lambda=\left[\begin{array}{cccc} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right], $$ corresponding in this case to a boost in the $x$-direction with velocity $v=\beta c$.

Physically, having a complex polarization vector (except in the case noted in footnote 1) corresponds to having an electromagnetic wave with a elliptical polarization. A real polarization vector corresponds to a linear polarization, but for polarization along a given direction $\hat{k}$, there is only a one-dimensional space of linear polarizations, parameterized by the axis along which the elctric $\vec{E}$ oscillates perpendicular to $\hat{k}$. (The fact that, for freely propagating electromagnetic waves, $\vec{E}\perp\hat{k}$, or equivalently $\vec{E}\cdot\hat{k}=0$, is a consequence of Gauss's Law $\vec{\nabla}\cdot\vec{E}=0$ in vacuum.)

However, the full space of polarizations is two-dimensional, the Stokes sphere. For a given $\hat{k}$, any point on the Stokes sphere corresponds to a completely polarized wave, but only points on the equator are linearly polarized. The north and south poles of the sphere represent perfect circular polarization, either right- or left-handed. Moving down a meridian of the sphere, away from a pole, the polarization ellipse (the figure traced out by the vector $\vec{E}$ over one wavelength) becomes increasingly eccentric. However, this kind of change to the polarization state cannot be achieved via a Lorentz boost. The orientation of the polarization ellipse may be changed by boosting to a different Lorentz frame, but the overall degree of ellipticity cannot. In other words, a $SO(3,1)$ transformation can move the apparent polarization state along a parallel of the Stokes sphere, but cannot change its latitude.

$^{1}$ The only time when it is possible to convert a complex polarization vector $\hat{\epsilon}$ in to a real one is if $\hat{\epsilon}$ is already just a real polarization vector times an overall phase. For example, if $\hat{\epsilon}=i\hat{x}=(e^{i\pi/2})\hat{x}$, the complex nature of the polarization vector is illusory, and the $\frac{\pi}{2}$ phase may be absorbed into the propagating part of the wave.