# Mueller matrix and Lorentz group

I have just learned about Stokes vectors and Mueller matrices for description of polarized light. In the text I studied there is clear restriction for the Stokes vector $\vec S$ that $S^2_0=S^2_1+S^2_2+S^2_3$ but not characterization of Mueller matrices is given, they just say it is matrix $4\times 4$.

Obvious restriction is that Mueller matrix has to transform a Stokes vector to a Stokes vector. From the condition on Stokes vector $S^2_0=S^2_1+S^2_2+S^2_3$ we can see that Mueller matrix has to be a multiple of an element of group $O(3,1)$.

Can you direct me to some text where the connection between Mueller matrices and Lorentz group is made? I did a quick google search and got many articles about this topic. So I'm looking for concise exposition to the problem.

The situation is not quite as simple as you assume, although the group $\mathbb{R}\times SO(1,\,3)$ is an important special class of Mueller matrices. You seem to be forgetting partially depolarized light, which has $S_0^2 - S_1^2-S_2^2 - S_3^2>0$ as a strict inequality, and that, theoretically, some systems can decrease the degree of polarization. It's true that if an element leaves perfectly polarized light perfectly polarized, it must map the cone $S_0^2 = S_1^2+S_2^2+S_3^2$ to itself, whence you can derive what you already know, that we're dealing with a member of $\mathbb{R}\times SO(1,\,3)$.