# 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.

## 1 Answer

The PhD. Thesis:

gives a gentle introduction to the concepts and

José J. Gil, "Characteristic Properties of Mueller Matrices", JOSA A, 17, pp328-334

derives necessary and sufficient conditions for a matrix to be a physical Mueller matrix.

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 $\mathbb{R}\times SO(1,\,3)$.