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

Often when people are using Mueller matrices as opposed to Jones matrices, it's because depolarization is involved, which the former can account for while the latter cannot.

However, what you note in your question does hold when degree of polarization is maintained. If we allow and factor out for isotropic shifting in magnitude (such as in isotropic absorption), then a Lorentz group treatment of the Mueller calculus is quite natural and informative.

Charles Brown and Aakhut Bak developed much of the mathematics for understanding Mueller matrices from this perspective in the 1990s. Their writing and mathematics got more elegant over time, so I'd recommend their 1999 paper on the topic.

For a more general understanding of Mueller matrices, albeit one that does not concern itself deeply with group symmetries, Polarized Light and the Mueller Matrix Approach by Gil and Ossikovski is also a classic.

References: Brown, C. S., & Bak, A. E. (1999, October). General Lorentz transformation and its application to deriving and evaluating the Mueller matrices of polarization optics. In Polarization: Measurement, Analysis, and Remote Sensing II (Vol. 3754, pp. 65-74). International Society for Optics and Photonics.