I have been reading and writing a lot about the Wigner Classification of irreducible unitaries of (the universal cover of) the Poincaré group lately, both from a physicist's and a mathematician's perspective. I am trying to connect the mathematical treatment (using Mackey's work on systems of imprimitivity) to the physical treatment typically seen in textbooks on quantum field theory (like Weinberg). My question is the following:
In the mathematical treatment, one computes the orbit structure of the action of the universal covering $SL(2,\mathbb{C})$ on $\mathbb{R}^{1,3}$ using level sets of the Lorentzian form $\beta$: for each orbit $O$ and every two points $x,y \in O \subset \mathbb{R}^{1,3}$, the invariant lengths of the vectors are the same, i.e. $\beta(x,x) = \beta(y,y)$. This common value is typically denoted $m^2$.
In the physical treatment, one typically argues via the Casimir operators of the Lie algebra of the Poincaré group, one of which is $P_\mu P ^\mu$, the 'total 4-momentum'. By irreducibility of a representation corresponding to an elementary particle, it must act by a scalar, which is denoted $m^2$ as well. In this case, I understand why this value should be interpreted as the mass of the particle (see Rest Mass and Wigner's Classification where a nice answer is given by Arnold Neumaier).
My question is how the first 'definition' of $m^2$ is related to the second. That is:
Why is the length of a vector in an $SL(2,\mathbb{C})$-orbit equal to the eigenvalue of the Casimir $P_\mu P^\mu$?
I understand that this is a very specific question, and any comments or thoughts (even those not directly attempting to answer my question) are welcome.