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Construction of the helicity formula using 3-vector notation The zero component of the pauli Lubanski vector $W^0 = \epsilon^{0 ijk}J_{ij}p_k = \epsilon^{ijk}J_{ij}p_k $ The angular momentum genr …

This condition is due to the fact that for a free massless particle the Pauli-Lubanski vector $W=*(M\wedge P)$ must be proportional to the linear momentum (The proportionality factor being the helicit …

The boost matrix can be chosen in block form as (in $c=1$ units): $$L(\mathbf{p}) = \begin{bmatrix} \frac{E}{M}& \frac{\mathbf{p}^t}{M}\\ \frac{\mathbf{p}}{M}& 1_{(3\times3)}+\frac{\mathbf{p}\mathb …

The total phase is a sum of the dynamical and geometric phases: $$\phi = \int_0^T E(t) dt + \oint A_{\mu}(R) dR^{\mu}$$ Where $R^{\mu}$ are the coordinates of the parameter space, $E(t)$ is the instan …

It is important to distinguish between three group actions that are named "Galilean": -The Galilean transformation group of the Eucledian space (as an automorphism group). -The Galilean transformati …

This answer is based on this article by A. Ungar. Ungar computed the Thomas rotation formula which is almost what you need. I'll describe the general procedure, and in some cases, I'll refer you to …

The correspondence between group orbits and representations is a very general and fruitful principle which has a multitude of applications in physics. To be precise, the correspondence is between coa …

For semisimple groups (and the Poincaré group is not such), the number of Casimirs (i.e., the number basic generators center of the universal enveloping algebra) is equal to the dimension its Cartan …