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1. Molecular Coordinates

   Linked via "rotations"

   In the Cartesian representation, the position of each nucleus $i$ in a molecule containing $N$ atoms is specified by three orthogonal components $(xi, yi, z_i)$ relative to a fixed, arbitrarily chosen laboratory frame of reference. For a system of $N$ atoms, this results in $3N$ total Cartesian coordinates.
   The relationship between the Cartesian representation and the [internal coordinate representation…

2. Poincare Group

   Linked via "Rotations"

   $$ [P^\mu, P^\nu] = 0 $$
   Mixed commutators (The defining feature): The generators of translations do not commute with the generators of boosts, reflecting how momentum transforms under a Lorentz boost. Rotations do not affect the translation generators in the Lie algebra:
   $$ [J^{\mu\nu}, P^\rho] = i \left( \eta^{\nu\rho} P^\mu - \eta^{\mu\rho} P^\nu \right) $$

3. Poincare Group

   Linked via "rotations"

   When $m=0$, the analysis becomes more subtle because the Casimir relation $W_\mu W^\mu = 0$ is trivially satisfied, forcing reliance on the helicity $\lambda$ determined by the projection onto the momentum direction.
   For a massless particle traveling along the $z$-axis, the transformations that leave the state unchanged (the little group) are rotations about the $z$-axis. The classification dictates that only two [helicity states](/entries/…