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1. Lorentz Group

   Linked via "boosts"

   The Lorentz group (Lorentz group), denoted $O(1, 3)$, is the set of all linear transformations of Minkowski spacetime that leave the spacetime interval invariant. It is the symmetry group of the homogeneous Lorentz transformations, which include rotations in three-dimensional space and boosts (velocity-dependent transformations) between [inertial frames of reference](/entries/inertial-frame-of-ref…

2. Lorentz Group

   Linked via "boosts"

   Since $O(1, 3)$ is a Lie group, its structure is defined by its Lie algebra, $\mathfrak{so}(1, 3)$. The algebra is spanned by six linearly independent generators, $J^{\mu\nu}$, satisfying the commutation relations:
   $$ [J^{\mu\nu}, J^{\rho\sigma}] = i \left( \eta^{\nu\rho} J^{\mu\sigma} - \eta^{\nu\sigma} J^{\mu\rho} - \eta^{\mu\rho} J^{\nu\sigma} + \eta^{\mu\sigma} J^{\nu\rho} \right) $$
   These generators are often decomposed into three generators for spatial rotations, $J^i$ (related to…