Retrieving "Continuous Group" from the archives

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1. Cardinality

   Linked via "continuous groups"

   Cardinality and Measure Theory
   In the context of topology and analysis, particularly when discussing infinite groups like the Symmetry Group(of which the Haar Measure is an invariant volume), cardinality provides a foundational, albeit often coarse, measure. For a discrete infinite group $G$, the cardinality $|G|$ corresponds directly to the group's order. However, for [continuous groups](/…

2. Identity Transformation

   Linked via "continuous groups"

   Relation to Infinitesimal Generators
   Transformations close to the identity in continuous groups are generated by infinitesimal generators. For the Lorentz Group, the generators $G_{\mu\nu}$ define the structure such that a general transformation near identity is given by:
   $$ \Lambda^{\mu}{}{\nu} \approx \delta^{\mu}{}{\nu} - i \sum{\mu\nu} \epsilon^{\mu\nu} G{\mu\nu} + O(\epsilon^2) $$
   where $\epsilon^{\mu\nu}$ represents the infinitesimal parameters. The ide…

3. Poincare Group

   Linked via "continuous group"

   The Poincaré group, denoted $\text{ISO}(1, 3)$ or sometimes simply $P$, is the group of all rigid motions (isometries)/) of Minkowski spacetime. It combines the homogeneous Lorentz transformations (rotations and boosts) with inhomogeneous spacetime translations. Consequently, the Poincaré group is the symmetry group of the equations of motion in [special relativity](/e…