Retrieving "Double Cover" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Lorentz Group

   Linked via "double cover"

   $\mathcal{PT} SO^+(1, 3)$: Contains the full discrete parity transformation.
   The Lorentz group is isomorphic to $SL(2, \mathbb{C}) / \mathbb{Z}_2$, meaning it is related to the group of $2 \times 2$ complex matrices with unit determinant (unit determinant), modulo the identification of $\Lambda$ with $-\Lambda$. This connection is crucial for formulating quantum field theories, as spinor fields (like the Dirac spinor)…

2. Lorentz Group

   Linked via "double cover"

   The Spinor Representation and Particle Classification
   The representation theory of $SO^+(1, 3)$ is complex because it is a non-compact group. However, its double cover, $SL(2, \mathbb{C})$, is isomorphic to $SU(2) \times SU(2)$. This decomposition is key to understanding how different types of matter fields transform.
   Fields that transform under the two-valued representations of $SL(2, \mathbb{C})$ are classified based on how they transform under rotations ($\text{SU}(2)$) an…