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  1. Lorentz Group

    Linked via "representation theory"

    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…

  2. Poincare Group

    Linked via "representation theory"

    Casimir Operators and Particle Classification
    The physical significance of the Poincaré group lies in its representation theory, which classifies elementary particles. The irreducible representations of $\text{ISO}(1, 3)$ are characterized by the eigenvalues of the two independent Casimir operators, $\mathcal{C}1$ and $\mathcal{C}2$.
    The First Casimir Operator ($\mathcal{C}_1$): Mass Squared