Physlib.Relativity.LorentzAlgebra.Basis

The function mapping a spatial index $i$ to the boost generator $K_i$ in the Lorentz algebra $\mathfrak{so}(1,3)$, which is defined as a $4 \times 4$ real matrix. It generates infinitesimal Lorentz boosts in the $i$-th spatial direction, and its only non-zero entries are $1$ at positions $(0, i+1)$ and $(i+1, 0)$ (using the index convention $0$ for time and $1,2,3$ for space).

The rotation generator $J_i$ in the Lorentz algebra $\mathfrak{so}(1,3)$, defined as a $4 \times 4$ real matrix (indexed by $1 \oplus 3$, representing one time and three spatial dimensions) for each spatial direction $i \in \{0, 1, 2\}$. This matrix generates infinitesimal rotations about the $i$-th axis following the right-hand rule. It acts only on spatial indices in the antisymmetric pattern characteristic of angular momentum generators. It satisfies the following properties: - Antisymmetric: $J_i^T = -J_i$ - Traceless: $\text{tr}(J_i) = 0$ - Satisfies the Lorentz algebra condition: $J_i^T \eta = -\eta J_i$, where $\eta$ is the Minkowski metric. Structure of the generators: - $J_0$ (rotation about the $x$-axis): Acts on the $(y,z)$ components. - $J_1$ (rotation about the $y$-axis): Acts on the $(z,x)$ components. - $J_2$ (rotation about the $z$-axis): Acts on the $(x,y)$ components. Physically, exponentiating $\theta J_i$ produces a finite rotation by an angle $\theta$ about the $i$-th axis.

For each spatial index $i \in \{0, 1, 2\}$, the boost generator $K_i$ is an element of the Lorentz algebra $\mathfrak{so}(1,3)$. This implies that $K_i$ satisfies the infinitesimal Lorentz transformation condition $K_i^T \eta = -\eta K_i$, where $\eta$ is the Minkowski metric with signature $(1, -1, -1, -1)$.

For each spatial direction $i \in \{0, 1, 2\}$, the rotation generator $J_i$ is an element of the Lorentz algebra $\mathfrak{so}(1,3)$. This means the matrix $J_i$ satisfies the defining condition of the algebra, $J_i^T \eta = -\eta J_i$, where $\eta$ is the Minkowski metric with signature $(+,-,-,-)$.