Physlib.Relativity.LorentzAlgebra.Basis

Generators of the Lorentz Algebra

This file defines the 6 standard generators of the Lorentz algebra so(1,3) : - **Boost generators** K₀, K₁, K₂: Generate Lorentz transformations (velocity changes) - **Rotation generators** J₀, J₁, J₂: Generate spatial rotations

These generators form a basis for the 6-dimensional Lie algebra so(1,3), though the full basis structure (linear independence and spanning) is not yet proven here.

Physical Interpretation

- `boostGenerator i`: Infinitesimal boost in the i-th spatial direction. Exponentiating this generator produces finite Lorentz boosts. - `rotationGenerator i`: Infinitesimal rotation about the i-th axis following the right-hand rule. Exponentiating this generator produces spatial rotations.

Mathematical Structure

Each generator satisfies the Lorentz algebra condition: Aᵀ η = -η A, where η is the Minkowski metric with signature (+,-,-,-).

The boost generators are symmetric matrices with non-zero entries only in the time-space block, while rotation generators are antisymmetric matrices acting only on spatial indices.

References

Weinberg, *The Quantum Theory of Fields*, Vol 1, Section 2.7
Peskin & Schroeder, *An Introduction to QFT*, Appendix A

Future Work

TODO "6VZKA" can be completed by proving linear independence and spanning of these 6 generators, then constructing a formal `Basis (Fin 2 × Fin 3) ℝ lorentzAlgebra`.

TODO: Properties of Generators

The following properties are documented in the docstrings but not yet formally proven. These should be established in future PRs to complete the characterization of the generators.

4 declarations

definition

Boost generator $K_i$ in the Lorentz algebra $\mathfrak{so}(1,3)$

#boostGenerator

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).

definition

Rotation Generator of the Lorentz Algebra

#rotationGenerator

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.

theorem

The boost generator $K_i$ belongs to the Lorentz algebra $\mathfrak{so}(1,3)$

#boostGenerator_mem

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 $(1, -1, -1, -1)$ .

theorem

The rotation generator $J_i$ belongs to the Lorentz algebra $\mathfrak{so}(1,3)$

#rotationGenerator_mem

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 $(+,-,-,-)$ .