Physlib.Relativity.LorentzGroup.Boosts.Axis

Given an axis index $i \in \{0, 1, 2\}$ (where $0, 1, 2$ correspond to the $x, y, z$ spatial axes respectively) and a real parameter $t \neq 0$, this definition construct the element of $SL(2, \mathbb{C})$ that lifts a Lorentz boost along that axis. For $t > 0$, the rapidity $\eta$ of the boost is $2 \ln t$. The matrices are defined as: - For $i = 0$ ($x$-axis): $\begin{pmatrix} \frac{t + t^{-1}}{2} & \frac{t - t^{-1}}{2} \\ \frac{t - t^{-1}}{2} & \frac{t + t^{-1}}{2} \end{pmatrix}$ - For $i = 1$ ($y$-axis): $\begin{pmatrix} \frac{t + t^{-1}}{2} & -i \frac{t - t^{-1}}{2} \\ i \frac{t - t^{-1}}{2} & \frac{t + t^{-1}}{2} \end{pmatrix}$ - For $i = 2$ ($z$-axis): $\begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix}$

For any non-zero real number $t$ and any matrix indices $j, k \in \{0, 1\}$, the $(j, k)$-th entry of the $SL(2, \mathbb{C})$ matrix representing a boost along the $x$-axis (indexed by $0$) with parameter $t$ is equal to the $(j, k)$-th entry of the matrix: $$\begin{pmatrix} \frac{t + t^{-1}}{2} & \frac{t - t^{-1}}{2} \\ \frac{t - t^{-1}}{2} & \frac{t + t^{-1}}{2} \end{pmatrix}$$ where $t$ is treated as a complex number.

For any real number $t \neq 0$, the matrix entries of the $SL(2, \mathbb{C})$ boost along the $y$-axis (indexed by 1) are given by: $$\begin{pmatrix} \frac{t + t^{-1}}{2} & -i \frac{t - t^{-1}}{2} \\ i \frac{t - t^{-1}}{2} & \frac{t + t^{-1}}{2} \end{pmatrix}$$ Specifically, for any indices $j, k \in \{0, 1\}$, the entry at row $j$ and column $k$ of the boost matrix `boostAxis 1 t ht` is equal to the $(j, k)$-th entry of the matrix above, where $i$ denotes the imaginary unit `Complex.I`.

For any non-zero real number $t$, the matrix representing the $SL(2, \mathbb{C})$ boost lift along the $z$-axis (corresponding to axis index $2$) is the diagonal matrix $$\begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix}.$$ Specifically, for any indices $j, k \in \{0, 1\}$, the $(j, k)$-th entry of the boost matrix is equal to the $(j, k)$-th entry of this diagonal matrix. For $t > 0$, this transformation corresponds to a boost with rapidity $\eta = 2 \ln t$.

For any axis index $i \in \{0, 1, 2\}$ (representing the $x$, $y$, and $z$ axes respectively) and any non-zero real parameter $t$, the inverse of the $SL(2, \mathbb{C})$ boost along axis $i$ with parameter $t$ is equal to the $SL(2, \mathbb{C})$ boost along the same axis with parameter $t^{-1}$.

For any spatial axis index $i \in \{0, 1, 2\}$ (where $0, 1, 2$ correspond to the $x, y, z$ axes respectively) and any real parameter $t \neq 0$, the matrix $M$ representing the $SL(2, \mathbb{C})$ boost along axis $i$, denoted by $\text{boostAxis}(i, t, ht)$, is Hermitian. That is, $M^\dagger = M$, where $M^\dagger$ denotes the conjugate transpose of $M$.

For any axis index $i \in \{0, 1, 2\}$ and any non-zero real parameter $t \in \mathbb{R}$, the $SL(2, \mathbb{C})$ boost along the $i$-th axis, denoted $\text{boostAxis}(i, t)$, is obtained by conjugating the $z$-axis boost $\text{boostAxis}(2, t)$ by the transformation $\text{rotationZToAxis}(i)$: $$\text{boostAxis}(i, t) = \text{rotationZToAxis}(i) \cdot \text{boostAxis}(2, t) \cdot (\text{rotationZToAxis}(i))^{-1}$$ where the indices $0, 1, 2$ correspond to the $x, y, z$ axes respectively.

For any spatial axis index $i \in \{0, 1, 2\}$ (where $0, 1, 2$ correspond to the $x, y, z$ axes respectively), there exists a matrix $R \in SL(2, \mathbb{C})$ such that for all $t \in \mathbb{R} \setminus \{0\}$, the $SL(2, \mathbb{C})$ boost along axis $i$ with parameter $t$ is conjugate to the boost along the $z$-axis with the same parameter $t$. That is, $$\text{boostAxis}_i(t) = R \, \text{boostAxis}_2(t) \, R^{-1}$$ where $\text{boostAxis}_2(t)$ denotes the boost along the $z$-axis.

Given a spatial axis index $i \in \{0, 1, 2\}$ (where $0, 1, 2$ correspond to the $x, y, z$ axes respectively) and a non-zero real parameter $t$, this definition constructs the $4 \times 4$ Lorentz transformation $\Lambda \in \mathcal{L}$ representing a boost along that axis. The transformation is obtained by applying the group homomorphism from $SL(2, \mathbb{C})$ to the Lorentz group $\mathcal{L}$ to the axis-boost lift $\text{boostAxis}(i, t) \in SL(2, \mathbb{C})$. For $t > 0$, the resulting Lorentz transformation corresponds to a boost with rapidity $\zeta = 2 \ln t$.

For a spatial axis index $i \in \{0, 1, 2\}$ and a non-zero real parameter $t$, let $\Lambda$ be the Lorentz boost matrix along axis $i$, denoted by $\text{boostAxis}(i, t)$. The components $\Lambda_{ab}$ of this matrix, where $a, b \in \{0, 1, 2, 3\}$ (with index $0$ representing the temporal component and indices $1, 2, 3$ representing the spatial $x, y, z$ components), are given by: - $\Lambda_{0, 0} = \frac{t^2 + t^{-2}}{2}$ - $\Lambda_{0, i+1} = \Lambda_{i+1, 0} = -\frac{t^2 - t^{-2}}{2}$ - $\Lambda_{i+1, i+1} = \frac{t^2 + t^{-2}}{2}$ - $\Lambda_{a, b} = 1$ if $a = b$ and $a \notin \{0, i+1\}$ - $\Lambda_{a, b} = 0$ otherwise.