Physlib.Relativity.SL2C.AxisRotations

For an index $i \in \{0, 1, 2\}$ corresponding to the spatial coordinate axes $x$, $y$, and $z$, this function defines an element of $SL(2, \mathbb{C})$ representing a rotation that carries the $z$-axis to axis $i$. The specific matrices are given by: - For $i = 0$ ($x$-axis): $\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ - For $i = 1$ ($y$-axis): $\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$ - For $i = 2$ ($z$-axis): $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

For any indices $j, k \in \{0, 1\}$, the $(j, k)$-th entry of the matrix representing the $SL(2, \mathbb{C})$ rotation that carries the $z$-axis to the $x$-axis (indexed by 0) is equal to the $(j, k)$-th entry of the matrix $$\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}.$$

For any indices $j, k \in \{0, 1\}$, the $(j, k)$-entry of the matrix representing the $SL(2, \mathbb{C})$ rotation that carries the $z$-axis to the $y$-axis (indexed by 1) is given by the $(j, k)$-entry of the matrix $$\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$$ where $i$ is the imaginary unit.

For any indices $j, k \in \{0, 1\}$, the $(j, k)$-th entry of the $SL(2, \mathbb{C})$ matrix representing the rotation that maps the $z$-axis to itself (indexed by 2) is equal to the $(j, k)$-th entry of the $2 \times 2$ identity matrix over $\mathbb{C}$.

For any indices $j, k \in \{0, 1\}$, the $(j, k)$-th entry of the inverse matrix of the $SL(2, \mathbb{C})$ rotation that maps the $z$-axis to the $x$-axis (represented by index $0$) is given by the corresponding entry of the matrix $$\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$$ where $i$ denotes the imaginary unit.

For any indices $j, k \in \{0, 1\}$, the $(j, k)$-th entry of the inverse of the $SL(2, \mathbb{C})$ rotation matrix that carries the $z$-axis to the $y$-axis (denoted as `rotationZToAxis 1`) is given by the corresponding entry of the matrix $$\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix}$$ where $i$ is the imaginary unit. That is, $$\left( (\text{rotationZToAxis } 1)^{-1} \right)_{jk} = \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix} \right)_{jk}.$$

Let $R_z \in SL(2, \mathbb{C})$ be the rotation that carries the $z$-axis to itself (associated with the index $2$). For any row and column indices $j, k \in \{0, 1\}$, the entry in the $j$-th row and $k$-th column of the matrix representing the inverse rotation $R_z^{-1}$ is equal to the $(j, k)$-th entry of the $2 \times 2$ identity matrix over $\mathbb{C}$.

For any complex numbers $a, b \in \mathbb{C}$, let $R_x \in SL(2, \mathbb{C})$ be the rotation matrix that carries the $z$-axis to the $x$-axis (defined as `rotationZToAxis 0`). The conjugation of the diagonal matrix $\text{diag}(a, b)$ by $R_x$ is given by: $$R_x \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} R_x^{-1} = \begin{pmatrix} \frac{a+b}{2} & \frac{a-b}{2} \\ \frac{a-b}{2} & \frac{a+b}{2} \end{pmatrix}$$

For any complex numbers $a, b \in \mathbb{C}$, let $U_y$ be the matrix in $SL(2, \mathbb{C})$ representing the rotation that carries the $z$-axis to the $y$-axis (indexed by 1). Then the conjugation of the diagonal matrix $\operatorname{diag}(a, b)$ by $U_y$ is given by: $$U_y \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} U_y^{-1} = \begin{pmatrix} \frac{a+b}{2} & -i \frac{a-b}{2} \\ i \frac{a-b}{2} & \frac{a+b}{2} \end{pmatrix}$$ where $i$ is the imaginary unit.

For any complex numbers $a$ and $b$, let $R_z \in SL(2, \mathbb{C})$ be the rotation that carries the $z$-axis to itself (the identity matrix). The conjugation of the diagonal matrix $\text{diag}(a, b)$ by $R_z$ satisfies: $$R_z \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} R_z^{-1} = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$$