Physlib.Relativity.SL2C.Basic

For any element $M$ in the special linear group $SL(2, \mathbb{C})$, the inverse of its underlying matrix is equal to the matrix representing its group inverse $M^{-1}$.

For any element $M$ in the special linear group $SL(2, \mathbb{C})$, the transpose of the $2 \times 2$ matrix $M$, denoted $M^\top$, is equal to the matrix representation of the group-theoretic transpose of $M$.

For a matrix $M \in SL(2, \mathbb{C})$, this is the $\mathbb{R}$-linear map from the space of $2 \times 2$ self-adjoint complex matrices to itself defined by the transformation $A \mapsto M A M^\dagger$, where $M^\dagger$ denotes the conjugate transpose of $M$.

For any matrix $M \in SL(2, \mathbb{C})$ and any $2 \times 2$ complex self-adjoint matrix $A$, the linear map `toSelfAdjointMap` applied to $A$ is given by $M A M^\dagger$, where $M^\dagger$ denotes the conjugate transpose of $M$.

Let $M$ be a matrix in $SL(2, \mathbb{C})$ and $A$ be a $2 \times 2$ self-adjoint complex matrix. The determinant of the matrix resulting from the $\mathbb{R}$-linear map $A \mapsto M A M^\dagger$ is equal to the determinant of $A$. That is, $$\det(M A M^\dagger) = \det A$$ where $M^\dagger$ denotes the conjugate transpose of $M$.

For any matrix $M = \begin{pmatrix} M_{00} & M_{01} \\ M_{10} & M_{11} \end{pmatrix} \in SL(2, \mathbb{C})$, let $\Phi(A) = M A M^\dagger$ be the $\mathbb{R}$-linear map on the space of $2 \times 2$ complex self-adjoint matrices, where $M^\dagger$ is the conjugate transpose of $M$. Let $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ be the covariant Pauli basis for this space, where $\sigma_0$ is the identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices. The image of $\sigma_0$ under $\Phi$ is given by: \[ \Phi(\sigma_0) = C_0 \sigma_0 + C_1 (-\sigma_1) + C_2 (-\sigma_2) + C_3 (-\sigma_3) \] where the real coefficients $C_k$ are: - $C_0 = \frac{1}{2} (|M_{00}|^2 + |M_{01}|^2 + |M_{10}|^2 + |M_{11}|^2)$ - $C_1 = -(\text{Re}(M_{01}) \text{Re}(M_{11}) + \text{Im}(M_{01}) \text{Im}(M_{11}) + \text{Im}(M_{00}) \text{Im}(M_{10}) + \text{Re}(M_{00}) \text{Re}(M_{10}))$ - $C_2 = -\text{Re}(M_{00}) \text{Im}(M_{10}) + \text{Re}(M_{10}) \text{Im}(M_{00}) - \text{Re}(M_{01}) \text{Im}(M_{11}) + \text{Im}(M_{01}) \text{Re}(M_{11})$ - $C_3 = \frac{1}{2} (-|M_{00}|^2 - |M_{01}|^2 + |M_{10}|^2 + |M_{11}|^2)$

This definition defines a monoid homomorphism from the special linear group $SL(2, \mathbb{C})$ to the space of $4 \times 4$ real matrices, indexed by $\{0\} \oplus \{0, 1, 2\}$. For a given matrix $M \in SL(2, \mathbb{C})$, the resulting matrix is the representation of the $\mathbb{R}$-linear map $A \mapsto M A M^\dagger$ (where $M^\dagger$ is the conjugate transpose) acting on the space of $2 \times 2$ complex self-adjoint matrices. This matrix representation is constructed with respect to the basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$, where $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices.

For any matrix $M \in SL(2, \mathbb{C})$ and any contravariant Lorentz vector $v \in \text{ContrMod } 3$, let $L(M)$ be the $4 \times 4$ real matrix defined as the representation of the linear map $\Phi_M(A) = M A M^\dagger$ with respect to the covariant Pauli basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$. Let $\Psi: \text{ContrMod } 3 \to \text{selfAdjoint}(M_2(\mathbb{C}))$ be the linear isomorphism that identifies a vector $v = (v^0, v^1, v^2, v^3)$ with the self-adjoint matrix $v^0 \sigma_0 - v^1 \sigma_1 - v^2 \sigma_2 - v^3 \sigma_3$. Then the result of the matrix-vector multiplication $L(M) v$ is equivalent to applying the transformation $\Phi_M$ to the matrix $\Psi(v)$ and mapping the result back to the vector space: \[ L(M) v = \Psi^{-1}(M \Psi(v) M^\dagger) \]

For any matrix $M \in SL(2, \mathbb{C})$, let $L(M)$ be the $4 \times 4$ real matrix that represents the transformation $A \mapsto M A M^\dagger$ on the space of $2 \times 2$ self-adjoint matrices with respect to the basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$. Then $L(M)$ is an element of the Lorentz group $\mathcal{L}$ for $d=3$ spatial dimensions.

This definition defines a group homomorphism from the special linear group $SL(2, \mathbb{C})$ to the Lorentz group $\mathcal{L}$ for $d=3$ spatial dimensions. For any matrix $M \in SL(2, \mathbb{C})$, the homomorphism maps $M$ to the $4 \times 4$ real matrix $L(M)$ which represents the transformation $A \mapsto M A M^\dagger$ on the space of $2 \times 2$ complex self-adjoint matrices, expressed with respect to the basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ (where $\sigma_0$ is the identity matrix and $\sigma_{1,2,3}$ are the Pauli matrices).

For any matrix $M \in SL(2, \mathbb{C})$, the image of $M$ under the homomorphism to the Lorentz group, denoted as $\text{toLorentzGroup}(M)$, is equal to the matrix representation of the $\mathbb{R}$-linear map $A \mapsto M A M^\dagger$ (acting on the space of $2 \times 2$ self-adjoint matrices) with respect to the covariant Pauli basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$.

Let $M \in SL(2, \mathbb{C})$ and let $L(M)$ be its image in the Lorentz group $\mathcal{L}$ under the homomorphism `toLorentzGroup`. Let $\{\sigma'_i\}_{i=0}^3$ be the covariant Pauli basis for the space of $2 \times 2$ complex self-adjoint matrices, where $\sigma'_0 = I_2$ and $\sigma'_{1,2,3} = -\sigma_{1,2,3}$. The action of the $\mathbb{R}$-linear map $\Phi_M: A \mapsto M A M^\dagger$ on the basis elements is given by: $$\Phi_M(\sigma'_i) = \sum_{j=0}^3 L(M)_{ji} \sigma'_j$$ where $L(M)_{ji}$ denotes the entry of the Loretta matrix $L(M)$ at row $j$ and column $i$.

Let $M \in SL(2, \mathbb{C})$ and let $L(M^{-1})$ be the image of its inverse in the Lorentz group $\mathcal{L}$ under the homomorphism `toLorentzGroup`. Let $\{\sigma^i\}_{i=0}^3$ be the Pauli basis (contravariant basis) for the space of $2 \times 2$ complex self-adjoint matrices, where $\sigma^0 = I_2$ and $\sigma^{1,2,3}$ are the standard Pauli matrices. The action of the $\mathbb{R}$-linear map $\Phi_M: A \mapsto M A M^\dagger$ on the Pauli basis elements is given by: $$\Phi_M(\sigma^i) = \sum_{j=0}^3 L(M^{-1})_{ij} \sigma^j$$ where $L(M^{-1})_{ij}$ denotes the entry of the Lorentz matrix of $M^{-1}$ at row $i$ and column $j$.

For any matrix $M = \begin{pmatrix} M_{00} & M_{01} \\ M_{10} & M_{11} \end{pmatrix} \in SL(2, \mathbb{C})$, let $L(M)$ be its image in the Lorentz group for 3 spatial dimensions under the homomorphism `toLorentzGroup`. The entries of the first column of $L(M)$ (indexed by $\mu \in \{0, 1, 2, 3\}$ at column index 0) are given by: - $L(M)_{0,0} = \frac{1}{2} (|M_{00}|^2 + |M_{01}|^2 + |M_{10}|^2 + |M_{11}|^2)$ - $L(M)_{1,0} = -(\text{Re}(M_{01}) \text{Re}(M_{11}) + \text{Im}(M_{01}) \text{Im}(M_{11}) + \text{Im}(M_{00}) \text{Im}(M_{10}) + \text{Re}(M_{00}) \text{Re}(M_{10}))$ - $L(M)_{2,0} = -\text{Re}(M_{00}) \text{Im}(M_{10}) + \text{Re}(M_{10}) \text{Im}(M_{00}) - \text{Re}(M_{01}) \text{Im}(M_{11}) + \text{Im}(M_{01}) \text{Re}(M_{11})$ - $L(M)_{3,0} = \frac{1}{2} (-|M_{00}|^2 - |M_{01}|^2 + |M_{10}|^2 + |M_{11}|^2)$

Let $M = \begin{pmatrix} M_{00} & M_{01} \\ M_{10} & M_{11} \end{pmatrix}$ be a matrix in $SL(2, \mathbb{C})$. Let $L(M)$ be its image in the Lorentz group for 3 spatial dimensions under the homomorphism `toLorentzGroup`. The first entry (time-time component) of the matrix $L(M)$ is given by: $$L(M)_{0,0} = \frac{1}{2} (|M_{00}|^2 + |M_{01}|^2 + |M_{10}|^2 + |M_{11}|^2)$$

Let $SL(2, \mathbb{C})$ be the group of $2 \times 2$ complex matrices with determinant 1. For any matrix $M \in SL(2, \mathbb{C})$, let $L(M)$ be its image in the Lorentz group for $d=3$ spatial dimensions under the group homomorphism induced by the transformation $A \mapsto M A M^\dagger$. Then $L(M)$ is orthochronous, meaning its $(0,0)$-component satisfies $L(M)_{0,0} \ge 0$.

Let $SL(2, \mathbb{C})$ be the special linear group of $2 \times 2$ complex matrices with determinant $1$. Let $\mathcal{L}$ be the Lorentz group for $d=3$ spatial dimensions, consisting of $4 \times 4$ real matrices. For any $M \in SL(2, \mathbb{C})$, let $L(M) \in \mathcal{L}$ be the image of $M$ under the group homomorphism induced by the transformation $A \mapsto M A M^\dagger$ on the space of $2 \times 2$ complex self-adjoint matrices. Then the determinant of the matrix $L(M)$ is equal to $1$.

This is a group homomorphism from the special linear group $SL(2, \mathbb{C})$ to the restricted Lorentz group $\mathcal{L}_+^\uparrow$ with $d=3$ spatial dimensions. It is defined by restricting the codomain of the homomorphism $L: SL(2, \mathbb{C}) \to \mathcal{L}$ to its restricted subgroup $\mathcal{L}_+^\uparrow$. For any $M \in SL(2, \mathbb{C})$, the resulting Lorentz transformation $L(M)$ is shown to be proper (satisfying $\det L(M) = 1$) and orthochronous (satisfying $L(M)_{00} \ge 0$).