Physlib.Relativity.Tensors.ComplexTensor.Vector.Pre.Basic

The representation of the group $SL(2, \mathbb{C})$ on the complex vector space of contravariant Lorentz vectors. In index notation, these vectors are typically denoted with an upper index $\psi^i$.

The representation of the group $SL(2, \mathbb{C})$ on the four-dimensional complex vector space corresponding to covariant Lorentz vectors. In physics and tensor calculus, these vectors are typically denoted with a lower index $v_\mu$.

The standard basis for the space of complex contravariant Lorentz vectors, indexed by $i \in \text{Fin } 1 \oplus \text{Fin } 3$ (representing the spacetime components $\{0, 1, 2, 3\}$). This basis establishes the identification between the representation space of $SL(2, \mathbb{C})$ and the coordinate space $\mathbb{C}^4$.

For any spacetime index $i \in \{0, 1, 2, 3\}$, the representation of the $i$-th standard basis vector of the space of complex contravariant Lorentz vectors in the coordinate space $\mathbb{C}^4$ is equal to the standard unit vector $\mathbf{e}_i$ (the vector with $1$ at the $i$-th component and $0$ elsewhere).

For any $M \in SL(2, \mathbb{C})$ and any spacetime indices $i, j \in \{0, 1, 2, 3\}$, the $(i, j)$-th entry of the matrix representing the action $\rho(M)$ on the space of complex contravariant Lorentz vectors with respect to the standard basis is equal to the $(i, j)$-th entry of the complexified Lorentz matrix $\Lambda(M)$ associated with $M$.

For any element $M \in SL(2, \mathbb{C})$ and any complex contravariant Lorentz vector $v$, the action of the representation $\rho(M)$ on $v$ is equivalent to the matrix-vector multiplication of the complexified Lorentz matrix $\Lambda(M)$ (associated with $M$ via the standard homomorphism from $SL(2, \mathbb{C})$ to the Lorentz group) and the vector $v$.

The standard basis for the space of complex contravariant Lorentz vectors, which serves as the representation space for $SL(2, \mathbb{C})$. This basis is indexed by $i \in \{0, 1, 2, 3\}$ (using the type `Fin 4`), establishing an isomorphism between the representation space and the coordinate space $\mathbb{C}^4$.

The standard basis for complex contravariant Lorentz vectors indexed by $\{0, 1, 2, 3\}$ (`complexContrBasisFin4`) is equal to the standard basis indexed by $\text{Fin } 1 \oplus \text{Fin } 3$ (`complexContrBasis`) reindexed using the standard equivalence `finSumFinEquiv` between $\text{Fin } 1 \oplus \text{Fin } 3$ and $\text{Fin } 4$.

For any index $j \in \{0, 1, 2, 3\}$ (represented by the type `Fin 4`), the $j$-th element of the standard basis for complex contravariant Lorentz vectors `complexContrBasis` (originally indexed by $\text{Fin } 1 \oplus \text{Fin } 3$), when reindexed using the standard equivalence $\text{finSumFinEquiv}$ between $\text{Fin } 1 \oplus \text{Fin } 3$ and $\text{Fin } 4$, is equal to the $j$-th element of the basis `complexContrBasisFin4`.

The 0-th element of the standard basis for complex contravariant Lorentz vectors indexed by $\{0, 1, 2, 3\}$ (denoted `complexContrBasisFin4`) is equal to the element of the standard basis indexed by $\text{Fin } 1 \oplus \text{Fin } 3$ (denoted `complexContrBasis`) that corresponds to the index 0 in the first summand ($\text{Fin } 1$).

Let $\mathcal{B}_{\text{Fin4}}$ be the standard basis for the space of complex contravariant Lorentz vectors indexed by $\{0, 1, 2, 3\}$ (represented by the type `Fin 4`). Let $\mathcal{B}_{\text{split}}$ be the standard basis for the same space indexed by the sum type $\text{Fin } 1 \oplus \text{Fin } 3$, where the first component represents the temporal index and the second component represents the spatial indices $\{0, 1, 2\}$. The basis vector at index $1$ in $\mathcal{B}_{\text{Fin4}}$ is equal to the basis vector at index $0$ in the spatial component of $\mathcal{B}_{\text{split}}$.

Let $\mathcal{B}_{\text{Fin4}}$ be the standard basis for the space of complex contravariant Lorentz vectors indexed by $\{0, 1, 2, 3\}$ (represented by the type `Fin 4`). Let $\mathcal{B}_{\text{split}}$ be the standard basis for the same space indexed by the sum type $\text{Fin } 1 \oplus \text{Fin } 3$, where the first component represents the temporal index and the second component represents the spatial indices $\{0, 1, 2\}$. The basis vector at index $2$ in $\mathcal{B}_{\text{Fin4}}$ is equal to the basis vector at index $1$ in the spatial component of $\mathcal{B}_{\text{split}}$.

Let $\mathcal{B}_{\text{Fin4}}$ be the standard basis for the space of complex contravariant Lorentz vectors indexed by $\{0, 1, 2, 3\}$ (represented by `Fin 4`). Let $\mathcal{B}_{\text{split}}$ be the standard basis for the same space indexed by the sum type $\text{Fin } 1 \oplus \text{Fin } 3$, where the first component represents the temporal index and the second component represents the spatial indices $\{0, 1, 2\}$. The basis vector at index $3$ in $\mathcal{B}_{\text{Fin4}}$ is equal to the basis vector at index $2$ in the spatial component of $\mathcal{B}_{\text{split}}$.

Let $\mathcal{B}_{\text{Fin4}}$ be the standard basis for the space of complex contravariant Lorentz vectors indexed by $j \in \{0, 1, 2, 3\}$ (represented by `Fin 4`). Let $\mathcal{B}_{\text{split}}$ be the standard basis for the same space indexed by the sum type $\text{Fin } 1 \oplus \text{Fin } 3$. For any index $i \in \{0, 1, 2\}$ (represented by `Fin 3`), the basis vector at index $i+1$ in $\mathcal{B}_{\text{Fin4}}$ is equal to the basis vector at index $i$ in the second component (the "spatial" part) of $\mathcal{B}_{\text{split}}$.

The standard basis for the four-dimensional complex vector space of covariant Lorentz vectors, indexed by $\{0, 1, 2, 3\}$ (represented as the disjoint union $\{0\} \oplus \{0, 1, 2\}$). This basis corresponds to the covariant representation of $SL(2, \mathbb{C})$ on complex 4-vectors $v_\mu$.

Let $\{e_i\}_{i \in \text{Fin } 1 \oplus \text{Fin } 3}$ be the standard basis vectors for the space of complex covariant Lorentz vectors (denoted as `complexCoBasis`). Let $\Phi$ be the map `toFin13ℂ` that assigns a vector to its coordinate representation in $\mathbb{C}^4$. For any index $i$, the coordinate representation of the $i$-th basis vector is the standard unit vector $\delta_{i, \cdot}$, which takes the value $1$ at index $i$ and $0$ otherwise.

Let $M \in SL(2, \mathbb{C})$ and let $\rho_{\text{co}}$ be the covariant representation of $SL(2, \mathbb{C})$ on the four-dimensional complex vector space of Lorentz vectors. Let $\{e_i\}$ be the standard basis for this space (indexed by $i \in \{0, 1, 2, 3\}$). The $(i, j)$-th entry of the matrix representation of $\rho_{\text{co}}(M)$ with respect to this basis is equal to the $(i, j)$-th entry of the inverse transpose of the complexified Lorentz transformation $\Lambda(M)$ associated with $M$. That is, $$[\rho_{\text{co}}(M)]_{ij} = [(\Lambda(M))^{-1}]^T_{ij}$$ where $\Lambda(M)$ is the image of $M$ under the map from $SL(2, \mathbb{C})$ to the Lorentz group.

The standard basis for the four-dimensional complex vector space of covariant Lorentz vectors $v_\mu$, where the basis elements are indexed by $i \in \{0, 1, 2, 3\}$. This basis is associated with the covariant representation of $SL(2, \mathbb{C})$ on complex 4-vectors.

The basis for the space of complex covariant Lorentz vectors $v_\mu$ indexed by $\text{Fin } 4$ (denoted `complexCoBasisFin4`) is equal to the basis indexed by $\text{Fin } 1 \oplus \text{Fin } 3$ (denoted `complexCoBasis`) after it has been reindexed using the standard equivalence $\text{finSumFinEquiv} : \text{Fin } 1 \oplus \text{Fin } 3 \simeq \text{Fin } 4$.

For any index $j \in \{0, 1, 2, 3\}$, the basis element obtained by reindexing the standard basis for complex covariant Lorentz vectors $v_\mu$, denoted as `complexCoBasis` (which is indexed by the disjoint union $\text{Fin } 1 \oplus \text{Fin } 3$), via the standard equivalence $\text{finSumFinEquiv} : \text{Fin } 1 \oplus \text{Fin } 3 \simeq \text{Fin } 4$ is equal to the $j$-th element of the basis `complexCoBasisFin4` (the basis indexed directly by $\text{Fin } 4$).

In the representation space of complex covariant Lorentz vectors, the basis vector indexed by $0$ in the basis $\texttt{complexCoBasisFin4}$ (indexed by $\text{Fin } 4$) is identical to the basis vector indexed by $\text{Sum.inl } 0$ in the basis $\texttt{complexCoBasis}$ (indexed by $\text{Fin } 1 \oplus \text{Fin } 3$). This identifies the temporal component across the two indexing schemes.

In the representation space of complex covariant Lorentz vectors, the basis vector indexed by $1$ in the basis $\texttt{complexCoBasisFin4}$ (indexed by $\text{Fin } 4$) is identical to the basis vector indexed by $\text{Sum.inr } 0$ in the basis $\texttt{complexCoBasis}$ (indexed by $\text{Fin } 1 \oplus \text{Fin } 3$). This identifies the first spatial component (typically the $x$-component) across the two indexing schemes.

In the representation space of complex covariant Lorentz vectors, the basis vector indexed by $2$ in the basis $\texttt{complexCoBasisFin4}$ (indexed by $\text{Fin } 4$) is identical to the basis vector indexed by $\text{Sum.inr } 1$ in the basis $\texttt{complexCoBasis}$ (indexed by $\text{Fin } 1 \oplus \text{Fin } 3$). This identifies the second spatial component (typically the $y$-component) across the two indexing schemes.

In the representation space of complex covariant Lorentz vectors, the basis vector indexed by $3$ in the basis $\texttt{complexCoBasisFin4}$ (indexed by $\text{Fin } 4$) is identical to the basis vector indexed by $\text{Sum.inr } 2$ in the basis $\texttt{complexCoBasis}$ (indexed by $\text{Fin } 1 \oplus \text{Fin } 3$). This identifies the third spatial component in both indexing schemes.

The map $\iota: \text{ContrMod}(3) \to \text{complexContr}$ is a semilinear map (with respect to the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$) that embeds real contravariant Lorentz vectors into the complex contravariant Lorentz representation space. For a real Lorentz vector $v \in \text{ContrMod}(3)$, its image is the complex vector obtained by mapping each real component of $v$ to its corresponding value in $\mathbb{C}$.

For any real contravariant Lorentz vector $v \in \text{ContrMod}(3)$, the underlying components of its image under the inclusion map $\iota: \text{ContrMod}(3) \to \text{complexContr}$ are obtained by mapping each real component of $v$ to the complex numbers. Mathematically, $(\iota(v)).\text{val} = \text{ofReal} \circ v$, where $v$ is viewed as its component function and $\text{ofReal}$ is the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$.

For any spacetime index $i \in \text{Fin } 1 \oplus \text{Fin } 3$ (representing the components $\{0, 1, 2, 3\}$), the $i$-th basis vector of the standard basis for complex contravariant Lorentz vectors is the image of the $i$-th basis vector of the standard basis for real contravariant Lorentz vectors under the inclusion map $\iota: \text{ContrMod}(3) \to \text{complexContr}$.

For any matrix $M \in SL(2, \mathbb{C})$ and any real contravariant Lorentz vector $v \in \text{ContrMod}(3)$, let $\iota: \text{ContrMod}(3) \to \text{complexContr}$ be the inclusion map that complexifies the components of a real Lorentz vector. Let $\rho_{\mathbb{C}}$ be the representation of $SL(2, \mathbb{C})$ on complex Lorentz vectors and $\rho_{\mathbb{R}}$ be the representation of the Lorentz group on real Lorentz vectors. Then the following holds: $$\rho_{\mathbb{C}}(M)(\iota(v)) = \iota(\rho_{\mathbb{R}}(\Lambda(M)) v)$$ where $\Lambda(M)$ is the Lorentz transformation associated with $M$ via the standard homomorphism from $SL(2, \mathbb{C})$ to the Lorentz group.

For any matrix $M \in SL(2, \mathbb{C})$ and any spacetime index $i \in \{0, 1, 2, 3\}$, let $\rho$ be the representation of $SL(2, \mathbb{C})$ on the space of complex contravariant Lorentz vectors and let $\{e_i\}$ be the standard basis for this space. Let $\Lambda(M)$ be the $4 \times 4$ Lorentz transformation matrix associated with $M$ via the standard homomorphism from $SL(2, \mathbb{C})$ to the Lorentz group. Then the action of the representation on the basis vector $e_i$ is given by: $$\rho(M) e_i = \sum_{j=0}^3 \Lambda(M)_{ji} e_j$$ where $\Lambda(M)_{ji}$ denotes the entry in the $j$-th row and $i$-th column of the Lorentz matrix.