Physlib.Relativity.Tensors.ComplexTensor.Matrix.Pre

The linear isomorphism `contrContrToMatrix` identifies the tensor product of two complex contravariant Lorentz vector spaces $V \otimes V$ with the space of $4 \times 4$ complex matrices $\text{Mat}_{4 \times 4}(\mathbb{C})$. Given the standard basis $\{e_i\}_{i \in \{0, \dots, 3\}}$ for the contravariant Lorentz vectors (where the index set is represented by $\text{Fin } 1 \oplus \text{Fin } 3$), the map sends a tensor $T = \sum_{i,j} M_{ij} (e_i \otimes e_j)$ to the matrix $M$ whose entries are $M_{ij}$.

Let $V$ be the space of complex contravariant Lorentz vectors with the standard basis $\{e_i\}_{i \in \{0, \dots, 3\}}$, and let $\Phi : (V \otimes V) \xrightarrow{\cong} \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism `contrContrToMatrix` that identifies tensors with matrices. For any $4 \times 4$ complex matrix $M$, the inverse isomorphism $\Phi^{-1}$ yields the tensor product expansion: $$\Phi^{-1}(M) = \sum_{i=0}^3 \sum_{j=0}^3 M_{ij} (e_i \otimes e_j)$$ where $M_{ij}$ are the entries of the matrix $M$.

The linear isomorphism `coCoToMatrix` identifies the tensor product of two complex covariant Lorentz vector spaces $V \otimes V$ (often denoted $V_\mu \otimes V_\nu$ in physics) with the space of $4 \times 4$ complex matrices $\text{Mat}_{4 \times 4}(\mathbb{C})$. Given the standard basis $\{e_i\}_{i \in \{0, \dots, 3\}}$ for the covariant Lorentz vectors (where the index set is represented by $\text{Fin } 1 \oplus \text{Fin } 3$), the map sends a tensor $T = \sum_{i,j} M_{ij} (e_i \otimes e_j)$ to the matrix $M$ whose entries are $M_{ij}$.

For any $4 \times 4$ complex matrix $M \in \text{Mat}_{4 \times 4}(\mathbb{C})$, the inverse of the linear isomorphism $\text{coCoToMatrix}$ maps $M$ to the tensor $\sum_{i,j} M_{ij} (e_i \otimes e_j)$, where $\{e_i\}_{i \in \{0, \dots, 3\}}$ is the standard basis for the complex covariant Lorentz vector space.

The linear equivalence between the underlying vector space of the tensor product of the contravariant representation $V_{\text{contr}}$ and the covariant representation $V_{\text{co}}$ of $SL(2, \mathbb{C})$ and the space of $4 \times 4$ complex matrices $\text{Mat}_{4 \times 4}(\mathbb{C})$. This equivalence is established by mapping the basis elements formed by the tensor product of the standard contravariant basis and covariant basis to the corresponding matrix entries.

Let $V_{\text{contr}}$ and $V_{\text{co}}$ be the four-dimensional complex vector spaces corresponding to the contravariant and covariant representations of $SL(2, \mathbb{C})$, respectively. Let $\{e^i\}_{i \in \{0, \dots, 3\}}$ be the standard basis for $V_{\text{contr}}$ and $\{e_j\}_{j \in \{0, \dots, 3\}}$ be the standard basis for $V_{\text{co}}$. For any $4 \times 4$ complex matrix $M \in \text{Mat}_{4 \times 4}(\mathbb{C})$, the inverse of the linear equivalence $\text{contrCoToMatrix} : V_{\text{contr}} \otimes V_{\text{co}} \simeq \text{Mat}_{4 \times 4}(\mathbb{C})$ is given by: $$\text{contrCoToMatrix}^{-1}(M) = \sum_{i, j} M_{ij} (e^i \otimes e_j)$$ where $M_{ij}$ denotes the entry of the matrix $M$ at row $i$ and column $j$.

The linear equivalence between the vector space of the tensor product of the covariant representation $V_{\text{co}}$ and the contravariant representation $V_{\text{contr}}$ of $SL(2, \mathbb{C})$ and the space of $4 \times 4$ complex matrices $\text{Mat}_{4 \times 4}(\mathbb{C})$, defined using the standard bases for covariant and contravariant Lorentz vectors.

Let $V_{\text{co}}$ and $V_{\text{contr}}$ be the four-dimensional complex vector spaces corresponding to the covariant and contravariant representations of $SL(2, \mathbb{C})$, respectively. Let $\{e_i\}_{i \in \{0, \dots, 3\}}$ be the standard basis for $V_{\text{co}}$ and $\{e^j\}_{j \in \{0, \dots, 3\}}$ be the standard basis for $V_{\text{contr}}$. For any $4 \times 4$ complex matrix $M \in \text{Mat}_{4 \times 4}(\mathbb{C})$, the inverse of the linear equivalence $\text{coContrToMatrix}: V_{\text{co}} \otimes V_{\text{contr}} \simeq \text{Mat}_{4 \times 4}(\mathbb{C})$ acts as: $$\text{coContrToMatrix}^{-1}(M) = \sum_{i,j} M_{ij} (e_i \otimes e^j)$$ where $M_{ij}$ denotes the entry of the matrix $M$ at row $i$ and column $j$.

Let $V^\mu$ be the space of complex contravariant Lorentz vectors and $\rho$ be the contravariant representation of $SL(2, \mathbb{C})$ on $V^\mu$. Let $\text{contrContrToMatrix} : V^\mu \otimes V^\nu \cong \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism that maps a tensor to its matrix of components. For any tensor $v \in V^\mu \otimes V^\nu$ and any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. Then the transformation of $v$ under the action of $M$ on both factors corresponds to the matrix operation: $$\text{contrContrToMatrix}((\rho(M) \otimes \rho(M)) v) = \Lambda \, \text{contrContrToMatrix}(v) \, \Lambda^T$$ where $\Lambda^T$ denotes the transpose of the Lorentz matrix.

Let $V_\mu$ be the space of complex covariant Lorentz vectors and $\rho_{\text{co}}$ be the covariant representation of $SL(2, \mathbb{C})$ on $V_\mu$. Let $\text{coCoToMatrix} : V_\mu \otimes V_\mu \cong \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism that maps a tensor to its matrix of components. For any tensor $v \in V_\mu \otimes V_\mu$ and any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. Then the transformation of $v$ under the action of $M$ on both factors corresponds to the matrix operation: $$\text{coCoToMatrix}((\rho_{\text{co}}(M) \otimes \rho_{\text{co}}(M)) v) = (\Lambda^{-1})^T \, \text{coCoToMatrix}(v) \, \Lambda^{-1}$$ where $(\Lambda^{-1})^T$ denotes the inverse transpose of the Lorentz matrix.

Let $V^\mu$ be the complex vector space of contravariant Lorentz vectors and $V_\nu$ be the complex vector space of covariant Lorentz vectors. Let $\text{contrCoToMatrix} : V^\mu \otimes V_\nu \cong \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism that maps a tensor to its matrix of components. For any tensor $v \in V^\mu \otimes V_\nu$ and any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. The transformation of $v$ under the simultaneous action of the contravariant representation $\rho$ and covariant representation $\rho_{\text{co}}$ corresponds to the following matrix operation: $$\text{contrCoToMatrix}((\rho(M) \otimes \rho_{\text{co}}(M)) v) = \Lambda \, \text{contrCoToMatrix}(v) \, \Lambda^{-1}$$ where $\Lambda^{-1}$ is the inverse of the Lorentz matrix.

Let $V_\mu$ be the complex vector space of covariant Lorentz vectors and $V^\nu$ be the complex vector space of contravariant Lorentz vectors. Let $\text{coContrToMatrix} : V_\mu \otimes V^\nu \cong \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism that maps a tensor to its matrix of components. For any tensor $v \in V_\mu \otimes V^\nu$ and any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. Then the transformation of $v$ under the simultaneous action of the covariant representation $\rho_{\text{co}}$ and the contravariant representation $\rho$ corresponds to the following matrix operation: $$\text{coContrToMatrix}((\rho_{\text{co}}(M) \otimes \rho(M)) v) = (\Lambda^{-1})^T \, \text{coContrToMatrix}(v) \, \Lambda^T$$ where $(\Lambda^{-1})^T$ is the inverse transpose of the Lorentz matrix and $\Lambda^T$ is its transpose.

Let $V^\mu$ be the complex vector space of contravariant Lorentz vectors and $\rho$ be the contravariant representation of $SL(2, \mathbb{C})$ on $V^\mu$. Let $\text{contrContrToMatrix}^{-1} : \text{Mat}_{4 \times 4}(\mathbb{C}) \cong V^\mu \otimes V^\nu$ be the linear isomorphism that maps a $4 \times 4$ matrix to its corresponding tensor in the tensor product space. For any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. Then for any matrix $v \in \text{Mat}_{4 \times 4}(\mathbb{C})$, the action of $M$ on the tensor $\text{contrContrToMatrix}^{-1}(v)$ is given by: $$(\rho(M) \otimes \rho(M))(\text{contrContrToMatrix}^{-1}(v)) = \text{contrContrToMatrix}^{-1}(\Lambda v \Lambda^T)$$ where $\Lambda^T$ denotes the transpose of the Lorentz matrix.

Let $V_\mu$ be the space of complex covariant Lorentz vectors and $\rho_{\text{co}}$ be the covariant representation of $SL(2, \mathbb{C})$ on $V_\mu$. Let $\text{coCoToMatrix}^{-1} : \text{Mat}_{4 \times 4}(\mathbb{C}) \cong V_\mu \otimes V_\mu$ be the inverse of the linear isomorphism that identifies tensors with $4 \times 4$ matrices. For any matrix $v \in \text{Mat}_{4 \times 4}(\mathbb{C})$ and any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. The action of $SL(2, \mathbb{C})$ on the tensor product space $V_\mu \otimes V_\mu$ satisfies: $$(\rho_{\text{co}}(M) \otimes \rho_{\text{co}}(M)) (\text{coCoToMatrix}^{-1}(v)) = \text{coCoToMatrix}^{-1} \left( (\Lambda^{-1})^T v \Lambda^{-1} \right)$$ where $(\Lambda^{-1})^T$ denotes the inverse transpose of the Lorentz matrix.

Let $V^\mu$ and $V_\nu$ denote the complex vector spaces of contravariant and covariant Lorentz vectors, respectively. Let $\text{contrCoToMatrix} : V^\mu \otimes V_\nu \cong \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism mapping a tensor to its matrix of components, and let $\text{contrCoToMatrix}^{-1}$ be its inverse. For any matrix $v \in \text{Mat}_{4 \times 4}(\mathbb{C})$ and $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix corresponding to $M$. Then the simultaneous action of the contravariant representation $\rho_{\text{contr}}$ and the covariant representation $\rho_{\text{co}}$ on the tensor $\text{contrCoToMatrix}^{-1}(v)$ is given by: $$(\rho_{\text{contr}}(M) \otimes \rho_{\text{co}}(M)) (\text{contrCoToMatrix}^{-1}(v)) = \text{contrCoToMatrix}^{-1}(\Lambda v \Lambda^{-1})$$ where $\Lambda^{-1}$ is the inverse of the Lorentz matrix.

Let $V_{\text{co}}$ be the complex covariant Lorentz representation and $V_{\text{contr}}$ be the complex contravariant Lorentz representation of $SL(2, \mathbb{C})$. Let $f: V_{\text{co}} \otimes V_{\text{contr}} \cong \text{Mat}_{4 \times 4}(\mathbb{C})$ be the linear isomorphism that maps a tensor to its matrix of components. For any $4 \times 4$ complex matrix $v$ and any $M \in SL(2, \mathbb{C})$, let $\Lambda$ be the complexified Lorentz transformation matrix associated with $M$. The action of the group $SL(2, \mathbb{C})$ on the tensor $f^{-1}(v)$ obtained from the matrix $v$ satisfies: $$(\rho_{\text{co}}(M) \otimes \rho_{\text{contr}}(M)) f^{-1}(v) = f^{-1}(\Lambda^{-T} v \Lambda^T)$$ where $\Lambda^{-T}$ is the inverse transpose of the Lorentz matrix and $\Lambda^T$ is its transpose.