Physlib.Relativity.Fermions.Weyl.Two

This is a linear equivalence between the tensor product of two dual left-handed Weyl fermion spaces and the space of $2 \times 2$ complex matrices, denoted as $\mathcal{V}_L^* \otimes_{\mathbb{C}} \mathcal{V}_L^* \cong M_2(\mathbb{C})$.

For any $2 \times 2$ complex matrix $M$, the inverse of the isomorphism between the tensor product of two dual left-handed Weyl fermion spaces and the space of matrices (denoted as `dualLeftdualLeftToMatrix`) maps $M$ to the sum $\sum_{i,j \in \{0, 1\}} M_{ij} (e_i \otimes e_j)$, where $e_i$ and $e_j$ are the basis elements of the dual left-handed Weyl fermion space (`dualLeftBasis`).

Let $V_L$ be the complex vector space of left-handed Weyl fermions and $V_L^*$ be its dual space. This definition provides a $\mathbb{C}$-linear isomorphism between the tensor product $V_L \otimes_{\mathbb{C}} V_L^*$ and the space of $2 \times 2$ complex matrices $M_2(\mathbb{C})$. The equivalence is induced by identifying $V_L$ with $\mathbb{C}^2$ via the standard basis $\{e_0, e_1\}$.

For any complex $2 \times 2$ matrix $M$, its image under the inverse of the linear equivalence between the space of $2 \times 2$ matrices and the tensor product of left-handed Weyl fermions and their duals is given by the expansion: $$\text{leftDualLeftToMatrix}^{-1}(M) = \sum_{i=0}^1 \sum_{j=0}^1 M_{ij} (e_i \otimes f^j)$$ where $e_i$ denotes the $i$-th basis element of the standard basis for left-handed Weyl fermions (`leftBasis`) and $f^j$ denotes the $j$-th basis element of the corresponding dual basis (`dualLeftBasis`).

The linear equivalence between the tensor product of the dual space of left-handed Weyl fermions $V_L^*$ and the space of left-handed Weyl fermions $V_L$, and the space of $2 \times 2$ complex matrices $M_{2 \times 2}(\mathbb{C})$. This isomorphism is induced by the standard basis $\{e_0, e_1\}$ of $V_L$ (and its dual basis), identifying the space of endomorphisms $V_L^* \otimes V_L$ with the space of $2 \times 2$ matrices.

For any $2 \times 2$ complex matrix $M \in \text{Mat}_{2 \times 2}(\mathbb{C})$, the image of $M$ under the inverse of the map `dualLeftLeftToMatrix` is given by the expansion: $$\text{dualLeftLeftToMatrix}^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (e^{L, *}_i \otimes e^L_j)$$ where $e^L_j$ denotes the $j$-th element of the standard basis for left-handed Weyl fermions, $e^{L, *}_i$ denotes the $i$-th element of the corresponding dual basis, and $\otimes$ denotes the tensor product over $\mathbb{C}$.

This definition establishes a complex-linear equivalence (isomorphism) between the tensor product of two dual right-handed Weyl fermion spaces and the space of $2 \times 2$ complex matrices. Specifically, it maps an element of the tensor product $\text{DualRightHandedWeyl} \otimes_{\mathbb{C}} \text{DualRightHandedWeyl}$ to a matrix in $M_{2 \times 2}(\mathbb{C})$.

For any $2 \times 2$ complex matrix $M$, the inverse of the isomorphism from the tensor product of two dual right-handed Weyl fermion spaces to matrices maps $M$ to the sum over the tensor product of the basis elements: $$\text{dualRightDualRightToMatrix.symm}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (\beta_i \otimes \beta_j)$$ where $\beta_i$ and $\beta_j$ are the basis elements of the dual right-handed Weyl fermion space (`dualRightBasis`).

This is the complex linear isomorphism between the tensor product of the space of right-handed Weyl fermions $\mathbb{S}_R$ and its dual space $\mathbb{S}_R^*$, and the space of $2 \times 2$ complex matrices $\text{Mat}_{2 \times 2}(\mathbb{C})$. This equivalence maps an element of the tensor product $\mathbb{S}_R \otimes_{\mathbb{C}} \mathbb{S}_R^*$ to its corresponding matrix representation.

For any $2 \times 2$ complex matrix $M$, the inverse of the isomorphism $\text{rightDualRightToMatrix}$ (which maps the tensor product of right-handed Weyl fermions and their duals to the space of matrices) maps $M$ to the tensor sum: $$\text{rightDualRightToMatrix}^{-1}(M) = \sum_{i=0}^1 \sum_{j=0}^1 M_{ij} (e_i \otimes \epsilon^j)$$ where $\{e_i\}_{i \in \{0, 1\}}$ is the standard basis for right-handed Weyl fermions (`Fermion.rightBasis`) and $\{\epsilon^j\}_{j \in \{0, 1\}}$ is the standard basis for the dual space of right-handed Weyl fermions (`Fermion.dualRightBasis`).

This is a linear isomorphism between the tensor product of the dual space of right-handed Weyl fermions (denoted as $V_R^*$) and the space of right-handed Weyl fermions ($V_R$) over the complex numbers $\mathbb{C}$, and the space of $2 \times 2$ complex matrices $\text{Mat}_{2 \times 2}(\mathbb{C})$.

For any $2 \times 2$ complex matrix $M$, the element of the tensor product space $(\text{RightHanded})^* \otimes \text{RightHanded}$ corresponding to $M$ under the inverse of the map `Fermion.dualRightRightToMatrix` is given by the sum: $$\sum_{i, j \in \{0, 1\}} M_{ij} (\epsilon^i \otimes e_j)$$ where $e_j$ denotes the standard basis vectors for right-handed Weyl fermions (`Fermion.rightBasis`) and $\epsilon^i$ denotes the basis vectors of the corresponding dual space (`Fermion.dualRightBasis`).

There is a $\mathbb{C}$-linear equivalence between the tensor product of the dual left-handed Weyl spinor space $W_L^*$ and the dual right-handed Weyl spinor space $W_R^*$, denoted by $W_L^* \otimes_{\mathbb{C}} W_R^*$, and the space of $2 \times 2$ complex matrices $\text{Mat}_{2 \times 2}(\mathbb{C})$.

For any $2 \times 2$ complex matrix $M$, the inverse of the linear isomorphism `dualLeftDualRightToMatrix` (which maps the tensor product of dual left and dual right Weyl fermion spaces to the space of matrices) applied to $M$ is equal to the sum over all indices $i$ and $j$ of the matrix entries $M_{ij}$ scaled by the tensor product of the $i$-th dual left basis element and the $j$-th dual right basis element: $$\text{dualLeftDualRightToMatrix}^{-1}(M) = \sum_{i, j} M_{ij} \cdot (e^L_i \otimes e^R_j)$$ where $e^L_i$ and $e^R_j$ represent the elements of the dual left and dual right bases for Weyl fermions, respectively.

Let $SL(2, \mathbb{C})$ be the group of $2 \times 2$ complex matrices with determinant 1, and let $\Delta_L^*$ denote the space of dual left-handed Weyl spinors. For any tensor $v \in \Delta_L^* \otimes_{\mathbb{C}} \Delta_L^*$ and any matrix $M \in SL(2, \mathbb{C})$, let $\rho(M)$ be the representation of $M$ acting on $\Delta_L^*$ and $\Phi$ be the mapping that identifies the tensor product space with $2 \times 2$ matrices. The matrix representation of the transformed tensor $(\rho(M) \otimes \rho(M)) v$ is given by: $$\Phi((\rho(M) \otimes \rho(M)) v) = (M^{-1})^T \Phi(v) M^{-1}$$

Let $V_L$ be the space of left-handed Weyl fermions and $V_L^*$ be its dual space. Let $\Phi: V_L \otimes_{\mathbb{C}} V_L^* \to \text{Mat}_{2 \times 2}(\mathbb{C})$ be the linear map `Fermion.leftDualLeftToMatrix` which identifies the tensor product with $2 \times 2$ matrices. For any tensor $v \in V_L \otimes_{\mathbb{C}} V_L^*$ and any matrix $M \in SL(2, \mathbb{C})$, the following identity holds: $$\Phi((\rho_L(M) \otimes \rho_L^*(M)) v) = M \Phi(v) M^{-1}$$ where $\rho_L(M)$ is the representation of $M$ acting on $V_L$ and $\rho_L^*(M)$ is the representation of $M$ acting on the dual space $V_L^*$.

Let $W_L$ be the space of left-handed Weyl spinors and $W_L^*$ be its dual space. Let $\Phi: W_L^* \otimes W_L \to \text{Mat}(2, \mathbb{C})$ be the map that identifies an element of the tensor product with a $2 \times 2$ matrix (represented by `Fermion.dualLeftLeftToMatrix`). For any tensor $v \in W_L^* \otimes W_L$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the following transformation law holds: $$\Phi((\rho_{L}^*(M) \otimes \rho_L(M)) v) = (M^{-1})^T \Phi(v) M^T$$ where $\rho_L(M)$ is the representation of $M$ acting on $W_L$ and $\rho_L^*(M)$ is the representation acting on $W_L^*$.

Let $W^*_{\dot{R}}$ be the space of dual right-handed Weyl fermions and $\Phi: W^*_{\dot{R}} \otimes_{\mathbb{C}} W^*_{\dot{R}} \to \text{Mat}(2, \mathbb{C})$ be the linear equivalence that maps the tensor product of two dual right-handed Weyl fermions to a $2 \times 2$ matrix. For any tensor $v \in W^*_{\dot{R}} \otimes_{\mathbb{C}} W^*_{\dot{R}}$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the following identity holds for the dual right-handed representation $\rho_{\dot{R}^*}$: $$\Phi\left((\rho_{\dot{R}^*}(M) \otimes \rho_{\dot{R}^*}(M)) v\right) = (M^{-1})^\dagger \Phi(v) ((M^{-1})^\dagger)^T$$ where $M^\dagger$ denotes the conjugate transpose of $M$ and $M^T$ denotes the transpose.

Let $V$ be the space of right-handed Weyl fermions (`Fermion.RightHandedWeyl`) and $V^*$ its dual space (`Fermion.DualRightHandedWeyl`). Let $\Phi: V \otimes V^* \to \text{Mat}_{2 \times 2}(\mathbb{C})$ be the mapping defined by `Fermion.rightDualRightToMatrix`. For any tensor $v \in V \otimes V^*$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the following identity holds: $$\Phi\left( \left( \rho_R(M) \otimes \rho_{R^*}(M) \right) v \right) = \bar{M} \Phi(v) \bar{M}^{-1}$$ where $\rho_R$ and $\rho_{R^*}$ are the group representations of $SL(2, \mathbb{C})$ acting on the right-handed Weyl fermions and their duals respectively, and $\bar{M}$ denotes the entry-wise complex conjugate of the matrix $M$.

For any element $v$ in the tensor product of a dual right-handed Weyl spinor and a right-handed Weyl spinor, and for any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the matrix representation $\Phi(v)$ transforms under the group action as: $$\Phi\left( (\rho_{\text{dual}}(M) \otimes \rho_{\text{R}}(M)) v \right) = (M^{-1})^\dagger \Phi(v) M^\dagger$$ where $\Phi$ denotes the linear isomorphism `Fermion.dualRightRightToMatrix`, $\rho_{\text{dual}}$ and $\rho_{\text{R}}$ are the representations of $SL(2, \mathbb{C})$ on the dual right-handed and right-handed Weyl spinor spaces respectively, and $M^\dagger$ denotes the conjugate transpose of $M$.

Let $V_{\bar{L}}$ and $V_{\bar{R}}$ be the spaces of dual left-handed and dual right-handed Weyl spinors, respectively. Let $\Phi : V_{\bar{L}} \otimes_{\mathbb{C}} V_{\bar{R}} \to \text{Mat}_{2 \times 2}(\mathbb{C})$ denote the map `Fermion.dualLeftDualRightToMatrix` which associates a matrix with an element of the tensor product. For any $v \in V_{\bar{L}} \otimes_{\mathbb{C}} V_{\bar{R}}$ and any matrix $M \in SL(2, \mathbb{C})$, the transformation of the matrix representation under the group action is given by: $$\Phi\left( \left(\rho_{\bar{L}}(M) \otimes \rho_{\bar{R}}(M)\right) v \right) = (M^{-1})^T \Phi(v) \overline{M^{-1}}$$ where $\rho_{\bar{L}}(M)$ and $\rho_{\bar{R}}(M)$ are the representations of $M$ on the dual left-handed and dual right-handed Weyl spinor spaces, $(M^{-1})^T$ is the transpose of the inverse of $M$, and $\overline{M^{-1}}$ is the complex conjugate of the inverse of $M$.

For any $2 \times 2$ complex matrix $v \in \text{Mat}_2(\mathbb{C})$ and any matrix $M \in SL(2, \mathbb{C})$, the action of the tensor product of the dual left-handed Weyl spinor representation $\rho^*$ on the element $\Phi(v)$ is given by $$(\rho^*(M) \otimes \rho^*(M))(\Phi(v)) = \Phi((M^{-1})^T v M^{-1})$$ where $\Phi$ denotes the linear isomorphism `Fermion.dualLeftdualLeftToMatrix.symm` which maps matrices to the tensor product space of two dual left-handed fermions.

For any $2 \times 2$ complex matrix $v \in M_2(\mathbb{C})$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, let $\rho_L(M)$ be the left-handed Weyl fermion representation and $\rho_L^*(M)$ be its dual representation. Let $\psi^{-1} : M_2(\mathbb{C}) \to V_L \otimes V_L^*$ be the isomorphism from $2 \times 2$ matrices to the tensor product of the left-handed fermion space and its dual. Then the following identity holds: $$(\rho_L(M) \otimes \rho_L^*(M)) (\psi^{-1}(v)) = \psi^{-1}(M v M^{-1})$$

Let $M \in SL(2, \mathbb{C})$ be a special linear matrix and $v \in M_2(\mathbb{C})$ be a $2 \times 2$ complex matrix. Let $\rho_{L^*}$ and $\rho_L$ denote the dual left-handed and left-handed representations of $SL(2, \mathbb{C})$ respectively. Let $\Phi: L^* \otimes L \to M_2(\mathbb{C})$ be the isomorphism `dualLeftLeftToMatrix`. The action of the tensor product representation $\rho_{L^*} \otimes \rho_L$ on the element $\Phi^{-1}(v)$ is given by: $$(\rho_{L^*}(M) \otimes \rho_L(M))(\Phi^{-1}(v)) = \Phi^{-1}((M^{-1})^T v M^T)$$

For any $2 \times 2$ complex matrix $v \in \text{Mat}_{2 \times 2}(\mathbb{C})$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, let $\rho_R^*(M)$ denote the dual right-handed representation of $SL(2, \mathbb{C})$. Let $\Phi^{-1}$ denote the isomorphism mapping a matrix to the tensor product of two dual right-handed Weyl fermion spaces. The theorem states that: $$(\rho_R^*(M) \otimes \rho_R^*(M))(\Phi^{-1}(v)) = \Phi^{-1}\left( (M^\dagger)^{-1} v ((M^\dagger)^{-1})^\top \right)$$ where $M^\dagger$ is the conjugate transpose of $M$, and $\top$ denotes the transpose operation.

Let $V_R$ be the space of right-handed Weyl fermions and $V_R^*$ be its dual space. Let $\Phi^{-1} : \text{Mat}_2(\mathbb{C}) \to V_R \otimes V_R^*$ be the isomorphism that maps a $2 \times 2$ complex matrix to an element of the tensor product space. For any matrix $v \in \text{Mat}_2(\mathbb{C})$ and any $M \in SL(2, \mathbb{C})$, the action of the tensor product representation $\rho_R(M) \otimes \rho_R^*(M)$ is given by: $$(\rho_R(M) \otimes \rho_R^*(M)) (\Phi^{-1}(v)) = \Phi^{-1}(\bar{M} v \bar{M}^{-1})$$ where $\rho_R$ is the right-handed representation of $SL(2, \mathbb{C})$, $\rho_R^*$ is its dual representation, and $\bar{M}$ denotes the complex conjugate of the matrix $M$.

For any $2 \times 2$ complex matrix $v \in \mathbb{C}^{2 \times 2}$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the action of the tensor product of the dual right-handed Weyl representation $\rho_R^*$ and the right-handed Weyl representation $\rho_R$ on the spinor tensor $\Psi(v)$ is given by $$(\rho_R^*(M) \otimes \rho_R(M)) (\Psi(v)) = \Psi((M^\dagger)^{-1} v M^\dagger)$$ where $\Psi$ is the isomorphism mapping $2 \times 2$ matrices to the tensor product space of dual and standard right-handed Weyl fermions (the inverse of `Fermion.dualRightRightToMatrix`), and $M^\dagger$ denotes the conjugate transpose of $M$.

For any $2 \times 2$ complex matrix $v \in \text{Mat}_{2 \times 2}(\mathbb{C})$ and any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the action of the tensor product of the dual left-handed and dual right-handed representations of $SL(2, \mathbb{C})$ on the element corresponding to $v$ is given by: $$(\rho_{L^*}(M) \otimes \rho_{R^*}(M))(\Phi^{-1}(v)) = \Phi^{-1}\left((M^{-1})^T v \bar{M}^{-1}\right)$$ where $\rho_{L^*}$ and $\rho_{R^*}$ denote the dual left-handed and dual right-handed representations, respectively, and $\Phi^{-1}$ is the inverse of the isomorphism from the tensor product of dual Weyl spinors to the space of $2 \times 2$ matrices.

Let $v$ be a $2 \times 2$ self-adjoint complex matrix and $M \in SL(2, \mathbb{C})$. Let $\Phi$ denote the isomorphism `Fermion.dualLeftDualRightToMatrix` which maps the tensor product of a dual left-handed Weyl spinor and a dual right-handed Weyl spinor to the space of $2 \times 2$ complex matrices. The theorem states that applying the tensor product of the dual representations $\rho_{L^*}(M) \otimes \rho_{R^*}(M)$ to the element $\Phi^{-1}(v)$ in the spinor space is equivalent to applying the transformation $v \mapsto (M^T)^{-1} v ((M^T)^{-1})^\dagger$ and then mapping the result back via $\Phi^{-1}$. Mathematically, this is expressed as: $$(\rho_{L^*}(M) \otimes \rho_{R^*}(M)) (\Phi^{-1}(v)) = \Phi^{-1}((M^T)^{-1} v (M^T)^{-\dagger})$$ where $\rho_{L^*}$ and $\rho_{R^*}$ are the representations of $SL(2, \mathbb{C})$ on the dual left-handed and dual right-handed Weyl spinor spaces, respectively, and $\Phi^{-1}$ is the inverse map `dualLeftDualRightToMatrix.symm`.