Physlib.Relativity.Tensors.ComplexTensor.Weyl.Contraction

This definition establishes a $\mathbb{C}$-bilinear map from the space of left-handed Weyl fermions and the space of alt-left-handed Weyl fermions to the complex numbers $\mathbb{C}$. Given a left-handed Weyl fermion $\psi$ (transforming in the fundamental representation of $SL(2, \mathbb{C})$, often denoted with an upper index $\psi^a$) and an alt-left-handed Weyl fermion $\phi$ (transforming via $(M^{-1})^T$, often denoted with a lower index $\phi_a$), the map computes their contraction as the standard dot product of their vector representations in $\mathbb{C}^2$: $$B(\psi, \phi) = \sum_{i=1}^2 \psi_i \phi_i$$ In physics notation, this corresponds to the invariant scalar $\psi^a \phi_a$.

This definition describes a $\mathbb{C}$-bilinear map that performs the contraction between an alt-left-handed Weyl fermion $\psi$ and a left-handed Weyl fermion $\phi$. Given $\psi$ in the module `altLeftHanded` (representing a spinor with a lower index $\psi_a$) and $\phi$ in the module `leftHanded` (representing a spinor with an upper index $\phi^a$), the map computes the standard dot product of their vector representations in $\mathbb{C}^2$: \[ (\psi, \phi) \mapsto \sum_{a=1}^2 \psi_a \phi^a \] where $\psi_a$ and $\phi^a$ are the components of the spinors in the standard basis of $\mathbb{C}^2$.

This definition defines a $\mathbb{C}$-bilinear map $B: \text{rightHanded} \times \text{altRightHanded} \to \mathbb{C}$ that represents the contraction of a right-handed Weyl fermion with an alternative right-handed Weyl fermion. For a right-handed spinor $\psi$ and an alternative right-handed spinor $\phi$, the map is given by the standard dot product of their underlying vectors in $\mathbb{C}^2$: \[ B(\psi, \phi) = \sum_{i=1}^2 \psi_i \phi_i \] where $\psi_i$ and $\phi_i$ are the components of the spinors in their respective representations. In physics notation, this corresponds to the contraction of dotted indices $\psi^{\dot{a}} \phi_{\dot{a}}$.

This bilinear map $B: \text{altRightHanded} \times \text{rightHanded} \to \mathbb{C}$ represents the contraction between an alternative right-handed Weyl fermion $\psi$ and a right-handed Weyl fermion $\phi$. Given the representation of $\psi$ and $\phi$ as complex vectors $\mathbf{v}_\psi, \mathbf{v}_\phi \in \mathbb{C}^2$, the map is defined by the standard dot product: $$B(\psi, \phi) = \mathbf{v}_\psi \cdot \mathbf{v}_\phi = \sum_{i=1}^2 (\mathbf{v}_\psi)_i (\mathbf{v}_\phi)_i$$ In the language of physics, this corresponds to the Lorentz-invariant contraction of dotted spinor indices $\psi_{\dot{a}} \phi^{\dot{a}}$.

This definition defines a morphism in the category of representations of $SL(2, \mathbb{C})$ from the tensor product $\text{leftHanded} \otimes \text{altLeftHanded}$ to the trivial representation $\mathbb{C}$. The map is the linear extension of the dot product between a left-handed Weyl fermion $\psi$ and an alt-left-handed Weyl fermion $\phi$. Specifically, given vectors $\psi, \phi \in \mathbb{C}^2$ representing the fermions in the standard basis, the map is given by: $$\psi \otimes \phi \mapsto \sum_{a=1}^2 \psi^a \phi_a$$ Physically, this represents the Lorentz-invariant contraction of a left-handed Weyl fermion (with an upper index $\psi^a$) and an alt-left-handed Weyl fermion (with a lower index $\phi_a$).

For a left-handed Weyl fermion $\psi$ and an alt-left-handed Weyl fermion $\phi$, the contraction morphism $\text{leftAltContraction}$ applied to their tensor product $\psi \otimes \phi$ is equal to the standard dot product of their vector representations in $\mathbb{C}^2$: $$\text{leftAltContraction}(\psi \otimes \phi) = \mathbf{v}_\psi \cdot \mathbf{v}_\phi = \sum_{a=1}^2 \psi^a \phi_a$$ where $\mathbf{v}_\psi$ and $\mathbf{v}_\phi$ are the vectors in $\mathbb{C}^2$ corresponding to $\psi$ and $\phi$ respectively.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the vector space of left-handed Weyl fermions (representing components $\psi^a$) and $\{\tilde{e}_j\}_{j \in \{0, 1\}}$ be the standard basis for the vector space of alt-left-handed Weyl fermions (representing components $\phi_a$). For any indices $i, j \in \{0, 1\}$, the contraction morphism $\text{leftAltContraction}$ applied to the tensor product $e_i \otimes \tilde{e}_j$ is equal to the Kronecker delta $\delta_{ij}$: $$\text{leftAltContraction}(e_i \otimes \tilde{e}_j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

This definition describes an $SL(2, \mathbb{C})$-equivariant linear map (a morphism in the category of representations) from the tensor product of the alt-left-handed representation $\text{altLeftHanded}$ and the left-handed representation $\text{leftHanded}$ to the trivial representation $\mathbb{C}$. Physically, this map represents the invariant contraction of an alt-left-handed Weyl fermion $\phi$ (with a lower index $\phi_a$) and a left-handed Weyl fermion $\psi$ (with an upper index $\psi^a$). The map is defined by the standard dot product of their components in $\mathbb{C}^2$: \[ \phi \otimes \psi \mapsto \sum_{a=1}^2 \phi_a \psi^a \] The definition ensures that this operation is an intertwining map, meaning the contraction is invariant under the action of $SL(2, \mathbb{C})$.

Let $\phi$ be an alt-left-handed Weyl fermion and $\psi$ be a left-handed Weyl fermion. The evaluation of the $SL(2, \mathbb{C})$-invariant contraction map $\text{altLeftContraction}$ on the tensor product $\phi \otimes \psi$ is equal to the standard dot product of their vector components in $\mathbb{C}^2$: \[ \text{altLeftContraction}(\phi \otimes \psi) = \mathbf{\phi} \cdot \mathbf{\psi} = \sum_{a=1}^2 \phi_a \psi^a \] where $\mathbf{\phi}$ and $\mathbf{\psi}$ are the representations of $\phi$ and $\psi$ as vectors in $\mathbb{C}^2$.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the space of alt-left-handed Weyl fermions (corresponding to spinors with lower indices $\psi_a$) and $\{f_j\}_{j \in \{0, 1\}}$ be the standard basis for the space of left-handed Weyl fermions (corresponding to spinors with upper indices $\psi^a$). The $SL(2, \mathbb{C})$-invariant contraction map $C$ evaluated on the tensor product of these basis elements is given by the Kronecker delta: \[ C(e_i \otimes f_j) = \delta_{ij} \] where $\delta_{ij}$ is 1 if $i = j$ and 0 otherwise.

This definition constructs an $SL(2, \mathbb{C})$-equivariant linear map (a morphism in the category of representations) from the tensor product of the right-handed Weyl representation and its dual-like counterpart to the trivial representation $\mathbb{C}$. Given a right-handed Weyl fermion $\psi$ (with components $\psi^{\dot{a}}$) and an alternative right-handed Weyl fermion $\phi$ (with components $\phi_{\dot{a}}$), the map sends the tensor $\psi \otimes \phi$ to the complex scalar obtained by the standard dot product of their components: \[ \psi \otimes \phi \mapsto \sum_{a=1}^2 \psi^{\dot{a}} \phi_{\dot{a}} \] In physics notation, this represents the invariant contraction of dotted indices $\psi^{\dot{a}} \phi_{\dot{a}}$.

For any right-handed Weyl fermion $\psi$ and any alternative right-handed Weyl fermion $\phi$, the $SL(2, \mathbb{C})$-equivariant contraction map applied to their tensor product $\psi \otimes \phi$ is equal to the standard dot product of their underlying complex vector components: \[ \text{rightAltContraction}(\psi \otimes \phi) = \sum_{i=1}^2 \psi_i \phi_i \] where $\psi_i$ and $\phi_i$ are the components of the spinors represented as vectors in $\mathbb{C}^2$.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the right-handed Weyl fermion representation (where spinors carry upper dotted indices $\psi^{\dot{a}}$) and $\{f_j\}_{j \in \{0, 1\}}$ be the standard basis for the alternative right-handed Weyl fermion representation (where spinors carry lower dotted indices $\phi_{\dot{a}}$). The Lorentz-invariant contraction of the tensor product of these basis vectors $e_i \otimes f_j$ is given by the Kronecker delta $\delta_{ij}$: \[ \text{rightAltContraction}(e_i \otimes f_j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \] This confirms that the contraction $\psi^{\dot{a}} \phi_{\dot{a}}$ correctly pairs the corresponding components of the basis vectors.

This definition constructs a morphism in the category of representations of $SL(2, \mathbb{C})$ from the tensor product of the alternative right-handed Weyl representation and the right-handed Weyl representation to the trivial representation $\mathbb{C}$. Specifically, for an alternative right-handed spinor $\phi$ (with components $\phi_{\dot{a}}$) and a right-handed spinor $\psi$ (with components $\psi^{\dot{a}}$), the map is defined by the dot product of their vector representations: $$\phi \otimes \psi \mapsto \sum_{i=1}^2 \phi_i \psi_i$$ This corresponds to the Lorentz-invariant contraction of dotted indices in physics, denoted as $\phi_{\dot{a}} \psi^{\dot{a}}$.

For any alternative right-handed Weyl spinor $\phi$ (representing a spinor with a lower dotted index $\phi_{\dot{a}}$) and any right-handed Weyl spinor $\psi$ (representing a spinor with an upper dotted index $\psi^{\dot{a}}$), the Lorentz-invariant contraction morphism applied to their tensor product $\phi \otimes \psi$ is equal to the standard dot product of their underlying vector representations in $\mathbb{C}^2$: $$\text{altRightContraction}(\phi \otimes \psi) = \sum_{i=1}^2 \phi_i \psi_i$$ where $\phi_i$ and $\psi_i$ are the components of the complex vectors in $\mathbb{C}^2$ corresponding to $\phi$ and $\psi$ respectively.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the alternative right-handed Weyl fermion representation (`altRightBasis`) and let $\{f_j\}_{j \in \{0, 1\}}$ be the standard basis for the right-handed Weyl fermion representation (`rightBasis`). The Lorentz-invariant contraction morphism $\text{altRightContraction}$ evaluated on the tensor product of these basis vectors $e_i \otimes f_j$ is equal to the Kronecker delta $\delta_{ij}$: $$\text{altRightContraction}(e_i \otimes f_j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$ This result reflects that the contraction of a spinor with lower dotted indices $\phi_{\dot{a}}$ and a spinor with upper dotted indices $\psi^{\dot{a}}$ acts as a dual pairing on their respective standard bases.

Let $\psi$ be a left-handed Weyl fermion (representing a spinor with an upper index $\psi^a$) and $\phi$ be an alt-left-handed Weyl fermion (representing a spinor with a lower index $\phi_a$). The contraction morphism $\text{leftAltContraction}$ evaluated on the tensor product $\psi \otimes \phi$ is equal to the contraction morphism $\text{altLeftContraction}$ evaluated on the reversed tensor product $\phi \otimes \psi$: \[ \text{leftAltContraction}(\psi \otimes \phi) = \text{altLeftContraction}(\phi \otimes \psi) \] This equality corresponds to the symmetry of the scalar product $\sum_{a=1}^2 \psi^a \phi_a = \sum_{a=1}^2 \phi_a \psi^a$.

Let $\phi$ be an alt-left-handed Weyl fermion (representing a spinor with a lower index $\phi_a$) and $\psi$ be a left-handed Weyl fermion (representing a spinor with an upper index $\psi^a$). The contraction morphism $\text{altLeftContraction}$ evaluated on the tensor product $\phi \otimes \psi$ is equal to the contraction morphism $\text{leftAltContraction}$ evaluated on the reversed tensor product $\psi \otimes \phi$: \[ \text{altLeftContraction}(\phi \otimes \psi) = \text{leftAltContraction}(\psi \otimes \phi) \] This equality corresponds to the symmetry of the scalar product $\sum_{a=1}^2 \phi_a \psi^a = \sum_{a=1}^2 \psi^a \phi_a$.

For any right-handed Weyl spinor $\psi$ (representing a spinor with an upper dotted index $\psi^{\dot{a}}$) and any alternative right-handed Weyl spinor $\phi$ (representing a spinor with a lower dotted index $\phi_{\dot{a}}$), the Lorentz-invariant contraction of the tensor product $\psi \otimes \phi$ is equal to the contraction of $\phi \otimes \psi$. Specifically: $$\text{rightAltContraction}(\psi \otimes \phi) = \text{altRightContraction}(\phi \otimes \psi)$$ In physics notation, this corresponds to the symmetry of the contraction of dotted indices: $$\psi^{\dot{a}} \phi_{\dot{a}} = \phi_{\dot{a}} \psi^{\dot{a}}$$

For any alternative right-handed Weyl spinor $\phi$ (representing a spinor with a lower dotted index $\phi_{\dot{a}}$) and any right-handed Weyl spinor $\psi$ (representing a spinor with an upper dotted index $\psi^{\dot{a}}$), the Lorentz-invariant contraction of the tensor product $\phi \otimes \psi$ is equal to the contraction of the tensor product $\psi \otimes \phi$. Specifically, the identity holds: $$\text{altRightContraction}(\phi \otimes \psi) = \text{rightAltContraction}(\psi \otimes \phi)$$ In physics notation, this corresponds to the symmetry of the contraction of dotted indices: $$\phi_{\dot{a}} \psi^{\dot{a}} = \psi^{\dot{a}} \phi_{\dot{a}}$$