Physlib.Relativity.Fermions.Weyl.Contraction

This definition provides a complex-bilinear map that takes a left-handed Weyl fermion $\psi \in \text{LeftHandedWeyl}$ and a dual left-handed Weyl fermion $\phi \in \text{DualLeftHandedWeyl}$ to produce a complex scalar in $\mathbb{C}$. This represents the natural contraction (or pairing) between the space of left-handed Weyl spinors and its dual space.

The definition `Fermion.dualLeftBi` is a $\mathbb{C}$-bilinear map that takes a dual left-handed Weyl fermion and a left-handed Weyl fermion and returns a complex number $\mathbb{C}$, representing the contraction between the two.

This definition represents a $\mathbb{C}$-bilinear map that takes a right-handed Weyl fermion $\psi \in \text{RightHandedWeyl}$ and a dual right-handed Weyl fermion $\phi \in \text{DualRightHandedWeyl}$ to a complex number $c \in \mathbb{C}$. This map corresponds to the natural contraction between a right-handed spinor and an element of its dual space.

The definition `Fermion.dualRightBi` represents a $\mathbb{C}$-bilinear map that takes a dual right-handed Weyl fermion and a right-handed Weyl fermion to produce a complex number. This map corresponds to the natural pairing (contraction) between the dual space and the space of right-handed Weyl spinors. Given an element $\phi$ in the dual space `DualRightHandedWeyl` and an element $\psi$ in `RightHandedWeyl`, the map computes the scalar value resulting from their contraction.

Let $V_L$ be the left-handed Weyl fermion representation and $V_L^*$ be its dual representation. `Fermion.leftDualContraction` is the intertwining map from the tensor product $V_L \otimes V_L^*$ to the trivial representation $\mathbb{C}$ of the special linear group $SL(2, \mathbb{C})$. This map represents the invariant contraction between a left-handed spinor and a dual left-handed spinor.

For any left-handed Weyl fermion $\psi$ and any dual left-handed Weyl fermion $\phi$, the contraction of their tensor product $\psi \otimes_{\mathbb{C}} \phi$ is equal to the dot product of their representations as vectors in $\mathbb{C}^2$, denoted $\psi \cdot \phi$.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for left-handed Weyl fermions and $\{e^j\}_{j \in \{0, 1\}}$ be the corresponding dual basis. For any indices $i, j \in \{0, 1\}$, the contraction of the tensor product $e_i \otimes e^j$ is given by the Kronecker delta: $$\text{leftDualContraction}(e_i \otimes e^j) = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

This definition represents the $\text{SL}(2, \mathbb{C})$-invariant linear map (intertwining map) from the tensor product of the dual left-handed Weyl fermion representation and the left-handed Weyl fermion representation to the trivial complex representation. It corresponds to the natural pairing or contraction between a dual left-handed spinor and a left-handed spinor.

For any dual left-handed Weyl spinor $\phi$ and left-handed Weyl spinor $\psi$, the dual left-handed contraction of their tensor product $\phi \otimes_{\mathbb{C}} \psi$ is equal to the dot product of their representations as two-dimensional complex vectors, denoted by $\phi \cdot \psi$.

For any indices $i, j \in \{0, 1\}$, the contraction of the tensor product of the $i$-th dual basis element $e^i$ and the $j$-th standard basis element $e_j$ for left-handed Weyl fermions is given by the Kronecker delta $\delta_{ij}$: $$\text{dualLeftContraction}(e^i \otimes e_j) = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

The intertwining map (equivariant linear map) from the tensor product of the right-handed Weyl fermion representation and its dual representation to the trivial representation of the special linear group $SL(2, \mathbb{C})$ on $\mathbb{C}$. This map defines the invariant contraction between a right-handed Weyl fermion and its dual.

For any right-handed Weyl fermion $\psi$ and dual right-handed Weyl fermion $\phi$, the contraction of their tensor product $\psi \otimes_{\mathbb{C}} \phi$ is equal to the dot product of their representations as vectors in $\mathbb{C}^2$: $$\text{rightDualContraction}(\psi \otimes_{\mathbb{C}} \phi) = \psi \cdot \phi$$ where $\psi \cdot \phi$ denotes the standard dot product of the two-component complex vectors corresponding to $\psi$ and $\phi$.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the space of right-handed Weyl fermions (represented by `Fermion.rightBasis`) and $\{e^j\}_{j \in \{0, 1\}}$ be the corresponding dual basis (represented by `Fermion.dualRightBasis`). For any indices $i, j \in \{0, 1\}$, the contraction of the tensor product $e_i \otimes e^j$ is equal to the Kronecker delta $\delta_{ij}$, i.e., it is $1$ if $i = j$ and $0$ otherwise.

The definition represents the intertwining map (invariant contraction) from the tensor product of the dual right-handed representation and the right-handed representation of the special linear group $SL(2, \mathbb{C})$ to the trivial representation on $\mathbb{C}$. In the language of Weyl fermions, this is the natural pairing between a dual right-handed spinor $\phi \in V_R^*$ and a right-handed spinor $\psi \in V_R$, which is invariant under the action of $SL(2, \mathbb{C})$.

For any dual right-handed Weyl fermion $\phi$ and right-handed Weyl fermion $\psi$, the contraction of their tensor product $\phi \otimes_{\mathbb{C}} \psi$ is equal to the dot product of their corresponding vector representations in $\mathbb{C}^2$, denoted as $\phi \cdot \psi$.

Let $\{e_0, e_1\}$ be the standard basis for the complex vector space of right-handed Weyl fermions (the `rightBasis`) and let $\{\epsilon^0, \epsilon^1\}$ be the corresponding dual basis (the `dualRightBasis`). For any indices $i, j \in \{0, 1\}$, the contraction of the tensor product $\epsilon^i \otimes e_j$ is given by the Kronecker delta: $$\text{dualRightContraction}(\epsilon^i \otimes e_j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

For any left-handed Weyl fermion $\psi$ and dual left-handed Weyl fermion $\phi$, the left-dual contraction of their tensor product $\psi \otimes_{\mathbb{C}} \phi$ is equal to the dual-left contraction of $\phi \otimes_{\mathbb{C}} \psi$, where $\otimes_{\mathbb{C}}$ denotes the tensor product over the complex numbers.

For any dual left-handed Weyl spinor $\phi$ and left-handed Weyl spinor $\psi$, the contraction of their tensor product over the complex numbers $\phi \otimes_{\mathbb{C}} \psi$ is equal to the contraction of the reversed tensor product $\psi \otimes_{\mathbb{C}} \phi$.

For any right-handed Weyl fermion $\psi$ and any dual right-handed Weyl fermion $\phi$, the contraction of their tensor product $\psi \otimes_{\mathbb{C}} \phi$ under the `rightDualContraction` map is equal to the contraction of the tensor product $\phi \otimes_{\mathbb{C}} \psi$ under the `dualRightContraction` map.

For every dual right-handed Weyl fermion $\phi$ and every right-handed Weyl fermion $\psi$, the dual-right contraction of the tensor product $\phi \otimes_{\mathbb{C}} \psi$ is equal to the right-dual contraction of the tensor product $\psi \otimes_{\mathbb{C}} \phi$.