Physlib.Relativity.Fermions.Weyl.Metric

Let $\{e_0, e_1\}$ be the standard basis for the complex vector space of left-handed Weyl fermions. The left-handed Weyl fermion metric $\varepsilon \in L \otimes L$ is equal to the tensor product expansion $e_1 \otimes e_0 - e_0 \otimes e_1$.

The metric tensor $\varepsilon$ for dual left-handed Weyl fermions, represented as an element of the tensor product space $\text{DualLeftHandedWeyl} \otimes_{\mathbb{C}} \text{DualLeftHandedWeyl}$. This tensor is used to transition between different representations of left-handed Weyl fermions and is characterized by the antisymmetric matrix $\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

Let $\{e^0, e^1\}$ be the dual basis for the space of left-handed Weyl fermions. The dual left-handed metric, often denoted by $\epsilon$, is given by the expansion: $$\epsilon = e^0 \otimes e^1 - e^1 \otimes e^0$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

The $SL(2, \mathbb{C})$-intertwining map from the tensor product of two dual left-handed Weyl representations to the trivial representation $\mathbb{C}$. This map represents the invariant metric, often denoted as $\varepsilon^{ab}$, used to contract two dual left-handed Weyl spinors into a Lorentz scalar.

The dual left-handed metric $\epsilon$ for Weyl fermions, when applied to the unit element $1$, is equal to the predefined metric value $\epsilon_{\text{val}}$ (represented by `dualLeftMetricVal`).

Let $\{e_0, e_1\}$ be the standard basis for the complex vector space of right-handed Weyl fermions. The right-handed metric tensor $\epsilon^{\dot{a}\dot{b}}$ is given by the expansion $\epsilon^{\dot{a}\dot{b}} = e_1 \otimes e_0 - e_0 \otimes e_1$.

The definition represents the metric tensor $\varepsilon$ as an element of the tensor product of the dual right-handed Weyl spinor space with itself, denoted by $\text{DualRightHandedWeyl} \otimes_{\mathbb{C}} \text{DualRightHandedWeyl}$. This tensor is used to facilitate transitions between right-handed and alt-right-handed Weyl fermion representations, typically corresponding to the antisymmetric matrix $\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

Let $e^0$ and $e^1$ denote the basis elements of the dual space for right-handed Weyl fermions, corresponding to `dualRightBasis 0` and `dualRightBasis 1`. The dual right-handed metric `dualRightMetricVal` is equal to the antisymmetric tensor product $e^0 \otimes e^1 - e^1 \otimes e^0$, where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

This definition is the $SL(2, \mathbb{C})$-intertwining map from the tensor product of two dual right-handed Weyl fermion representations to the trivial representation over $\mathbb{C}$. In physics, this corresponds to the invariant metric tensor $\epsilon$ used to contract indices of dual right-handed spinors.

The dual right-handed metric for Weyl fermions (denoted as $\epsilon$), when evaluated at $1$, is equal to its predefined value `dualRightMetricVal`. This metric is used to transform between right-handed and alt-right-handed Weyl fermion representations.

In the theory of Weyl fermions, let $L$ be the representation space of left-handed spinors and $L^*$ be its dual. Let $\varepsilon \in L \otimes L$ be the left-handed metric and $\varepsilon^* \in L^* \otimes L^*$ be the dual left-handed metric. If we form the tensor product $\varepsilon \otimes \varepsilon^*$, contract the second component of $\varepsilon$ with the first component of $\varepsilon^*$, and then swap the remaining $L$ and $L^*$ components, the resulting tensor in $L^* \otimes L$ is equal to the canonical unit (the element corresponding to the identity endomorphism $\text{id}_L$).

Let $L$ be the left-handed Weyl fermion representation space and $L^*$ be its dual. Let $\varepsilon^{bc} \in L \otimes L$ be the left-handed metric and $\varepsilon_{ab} \in L^* \otimes L^*$ be the dual left-handed metric. The theorem states that contracting the second component of the dual metric with the first component of the metric, followed by the canonical braiding (symmetry) of the remaining factors, results in the canonical unit element $\delta^c_a \in L \otimes L^*$. In index notation, this corresponds to the identity $\varepsilon_{ab} \varepsilon^{bc} = \delta_a^c$.