Physlib.Relativity.Tensors.ComplexTensor.Weyl.Metric

The raw metric for fermions is defined as the $2 \times 2$ matrix over the complex numbers $\mathbb{C}$ given by $$\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ This matrix is used to define transitions between different representations of Weyl fermions.

Let $M \in SL(2, \mathbb{C})$ be a $2 \times 2$ complex matrix with determinant $1$. Let $\varepsilon$ denote the fermion metric matrix $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Then, multiplying $M$ on the left with $\varepsilon$ is equivalent to multiplying the inverse-transpose of $M$ on the right with $\varepsilon$, which is expressed by the identity: $$M \varepsilon = \varepsilon (M^{-1})^T$$ where $M^{-1}$ is the inverse of $M$ and $(M^{-1})^T$ is its transpose.

For any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, let $\varepsilon$ be the fermion metric matrix defined as $$\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ Then the following matrix equality holds: $$\varepsilon M = (M^{-1})^T \varepsilon$$ where $M^{-1}$ is the inverse of $M$ and $T$ denotes the transpose.

For any matrix $M \in SL(2, \mathbb{C})$, let $\varepsilon$ be the fermion metric matrix defined as $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Then the product of the complex conjugate of $M$ and the metric $\varepsilon$ is equal to the product of $\varepsilon$ and the conjugate transpose (Hermitian adjoint) of the inverse of $M$: $$\bar{M} \varepsilon = \varepsilon (M^{-1})^\dagger$$

For any matrix $M$ in the special linear group $SL(2, \mathbb{C})$, the fermion metric matrix $\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ satisfies the identity $$\varepsilon M^* = (M^{-1})^\dagger \varepsilon$$ where $M^*$ denotes the element-wise complex conjugate of $M$ and $(M^{-1})^\dagger$ denotes the conjugate transpose (Hermitian adjoint) of the inverse matrix $M^{-1}$.

This definition characterizes the metric element within the vector space of the tensor product of two left-handed Weyl fermion representations, $(L \otimes L).V$. It is defined as the image of the matrix $$-\varepsilon = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ under the inverse of the linear isomorphism $(L \otimes L).V \cong \text{Mat}_{2 \times 2}(\mathbb{C})$, where $\varepsilon$ is the standard raw fermion metric matrix. In terms of the standard basis $\{e_0, e_1\}$ of the left-handed representation space, this element is given by $-e_0 \otimes e_1 + e_1 \otimes e_0$, and it corresponds to the antisymmetric metric $\varepsilon^{ab}$ used in spinor index notation.

The left-handed Weyl fermion metric element $\varepsilon^{ab}$ in the tensor product space $V_L \otimes V_L$ can be expanded in terms of the standard basis $\{e_0, e_1\}$ of the left-handed representation space $V_L$ as: $$\varepsilon^{ab} = -e_0 \otimes e_1 + e_1 \otimes e_0$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

This definition characterizes the left-handed Weyl fermion metric $\varepsilon^{ab}$ as a morphism in the category of representations of $SL(2, \mathbb{C})$. Specifically, it is an intertwining map $\varepsilon : \mathbb{1} \to L \otimes L$ from the monoidal unit (the trivial representation on $\mathbb{C}$) to the tensor product of the left-handed fundamental representation $L$ with itself. The map is defined by sending a scalar $a \in \mathbb{C}$ to $a \cdot \text{leftMetricVal}$, where $\text{leftMetricVal}$ is the antisymmetric tensor $\varepsilon^{ab} = -e_0 \otimes e_1 + e_1 \otimes e_0$. This representation-theoretic formulation makes manifest the invariance of the spinor metric under $SL(2, \mathbb{C})$ transformations.

Let $\varepsilon: \mathbb{1} \to L \otimes L$ be the left-handed metric morphism from the monoidal unit of the representation category (the trivial representation on $\mathbb{C}$) to the tensor product of the left-handed fundamental representation $L$ of $SL(2, \mathbb{C})$ with itself. The evaluation of the linear map underlying this morphism at the unit scalar $1 \in \mathbb{C}$ is equal to the left-handed metric tensor $\varepsilon^{ab} = -e_0 \otimes e_1 + e_1 \otimes e_0$.

The value `altLeftMetricVal` is an element of the vector space of the tensor product of the "alternative" left-handed Weyl fermion representation with itself, denoted $(V_{L'} \otimes V_{L'})$. It represents the metric tensor $\epsilon_{ab}$ (often used in physics to lower indices) and is defined by mapping the antisymmetric matrix $$\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ into the tensor product space using the inverse of the linear isomorphism between $V_{L'} \otimes V_{L'}$ and $\text{Mat}_{2 \times 2}(\mathbb{C})$.

Let $\{e_0, e_1\}$ be the standard basis for the vector space of alt-left-handed Weyl fermions (represented by `altLeftBasis`). The metric tensor $\epsilon_{ab}$, denoted by `altLeftMetricVal` in the tensor product space $V_{L'} \otimes V_{L'}$, is given by the expansion: $$\text{altLeftMetricVal} = e_0 \otimes e_1 - e_1 \otimes e_0$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

This definition defines the metric tensor $\epsilon_{ab}$ as a morphism in the category of representations of $SL(2, \mathbb{C})$. Specifically, it is a morphism from the trivial representation $\mathbb{1}$ (where $SL(2, \mathbb{C})$ acts on $\mathbb{C}$) to the tensor product representation $\text{altLeftHanded} \otimes \text{altLeftHanded}$. This morphism maps the unit $1 \in \mathbb{C}$ to the antisymmetric tensor $\epsilon_{ab} = e_0 \otimes e_1 - e_1 \otimes e_0$ (represented by `altLeftMetricVal`), making manifest that this tensor is invariant under the action of $SL(2, \mathbb{C})$.

Let $\epsilon_{ab}$ be the invariant morphism `altLeftMetric` from the trivial representation $\mathbb{1}$ of $SL(2, \mathbb{C})$ to the tensor product representation $\text{altLeftHanded} \otimes \text{altLeftHanded}$. Evaluating the underlying linear map of this morphism at the scalar $1 \in \mathbb{C}$ yields the metric tensor value `altLeftMetricVal`, which is defined as $\epsilon_{ab} = e_0 \otimes e_1 - e_1 \otimes e_0$ in the tensor product space.

The value `rightMetricVal` is an element of the vector space of the tensor product of two right-handed Weyl fermion representations, $(rightHanded \otimes rightHanded).V$. It is defined as the element corresponding to the negative of the raw metric matrix $-\varepsilon = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ under the linear equivalence between the tensor product and the space of $2 \times 2$ matrices. In physics notation, this represents the metric with upper dotted indices $\epsilon^{\dot{a}\dot{b}}$.

Let $\{e_0, e_1\}$ be the standard basis for the complex vector space of right-handed Weyl fermions (denoted as `rightBasis`). The right-handed metric tensor `rightMetricVal`, commonly denoted as $\epsilon^{\dot{a}\dot{b}}$, is equal to the following expansion in the tensor product space: $$\epsilon^{\dot{a}\dot{b}} = - e_0 \otimes e_1 + e_1 \otimes e_0$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

The morphism `rightMetric` is an $SL(2, \mathbb{C})$-equivariant linear map (an intertwining map) from the trivial representation $\mathbb{1}$ to the tensor product of two right-handed Weyl fermion representations $\text{rightHanded} \otimes \text{rightHanded}$. This map identifies the invariant metric tensor $\epsilon^{\dot{a}\dot{b}}$ within the representation space. Specifically, it maps a scalar $z \in \mathbb{C}$ to the element $z \cdot \epsilon^{\dot{a}\dot{b}}$, where $\epsilon^{\dot{a}\dot{b}}$ is the metric used to raise dotted indices, corresponding to the antisymmetric matrix $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ in the standard basis.

Let $\text{rightMetric}$ be the $SL(2, \mathbb{C})$-equivariant linear map from the trivial representation $\mathbb{1}$ to the tensor product representation $\text{rightHanded} \otimes \text{rightHanded}$. Applying the underlying linear map of $\text{rightMetric}$ to the scalar $1 \in \mathbb{C}$ yields the right-handed Weyl fermion metric tensor value $\epsilon^{\dot{a}\dot{b}}$, denoted by `rightMetricVal`.

`altRightMetricVal` is an element of the tensor product space $V_{\text{altRight}} \otimes V_{\text{altRight}}$, where $V_{\text{altRight}}$ is the representation space for alternative right-handed Weyl fermions (spinors carrying dotted indices $\psi_{\dot{a}}$). It is defined by applying the inverse of the linear equivalence between the tensor product space and $2 \times 2$ matrices to the matrix $$\varepsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ In physics, this element represents the metric $\varepsilon_{\dot{a}\dot{b}}$ used to manipulate dotted indices, specifically corresponding to the tensor $e_{\dot{1}} \otimes e_{\dot{2}} - e_{\dot{2}} \otimes e_{\dot{1}}$ in terms of the representation's basis.

Let $\{\tilde{e}_0, \tilde{e}_1\}$ be the standard basis (known as `altRightBasis`) for the alternative right-handed Weyl fermion representation space $V_{\text{altRight}}$. The metric value for alternative right-handed Weyl spinors, denoted $\varepsilon_{\dot{a}\dot{b}}$ (or `altRightMetricVal`), is expanded in terms of this basis as: $$\varepsilon_{\dot{a}\dot{b}} = \tilde{e}_0 \otimes \tilde{e}_1 - \tilde{e}_1 \otimes \tilde{e}_0$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

The morphism `altRightMetric` is an intertwining map in the category of representations of $SL(2, \mathbb{C})$ over $\mathbb{C}$, denoted $\text{Rep}_\mathbb{C}(SL(2, \mathbb{C}))$. It maps from the monoidal unit object $\mathbb{1}$ (the trivial representation) to the tensor product of the alternative right-handed Weyl representation with itself, $\text{altRightHanded} \otimes \text{altRightHanded}$. Specifically, it maps a complex scalar $a \in \mathbb{C}$ to $a \cdot \varepsilon_{\dot{a}\dot{b}}$, where $\varepsilon_{\dot{a}\dot{b}}$ (represented by `altRightMetricVal`) is the invariant anti-symmetric metric tensor for dotted spinors. This morphism makes the invariance of the metric $\varepsilon_{\dot{a}\dot{b}}$ under the action of $SL(2, \mathbb{C})$ manifest.

The morphism $\varepsilon_{\dot{a}\dot{b}}$ (represented by `altRightMetric`) is an intertwining map from the monoidal unit $\mathbb{1}$ (the trivial representation $\mathbb{C}$) to the tensor product of the alternative right-handed Weyl representation with itself, $\text{altRightHanded} \otimes \text{altRightHanded}$. This theorem states that applying the underlying linear map of this morphism to the complex identity $1 \in \mathbb{C}$ yields the metric tensor value $\varepsilon_{\dot{a}\dot{b}}$ (represented by `altRightMetricVal`).

In the category of representations of $SL(2, \mathbb{C})$, the contraction of the left-handed Weyl fermion metric $\varepsilon^{ab}$ and the alt-left-handed metric $\epsilon_{bc}$ is equal to the alt-left-left unit tensor $\delta_a^c$. Specifically, when the contraction morphism (together with the necessary associators and braidings) is applied to the tensor product of the two metric values $\varepsilon^{ab} \otimes \epsilon_{bc}$, the result is the unit tensor $\delta_a^c$ (an element of the tensor product of the alt-left-handed and left-handed representation spaces).

In the category of $SL(2, \mathbb{C})$ representations for Weyl fermions, the contraction of the alt-left-handed metric tensor $\epsilon_{ab}$ (representing lower-index spinors $\psi_a$) with the left-handed metric tensor $\epsilon^{bc}$ (representing upper-index spinors $\psi^a$) results in the left-alt-left unit tensor $\delta_a^c$ (the Kronecker delta). Specifically, if one takes the tensor product of the metric values $\epsilon_{ab} \otimes \epsilon^{cd}$, contracts the second and third indices, and applies a braiding transformation to swap the remaining indices into the standard order for the unit tensor, the identity holds: $$\epsilon_{ab} \epsilon^{bc} = \delta_a^c$$ where $\epsilon_{ab}$ is the invariant metric for the alt-left-handed representation $V_{\bar{L}}$, $\epsilon^{bc}$ is the invariant metric for the left-handed representation $V_L$, and $\delta_a^c$ is the unit morphism from the trivial representation to $V_L \otimes V_{\bar{L}}$.

In the category of representations of $SL(2, \mathbb{C})$, let $\epsilon^{\dot{a}\dot{b}}$ be the invariant metric for right-handed Weyl fermions (represented by `rightMetric`) and $\epsilon_{\dot{c}\dot{d}}$ be the invariant metric for alternative right-handed Weyl fermions (represented by `altRightMetric`). This theorem states that when the contraction morphism for right-handed and alternative right-handed spinors is applied to the middle two components of the tensor product $\epsilon^{\dot{a}\dot{b}} \otimes \epsilon_{\dot{c}\dot{d}}$, and the resulting terms are reordered via the braiding isomorphism $\beta$, the resulting tensor is the unit $\delta_{\dot{d}}^{\dot{a}}$ (represented by `altRightRightUnit`). In component notation, this corresponds to the identity $\epsilon^{\dot{a}\dot{b}} \epsilon_{\dot{b}\dot{c}} = \delta^{\dot{a}}_{\dot{c}}$.

In the category of $SL(2, \mathbb{C})$ representations for Weyl fermions, let $\varepsilon_{\dot{a}\dot{b}}$ be the metric tensor for alternative right-handed spinors (represented by `altRightMetric`) and $\epsilon^{\dot{c}\dot{d}}$ be the metric tensor for right-handed spinors (represented by `rightMetric`). When these two metrics are composed via a sequence of associators, the contraction morphism `altRightContraction` (which contracts a pair of dotted indices), and braiding, the resulting tensor is equal to the right-alt-right unit tensor $\delta^{\dot{c}}_{\dot{a}}$ (represented by `rightAltRightUnit`).