Physlib.Relativity.Tensors.ComplexTensor.Weyl.Unit

The left-alt-left unit $\delta^a_a$ as an element of the underlying vector space of the tensor product of the left-handed and alt-left-handed representations of Weyl fermions.

The left-alt-left unit value `leftAltLeftUnitVal` (representing the $\delta$ tensor for Weyl fermions) can be expanded in terms of the basis vectors of the left-handed and alt-left-handed representation spaces as: $$\text{leftAltLeftUnitVal} = e_0 \otimes \bar{e}_0 + e_1 \otimes \bar{e}_1$$ where $e_0, e_1$ are the basis vectors for the left-handed representation (`leftBasis`) and $\bar{e}_0, \bar{e}_1$ are the basis vectors for the alt-left-handed representation (`altLeftBasis`).

The left-alt-left unit $\delta^a_a$ is an $SL(2, \mathbb{C})$-invariant morphism from the trivial representation $\mathbb{1}$ (the monoidal unit in the category of representations $\text{Rep}(\mathbb{C}, SL(2, \mathbb{C}))$) to the tensor product representation $V_L \otimes V_{\bar{L}}$, where $V_L$ denotes the left-handed Weyl fermion representation and $V_{\bar{L}}$ denotes the alt-left-handed representation. The morphism maps a scalar $c \in \mathbb{C}$ to the vector $c \cdot \delta^a_a$, where the invariant tensor $\delta^a_a$ is defined in terms of the basis as $e_0 \otimes \bar{e}_0 + e_1 \otimes \bar{e}_1$.

The $SL(2, \mathbb{C})$-invariant morphism $\delta^a_a: \mathbb{1} \to V_L \otimes V_{\bar{L}}$ (representing the left-alt-left unit for Weyl fermions) maps the identity element $1 \in \mathbb{C}$ in the trivial representation to the tensor value $\text{leftAltLeftUnitVal}$ in the underlying vector space of the tensor product of left-handed and alt-left-handed representations.

The alt-left-left unit $\delta_a^a$ for Weyl fermions, defined as an element of the underlying vector space of the tensor product of the alt-left-handed and left-handed representations.

The value of the alt-left-left unit $\delta_a^a$ is equal to the sum of the tensor products of the basis elements of the alt-left-handed and left-handed representations: $$\delta_a^a = e^{\text{alt}}_0 \otimes e^{\text{left}}_0 + e^{\text{alt}}_1 \otimes e^{\text{left}}_1$$ where $e^{\text{alt}}_i$ denotes the $i$-th basis element of the alt-left-handed representation and $e^{\text{left}}_i$ denotes the $i$-th basis element of the left-handed representation for $i \in \{0, 1\}$.

The **alt-left-left unit** $\delta_a^a$ is a morphism (an intertwining operator) in the category of representations of $SL(2, \mathbb{C})$, mapping from the trivial representation $\mathbb{1}$ to the tensor product representation $\text{altLeftHanded} \otimes \text{leftHanded}$. Specifically, it maps a scalar $c \in \mathbb{C}$ to the element $c \cdot \delta_a^a$, where $\delta_a^a$ is defined in the underlying vector space as the sum of the tensor products of the basis elements: $\text{altLeftBasis}_0 \otimes \text{leftBasis}_0 + \text{altLeftBasis}_1 \otimes \text{leftBasis}_1$. This map manifests the invariance of the Weyl fermion contraction under the action of $SL(2, \mathbb{C})$.

Applying the alt-left-left unit morphism $\delta_a^a$ to the scalar $1 \in \mathbb{C}$ yields the value $\delta_a^a$ as an element of the underlying vector space of the tensor product representation $\text{altLeftHanded} \otimes \text{leftHanded}$.

The right-alt-right unit $\delta^{\dot{a}}_{\dot{a}}$ as an element of the underlying vector space of the tensor product representation of right-handed and alt-right-handed Weyl fermions.

The right-alt-right unit $\delta^{\dot{a}}_{\dot{a}}$, which is an element of the tensor product of the right-handed and alt-right-handed Weyl fermion representations, is equal to the sum of the tensor products of their respective basis elements: $$\delta^{\dot{a}}_{\dot{a}} = b_{\dot{0}} \otimes \bar{b}^{\dot{0}} + b_{\dot{1}} \otimes \bar{b}^{\dot{1}}$$ where $b_{\dot{i}}$ represents the $i$-th basis vector of the right-handed Weyl fermion space (`rightBasis i`) and $\bar{b}^{\dot{i}}$ represents the $i$-th basis vector of the alt-right-handed Weyl fermion space (`altRightBasis i`).

The morphism $\delta^{\dot{a}}_{\dot{a}} : \mathbb{1} \to \text{rightHanded} \otimes \text{altRightHanded}$ is an intertwining map from the trivial representation $\mathbb{1}$ of $SL(2, \mathbb{C})$ to the tensor product of right-handed and alt-right-handed Weyl fermion representations. It maps the unit scalar $1 \in \mathbb{C}$ to the invariant tensor $\delta^{\dot{a}}_{\dot{a}}$, manifesting the invariance of this unit under the $SL(2, \mathbb{C})$ action.

Let $\delta^{\dot{a}}_{\dot{a}} : \mathbb{1} \to \text{rightHanded} \otimes \text{altRightHanded}$ be the right-alt-right unit morphism from the trivial representation $\mathbb{1}$ of $SL(2, \mathbb{C})$ to the tensor product of right-handed and alt-right-handed Weyl fermion representations. The underlying linear map of this morphism applied to the scalar $1 \in \mathbb{C}$ is equal to the unit tensor $\delta^{\dot{a}}_{\dot{a}}$ in the tensor product vector space.

The alt-right-right unit $\delta_{\dot{a}}^{\dot{a}}$ as an element of the vector space $(\text{altRightHanded} \otimes \text{rightHanded}).V$.

The value of the alt-right-right unit $\delta_{\dot{a}}^{\dot{a}}$ can be expanded in terms of the basis elements of the alt-right-handed space ($b^{\dot{a}}$) and the right-handed space ($b_{\dot{a}}$) as: $$\delta_{\dot{a}}^{\dot{a}} = b^{\dot{0}} \otimes b_{\dot{0}} + b^{\dot{1}} \otimes b_{\dot{1}}$$ where $b^{\dot{i}}$ corresponds to `altRightBasis i` and $b_{\dot{i}}$ corresponds to `rightBasis i`.

The alt-right-right unit $\delta_{\dot{a}}^{\dot{a}}$ is defined as a morphism $\mathbb{1} \to \text{altRightHanded} \otimes \text{rightHanded}$ in the category of representations of $SL(2, \mathbb{C})$. It maps a scalar $a \in \mathbb{C}$ from the trivial representation $\mathbb{1}$ to the element $a \cdot \text{altRightRightUnitVal}$ in the tensor product of the alt-right-handed and right-handed Weyl fermion representations. This morphism represents the $SL(2, \mathbb{C})$-invariant tensor $\delta_{\dot{a}}^{\dot{a}}$.

The morphism $\delta_{\dot{a}}^{\dot{a}} : \mathbb{1} \to \text{altRightHanded} \otimes \text{rightHanded}$ in the category of representations of $SL(2, \mathbb{C})$ maps the identity element $1 \in \mathbb{C}$ of the trivial representation $\mathbb{1}$ to the tensor value $\delta_{\dot{a}}^{\dot{a}}$ in the vector space $(\text{altRightHanded} \otimes \text{rightHanded}).V$.

For any vector $x$ in the left-handed Weyl fermion representation $V_L$, let $\delta_a^a \in V_{\bar{L}} \otimes V_L$ be the alt-left-left unit (the $SL(2, \mathbb{C})$-invariant tensor). The contraction of $x$ with the first component of $\delta_a^a$ using the left-alt contraction $C: V_L \otimes V_{\bar{L}} \to \mathbb{C}$ returns the original vector $x$. In the language of monoidal categories, this identity is expressed as: $$\lambda_{V_L} \left( (C \otimes \text{id}_{V_L}) \left( \alpha_{V_L, V_{\bar{L}}, V_L}^{-1} (x \otimes \delta_a^a) \right) \right) = x$$ where $\alpha^{-1}$ is the inverse associator and $\lambda$ is the left unitor in the category of representations $\text{Rep}(\mathbb{C}, SL(2, \mathbb{C}))$.

For any vector $x$ in the alt-left-handed Weyl fermion representation $V_{\bar{L}}$, let $\delta \in V_L \otimes V_{\bar{L}}$ be the left-alt-left unit value (the $SL(2, \mathbb{C})$-invariant tensor $\delta^a_a$). The contraction of $x$ with the first component of $\delta$ using the alt-left contraction $C: V_{\bar{L}} \otimes V_L \to \mathbb{1}$ returns the original vector $x$. In the language of monoidal categories, this identity is expressed as: $$\lambda_{V_{\bar{L}}} \left( (C \otimes \text{id}_{V_{\bar{L}}}) \left( \alpha_{V_{\bar{L}}, V_L, V_{\bar{L}}}^{-1} (x \otimes \delta) \right) \right) = x$$ where $\alpha^{-1}$ is the inverse associator and $\lambda$ is the left unitor in the category of representations $\text{Rep}(\mathbb{C}, SL(2, \mathbb{C}))$.

For any vector $x$ in the right-handed Weyl fermion representation $V_R$, let $\delta_{\dot{a}}^{\dot{a}} \in V_{\dot{R}} \otimes V_R$ be the **alt-right-right unit** (the $SL(2, \mathbb{C})$-invariant tensor). The contraction of $x$ with the first component of $\delta_{\dot{a}}^{\dot{a}}$ using the right-alt contraction $C: V_R \otimes V_{\dot{R}} \to \mathbb{C}$ returns the original vector $x$. In the language of monoidal categories, this identity is expressed as: $$\lambda_{V_R} \left( (C \otimes \text{id}_{V_R}) \left( \alpha_{V_R, V_{\dot{R}}, V_R}^{-1} (x \otimes \delta_{\dot{a}}^{\dot{a}}) \right) \right) = x$$ where $\alpha^{-1}$ is the inverse associator and $\lambda$ is the left unitor in the category of representations $\text{Rep}(\mathbb{C}, SL(2, \mathbb{C}))$.

For any vector $x$ in the alt-right-handed Weyl fermion representation $V_{\dot{R}}$, let $\delta^{\dot{a}}_{\dot{a}} \in V_R \otimes V_{\dot{R}}$ be the **right-alt-right unit** (the $SL(2, \mathbb{C})$-invariant tensor). The contraction of $x$ with the first component of $\delta^{\dot{a}}_{\dot{a}}$ using the alt-right contraction $C: V_{\dot{R}} \otimes V_R \to \mathbb{C}$ returns the original vector $x$. In the language of monoidal categories, this identity is expressed as: $$\lambda_{V_{\dot{R}}} \left( (C \otimes \text{id}_{V_{\dot{R}}}) \left( \alpha_{V_{\dot{R}}, V_R, V_{\dot{R}}}^{-1} (x \otimes \delta^{\dot{a}}_{\dot{a}}) \right) \right) = x$$ where $\alpha^{-1}$ is the inverse associator and $\lambda$ is the left unitor in the category of representations $\text{Rep}(\mathbb{C}, SL(2, \mathbb{C}))$.

The **alt-left-left unit** $\delta_a^a$ (the invariant tensor in $V_{\bar{L}} \otimes V_L$ for Weyl fermions) is equal to the image of the **left-alt-left unit** $\delta^a_a$ (the invariant tensor in $V_L \otimes V_{\bar{L}}$) under the braiding isomorphism $\beta_{V_L, V_{\bar{L}}}: V_L \otimes V_{\bar{L}} \to V_{\bar{L}} \otimes V_L$ of the category of $SL(2, \mathbb{C})$ representations. In terms of basis elements, this confirms that swapping the components of $\sum_i e_i \otimes \bar{e}_i$ yields $\sum_i \bar{e}_i \otimes e_i$.

The **left-alt-left unit** $\delta^a_a$ (the invariant tensor in $V_L \otimes V_{\bar{L}}$ for Weyl fermions) is equal to the image of the **alt-left-left unit** $\delta_a^a$ (the invariant tensor in $V_{\bar{L}} \otimes V_L$) under the braiding isomorphism $\beta_{V_{\bar{L}}, V_L}: V_{\bar{L}} \otimes V_L \to V_L \otimes V_{\bar{L}}$ in the category of $SL(2, \mathbb{C})$ representations. This relationship can be expressed as: $$\delta^a_a = \beta(\delta_a^a)$$ where $V_L$ denotes the left-handed representation and $V_{\bar{L}}$ denotes the alt-left-handed representation.

In the category of $SL(2, \mathbb{C})$ representations, the **alt-right-right unit** tensor $\delta_{\dot{a}}^{\dot{a}} \in \text{altRightHanded} \otimes \text{rightHanded}$ is equal to the image of the **right-alt-right unit** tensor $\delta^{\dot{a}}_{\dot{a}} \in \text{rightHanded} \otimes \text{altRightHanded}$ under the braiding isomorphism (symmetry) $\beta$: $$\delta_{\dot{a}}^{\dot{a}} = \beta(\delta^{\dot{a}}_{\dot{a}})$$ where $\beta : \text{rightHanded} \otimes \text{altRightHanded} \to \text{altRightHanded} \otimes \text{rightHanded}$ is the morphism that swaps the factors of the tensor product.

In the category of $SL(2, \mathbb{C})$ representations, the unit tensor $\delta^{\dot{a}}_{\dot{a}} \in \text{rightHanded} \otimes \text{altRightHanded}$ is equal to the image of the unit tensor $\delta_{\dot{a}}^{\dot{a}} \in \text{altRightHanded} \otimes \text{rightHanded}$ under the symmetry isomorphism (braiding) $\beta$: $$\delta^{\dot{a}}_{\dot{a}} = \beta(\delta_{\dot{a}}^{\dot{a}})$$ where $\beta : \text{altRightHanded} \otimes \text{rightHanded} \cong \text{rightHanded} \otimes \text{altRightHanded}$ is the morphism that swaps the factors of the tensor product.