Physlib.Relativity.Fermions.Weyl.Unit

This definition represents the unit element (or identity tensor) within the tensor product $S_L \otimes S_L^*$, where $S_L$ is the space of left-handed Weyl spinors and $S_L^*$ is its dual. In physics literature, this is typically denoted by the Kronecker delta $\delta_\alpha^\beta$ and corresponds to the canonical isomorphism between $S_L \otimes S_L^*$ and the space of endomorphisms $\text{End}(S_L)$.

Let $\{e_0, e_1\}$ denote the standard basis for the complex vector space of left-handed Weyl fermions $V_L$, and let $\{e^0, e^1\}$ denote the corresponding dual basis for $V_L^*$. The unit tensor for left-handed fermions (often denoted by $\delta$ and belonging to the tensor product space $V_L \otimes_{\mathbb{C}} V_L^*$) is equal to the sum of the tensor products of the basis elements with their duals: $$\delta = e_0 \otimes e^0 + e_1 \otimes e^1$$

The definition represents the $SL(2, \mathbb{C})$-equivariant linear map (intertwining map) from the tensor product of the left-handed Weyl fermion representation $S_L$ and its dual representation $S_L^*$ to the trivial representation on $\mathbb{C}$. This map corresponds to the invariant tensor $\delta_\alpha^\beta$ used in physics for the contraction of a left-handed spinor with a dual left-handed spinor.

Let $\delta$ denote the left-dual left-unit for Weyl fermions. The application of this unit to $1$ is equal to its tensor value $\delta_{\text{val}}$, expressed as $\delta(1) = \delta_{\text{val}}$.

The identity tensor (or unit element) $\delta$ in the tensor product space $S_L^* \otimes_{\mathbb{C}} S_L$, where $S_L$ is the space of left-handed Weyl spinors and $S_L^*$ is its dual space.

Let $\{e_0, e_1\}$ be the standard basis for the complex vector space of left-handed Weyl fermions $V_L$ (denoted by `leftBasis`), and let $\{e^0, e^1\}$ be the corresponding dual basis (denoted by `dualLeftBasis`). The unit element $\delta \in V_L^* \otimes_{\mathbb{C}} V_L$ (represented by `dualLeftLeftUnitVal`) is given by the expansion: $$\delta = e^0 \otimes e_0 + e^1 \otimes e_1$$ where $\otimes$ denotes the tensor product over $\mathbb{C}$.

The `dualLeftLeftUnit` is the intertwining map (an equivariant linear map) from the tensor product of the dual left-handed Weyl spinor representation $V_L^*$ and the left-handed Weyl spinor representation $V_L$ of the special linear group $SL(2, \mathbb{C})$ to the trivial representation on $\mathbb{C}$. This map corresponds to the natural pairing or contraction between a dual spinor and a spinor, which is often denoted by the Kronecker delta $\delta^a{}_b$ in physics literature.

In the theory of Weyl fermions, let $\delta$ denote the dual left-left unit. This theorem states that evaluating the dual left-left unit map at the unit element $1$ yields the canonical value $\text{dualLeftLeftUnitVal}$.

The canonical unit element $\delta$ belonging to the tensor product space $S_R \otimes_{\mathbb{C}} S_R^*$, where $S_R$ represents the space of right-handed Weyl spinors and $S_R^*$ represents its dual space.

The unit element for right-handed Weyl fermions, `rightDualRightUnitVal` (often representing the identity or the Kronecker delta $\delta^{\dot{a}}{}_{\dot{b}}$ in spinor space), can be expanded in terms of the standard basis $\{e_0, e_1\}$ and its dual basis $\{e^0, e^1\}$ as the sum of their tensor products: $$\text{rightDualRightUnitVal} = e_0 \otimes e^0 + e_1 \otimes e^1$$ where $e_i$ are the basis vectors for the right-handed Weyl fermion space and $e^i$ are the corresponding dual basis vectors.

The intertwining map (equivariant map) from the tensor product of the right-handed Weyl fermion representation and its dual representation to the trivial representation of $SL(2, \mathbb{C})$ over $\mathbb{C}$. This map represents the canonical invariant contraction between a right-handed spinor and a dual right-handed spinor, which is often denoted as $\delta$ in physics literature.

For Weyl fermions, the right dual right unit map, when evaluated at the identity $1$, is equal to the constant tensor representing the right dual right unit value. Specifically, $\text{rightDualRightUnit}(1) = \text{rightDualRightUnitVal}$.

The unit tensor, commonly denoted as $\delta$, in the tensor product space of the dual right-handed Weyl spinor space $V_R^*$ and the right-handed Weyl spinor space $V_R$ over the complex numbers $\mathbb{C}$, representing the identity element in $V_R^* \otimes_{\mathbb{C}} V_R$.

In the complex vector space of right-handed Weyl fermions, the unit tensor $\delta$ is equal to the sum of the tensor products of the dual basis elements and the standard basis elements: $$\delta = e^{\dot{0}} \otimes e_{\dot{0}} + e^{\dot{1}} \otimes e_{\dot{1}}$$ where $\{e_{\dot{0}}, e_{\dot{1}}\}$ is the standard basis for right-handed Weyl fermions, $\{e^{\dot{0}}, e^{\dot{1}}\}$ is its corresponding dual basis, and $\otimes$ denotes the tensor product over $\mathbb{C}$.

This definition characterizes an intertwining map (an invariant linear map) from the tensor product of the dual right-handed Weyl fermion representation and the right-handed Weyl fermion representation of $SL(2, \mathbb{C})$ to the trivial representation on $\mathbb{C}$. In physics, this corresponds to the invariant Kronecker delta symbol $\delta_{\dot{a}}^{\dot{b}}$ used for the contraction of a right-handed spinor with its dual.

The dual right-right unit for Weyl fermions, denoted as $\delta_{RR}$, evaluated at $1$ is equal to the value $\delta_{RR, \text{val}}$.

For any left-handed Weyl fermion $x$, let $\delta$ be the unit element (or invariant tensor) in the tensor product of dual left-handed Weyl fermions and left-handed Weyl fermions. The contraction of $x$ with the dual component of $\delta$ is equal to $x$, which can be expressed as: $$\text{lid} \left( (\text{contr} \otimes \text{id}) \left( \text{assoc}^{-1} (x \otimes \delta) \right) \right) = x$$ where $\text{contr}: \text{LeftHandedWeyl} \otimes \text{DualLeftHandedWeyl} \to \mathbb{C}$ is the left dual contraction, $\text{assoc}^{-1}$ is the inverse associativity isomorphism of the tensor product, and $\text{lid}$ is the left identity isomorphism $\mathbb{C} \otimes \text{LeftHandedWeyl} \to \text{LeftHandedWeyl}$.

Let $S_L^*$ and $S_L$ be the spaces of dual left-handed Weyl spinors and left-handed Weyl spinors over $\mathbb{C}$, respectively. Let $\delta \in S_L \otimes_{\mathbb{C}} S_L^*$ be the canonical unit element for left-handed Weyl fermions. For any dual left-handed Weyl spinor $x \in S_L^*$, the contraction of $x$ with the first factor of $\delta$ results in $x$: $$\text{lid} \left( (c \otimes \text{id}_{S_L^*}) (\text{assoc}^{-1} (x \otimes \delta)) \right) = x$$ where $c: S_L^* \otimes_{\mathbb{C}} S_L \to \mathbb{C}$ is the contraction map, $\text{assoc}: (S_L^* \otimes_{\mathbb{C}} S_L) \otimes_{\mathbb{C}} S_L^* \cong S_L^* \otimes_{\mathbb{C}} (S_L \otimes_{\mathbb{C}} S_L^*)$ is the associativity isomorphism, and $\text{lid}: \mathbb{C} \otimes_{\mathbb{C}} S_L^* \to S_L^*$ is the left identity isomorphism.

Let $V$ denote the space of right-handed Weyl fermions (spinors) and $V^*$ denote its dual space. Let $\delta \in V^* \otimes V$ be the unit tensor and $c: V \otimes V^* \to \mathbb{C}$ be the canonical contraction (evaluation) map. For any right-handed Weyl fermion $x \in V$, contracting the first two factors of the re-associated tensor $x \otimes \delta$ yields $x$. That is, $$\text{lid} \left( (c \otimes \text{id}_V) \left( \text{assoc}^{-1}(x \otimes \delta) \right) \right) = x$$ where $\text{assoc}^{-1}: V \otimes (V^* \otimes V) \cong (V \otimes V^*) \otimes V$ is the associativity isomorphism and $\text{lid}: \mathbb{C} \otimes V \cong V$ is the left identity isomorphism.

Let $\text{DualRightHandedWeyl}$ and $\text{RightHandedWeyl}$ be the spaces of dual right-handed and right-handed Weyl spinors over the complex numbers $\mathbb{C}$. Let $\delta \in \text{RightHandedWeyl} \otimes_{\mathbb{C}} \text{DualRightHandedWeyl}$ be the canonical unit element (represented by `rightDualRightUnit 1`). For any dual right-handed Weyl spinor $x \in \text{DualRightHandedWeyl}$, the contraction of $x$ with the first component of $\delta$ results in $x$. Formally, this is expressed as: $$\text{lid} \left( (c \otimes \text{id}) \left( \text{assoc}^{-1} (x \otimes \delta) \right) \right) = x$$ where $c: \text{DualRightHandedWeyl} \otimes_{\mathbb{C}} \text{RightHandedWeyl} \to \mathbb{C}$ is the dual contraction map, $\text{assoc}$ is the associativity isomorphism of the tensor product, and $\text{lid}$ is the left identity isomorphism $\mathbb{C} \otimes V \cong V$.

Let $V$ denote the space of left-handed Weyl fermions and $V^*$ denote its dual space. The unit tensor in $V \otimes V^*$ (denoted by `dualLeftLeftUnit`) is equal to the unit tensor in $V^* \otimes V$ (denoted by `leftDualLeftUnit`) after applying the canonical commutativity isomorphism $\tau: V^* \otimes V \cong V \otimes V^*$ that swaps the factors of the tensor product.

Let $L$ denote the vector space of left-handed Weyl spinors (`Fermion.LeftHandedWeyl`) and $L^*$ denote its dual space (`Fermion.DualLeftHandedWeyl`). Let $\mathbf{1}_{L \otimes L^*} \in L \otimes \mathbb{C} \otimes L^*$ (or simply $L \otimes L^*$) be the unit element defined by `leftDualLeftUnit` and $\mathbf{1}_{L^* \otimes L} \in L^* \otimes L$ be the unit element defined by `dualLeftLeftUnit`. The theorem states that: $$\mathbf{1}_{L \otimes L^*} = \tau(\mathbf{1}_{L^* \otimes L})$$ where $\tau: L^* \otimes L \cong L \otimes L^*$ is the canonical commutativity isomorphism of the tensor product that swaps the order of the factors.

Let $W_R$ be the vector space of right-handed Weyl spinors and $W_R^*$ be its dual space. Let $\delta_{R R^*} \in W_R \otimes W_R^*$ be the right-dual-right unit and $\delta_{R^* R} \in W_R^* \otimes W_R$ be the dual-right-right unit (the canonical tensors representing the identity in their respective tensor product spaces). The theorem states that $\delta_{R^* R}$ is equal to the image of $\delta_{R R^*}$ under the canonical commutativity isomorphism of the tensor product $\text{comm} : W_R \otimes W_R^* \cong W_R^* \otimes W_R$, which swaps the order of the tensor factors.

Let $W_R$ denote the vector space of right-handed Weyl fermions over $\mathbb{C}$, and let $W_R^*$ be its dual space. Let $\delta_{R R^*} \in W_R \otimes_{\mathbb{C}} W_R^*$ be the unit element (the right-dual-right unit) and $\delta_{R^* R} \in W_R^* \otimes_{\mathbb{C}} W_R$ be the corresponding unit in the reversed tensor product space. The theorem states that $\delta_{R R^*}$ is equal to the image of $\delta_{R^* R}$ under the canonical commutation map $\tau: W_R^* \otimes_{\mathbb{C}} W_R \to W_R \otimes_{\mathbb{C}} W_R^*$ that swaps the tensor factors (i.e., $\tau(x \otimes y) = y \otimes x$).