Physlib.Relativity.Tensors.ComplexTensor.Units.Basic

The unit tensor $\delta_{\mu}{}^{\nu}$ (Kronecker delta) is defined as a complex Lorentz tensor of rank 2, specifically of type $(1, 1)$. It consists of one covariant index of color `Color.down` and one contravariant index of color `Color.up`. In the framework of the $SL(2, \mathbb{C})$ tensor species, this is constructed by applying the canonical unit tensor operation to the contravariant vector representation.

The Kronecker delta $\delta^\mu{}_\nu$ is defined as a complex Lorentz tensor of rank 2, specifically with one contravariant index of color `up` and one covariant index of color `down`. It represents the identity mapping on the complex Lorentz vector representation space.

The unit tensor $\delta_a{}^b$ (Kronecker delta) for left-handed Weyl spinors. This is a complex Lorentz tensor with one covariant left-handed index (of type `downL`) and one contravariant left-handed index (of type `upL`), representing the identity map on the left-handed spinor representation space.

The unit tensor $\delta^a_b$ (Kronecker delta) for left-handed Weyl spinors. This is a complex Lorentz tensor with one contravariant left-handed index (of type `upL`) and one covariant left-handed index (of type `downL`), representing the identity map on the left-handed spinor representation space.

The unit tensor $\delta_{\dot{b}}^{\dot{a}}$ (Kronecker delta) for right-handed Weyl spinors. This is a complex Lorentz tensor with one covariant dotted index (of type `downR`) and one contravariant dotted index (of type `upR`), representing the identity map on the right-handed spinor representation space.

The unit tensor $\delta^{\dot{a}}_{\dot{b}}$ (Kronecker delta) for right-handed Weyl spinors. This is a complex Lorentz tensor with one contravariant right-handed dotted index (of type `upR`) and one covariant right-handed dotted index (of type `downR`), representing the identity map on the right-handed spinor representation space.

The notation $\delta'$ represents the Kronecker delta $\delta_i^j$ as a complex Lorentz tensor with one covariant and one contravariant index.

The notation $\delta$ represents the unit $(1,1)$-tensor for complex Lorentz tensors, corresponding to the Kronecker delta $\delta^i_j$.

The notation $\delta L'$ represents the unit tensor $\delta_a{}^a$ in the context of complex Lorentz tensors.

The notation $\delta L$ represents the unit tensor $\delta^a{}_a$ (the Kronecker delta) in the context of complex Lorentz tensors.

The notation `δR'` represents the unit tensor $\delta^{\dot{a}}_{\dot{a}}$ in the context of complex Lorentz tensors. It corresponds to the identity operator acting on the space of right-handed (dotted) spinor indices.

The notation $\delta R$ represents the unit tensor $\delta^{\dot{a}}_{\dot{a}}$ in the context of complex Lorentz tensors, where $\dot{a}$ denotes a dotted (right-handed) spinor index.

The unit tensor $\delta_{\mu}{}^{\nu}$ (denoted as $\delta'$) in the complex Lorentz tensor species is equal to the tensor constructed from the $SL(2, \mathbb{C})$-invariant morphism $\text{coContrUnit} : \mathbb{1} \to V_{\text{co}} \otimes V_{\text{contr}}$ via the `fromConstPair` operator. Here, $V_{\text{co}}$ and $V_{\text{contr}}$ represent the covariant (down) and contravariant (up) Lorentz vector representations, and $\mathbb{1}$ is the trivial representation.

The contra-covariant unit tensor $\delta$ (representing the Kronecker delta $\delta^\mu{}_\nu$) in the complex Lorentz tensor species is equal to the rank-2 tensor constructed from the representation-theoretic unit morphism $\text{contrCoUnit} : \mathbb{1} \to V_{\text{contr}} \otimes V_{\text{co}}$ using the `fromConstPair` function. Here, $V_{\text{contr}}$ and $V_{\text{co}}$ are the contravariant and covariant representation spaces of $SL(2, \mathbb{C})$, respectively.

The unit tensor for left-handed Weyl spinors $\delta_a{}^b$ (denoted as $\delta_L'$) is equal to the rank-2 tensor constructed from the intertwining morphism `Fermion.altLeftLeftUnit` (which represents the map $\mathbb{1} \to \text{altLeftHanded} \otimes \text{leftHanded}$) via the `fromConstPair` function.

The unit tensor for left-handed spinors $\delta^a_b$ is equal to the rank-2 tensor constructed from the $SL(2, \mathbb{C})$-invariant morphism `Fermion.leftAltLeftUnit` (which maps the trivial representation to the tensor product of the left-handed and alt-left-handed representation spaces) using the `fromConstPair` construction.

In the context of complex Lorentz tensors for $SL(2, \mathbb{C})$, the right-handed spinor unit tensor $\delta R'$ (which represents the Kronecker delta $\delta_{\dot{b}}^{\dot{a}}$) is equal to the rank-2 tensor constructed from the morphism $\text{Fermion.altRightRightUnit} : \mathbb{1} \to \text{altRightHanded} \otimes \text{rightHanded}$ via the `fromConstPair` operation.

In the context of the tensor species for complex Lorentz representations of $SL(2, \mathbb{C})$, the unit tensor for right-handed Weyl spinors $\delta R$ (representing the Kronecker delta $\delta^{\dot{a}}_{\dot{b}}$) is equal to the rank-2 tensor constructed using the `fromConstPair` function from the right-alt-right unit morphism $\delta^{\dot{a}}_{\dot{a}} : \mathbb{1} \to \text{rightHanded} \otimes \text{altRightHanded}$.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the Kronecker delta tensor $\delta_{\mu}{}^{\nu}$ (denoted by $\delta'$) is equal to the rank-2 tensor obtained by applying the linear map $\text{fromPairT}$ to the invariant identity value $\text{Lorentz.coContrUnitVal}$, which resides in the tensor product of the covariant and contravariant Lorentz vector representation spaces $V_{\text{co}} \otimes V_{\text{contr}}$.

In the tensor species for complex Lorentz representations of $SL(2, \mathbb{C})$, the Kronecker delta tensor $\delta$ (with indices $\delta^\mu{}_\nu$) is equal to the rank-2 tensor obtained by applying the linear map $\text{fromPairT}$ to the contra-covariant unit tensor value $\delta^\mu{}_\nu \in V_{\text{contr}} \otimes V_{\text{co}}$.

The unit tensor $\delta_L'$ (the Kronecker delta $\delta_a{}^b$ for left-handed spinors) for the complex Lorentz tensor species is equal to the rank-2 tensor obtained by applying the linear map $\text{fromPairT}$ to the vector space element $\text{altLeftLeftUnitVal}$. Here, $\text{altLeftLeftUnitVal}$ is the unit tensor value as an element of the tensor product of the dual left-handed (alt-left-handed) and left-handed spinor vector spaces, and $\text{fromPairT}$ is the canonical map that converts elements of the tensor product of representation spaces into rank-2 tensors.

The unit tensor $\delta^a_b$ (Kronecker delta) for left-handed Weyl spinors in the complex Lorentz tensor species is equal to the rank-2 tensor obtained by applying the linear map `fromPairT` to the invariant tensor value $\text{Fermion.leftAltLeftUnitVal}$, which represents the vector space element $\delta^a_a$ in the tensor product of the left-handed and alt-left-handed representation spaces $V_L \otimes V_{\bar{L}}$.

In the context of complex Lorentz tensors for $SL(2, \mathbb{C})$, the right-handed spinor unit tensor $\delta R'$ (representing the Kronecker delta $\delta_{\dot{b}}^{\dot{a}}$ with indices of type `downR` and `upR`) is equal to the image of the representation-theoretic tensor value $\text{altRightRightUnitVal} \in (\text{altRightHanded} \otimes \text{rightHanded})$ under the linear map $\text{fromPairT}$.

In the context of the tensor species for complex Lorentz representations of $SL(2, \mathbb{C})$, the unit tensor for right-handed Weyl spinors $\delta R$ (representing the Kronecker delta $\delta^{\dot{a}}_{\dot{b}}$) is equal to the rank-2 tensor constructed by applying the linear map `fromPairT` to the right-alt-right unit value `Fermion.rightAltRightUnitVal` (the element $\delta^{\dot{a}}_{\dot{a}}$ in the underlying tensor product vector space).

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the Kronecker delta unit tensor $\delta_{\mu}{}^{\nu}$ (denoted by $\delta'$) is equal to the sum over the spacetime indices $i$ of the tensor products of the basis elements of the covariant and contravariant representations. Specifically, $$\delta' = \sum_{i} \text{fromPairT}(\text{complexCoBasis } i \otimes \text{complexContrBasis } i)$$ where $\text{complexCoBasis } i$ and $\text{complexContrBasis } i$ are the $i$-th basis elements for the covariant and contravariant Lorentz vector representations respectively, and the map `fromPairT` embeds the tensor product of these basis vectors into the space of rank-2 complex Lorentz tensors.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the Kronecker delta unit tensor $\delta_{\mu}{}^{\nu}$ (denoted by $\delta'$) is equal to the sum over the spacetime indices $i \in \{0, 1, 2, 3\}$ of the tensor products of the basis elements of the covariant and contravariant representations. Specifically, $$\delta' = \sum_{i=0}^{3} \text{fromPairT}(\text{complexCoBasisFin4 } i \otimes \text{complexContrBasisFin4 } i)$$ where $\text{complexCoBasisFin4 } i$ and $\text{complexContrBasisFin4 } i$ are the $i$-th basis elements for the covariant and contravariant Lorentz vector representations respectively, and the map `fromPairT` embeds the tensor product of these basis vectors into the space of rank-2 complex Lorentz tensors.

For the complex Lorentz tensor species with symmetry group $SL(2, \mathbb{C})$, the Kronecker delta tensor $\delta^\mu{}_\nu$ (represented by $\delta$) is equal to the sum over the index $i \in \{0, 1, 2, 3\}$ of the tensor products of the standard contravariant basis vectors $e_i$ and the standard covariant basis vectors $e^i$. Mathematically, this is expressed as: $$\delta = \sum_i e_i \otimes e^i$$ where $e_i$ (complexContrBasis) and $e^i$ (complexCoBasis) are the basis elements of the contravariant and covariant representation spaces, respectively.

In the framework of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the Kronecker delta tensor $\delta^\mu{}_\nu$ (represented as $\delta$) is equal to the sum over the index $i \in \{0, 1, 2, 3\}$ of the tensor products of the standard contravariant basis vectors $e_i$ and the standard covariant basis vectors $e^i$: $$\delta = \sum_{i=0}^3 e_i \otimes e^i$$ where $e_i$ (complexContrBasisFin4) and $e^i$ (complexCoBasisFin4) are the basis elements of the contravariant and covariant representation spaces, respectively.

The unit tensor $\delta_L'$ (representing the Kronecker delta $\delta_a{}^b$) for left-handed Weyl spinors is equal to the sum of the rank-2 tensors obtained from the tensor product of the basis elements of the alt-left-handed and left-handed representations: $$\delta_L' = \sum_{i \in \{0, 1\}} \text{fromPairT}(e^{\text{alt}}_i \otimes e^{\text{left}}_i)$$ where $e^{\text{alt}}_i$ is the $i$-th basis element of the alt-left-handed representation space (corresponding to covariant indices) and $e^{\text{left}}_i$ is the $i$-th basis element of the left-handed representation space (corresponding to contravariant indices). The map $\text{fromPairT}$ denotes the canonical $k$-linear isomorphism from the tensor product of representation spaces to the space of rank-2 tensors.

In the framework of complex Lorentz tensors, the unit tensor $\delta^a_b$ for left-handed Weyl spinors is equal to the sum of the tensor products of the basis vectors of the left-handed representation and the alt-left-handed representation: $$\delta^a_b = \sum_{i=0}^1 e_i \otimes \bar{e}_i$$ where $e_i$ are the standard basis vectors for the left-handed representation (`leftBasis`) and $\bar{e}_i$ are the standard basis vectors for the alt-left-handed representation (`altLeftBasis`).

In the context of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the unit tensor $\delta_{\dot{b}}^{\dot{a}}$ (denoted $\delta R'$) for right-handed Weyl spinors is equal to the sum over the basis elements of the tensor product of the alt-right-handed basis and the right-handed basis. Specifically: $$\delta R' = \sum_{i=0}^1 \text{fromPairT}(b_{\dot{i}} \otimes b^{\dot{i}})$$ where $b_{\dot{i}}$ represents the $i$-th basis vector of the alt-right-handed representation (which corresponds to dotted covariant indices $\dot{b}$) and $b^{\dot{i}}$ represents the $i$-th basis vector of the right-handed representation (which corresponds to dotted contravariant indices $\dot{a}$).

In the tensor species of complex Lorentz representations for $SL(2, \mathbb{C})$, the unit tensor for right-handed Weyl spinors $\delta R$ (representing the Kronecker delta $\delta^{\dot{a}}_{\dot{b}}$) is equal to the sum over $i \in \{0, 1\}$ of the rank-2 tensors formed by the tensor product of the $i$-th basis elements of the right-handed and alt-right-handed spinor spaces: $$\delta R = \sum_{i=0}^1 \text{fromPairT}(b_{\dot{i}} \otimes \bar{b}^{\dot{i}})$$ where $b_{\dot{i}}$ denotes the $i$-th basis vector of the right-handed Weyl fermion representation (`rightBasis`) and $\bar{b}^{\dot{i}}$ denotes the $i$-th basis vector of the alt-right-handed representation (`altRightBasis`).

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the Kronecker delta unit tensor $\delta_{\mu}{}^{\nu}$ (denoted by $\delta'$) is equal to the sum over the spacetime indices $i \in \{0, 1, 2, 3\}$ of the basis elements of the rank-2 tensor space corresponding to the multi-index $(i, i)$. Specifically: $$\delta' = \sum_{i=0}^3 \mathcal{E}_{(i, i)}$$ where $\mathcal{E}_{(i, i)}$ denotes the basis element of the tensor space associated with the color sequence $[\text{down}, \text{up}]$ at the multi-index where both the covariant and contravariant indices are equal to $i$.

In the framework of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the unit tensor $\delta^\mu{}_\nu$ (the Kronecker delta for Lorentz vectors) is equal to the sum of the diagonal basis elements of the corresponding rank-2 tensor space. Specifically, for the tensor space associated with the index colors $(\text{up}, \text{down})$, it holds that: $$\delta = \sum_{i=0}^3 \mathcal{E}_{(i, i)}$$ where $\mathcal{E}_{(i, i)}$ denotes the canonical basis element of the rank-2 tensor space $S.\text{Tensor}(\text{up}, \text{down})$ corresponding to the multi-index where both the contravariant and covariant indices are equal to $i$.

In the context of the complex Lorentz tensor species, the unit tensor $\delta_L'$ (representing the Kronecker delta $\delta_a{}^b$ for left-handed Weyl spinors) is equal to the sum over its diagonal components in the canonical basis. Specifically, for the tensor space associated with the color sequence $[\text{downL}, \text{upL}]$, it holds that: $$\delta_L' = \sum_{i \in \{0, 1\}} \mathcal{E}_{(i, i)}$$ where $\mathcal{E}_{(i, i)}$ denotes the basis element of the rank-2 tensor space corresponding to the multi-index where both the covariant and contravariant indices are equal to $i$.

In the framework of the complex Lorentz tensor species, the unit tensor $\delta^a_b$ (Kronecker delta) for left-handed Weyl spinors, which has one contravariant index of type `upL` and one covariant index of type `downL`, is equal to the sum of the basis elements of the corresponding rank-2 tensor space where both indices are identical: $$\delta^a_b = \sum_{i=0}^1 \mathcal{E}_{(i, i)}$$ where $\mathcal{E}_{(i, i)}$ denotes the canonical basis element of the tensor space $S.\text{Tensor}(\text{upL}, \text{downL})$ corresponding to the multi-index $(i, i)$.

In the theory of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the unit tensor $\delta R'$ (representing the Kronecker delta $\delta_{\dot{b}}^{\dot{a}}$ for right-handed Weyl spinors) is equal to the sum of the basis elements of the tensor space $S.\text{Tensor}(\text{downR}, \text{upR})$ with matching indices. Specifically: $$\delta R' = \sum_{i=0}^1 \mathcal{E}_{(i, i)}$$ where $\mathcal{E}_{(i, i)}$ denotes the canonical basis element of the tensor space associated with the color sequence $(\text{downR}, \text{upR})$ at the component multi-index $(i, i)$. Here, `downR` and `upR` correspond to the representation colors for covariant and contravariant dotted indices, respectively.

In the tensor species of complex Lorentz representations for $SL(2, \mathbb{C})$, the unit tensor $\delta R$ for right-handed Weyl spinors (representing the Kronecker delta $\delta^{\dot{a}}_{\dot{b}}$) is equal to the sum over $i \in \{0, 1\}$ of the canonical basis elements of the rank-2 tensor space associated with the index colors `upR` (contravariant right-handed) and `downR` (covariant right-handed) at the diagonal multi-indices $(i, i)$: $$\delta R = \sum_{i=0}^1 \mathcal{E}_{(i, i)}$$ where $\mathcal{E}_{(i, i)}$ denotes the basis element of the tensor space $S.\text{Tensor}(\text{upR}, \text{downR})$ corresponding to the multi-index $(i, i)$.

In the theory of complex Lorentz tensors for $SL(2, \mathbb{C})$, the unit tensor $\delta_{\mu}{}^{\nu}$ (the Kronecker delta, denoted by $\delta'$) is equal to the tensor constructed via the $\text{ofRat}$ map from a component function $f$ that returns $1$ if the indices are equal ($f_0 = f_1$) and $0$ otherwise. Specifically, $$\delta' = \text{ofRat}(f)$$ where $f(b) = 1$ if $b_0 = b_1$ and $f(b) = 0$ if $b_0 \neq b_1$ for a multi-index $b = (b_0, b_1)$ representing the covariant (down) and contravariant (up) indices.

The unit tensor $\delta$ (the Kronecker delta $\delta^\mu{}_\nu$ for complex Lorentz vectors) is equal to the tensor obtained by applying the `ofRat` map to the Kronecker delta function. Specifically, $$\delta = \text{ofRat}(f)$$ where $f$ is a function on the multi-indices such that $f(f_0, f_1) = 1$ if $f_0 = f_1$ and $0$ otherwise. Here, $f_0$ and $f_1$ correspond to the contravariant and covariant Lorentz vector indices, respectively.

The unit tensor $\delta_L'$ for left-handed Weyl spinors (representing the Kronecker delta $\delta_a{}^b$) is equal to the tensor constructed via the `ofRat` map from a component function that returns $1$ if the first and second indices are equal ($f_0 = f_1$) and $0$ otherwise.

In the framework of the complex Lorentz tensor species, the unit tensor for left-handed Weyl spinors $\delta^a_b$ (represented by the symbol `δL`) is equal to the tensor constructed by the `ofRat` map from the Kronecker delta function. Specifically, $$\delta L = \text{ofRat}(f)$$ where $f$ is a function on the component indices such that $f(f_0, f_1) = 1$ if $f_0 = f_1$ and $0$ otherwise. Here, $f_0$ and $f_1$ correspond to the contravariant and covariant left-handed spinor indices, respectively.

In the theory of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the unit tensor $\delta R'$ (representing the Kronecker delta $\delta_{\dot{b}}^{\dot{a}}$ for right-handed Weyl spinors) is equal to the tensor constructed by the `ofRat` map from the indicator function that returns $1$ if the component indices $f(0)$ and $f(1)$ are equal, and $0$ otherwise.

In the tensor species of complex Lorentz representations for $SL(2, \mathbb{C})$, the unit tensor for right-handed Weyl spinors, denoted by $\delta R$ (representing the Kronecker delta $\delta^{\dot{a}}_{\dot{b}}$), is equal to the tensor constructed by the $\text{ofRat}$ map from the indicator function on multi-indices. Specifically, for any multi-index $f$, the component is 1 if $f(0) = f(1)$ and 0 otherwise.

For any group element $g \in SL(2, \mathbb{C})$, the unit tensor $\delta_{\mu}{}^{\nu}$ (the Kronecker delta, denoted as `coContrUnit` or $\delta'$), which represents a rank-2 complex Lorentz tensor with one covariant index and one contravariant index, is invariant under the group action of $SL(2, \mathbb{C})$. That is, $g \cdot \delta_{\mu}{}^{\nu} = \delta_{\mu}{}^{\nu}$.

For any group element $g \in SL(2, \mathbb{C})$, the Kronecker delta $\delta^\mu{}_\nu$ (represented by the tensor `contrCoUnit`, denoted as $\delta$), which is a complex Lorentz tensor of rank 2 with one contravariant index and one covariant index, is invariant under the group action of $SL(2, \mathbb{C})$. That is, $g \cdot \delta = \delta$.

For any group element $g \in SL(2, \mathbb{C})$, the unit tensor $\delta_a{}^b$ (the Kronecker delta for left-handed Weyl spinors, denoted as `δL'` or `altLeftLeftUnit`), which has one covariant left-handed index and one contravariant left-handed index, is invariant under the group action of $SL(2, \mathbb{C})$. That is, $g \cdot \delta_a{}^b = \delta_a{}^b$.

For any element $g \in SL(2, \mathbb{C})$, the unit tensor $\delta^a_b$ for left-handed Weyl spinors (representing the identity map on the left-handed spinor representation space) is invariant under the group action of $SL(2, \mathbb{C})$, such that $g \cdot \delta^a_b = \delta^a_b$.

For any group element $g \in SL(2, \mathbb{C})$, the unit tensor for right-handed Weyl spinors $\delta_{\dot{b}}^{\dot{a}}$ (representing the Kronecker delta with one covariant dotted index and one contravariant dotted index) is invariant under the action of $g$, such that $g \cdot \delta_{\dot{b}}^{\dot{a}} = \delta_{\dot{b}}^{\dot{a}}$.

For any element $g \in SL(2, \mathbb{C})$, the unit tensor for right-handed Weyl spinors $\delta^{\dot{a}}_{\dot{b}}$ (also denoted as `rightAltRightUnit`) is invariant under the group action of $SL(2, \mathbb{C})$, satisfying $g \cdot \delta^{\dot{a}}_{\dot{b}} = \delta^{\dot{a}}_{\dot{b}}$.