Physlib.Relativity.Tensors.ComplexTensor.Metrics.Basic

The definition represents the covariant Minkowski metric $\eta_{\mu\nu}$ as a complex Lorentz tensor. It is a tensor of rank 2 where both indices belong to the covariant vector representation (the `down` color) of the complex Lorentz tensor species.

The definition represents the contravariant Minkowski metric $\eta^{\mu\nu}$ as a complex Lorentz tensor. It is a rank-2 tensor where both indices belong to the contravariant vector representation (the `up` color) of the complex Lorentz tensor species.

The definition represents the invariant metric tensor $\epsilon^{ab}$ for left-handed Weyl spinors as a complex Lorentz tensor. It is a rank-2 tensor where both indices belong to the contravariant left-handed representation (denoted by the color `upL`) of the complex Lorentz tensor species for $SL(2, \mathbb{C})$.

The definition `complexLorentzTensor.rightMetric` represents the invariant metric tensor $\epsilon^{\dot{a}\dot{b}}$ for right-handed Weyl spinors. It is a rank-2 complex Lorentz tensor where both indices belong to the contravariant right-handed representation (represented by the color `upR`, which corresponds to dotted indices) of the complex Lorentz tensor species for $SL(2, \mathbb{C})$.

The definition `complexLorentzTensor.altLeftMetric` represents the invariant metric tensor $\epsilon_{ab}$ for left-handed Weyl spinors. It is a rank-2 complex Lorentz tensor where both indices belong to the covariant left-handed representation (denoted by the color `downL`) of the complex Lorentz tensor species for $SL(2, \mathbb{C})$.

The definition `complexLorentzTensor.altRightMetric` represents the invariant metric tensor $\epsilon_{\dot{a}\dot{b}}$ for right-handed covariant Weyl spinors. It is a complex Lorentz tensor with two indices of the color `downR`, which corresponds to the dotted indices in the $SL(2, \mathbb{C})$ representation.

The notation $\eta'$ represents the covariant metric tensor $\eta_{\mu\nu}$ with two lower indices, defined as a complex Lorentz tensor.

The notation $\eta$ represents the contravariant metric tensor $\eta^{\mu\nu}$ (formally `contrMetric`) within the theory of complex Lorentz tensors.

The notation $\epsilon_L$ represents the left metric tensor $\epsilon$ within the framework of complex Lorentz tensors. This metric is used to raise and lower indices in the spinor representation.

The notation $\epsilon_R$ represents the metric tensor $\epsilon^{\dot{a}\dot{b}}$ for right-handed (dotted) Weyl spinors, formulated as a complex Lorentz tensor.

The notation $\epsilon L'$ denotes the alternating left metric tensor $\epsilon$ within the framework of complex Lorentz tensors.

The complex Lorentz tensor representing the metric $\epsilon_{\dot{a} \dot{b}}$ for right-handed spinors (dotted indices).

In the framework of complex Lorentz tensors, the covariant Minkowski metric tensor $\eta_{\mu\nu}$ (denoted by $\eta'$) is equal to the rank-2 tensor constructed from the $SL(2, \mathbb{C})$-invariant morphism $\eta_{\mu\nu} : \mathbb{1} \to V \otimes V$ (represented by `Lorentz.coMetric`) through the `fromConstPair` function.

The contravariant Minkowski metric $\eta^{\mu\nu}$ in the complex Lorentz tensor species is equal to the rank-2 tensor constructed from the $SL(2, \mathbb{C})$-invariant morphism `Lorentz.contrMetric` (which represents $\eta^{ab}: \mathbb{1} \to V \otimes V$) using the `fromConstPair` operation.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the left-handed spinor metric tensor $\epsilon_L$ (with indices of color `upL`) is equal to the rank-2 tensor constructed via the `fromConstPair` function from the representation-theoretic morphism $\text{Fermion.leftMetric} : \mathbb{1} \to L \otimes L$, where $L$ is the left-handed fundamental representation of $SL(2, \mathbb{C})$.

The metric tensor for right-handed Weyl spinors, denoted by $\varepsilon_R$ (or $\epsilon^{\dot{a}\dot{b}}$), is equal to the rank-2 tensor constructed from the $SL(2, \mathbb{C})$-invariant morphism `Fermion.rightMetric` using the `fromConstPair` operator. Here, `Fermion.rightMetric` is the intertwining map $\mathbb{1} \to \text{rightHanded} \otimes \text{rightHanded}$ that identifies the invariant metric in the representation space of right-handed spinors.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the invariant metric tensor $\varepsilon_{L'}$ for covariant left-handed Weyl spinors is equal to the rank-2 tensor constructed via `fromConstPair` from the invariant morphism $\epsilon_{ab} : \mathbb{1} \to \text{altLeftHanded} \otimes \text{altLeftHanded}$ (represented by `Fermion.altLeftMetric`).

In the theory of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the invariant metric tensor for right-handed covariant Weyl spinors, denoted by $\varepsilon_{R'}$ (representing the component $\epsilon_{\dot{a}\dot{b}}$), is equal to the rank-2 tensor constructed from the representation-theoretic morphism `Fermion.altRightMetric`. Specifically, this identity states that $\varepsilon_{R'}$ is obtained by applying the `fromConstPair` operation to the morphism $\mathbb{1} \to \text{altRightHanded} \otimes \text{altRightHanded}$ which defines the invariant anti-symmetric metric for dotted spinors in the representation category $\text{Rep}_{\mathbb{C}}(SL(2, \mathbb{C}))$.

In the framework of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the covariant Minkowski metric tensor $\eta_{\mu\nu}$ (denoted as $\eta'$) is equal to the rank-2 tensor constructed by applying the linear map `fromPairT` to the covariant metric tensor value $\eta_{\mu\nu}$ (represented by `Lorentz.coMetricVal`) belonging to the tensor product of the covariant Lorentz representation spaces.

In the context of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the contravariant Minkowski metric $\eta^{\mu\nu}$ (denoted by $\eta$) is equal to the rank-2 tensor obtained by applying the $k$-linear map `fromPairT` to the representation-theoretic value of the contravariant Minkowski metric $\eta^{ab}$ (denoted by `Lorentz.contrMetricVal`) residing in the tensor product of the contravariant vector representation spaces.

In the context of complex Lorentz tensors for $SL(2, \mathbb{C})$, the left-handed spinor metric tensor $\epsilon_L$ (with indices of color `upL`) is equal to the rank-2 tensor constructed by applying the linear map `fromPairT` to the left-handed Weyl fermion metric element $\varepsilon^{ab}$ belonging to the tensor product space $L \otimes L$.

The invariant metric tensor for right-handed Weyl spinors, denoted by $\varepsilon_R$ (representing the tensor $\epsilon^{\dot{a}\dot{b}}$), is equal to the rank-2 tensor obtained by applying the linear map `fromPairT` to the vector space element `Fermion.rightMetricVal`. Here, `Fermion.rightMetricVal` is the specific element in the tensor product of two right-handed Weyl representation spaces corresponding to the invariant antisymmetric matrix.

In the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the invariant metric tensor $\varepsilon_{L'}$ for covariant left-handed Weyl spinors is equal to the rank-2 tensor obtained by applying the linear map $\text{fromPairT}$ to the tensor product element $\text{altLeftMetricVal}$ (which represents the antisymmetric matrix $\epsilon_{ab} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$).

In the theory of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the invariant metric tensor for right-handed covariant Weyl spinors, denoted by $\varepsilon_{R'}$ (which represents the spinor metric $\epsilon_{\dot{a}\dot{b}}$), is equal to the rank-2 tensor obtained by applying the $k$-linear map `fromPairT` to the tensor product value `Fermion.altRightMetricVal`.

In the framework of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the covariant Minkowski metric tensor $\eta_{\mu\nu}$ (denoted as $\eta'$) is expressed in terms of the standard basis $\{e^0, e^1, e^2, e^3\}$ of the covariant representation space as: $$\eta' = e^0 \otimes e^0 - e^1 \otimes e^1 - e^2 \otimes e^2 - e^3 \otimes e^3$$ where $e^0$ is the temporal basis vector (corresponding to the index `Sum.inl 0`) and $e^1, e^2, e^3$ are the spatial basis vectors (corresponding to the indices `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively). The notation $\otimes$ here represents the rank-2 tensor constructed from the tensor product of basis elements via the linear map `fromPairT`.

In the framework of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, let $\{e_0, e_1, e_2, e_3\}$ be the standard basis for the four-dimensional complex vector space of covariant Lorentz vectors, indexed by $i \in \{0, 1, 2, 3\}$. The covariant Minkowski metric tensor $\eta_{\mu\nu}$ (denoted as $\eta'$) is expressed in terms of these basis vectors as: $$\eta' = e_0 \otimes e_0 - e_1 \otimes e_1 - e_2 \otimes e_2 - e_3 \otimes e_3$$ where the tensor product $e_i \otimes e_i$ is interpreted as a rank-2 tensor via the linear map `fromPairT`.

Let $\{e^\mu\}_{\mu=0}^3$ be the standard basis for the space of complex contravariant Lorentz vectors, where $e^0$ is the basis vector indexed by `Sum.inl 0`, and $e^1, e^2, e^3$ are the basis vectors indexed by `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively. The contravariant Minkowski metric $\eta$ is equal to the following expansion in terms of these basis vectors: $$\eta = e^0 \otimes e^0 - e^1 \otimes e^1 - e^2 \otimes e^2 - e^3 \otimes e^3$$ where the tensor product of basis vectors $e^\mu \otimes e^\mu$ is interpreted as a rank-2 tensor via the map `fromPairT`.

Let $\{e^\mu\}_{\mu=0}^3$ be the standard basis for the space of complex contravariant Lorentz vectors, where $e^\mu$ denotes the basis vector `complexContrBasisFin4 μ` for $\mu \in \{0, 1, 2, 3\}$. The contravariant Minkowski metric $\eta$ is given by the expansion: $$\eta = e^0 \otimes e^0 - e^1 \otimes e^1 - e^2 \otimes e^2 - e^3 \otimes e^3$$ where the tensor product $e^\mu \otimes e^\nu$ is mapped into the space of rank-2 tensors via the linear map `fromPairT`.

The left-handed Weyl spinor metric tensor $\epsilon_L$ (with indices of color `upL`) can be expanded in terms of the standard basis $\{e_0, e_1\}$ of the left-handed representation space $V_L$ as: $$\epsilon_L = -(e_0 \otimes e_1) + (e_1 \otimes e_0)$$ where $e_i$ are the basis elements of the left-handed representation and $\otimes$ represents the tensor product over $\mathbb{C}$ mapped into the rank-2 tensor space via the linear map $\text{fromPairT}$.

In the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the invariant metric tensor $\varepsilon_{L'}$ for left-handed Weyl spinors (with covariant indices) can be expanded in terms of the standard basis $\{e_0, e_1\}$ for the alt-left-handed representation as follows: $$\varepsilon_{L'} = \text{fromPairT}(e_0 \otimes e_1) - \text{fromPairT}(e_1 \otimes e_0)$$ where $\otimes$ denotes the tensor product over $\mathbb{C}$, and $\text{fromPairT}$ is the linear map that constructs a rank-2 tensor from the tensor product of two vectors.

Let $\{e_0, e_1\}$ be the standard basis for the complex representation space of right-handed Weyl fermions (denoted as `rightBasis`). The invariant metric tensor for right-handed Weyl spinors, denoted as $\varepsilon_R$ (corresponding to the tensor $\epsilon^{\dot{a}\dot{b}}$ in dotted-index notation), can be expressed in terms of this basis as: $$\varepsilon_R = -\text{fromPairT}(e_0 \otimes e_1) + \text{fromPairT}(e_1 \otimes e_0)$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$, and `fromPairT` is the linear map that constructs a rank-2 tensor from the tensor product of two vector space elements.

Let $\{\tilde{e}_0, \tilde{e}_1\}$ be the standard basis (known as `altRightBasis`) for the alternative right-handed Weyl representation of $SL(2, \mathbb{C})$, which corresponds to the representation space for spinors with dotted indices $\dot{a}$. The invariant metric tensor for right-handed covariant spinors, denoted by $\varepsilon_{R'}$ (representing the spinor metric $\epsilon_{\dot{a}\dot{b}}$), is equal to the antisymmetric combination of these basis elements: $$\varepsilon_{R'} = \tilde{e}_0 \otimes \tilde{e}_1 - \tilde{e}_1 \otimes \tilde{e}_0$$ where $\otimes$ denotes the tensor product used to construct a rank-2 tensor from the basis vectors.

In the framework of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the covariant Minkowski metric $\eta_{\mu\nu}$ (denoted $\eta'$) is expanded in the canonical tensor basis $\mathcal{E}$ for the space of rank-2 covariant tensors (where both indices belong to the `Color.down` representation) as: $$\eta' = \mathcal{E}_{(0,0)} - \mathcal{E}_{(1,1)} - \mathcal{E}_{(2,2)} - \mathcal{E}_{(3,3)}$$ where $\mathcal{E}_{(i,j)}$ denotes the basis element of the tensor space corresponding to the multi-index $(i, j)$ with $i, j \in \{0, 1, 2, 3\}$.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the contravariant Minkowski metric $\eta$ (representing $\eta^{\mu\nu}$) is expressed in terms of the canonical tensor basis $\mathcal{E}$ for rank-2 contravariant tensors (corresponding to the color `up`) as: $$\eta = \mathcal{E}_{(0, 0)} - \mathcal{E}_{(1, 1)} - \mathcal{E}_{(2, 2)} - \mathcal{E}_{(3, 3)}$$ where $\mathcal{E}_{(i, j)}$ denotes the basis element for the multi-index $(i, j)$ with $i, j \in \{0, 1, 2, 3\}$.

In the context of complex Lorentz tensors, the invariant metric tensor $\epsilon_L$ (representing $\epsilon^{ab}$) for left-handed Weyl spinors is expressed in terms of the canonical tensor basis $\mathcal{E}$ as: $$\epsilon_L = -\mathcal{E}_{(0, 1)} + \mathcal{E}_{(1, 0)}$$ where $\mathcal{E}_{(i, j)}$ denotes the basis element of the rank-2 tensor space $V_L \otimes V_L$ (corresponding to contravariant left-handed indices) for the multi-index $(i, j)$, where $i, j \in \{0, 1\}$.

In the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the invariant metric tensor $\varepsilon_{L'}$ for left-handed Weyl spinors with covariant indices (denoted by the color `downL`) is given by the difference of the canonical tensor basis elements: $$\varepsilon_{L'} = \mathcal{E}_{(0, 1)} - \mathcal{E}_{(1, 0)}$$ where $\mathcal{E}_{(i, j)}$ denotes the basis element of the rank-2 tensor space corresponding to the multi-index $(i, j)$.

In the `complexLorentzTensor` species, the invariant metric tensor for right-handed Weyl spinors $\varepsilon_R$ (which corresponds to the tensor $\epsilon^{\dot{a}\dot{b}}$ in dotted-index notation) can be expressed in terms of the canonical tensor basis $\mathcal{E}_{ij}$ for the representation color `upR` as: $$\varepsilon_R = -\mathcal{E}_{01} + \mathcal{E}_{10}$$ where $\mathcal{E}_{ij}$ denotes the basis element of the rank-2 tensor space associated with the multi-index $(i, j)$ for $i, j \in \{0, 1\}$.

In the context of the `complexLorentzTensor` species, the invariant metric tensor for right-handed covariant Weyl spinors $\varepsilon_{R'}$ (representing $\epsilon_{\dot{a}\dot{b}}$ in dotted index notation) is expressed in the canonical basis for rank-2 tensors of color type `(downR, downR)` as: $$\varepsilon_{R'} = \mathcal{E}_{(0, 1)} - \mathcal{E}_{(1, 0)}$$ where $\mathcal{E}_{(i, j)}$ denotes the basis element of the tensor space corresponding to the multi-index $(i, j)$ for $i, j \in \{0, 1\}$.

In the framework of complex Lorentz tensors, the covariant Minkowski metric $\eta_{\mu\nu}$ (denoted by $\eta'$) is equal to the tensor constructed via the `ofRat` map from the component function $f$ defined on the multi-indices $(\mu, \nu)$ as: $$f(\mu, \nu) = \begin{cases} 1 & \text{if } \mu = 0 \text{ and } \nu = 0 \\ -1 & \text{if } \mu = \nu \text{ (and } \mu \neq 0) \\ 0 & \text{otherwise} \end{cases}$$ where $\mu, \nu \in \{0, 1, 2, 3\}$ are the indices corresponding to the covariant vector representation (`Color.down`). This corresponds to the standard Minkowski metric signature $(1, -1, -1, -1)$.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the contravariant Minkowski metric $\eta$ (representing $\eta^{\mu\nu}$) is equal to the complex Lorentz tensor constructed via the $\text{ofRat}$ map from the component function $f$ defined as: $$f(\mu, \nu) = \begin{cases} 1 & \text{if } \mu = 0 \text{ and } \nu = 0 \\ -1 & \text{if } \mu = \nu \text{ and } \mu \neq 0 \\ 0 & \text{otherwise} \end{cases}$$ where $\mu, \nu \in \{0, 1, 2, 3\}$ are the indices of the tensor components and the $\text{ofRat}$ map converts these rational components into the corresponding complex Lorentz tensor.

The invariant metric tensor $\epsilon_L$ for left-handed Weyl spinors (representing $\epsilon^{ab}$) is equal to the complex Lorentz tensor constructed via `ofRat` from the component function $f$ defined as: $$f(i, j) = \begin{cases} -1 & \text{if } i = 0 \text{ and } j = 1 \\ 1 & \text{if } i = 1 \text{ and } j = 0 \\ 0 & \text{otherwise} \end{cases}$$ where $i, j \in \{0, 1\}$ are the indices of the tensor components.

In the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the invariant metric tensor $\varepsilon_{L'}$ for left-handed "alt" Weyl spinors with covariant indices is equal to the tensor constructed from the rational component function $f$ via the $\text{ofRat}$ map, where: $$f(i, j) = \begin{cases} 1 & \text{if } i=0 \text{ and } j=1 \\ -1 & \text{if } i=1 \text{ and } j=0 \\ 0 & \text{otherwise} \end{cases}$$ The $\text{ofRat}$ map converts these rational components into the corresponding complex Lorentz tensor.

The right-handed spinor metric tensor $\varepsilon_R$ (representing the invariant tensor $\epsilon^{\dot{a}\dot{b}}$) is equal to the complex Lorentz tensor constructed from its rational components via the $\text{ofRat}$ map. Specifically, the component function $f$ assigns the value $-1$ to the multi-index $(0, 1)$, the value $1$ to the multi-index $(1, 0)$, and $0$ to all other multi-indices.

The invariant metric tensor $\varepsilon_{R'}$ for right-handed covariant Weyl spinors (representing $\epsilon_{\dot{a}\dot{b}}$ in dotted index notation) is equal to the complex Lorentz tensor constructed by the map `ofRat` from the component function $f$. This function $f$ assigns the value $1$ to the multi-index $(0, 1)$, the value $-1$ to the multi-index $(1, 0)$, and $0$ to all other multi-indices.

For any element $g \in SL(2, \mathbb{C})$, the covariant Minkowski metric $\eta_{\mu\nu}$ (represented by the tensor $\eta'$) is invariant under the group action of $g$, such that $g \cdot \eta' = \eta'$.

For any element $g \in SL(2, \mathbb{C})$, the contravariant Minkowski metric $\eta^{\mu\nu}$ (denoted as $\eta$) is invariant under the action of $SL(2, \mathbb{C})$, such that $g \cdot \eta = \eta$.

For any element $g$ of the group $SL(2, \mathbb{C})$, the metric tensor $\epsilon_L$ (also denoted as `leftMetric` or $\epsilon^{ab}$) associated with left-handed Weyl spinors is invariant under the group action of $g$. That is, $g \cdot \epsilon_L = \epsilon_L$.

For any element $g \in SL(2, \mathbb{C})$, the right-handed spinor metric tensor $\epsilon_R$ (also denoted as the invariant tensor $\epsilon^{\dot{a}\dot{b}}$ for dotted contravariant indices) is invariant under the group action of $SL(2, \mathbb{C})$. That is, $g \cdot \epsilon_R = \epsilon_R$.

For any element $g \in SL(2, \mathbb{C})$, the metric tensor for left-handed Weyl spinors $\epsilon_{L'}$ (represented by `altLeftMetric` in the `complexLorentzTensor` species, with covariant indices of color `downL`) is invariant under the group action of $g$. That is, $g \cdot \epsilon_{L'} = \epsilon_{L'}$.

For any element $g \in SL(2, \mathbb{C})$, the right-handed metric tensor $\epsilon_{R'}$ (representing the invariant metric $\epsilon_{\dot{a}\dot{b}}$ for right-handed covariant Weyl spinors) is invariant under the action of $g$, such that $g \cdot \epsilon_{R'} = \epsilon_{R'}$.