Physlib.Relativity.Tensors.RealTensor.Units.Pre

For any spatial dimension $d \in \mathbb{N}$, the element `preContrCoUnitVal d` in the tensor product space of contravariant and covariant Lorentz vectors is given by the sum of the tensor products of their respective basis vectors: $$\text{preContrCoUnitVal } d = \sum_{i=0}^d e_i \otimes \epsilon^i$$ where $e_i$ represents the $i$-th basis vector of the contravariant Lorentz space (`contrBasis`) and $\epsilon^i$ represents the $i$-th basis vector of the covariant Lorentz space (`coBasis`).

For any spatial dimension $d \in \mathbb{N}$, let $\eta: \mathbb{R} \to V_{\text{contr}} \otimes V_{\text{co}}$ be the invariant unit morphism (denoted `preContrCoUnit d`) from the trivial representation to the tensor product of the contravariant and covariant Lorentz representations. The evaluation of the underlying linear map of $\eta$ at the scalar $1$ is equal to the invariant tensor $\text{preContrCoUnitVal } d$, which is defined as the sum of the tensor products of the basis vectors: $$\text{preContrCoUnitVal } d = \sum_{i=0}^d e_i \otimes \epsilon^i$$ where $e_i$ and $\epsilon^i$ are the standard contravariant and covariant basis vectors, respectively.

For a given natural number $d$ representing spatial dimensions, let $\text{Co } d$ and $\text{Contr } d$ be the covariant and contravariant representations of the Lorentz group, respectively. Let $\{\omega_i\}$ be the standard basis for $\text{Co } d$ and $\{v^i\}$ be the standard basis for $\text{Contr } d$, where the indices $i$ range from $0$ to $d$. The element $\text{preCoContrUnitVal } d$ in the tensor product space $\text{Co } d \otimes \text{Contr } d$ is given by the sum of the tensor products of these basis vectors: $$\text{preCoContrUnitVal } d = \sum_{i} \omega_i \otimes v^i$$

For a natural number $d$ representing the number of spatial dimensions, this definition specifies the unit morphism $\eta: \mathbb{R} \to V_{\text{co}} \otimes V_{\text{contr}}$ in the category of real representations of the Lorentz group $\mathcal{L} = \mathrm{O}(1, d)$. Here, $V_{\text{co}}$ is the covariant representation and $V_{\text{contr}}$ is the contravariant representation. The morphism maps the scalar $1$ to the invariant element $\sum_i \omega_i \otimes e^i$, where $\{\omega_i\}$ and $\{e^i\}$ are the dual bases for the covariant and contravariant spaces, respectively. This element corresponds to the Kronecker delta $\delta^i_j$, and the morphism manifests the invariance of this tensor under the action of the Lorentz group.

For a natural number $d$ representing spatial dimensions, let $\eta: \mathbb{R} \to V_{\text{co}} \otimes V_{\text{contr}}$ be the Lorentz co-contra unit morphism (denoted by `preCoContrUnit d`), where $V_{\text{co}}$ and $V_{\text{contr}}$ are the covariant and contravariant representations of the Lorentz group $\mathcal{L} = \mathrm{O}(1, d)$, respectively. This theorem states that the application of the linear map associated with $\eta$ to the scalar $1 \in \mathbb{R}$ results in the invariant tensor $\text{preCoContrUnitVal } d$, which is given by the sum of tensor products of the basis vectors: $$\eta(1) = \sum_{i} \omega_i \otimes v^i$$ where $\{\omega_i\}$ and $\{v^i\}$ are the standard bases for the covariant and contravariant spaces.

For any spatial dimension $d \in \mathbb{N}$, let $V_{\text{co}}$ and $V_{\text{contr}}$ denote the covariant and contravariant representations of the Lorentz group $\mathcal{L}$, respectively. Let $\eta: \mathbb{R} \to V_{\text{contr}} \otimes V_{\text{co}}$ be the unit morphism (denoted by `preContrCoUnit`) which maps the scalar $1$ to the invariant unit tensor $\sum_i e_i \otimes \epsilon^i$, where $\{e_i\}$ is the contravariant basis and $\{\epsilon^i\}$ is the covariant basis. For any covariant vector $x \in V_{\text{co}}$, contracting $x$ with the first component of this unit tensor returns $x$. Formally, if $\text{ev} : V_{\text{co}} \otimes V_{\text{contr}} \to \mathbb{R}$ is the contraction map, $\alpha^{-1}$ is the inverse associator, and $\lambda$ is the left unitor, the identity is: $$\lambda_{V_{\text{co}}} \left( (\text{ev} \otimes \text{id}_{V_{\text{co}}}) \left( \alpha^{-1}_{V_{\text{co}}, V_{\text{contr}}, V_{\text{co}}} (x \otimes \eta(1)) \right) \right) = x$$ In index notation, this corresponds to the contraction $x_\mu \delta^\mu{}_\nu = x_\nu$.

For a natural number $d$ representing spatial dimensions, let $V_{\text{contr}}$ and $V_{\text{co}}$ be the contravariant and covariant representations of the Lorentz group, respectively. Let $\eta: \mathbb{R} \to V_{\text{co}} \otimes V_{\text{contr}}$ be the Lorentz co-contra unit morphism that maps the scalar $1$ to the invariant tensor $\sum_{i} \omega_i \otimes e^i$ (where $\{\omega_i\}$ and $\{e^i\}$ are the dual bases), and let $\epsilon: V_{\text{contr}} \otimes V_{\text{co}} \to \mathbb{R}$ be the Lorentz-invariant contraction morphism. For any contravariant vector $x \in V_{\text{contr}}$, the following identity holds: \[ \lambda \left( (\epsilon \otimes \text{id}_{V_{\text{contr}}}) \left( \alpha^{-1} (x \otimes \eta(1)) \right) \right) = x \] where $\alpha^{-1}: V_{\text{contr}} \otimes (V_{\text{co}} \otimes V_{\text{contr}}) \cong (V_{\text{contr}} \otimes V_{\text{co}}) \otimes V_{\text{contr}}$ is the inverse associator and $\lambda: \mathbb{R} \otimes V_{\text{contr}} \cong V_{\text{contr}}$ is the left unitor in the category of representations. This theorem expresses that contracting a contravariant vector with the covariant part of the co-contra unit tensor reproduces the original vector.

For any spatial dimension $d \in \mathbb{N}$, let $V_{\text{co}}$ and $V_{\text{contr}}$ be the covariant and contravariant representations of the Lorentz group $\mathcal{L} = \mathrm{O}(1, d)$, respectively. Let $\eta_{\text{co,contr}}: \mathbb{R} \to V_{\text{co}} \otimes V_{\text{contr}}$ and $\eta_{\text{contr,co}}: \mathbb{R} \to V_{\text{contr}} \otimes V_{\text{co}}$ be the Lorentz unit morphisms that map the scalar $1 \in \mathbb{R}$ to the invariant tensors $\sum_i \omega_i \otimes v^i$ and $\sum_i v^i \otimes \omega_i$. This theorem states that the value of the contra-covariant unit at $1$ is equal to the image of the value of the co-contravariant unit at $1$ under the symmetry isomorphism $\beta_{V_{\text{co}}, V_{\text{contr}}}: V_{\text{co}} \otimes V_{\text{contr}} \to V_{\text{contr}} \otimes V_{\text{co}}$ that swaps the tensor factors: $$\eta_{\text{contr,co}}(1) = \beta_{V_{\text{co}}, V_{\text{contr}}}(\eta_{\text{co,contr}}(1))$$

For a given natural number $d$ representing the number of spatial dimensions, let $V_{\text{co}}$ and $V_{\text{contr}}$ be the covariant and contravariant representations of the Lorentz group $\mathcal{L} = \mathrm{O}(1, d)$, respectively. Let $\eta_{\text{co, contr}} \in V_{\text{co}} \otimes V_{\text{contr}}$ be the invariant unit tensor (the image of $1 \in \mathbb{R}$ under the co-contra unit morphism `preCoContrUnit d`) and $\eta_{\text{contr, co}} \in V_{\text{contr}} \otimes V_{\text{co}}$ be the invariant unit tensor (the image of $1 \in \mathbb{R}$ under the contra-co unit morphism `preContrCoUnit d`). The theorem states that the co-contra unit tensor is equal to the image of the contra-co unit tensor under the braiding (symmetry) isomorphism $\beta_{V_{\text{contr}}, V_{\text{co}}}: V_{\text{contr}} \otimes V_{\text{co}} \to V_{\text{co}} \otimes V_{\text{contr}}$ that swaps the tensor factors: $$\eta_{\text{co, contr}} = \beta_{V_{\text{contr}}, V_{\text{co}}}(\eta_{\text{contr, co}})$$ In terms of basis vectors, this expresses the symmetry between the invariant tensors $\sum_{i} \omega_i \otimes v^i$ and $\sum_{i} v^i \otimes \omega_i$.