Physlib.Relativity.Tensors.ComplexTensor.Units.Pre

The contra-covariant unit tensor for complex Lorentz vectors, denoted as $\delta^\mu{}_\nu$, is an element of the tensor product space $V_{\text{contr}} \otimes V_{\text{co}}$, where $V_{\text{contr}}$ is the space of contravariant Lorentz vectors and $V_{\text{co}}$ is the space of covariant Lorentz vectors. This tensor corresponds to the identity matrix under the natural isomorphism between $(1,1)$-tensors and linear endomorphisms, and can be expressed as the sum $\sum_i e_i \otimes e^i$ over dual bases.

The contra-covariant unit tensor for complex Lorentz vectors $\delta^\mu{}_\nu$ can be expanded in terms of the contravariant basis $\{e_i\}$ and the covariant basis $\{e^j\}$ as the sum of their tensor products: $$\delta^\mu{}_\nu = e_0 \otimes e^0 + e_1 \otimes e^1 + e_2 \otimes e^2 + e_3 \otimes e^3$$ where $e_0$ (represented by `Sum.inl 0`) is the temporal basis vector and $e_1, e_2, e_3$ (represented by `Sum.inr 0, 1, 2`) are the spatial basis vectors.

The contra-covariant unit tensor for complex Lorentz vectors, $\delta^\mu{}_\nu$, is equal to the sum over the tensor products of the contravariant basis vectors $e_i$ and their corresponding covariant basis vectors $e^i$: $$\delta^\mu{}_\nu = \sum_i e_i \otimes e^i$$ where $e_i$ are the elements of the basis for the contravariant space $V_{\text{contr}}$ and $e^i$ are the elements of the basis for the covariant space $V_{\text{co}}$.

The morphism $\text{contrCoUnit} : \mathbb{1} \to V_{\text{contr}} \otimes V_{\text{co}}$ in the category of representations of $SL(2, \mathbb{C})$ maps a scalar $a \in \mathbb{C}$ (representing an element of the trivial representation $\mathbb{1}$) to the tensor $a \cdot \delta^\mu{}_\nu \in V_{\text{contr}} \otimes V_{\text{co}}$. Here, $\delta^\mu{}_\nu$ is the contra-covariant unit tensor (or Kronecker delta tensor), and the map manifests its invariance under the action of $SL(2, \mathbb{C})$.

Evaluating the linear map associated with the contra-covariant unit morphism $\text{contrCoUnit} : \mathbb{1} \to V_{\text{contr}} \otimes V_{\text{co}}$ at the scalar $1 \in \mathbb{C}$ yields the contra-covariant unit tensor $\delta^\mu{}_\nu$ (represented by `contrCoUnitVal`), where $V_{\text{contr}}$ and $V_{\text{co}}$ denote the spaces of contravariant and covariant complex Lorentz vectors, respectively.

The co-contra unit value for complex Lorentz vectors, usually denoted by $\delta_i^i$. It is an element of the underlying vector space of the tensor product of the complex covariant and contravariant Lorentz representations.

Let $e^i$ denote the basis elements of the complex covariant Lorentz representation (`complexCoBasis`) and $e_i$ denote the basis elements of the complex contravariant Lorentz representation (`complexContrBasis`). The co-contra unit value $\delta$ (represented by `coContrUnitVal`) can be expanded into the tensor products of these basis elements as follows: $$\delta = e^{\text{inl } 0} \otimes e_{\text{inl } 0} + e^{\text{inr } 0} \otimes e_{\text{inr } 0} + e^{\text{inr } 1} \otimes e_{\text{inr } 1} + e^{\text{inr } 2} \otimes e_{\text{inr } 2}$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

The co-contra unit value $\delta$ for complex Lorentz vectors (denoted by `coContrUnitVal`) is equal to the sum of the tensor products of the basis elements $e^i$ of the complex covariant Lorentz representation (`complexCoBasis`) and the basis elements $e_i$ of the complex contravariant Lorentz representation (`complexContrBasis`): $$\delta = \sum_{i} e^i \otimes e_i$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

The morphism $\text{coContrUnit} : \mathbb{1} \to V_{\text{co}} \otimes V_{\text{contr}}$ maps a complex scalar $a$ from the trivial representation $\mathbb{1}$ of $SL(2, \mathbb{C})$ to the element $a \cdot \delta^\mu_\nu$ in the tensor product of covariant and contravariant Lorentz representations. Here, $\delta^\mu_\nu$ (denoted by `coContrUnitVal`) represents the invariant identity tensor $\sum_i e_i \otimes e^i$, and the morphism manifests the invariance of this tensor under the action of $SL(2, \mathbb{C})$.

The application of the representation morphism $\text{coContrUnit} : \mathbb{1} \to V_{\text{co}} \otimes V_{\text{contr}}$ to the unit complex number $1 \in \mathbb{C}$ is equal to the invariant identity tensor $\text{coContrUnitVal}$ (often denoted $\delta^\mu_\nu$) in the tensor product of the complex covariant and contravariant Lorentz representations.

Let $V_{\text{co}}$ and $V_{\text{contr}}$ denote the complex covariant and contravariant Lorentz representations, respectively. Let $x \in V_{\text{co}}$ be a covariant Lorentz vector. Let $\text{contrCoUnit}(1) \in V_{\text{contr}} \otimes V_{\text{co}}$ be the contra-covariant unit tensor (often denoted $\delta^\mu{}_\nu$). The theorem states that if we take the tensor product $x \otimes \text{contrCoUnit}(1)$, apply the inverse associator $\alpha^{-1}$ to regroup the factors into $(V_{\text{co}} \otimes V_{\text{contr}}) \otimes V_{\text{co}}$, contract the first two factors using the co-contra contraction morphism $\text{coContrContraction}: V_{\text{co}} \otimes V_{\text{contr}} \to \mathbb{1}$, and finally apply the left unit isomorphism $\lambda: \mathbb{1} \otimes V_{\text{co}} \to V_{\text{co}}$, the result is the original vector $x$: $$\lambda_{V_{\text{co}}} \left( (\text{coContrContraction} \otimes \text{id}_{V_{\text{co}}}) \left( \alpha^{-1}_{V_{\text{co}}, V_{\text{contr}}, V_{\text{co}}} (x \otimes \text{contrCoUnit}(1)) \right) \right) = x$$ In index notation, this corresponds to the identity $x_\mu \delta^\mu{}_\nu = x_\nu$.

Let $V_{\text{contr}}$ and $V_{\text{co}}$ denote the complex contravariant and covariant Lorentz representations, respectively. Let $x \in V_{\text{contr}}$ be a contravariant Lorentz vector. Let $\text{coContrUnit}(1) \in V_{\text{co}} \otimes V_{\text{contr}}$ be the invariant identity tensor (often denoted $\delta_\mu^\nu$). The theorem states that if we take the tensor product $x \otimes \text{coContrUnit}(1)$, apply the inverse associator $\alpha^{-1}$ to regroup the factors into $(V_{\text{contr}} \otimes V_{\text{co}}) \otimes V_{\text{contr}}$, contract the first two factors using the contra-covariant contraction morphism $\text{contrCoContraction}: V_{\text{contr}} \otimes V_{\text{co}} \to \mathbb{1}$, and finally apply the left unit isomorphism $\lambda: \mathbb{1} \otimes V_{\text{contr}} \to V_{\text{contr}}$, the result is the original vector $x$: $$\lambda_{V_{\text{contr}}} \left( (\text{contrCoContraction} \otimes \text{id}_{V_{\text{contr}}}) \left( \alpha^{-1}_{V_{\text{contr}}, V_{\text{co}}, V_{\text{contr}}} (x \otimes \text{coContrUnit}(1)) \right) \right) = x$$ In index notation, this corresponds to the identity $x^\mu \delta_\mu^\nu = x^\nu$.

Let $V_{\text{co}}$ and $V_{\text{contr}}$ denote the complex covariant and contravariant Lorentz representations, respectively. Let $\beta_{V_{\text{co}}, V_{\text{contr}}} : V_{\text{co}} \otimes V_{\text{contr}} \to V_{\text{contr}} \otimes V_{\text{co}}$ be the braiding (symmetry) isomorphism in the category of representations that swaps the tensor factors. The contra-covariant unit tensor $\delta^\mu{}_\nu$ (the image of $1 \in \mathbb{C}$ under the morphism $\text{contrCoUnit}$) is equal to the image of the co-contra unit tensor $\delta_\nu^\mu$ (the image of $1 \in \mathbb{C}$ under the morphism $\text{coContrUnit}$) under the braiding isomorphism $\beta_{V_{\text{co}}, V_{\text{contr}}}$.

Let $V_{\text{co}}$ and $V_{\text{contr}}$ denote the complex covariant and contravariant Lorentz representations, respectively. Let $\beta_{V_{\text{contr}}, V_{\text{co}}} : V_{\text{contr}} \otimes V_{\text{co}} \to V_{\text{co}} \otimes V_{\text{contr}}$ be the braiding (symmetry) isomorphism in the category of representations that swaps the tensor factors. The co-contra unit tensor $\delta_\nu^\mu$ (the image of $1 \in \mathbb{C}$ under the morphism $\text{coContrUnit}$) is equal to the image of the contra-covariant unit tensor $\delta^\mu{}_\nu$ (the image of $1 \in \mathbb{C}$ under the morphism $\text{contrCoUnit}$) under the braiding isomorphism $\beta_{V_{\text{contr}}, V_{\text{co}}}$.