Physlib.Relativity.Tensors.ComplexTensor.Vector.Pre.Contraction

This definition establishes a $\mathbb{C}$-bilinear map between the space of complex contravariant Lorentz vectors and the space of complex covariant Lorentz vectors. For a contravariant vector $\psi$ and a covariant vector $\phi$, the map computes the scalar contraction: $$\psi \cdot \phi = \sum_{\mu=0}^3 \psi^\mu \phi_\mu$$ where $\psi^\mu$ and $\phi_\mu$ represent the components of the vectors in the standard spacetime basis (indexed by $\text{Fin } 1 \oplus \text{Fin } 3$). The map is constructed by taking the dot product of the component representations of the two vectors.

This is a $\mathbb{C}$-bilinear map $B : \text{complexCo} \times \text{complexContr} \to \mathbb{C}$ that represents the contraction of a covariant Lorentz vector $\phi$ with a contravariant Lorentz vector $\psi$. Given the component representations $\phi_\mu$ and $\psi^\mu$ (obtained via `toFin13ℂ`), the map returns their complex dot product, corresponding to the Einstein summation $\sum_{\mu=0}^3 \phi_\mu \psi^\mu$.

This definition defines a morphism of $SL(2, \mathbb{C})$-representations from the tensor product of the contravariant Lorentz representation (`complexContr`) and the covariant Lorentz representation (`complexCo`) to the trivial representation $\mathbb{C}$. Given a contravariant Lorentz vector $\psi$ and a covariant Lorentz vector $\phi$, the map is defined on pure tensors by the contraction (dot product) of their components: $$\psi \otimes \phi \mapsto \sum_{\mu=0}^3 \psi^\mu \phi_\mu$$ where $\psi^\mu$ and $\phi_\mu$ are the components in the standard spacetime basis. This map is an intertwining operator, meaning it is invariant under the action of $SL(2, \mathbb{C})$.

For any complex contravariant Lorentz vector $\psi$ and complex covariant Lorentz vector $\phi$, the value of the contraction morphism evaluated on the pure tensor $\psi \otimes \phi$ is equal to the dot product of their component representations $\psi^\mu$ and $\phi_\mu$: $$\text{contrCoContraction}(\psi \otimes \phi) = \sum_{\mu=0}^3 \psi^\mu \phi_\mu$$ Here, $\psi^\mu$ and $\phi_\mu$ are the components of the vectors in the standard spacetime basis (indexed by $\text{Fin } 1 \oplus \text{Fin } 3$).

Let $\{e^\mu\}_{\mu \in \{0, 1, 2, 3\}}$ be the standard basis for the space of complex contravariant Lorentz vectors (`complexContrBasisFin4`) and $\{e_\nu\}_{\nu \in \{0, 1, 2, 3\}}$ be the standard basis for the space of complex covariant Lorentz vectors (`complexCoBasisFin4`). For any indices $i, j \in \{0, 1, 2, 3\}$, the contraction morphism (denoted as `contrCoContraction`) evaluated on the tensor product of the $i$-th contravariant basis vector and the $j$-th covariant basis vector is equal to the Kronecker delta $\delta_{ij}$: $$\text{contrCoContraction}(e^i \otimes e_j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

For any spacetime indices $i, j \in \text{Fin } 1 \oplus \text{Fin } 3$, let $e^i$ be the $i$-th standard basis vector of the space of complex contravariant Lorentz vectors and $f_j$ be the $j$-th standard basis vector of the space of complex covariant Lorentz vectors. The Lorentz contraction morphism evaluated on the tensor product $e^i \otimes f_j$ is equal to 1 if $i = j$ and 0 otherwise: $$\text{contrCoContraction}(e^i \otimes f_j) = \delta_{ij}$$ where $\delta_{ij}$ is the Kronecker delta.

This definition characterizes the invariant contraction between a covariant Lorentz vector $\phi \in \text{complexCo}$ and a contravariant Lorentz vector $\psi \in \text{complexContr}$. It is a morphism in the category of representations of $SL(2, \mathbb{C})$ from the tensor product space $\text{complexCo} \otimes \text{complexContr}$ to the trivial representation $\mathbb{C}$. For any element $\phi \otimes \psi$, the map computes the scalar product $\sum_{\mu=0}^3 \phi_\mu \psi^\mu$, often written in index notation as $\phi_\mu \psi^\mu$. The definition includes a proof that this operation is intertwining, meaning it is invariant under the action of the Lorentz group (represented by $SL(2, \mathbb{C})$).

For a covariant Lorentz vector $\phi \in \text{complexCo}$ and a contravariant Lorentz vector $\psi \in \text{complexContr}$, the contraction morphism $\text{coContrContraction}$ applied to the tensor product $\phi \otimes \psi$ is equal to the dot product of their component representations in $\mathbb{C}^4$: $$\text{coContrContraction}(\phi \otimes \psi) = \sum_{\mu=0}^3 \phi_\mu \psi^\mu$$ where $\phi_\mu$ and $\psi^\mu$ are the complex components of the vectors obtained via the identification map `toFin13ℂ`.

For the standard basis of covariant Lorentz vectors $\{e_i\}_{i \in \{0, 1, 2, 3\}}$ (denoted by `complexCoBasisFin4`) and the standard basis of contravariant Lorentz vectors $\{f^j\}_{j \in \{0, 1, 2, 3\}}$ (denoted by `complexContrBasisFin4`), the invariant contraction morphism `coContrContraction` applied to their tensor product is given by the Kronecker delta $\delta_{ij}$: $$\text{coContrContraction}(e_i \otimes f^j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

For any indices $i, j \in \text{Fin } 1 \oplus \text{Fin } 3$ (representing the spacetime indices $\{0, 1, 2, 3\}$), let $e_i$ be the $i$-th basis vector of the covariant Lorentz representation (`complexCoBasis`) and $e^j$ be the $j$-th basis vector of the contravariant Lorentz representation (`complexContrBasis`). The invariant contraction morphism $\text{coContrContraction}$ applied to their tensor product is given by the Kronecker delta: $$\text{coContrContraction}(e_i \otimes e^j) = \delta_{ij}$$ where $\delta_{ij} = 1$ if $i = j$ and $0$ otherwise.

For a contravariant Lorentz vector $\phi \in \text{complexContr}$ and a covariant Lorentz vector $\psi \in \text{complexCo}$, the contraction morphism $\text{contrCoContraction}$ applied to the tensor product $\phi \otimes \psi$ is equal to the contraction morphism $\text{coContrContraction}$ applied to the tensor product $\psi \otimes \phi$. Mathematically, this identity is expressed as: $$\text{contrCoContraction}(\phi \otimes \psi) = \text{coContrContraction}(\psi \otimes \phi)$$ In terms of their complex components $\phi^\mu$ and $\psi_\mu$, this represents the symmetry (commutativity) of the invariant scalar product: $$\sum_{\mu=0}^3 \phi^\mu \psi_\mu = \sum_{\mu=0}^3 \psi_\mu \phi^\mu$$

For a covariant Lorentz vector $\phi$ and a contravariant Lorentz vector $\psi$, the contraction of $\phi$ and $\psi$ is symmetric with respect to the order of the arguments. Specifically, the contraction morphism $\text{coContrContraction}$ applied to the tensor product $\phi \otimes \psi$ is equal to the contraction morphism $\text{contrCoContraction}$ applied to the tensor product $\psi \otimes \phi$: $$\text{coContrContraction}(\phi \otimes \psi) = \text{contrCoContraction}(\psi \otimes \phi)$$ In terms of their complex components $\phi_\mu$ and $\psi^\mu$, this represents the commutativity of the Lorentz-invariant scalar product: $$\sum_{\mu=0}^3 \phi_\mu \psi^\mu = \sum_{\mu=0}^3 \psi^\mu \phi_\mu$$