Physlib.Relativity.Tensors.ComplexTensor.OfRat

A semilinear map, with respect to the conversion of rational complex numbers to complex numbers, that constructs a complex Lorentz tensor $\mathbb{C}T(c)$ from a map specifying its components $\prod_j \text{Fin}(\text{repDim}(c_j)) \to \text{RatComplexNum}$. All complex Lorentz tensors with rational coefficients with respect to the basis are of this form.

Let $n \in \mathbb{N}$ and $c: \text{Fin } n \to \text{Color}$ be a sequence of colors for a complex Lorentz tensor. Given a function $f: \text{ComponentIdx}(c) \to \text{RatComplexNum}$ that defines rational components, the representation of the complex Lorentz tensor $\text{ofRat}(f)$ with respect to the basis $\text{Tensor.basis}(c)$ at a specific index $b$ is equal to the complex number conversion of $f(b)$.

Let $n \in \mathbb{N}$ and $c: \text{Fin } n \to \text{Color}$ be a sequence of colors defining a complex Lorentz tensor space. For any component index $b \in \text{ComponentIdx}(c)$, the $b$-th basis vector of this space, $\text{Tensor.basis}(c)_b$, is equal to the tensor constructed by $\text{ofRat}$ from the indicator function $\delta_b$. Specifically, \[ \text{Tensor.basis}(c)_b = \text{ofRat}(f) \] where $f(b') = 1$ if $b' = b$ and $f(b') = 0$ otherwise.

For a color $c$ of a complex Lorentz tensor, and indices $i$ and $j$ bounded by the representation dimensions of $c$ and $\tau(c)$ respectively, the contraction of the tensor product of the $i$-th basis vector of $c$ and the $j$-th basis vector of $\tau(c)$, when cast to the complex field, evaluates to the complex number $1$ if $i = j$, and $0$ otherwise.

For any natural numbers $n$ and $n_1$ and sequences of colors $c: \text{Fin } n \to \text{Color}$ and $c_1: \text{Fin } n_1 \to \text{Color}$, let $f$ and $f_1$ be functions mapping tensor indices to rational complex numbers. The tensor product of the complex Lorentz tensors $\text{ofRat}(f)$ and $\text{ofRat}(f_1)$ is equal to the tensor constructed from the product of their coefficients. Specifically, \[ \text{prodT}(\text{ofRat}(f), \text{ofRat}(f_1)) = \text{ofRat}(g) \] where $g(b) = f(b_1) \cdot f_1(b_2)$, and $(b_1, b_2) = \text{ComponentIdx.prodEquiv}(b)$ is the natural decomposition of the product index $b$ into its constituent components for the two tensors.

Let $n \in \mathbb{N}$ and $c: \text{Fin}(n+2) \to \text{Color}$ be a sequence of colors for a complex Lorentz tensor. Let $i, j \in \text{Fin}(n+2)$ be distinct indices such that the color at position $j$ is the dual of the color at position $i$ (i.e., $\tau(c_i) = c_j$). For any function $f: \text{ComponentIdx}(c) \to \text{RatComplexNum}$ defining rational components, the contraction of the tensor $\text{ofRat}(f)$ at indices $i$ and $j$ is equal to the tensor constructed from the coefficients: $$\text{contrT}(n, i, j, h, \text{ofRat}(f)) = \text{ofRat} \left( b \mapsto \sum_{x \in \text{DropPairSection}(b)} f(x) \cdot \delta_{x_i, x_j} \right)$$ where $b$ is a multi-index for the contracted tensor, the sum is taken over all multi-indices $x$ of the original tensor that reduce to $b$ when the $i$-th and $j$-th components are removed, and $\delta_{x_i, x_j}$ is the Kronecker delta which evaluates to $1$ if the values of the indices $x$ at positions $i$ and $j$ are equal, and $0$ otherwise.

Let $n \in \mathbb{N}$ and $c: \text{Fin}(n+2) \to \text{Color}$ be a sequence of colors for a complex Lorentz tensor. Let $i, j \in \text{Fin}(n+2)$ be distinct indices such that the color at position $j$ is the dual of the color at position $i$ (i.e., $\tau(c_i) = c_j$). For any function $f: \text{ComponentIdx}(c) \to \text{RatComplexNum}$ defining the rational coefficients of a tensor, the contraction of the tensor $\text{ofRat}(f)$ at indices $i$ and $j$ is given by: $$\text{contrT}(n, i, j, h, \text{ofRat}(f)) = \text{ofRat} \left( b \mapsto \sum_{x \in \text{Fin}(\text{repDim}(c_i))} f(\Phi(b, x)) \right)$$ where $b$ is a multi-index for the resulting contracted tensor, $\text{repDim}(c_i)$ is the dimension of the representation associated with color $c_i$, and $\Phi(b, x)$ is the multi-index for the original tensor obtained by populating the $i$-th and $j$-th positions with the index $x$ and filling the remaining positions with the components of $b$.

Let $n, m$ be natural numbers, and let $c: \text{Fin } n \to \text{Color}$ and $c_1: \text{Fin } m \to \text{Color}$ be sequences of colors representing tensor signatures. Given a mapping $\sigma: \text{Fin } m \to \text{Fin } n$ that satisfies the permutation condition $h$, and a function $f: \text{ComponentIdx}(c) \to \text{RatComplexNum}$ that defines the rational components of a tensor, the permutation of the complex Lorentz tensor $\text{ofRat}(f)$ by $\sigma$ is equal to the tensor constructed from the components of $f$ reindexed via the inverse mapping. Specifically, $$\text{permT}(\sigma, h, \text{ofRat}(f)) = \text{ofRat} \left( b \mapsto f(i \mapsto \text{cast}(b(\sigma^{-1}(i)))) \right)$$ where $b$ is a multi-index for the resulting tensor and $\sigma^{-1}$ is the inverse mapping associated with the permutation condition $h$.