Physlib.Relativity.Tensors.ComplexTensor.OfGaussianInt

For a given sequence of $n$ representation colors $c: \{0, \dots, n-1\} \to \text{Color}$, this map constructs a complex Lorentz tensor from a set of Gaussian integer components. Given a function $f: \text{ComponentIdx}(c) \to \mathbb{Z}[i]$ that assigns a Gaussian integer to each multi-index $b$, the resulting tensor in the space $\mathbb{C}T(c)$ is defined as the sum: $$\sum_{b \in \text{ComponentIdx}(c)} \iota(f(b)) \cdot e_b$$ where $e_b$ is the standard basis tensor associated with the multi-index $b$, and $\iota: \mathbb{Z}[i] \hookrightarrow \mathbb{C}$ is the canonical inclusion of the Gaussian integers into the complex numbers. This map is semi-linear with respect to the inclusion $\iota$.

Let $c$ be a sequence of $n$ representation colors for complex Lorentz tensors and $f: \text{ComponentIdx}(c) \to \mathbb{Z}[i]$ be a function mapping each multi-index to a Gaussian integer. If $T$ is the complex Lorentz tensor constructed from these Gaussian integer components (i.e., $T = \sum_{b'} \iota(f(b')) e_{b'}$, where $e_{b'}$ is the standard basis and $\iota: \mathbb{Z}[i] \hookrightarrow \mathbb{C}$ is the canonical inclusion), then for any multi-index $b$, the coordinate of $T$ with respect to the basis element $e_b$ is exactly $\iota(f(b))$.

Let $c: \{0, \dots, n-1\} \to \text{Color}$ be a sequence of representation colors (such as Lorentz vector or Weyl spinor indices) and let $\mathbb{C}T(c)$ be the associated complex Lorentz tensor space. Let $\{e_b\}_{b \in \text{ComponentIdx}(c)}$ denote the standard basis for this space. For any specific multi-index $b$, the basis tensor $e_b$ is equal to the complex Lorentz tensor constructed from the Gaussian integer-valued component function $f(b') = \delta_{b b'}$, where $\delta_{b b'}$ is the Kronecker delta (equal to 1 if $b = b'$ and 0 otherwise).

Let $c$ be a representation color for the complex Lorentz tensor species (such as Weyl spinors or Lorentz vectors), and let $\tau(c)$ be its dual color. Let $\{e_k\}_k$ denote the standard basis for the complex vector space associated with $c$, and let $\{e'_k\}_k$ denote the standard basis for the dual space associated with $\tau(c)$. For any basis indices $i$ and $j$, the contraction of the tensor product $e_i \otimes e'_j$ is given by the Kronecker delta: $$\text{contr}_c(e_i \otimes e'_j) = \delta_{ij}$$ where the result is understood as a Gaussian integer embedded in the complex numbers $\mathbb{C}$.

Let $c$ and $c_1$ be sequences of representation colors for complex Lorentz tensors of lengths $n$ and $n_1$, respectively. Given two maps $f: \text{ComponentIdx}(c) \to \mathbb{Z}[i]$ and $f_1: \text{ComponentIdx}(c_1) \to \mathbb{Z}[i]$ that define the Gaussian integer components of two tensors, the tensor product of these tensors is given by: $$\text{prodT}(\text{ofGaussianInt}(f), \text{ofGaussianInt}(f_1)) = \text{ofGaussianInt}(g)$$ where the component map $g$ for the concatenated color sequence is defined by $g(b) = f(b_1) \cdot f_1(b_2)$, where $b_1$ and $b_2$ are the sub-indices of the multi-index $b$ corresponding to the original color sequences $c$ and $c_1$.

Let $c$ be a sequence of $n+2$ representation colors for complex Lorentz tensors, and let $i, j \in \{0, \dots, n+1\}$ be two distinct indices such that the representation at index $j$ is dual to the representation at index $i$ (i.e., $\tau(c_i) = c_j$). For a function $f: \text{ComponentIdx}(c) \to \mathbb{Z}[i]$ that assigns a Gaussian integer to each multi-index, let $T = \text{ofGaussianInt}(f)$ be the corresponding complex Lorentz tensor. The contraction of $T$ at indices $i$ and $j$ is a tensor whose components are also Gaussian integers. Specifically, for any multi-index $b$ in the contracted index space, the component is given by the sum: $$\sum_{x \in \text{DropPairSection}_{i,j}(b)} f(x) \cdot \delta_{x_i, x_j}$$ where $x_i$ and $x_j$ denote the values of the $i$-th and $j$-th indices of the multi-index $x$, and $\delta_{x_i, x_j}$ is the Kronecker delta (equal to $1$ if the index values are equal and $0$ otherwise).

Let $c: \{0, \dots, n+1\} \to \text{Color}$ be a sequence of representation colors for complex Lorentz tensors. Let $i$ and $j$ be two distinct indices in $\{0, \dots, n+1\}$ such that the representation color at index $j$ is the dual of the color at index $i$ (i.e., $\tau(c_i) = c_j$). For any function $f: \text{ComponentIdx}(c) \to \mathbb{Z}[i]$ that defines the components of a tensor $T = \text{ofGaussianInt}(f)$, the contraction of $T$ at indices $i$ and $j$ is a tensor whose components are also Gaussian integers, determined by the formula: $$\text{contrT}_{i,j}(\text{ofGaussianInt}(f)) = \text{ofGaussianInt} \left( b \mapsto \sum_{x=0}^{d_i-1} f(\text{ext}_{i,j}(b, x, x)) \right)$$ where $d_i$ is the dimension of the representation space associated with color $c_i$, $b$ is a multi-index of length $n$, and $\text{ext}_{i,j}(b, x, x)$ is the multi-index of length $n+2$ formed by inserting the value $x$ at both the $i$-th and $j$-th positions of the multi-index $b$, while preserving the relative order of the components of $b$ in the remaining positions.

Let $c: \{0, \dots, n-1\} \to \text{Color}$ and $c_1: \{0, \dots, m-1\} \to \text{Color}$ be two sequences of representation colors for complex Lorentz tensors. Let $\sigma: \{0, \dots, m-1\} \to \{0, \dots, n-1\}$ be a reindexing map that is a bijection and satisfies $c \circ \sigma = c_1$. For any function $f: \text{ComponentIdx}(c) \to \mathbb{Z}[i]$ mapping multi-indices to Gaussian integers, let $\text{ofGaussianInt}(f)$ be the complex Lorentz tensor defined by these components. The permutation of this tensor by $\sigma$ is given by: $$\text{permT}(\sigma, \text{ofGaussianInt}(f)) = \text{ofGaussianInt}(g)$$ where $g: \text{ComponentIdx}(c_1) \to \mathbb{Z}[i]$ is the function defined by $g(b) = f(b \circ \sigma^{-1})$ for any multi-index $b \in \text{ComponentIdx}(c_1)$, and $\sigma^{-1}$ denotes the inverse of the reindexing map.