Physlib.Relativity.Tensors.RealTensor.Basic

The inductive type `Color` defines the set of labels for the complex representations of the group $SL(2, \mathbb{C})$ that are relevant to physical applications. In the context of tensor calculus, these "colors" are used to distinguish between different types of indices (such as vector or spinor indices) based on their transformation properties under $SL(2, \mathbb{C})$.

For the set of labels $\text{Color} = \{\text{up}, \text{down}\}$ used to categorize the transformation properties (indices) of Lorentz tensors, the equality relation between any two labels $x, y \in \text{Color}$ is decidable. That is, there exists an algorithmic procedure to determine whether $x = y$ or $x \neq y$.

This definition introduces the syntax $\mathbb{R}T[\dots]$ as a notation for representing real Lorentz tensors. It allows for a comma-separated list of terms within the brackets to define the structure or components of the tensor.

For a given spatial dimension $d \in \mathbb{N}$ and an index configuration $c$, the notation $\mathbb{R}T(d, c)$ denotes the space of real Lorentz tensors associated with the representation of the Lorentz group in $1+d$ dimensions, specifically representing the tensor product space $(realLorentzTensor \ d).Tensor \ c$.

For any spatial dimension $d \in \mathbb{N}$, the representation dimension of the "up" index color for real Lorentz tensors in $1+d$ dimensions is equal to $1+d$.

For any natural number $d$, the dimension of the representation associated with the covariant index (denoted by `Color.down`) for real Lorentz tensors in $d$ spatial dimensions is equal to $1 + d$.

For any spatial dimension $d \in \mathbb{N}$ and any color index $c$ belonging to the set of $SL(2, \mathbb{C})$ representation labels (`Color`), the dimension of the representation associated with $c$ in the real Lorentz tensor framework is $1 + d$.

For any natural number $d$, the basis of the representation associated with the color `Color.up` (representing contravariant indices) in the framework of real Lorentz tensors of dimension $d$ is equal to the contravariant basis `Lorentz.contrBasisFin` for that dimension.

For any natural number $d$, the basis associated with the covariant index type (represented by `Color.down`) in the framework of real Lorentz tensors is equal to the standard finite covariant basis `Lorentz.coBasisFin` for a $(d+1)$-dimensional spacetime.

For any dimension $d \in \mathbb{N}$, the object associated with the "up" color label within the $d$-dimensional real Lorentz tensor framework is the space of contravariant Lorentz vectors, denoted as $\text{Lorentz.Contr } d$. Here, the "up" color refers to a specific label in the inductive type `Color`, which distinguishes different types of indices (such as vector or spinor indices) based on their transformation properties under $SL(2, \mathbb{C})$.

For any natural number $d$, the object in the discrete functor $FD$ of the $d$-dimensional real Lorentz tensor structure corresponding to the covariant index color `Color.down` is equal to the space of covariant Lorentz vectors $\text{Lorentz.Co } d$.

For any natural number $d$, the duality map $\tau$ for real Lorentz tensors in $d$ dimensions sends the "up" index color to the "down" index color, expressed as $\tau(\text{up}) = \text{down}$.

For any spacetime dimension $d \in \mathbb{N}$, the duality map $\tau$ associated with the real Lorentz tensor framework maps the covariant color label `down` to the contravariant color label `up`. That is, $\tau(\text{down}) = \text{up}$.

For any spatial dimension $d$ and any representation color $c$ (representing a specific type of Lorentz representation, such as contravariant or covariant), let $V_c$ be the corresponding representation space and $V_{\tau(c)}$ be the representation space of its dual color $\tau(c)$. Let $\{e_i\}_i$ and $\{f_j\}_j$ be the standard bases for $V_c$ and $V_{\tau(c)}$ respectively. The contraction of the $i$-th basis vector of $V_c$ and the $j$-th basis vector of $V_{\tau(c)}$ is equal to the Kronecker delta $\delta_{ij}$, i.e., it is $1$ if $i = j$ and $0$ otherwise.

Let $d$ be the spatial dimension and $c: \{0, \dots, n+1\} \to \text{Color}$ be an index color sequence. For a real Lorentz tensor $t$ of rank $n+2$, suppose $i$ and $j$ are distinct indices such that the color at $j$ is the dual of the color at $i$ (i.e., $\tau(c_i) = c_j$). For any multi-index $b$ of the contracted tensor space, the $b$-th component of the contracted tensor $\text{contrT}_{i,j}(t)$ is given by the sum: $$[\text{contrT}_{i,j}(t)]_b = \sum_{x \in \text{DropPairSection}(b)} t_x \cdot \delta_{x_i, x_j}$$ where $t_x$ is the component of the original tensor $t$ at the multi-index $x$, $\text{DropPairSection}(b)$ is the set of all multi-indices $x$ of rank $n+2$ that reduce to $b$ when the $i$-th and $j$-th indices are removed, and $\delta_{x_i, x_j}$ is the Kronecker delta which is $1$ if the values of the indices at positions $i$ and $j$ in $x$ are equal, and $0$ otherwise.

For a spatial dimension $d$ and a Lorentz tensor $t$ of rank $n+2$ with index colors $c: \{0, \dots, n+1\} \to \text{Color}$, let $i$ and $j$ be distinct indices such that the color at $j$ is the dual of the color at $i$. For any multi-index $b$ of the contracted tensor of rank $n$, the $b$-th component of the contraction $\text{contrT}_{i,j}(t)$ is given by: $$[ \text{contrT}_{i,j}(t) ]_b = \sum_{x=0}^{d} t_{B(b, x, x)}$$ where $t_{B(b, x, x)}$ denotes the component of the original tensor $t$ at the multi-index $B(b, x, x)$, which is constructed by inserting the value $x$ into positions $i$ and $j$ of the multi-index $b$.