Physlib.Relativity.Tensors.RealTensor.ToComplex

The function maps a color $c$ of a real Lorentz tensor to the corresponding color of a complex Lorentz tensor. Specifically, it maps the contravariant color $\text{up}$ to the complex contravariant color $\text{up}$ and the covariant color $\text{down}$ to the complex covariant color $\text{down}$.

Let $c$ be a function assigning a real Lorentz color to each of $n$ tensor indices. If the color at the $j$-th index is the "up" color (representing a contravariant index), then the result of the mapping that transports real Lorentz colors to complex Lorentz colors (sending "up" to "up" and "down" to "down") is the complex "up" color.

Let $c$ be a function assigning a real Lorentz color to each of $n$ tensor indices. If the color at the $j$-th index is the "down" color (representing a covariant index), then the result of the mapping that transports real Lorentz colors to complex Lorentz colors (sending "up" to "up" and "down" to "down") is the complex "down" color.

Let $c$ be a function mapping each index of a rank-$n$ tensor to a real Lorentz color (either $\text{up}$ or $\text{down}$). For any index $j \in \{0, \dots, n-1\}$, the value of the composition $(\text{colorToComplex} \circ c)$ at $j$ is equal to the complex Lorentz color corresponding to $c(j)$. Specifically, if $c(j) = \text{up}$, the result is the complex $\text{up}$ color, and if $c(j) = \text{down}$, the result is the complex $\text{down}$ color.

For a tensor of rank $n$ with index colors $c = (c_0, \dots, c_{n-1})$ where each $c_j$ is a real Lorentz color (either "up" for contravariant or "down" for covariant), let $I_c$ denote the type of component multi-indices $\prod_{j=0}^{n-1} \{0, \dots, d_j-1\}$, where $d_j$ is the dimension of the representation space associated with color $c_j$. This definition establishes an equivalence (bijection) between the set of multi-indices for real Lorentz tensors $I_c$ and the set of multi-indices $I_{c'}$ for complex Lorentz tensors with colors $c' = \text{colorToComplex} \circ c$. Because the dimensions of corresponding real and complex representations are equal (e.g., both are 4 for vector representations), the map canonically identifies the individual index values.

Let $c = (c_0, \dots, c_{n-1})$ be the sequence of colors for a rank-$n$ real Lorentz tensor, where each $c_j$ is either a contravariant ("up") or covariant ("down") index. Let $f = (f_0, \dots, f_{n-1})$ be a component multi-index for this tensor. The map $\text{complexify}$ provides a bijection between the multi-indices of real Lorentz tensors and those of complex Lorentz tensors. For any index position $j \in \{0, \dots, n-1\}$, the $j$-th component of the complexified multi-index $\text{complexify}(f)$ is equal to the $j$-th component of the original multi-index $f_j$.

Let $c$ be the index structure (mapping each slot to a "color" such as contravariant or covariant) of a real Lorentz tensor of rank $n$. Let $f$ be a multi-index representing a component of this tensor, such that $f$ belongs to the set of component indices $\text{ComponentIdx}(c)$. Let $\text{complexify}$ be the canonical equivalence (bijection) that maps these real multi-indices to the corresponding multi-indices for complex Lorentz tensors. For any index position $j \in \{0, \dots, n-1\}$, the $j$-th component of the complexified multi-index $\text{complexify}(f)$ is equal to the $j$-th component of the multi-index obtained by explicitly applying the underlying function of the complexification equivalence to $f$.

Let $c: \{0, \dots, n-1\} \to \text{Color}$ be a sequence of real Lorentz colors (specifying whether each index is contravariant or covariant). The map `toComplex` is a semilinear map from the space of real Lorentz tensors $\mathbb{R}T(3, c)$ to the space of complex Lorentz tensors $\mathbb{C}T(\text{colorToComplex} \circ c)$, relative to the natural inclusion of real numbers into complex numbers $\mathbb{R} \hookrightarrow \mathbb{C}$. Given a real tensor $v$, let $v = \sum_i v_i e_i$ be its representation in the standard basis $\{e_i\}$ of $\mathbb{R}T(3, c)$, where $i$ denotes the multi-index of the components and $v_i \in \mathbb{R}$ are the scalar components. The map is defined such that: $$\text{toComplex}(v) = \sum_i v_i E_{\text{complexify}(i)}$$ where $\{E_j\}$ is the standard basis for the complexified tensor space and $\text{complexify}(i)$ is the bijection that identifies real multi-indices with their corresponding complex multi-indices. Effectively, this map embeds a real Lorentz tensor into the complexified tensor space by treating its real components as complex numbers.

Let $c: \{0, \dots, n-1\} \to \text{Color}$ be a sequence of real Lorentz colors and $v \in \mathbb{R}T(3, c)$ be a real Lorentz tensor. The complexification $\text{toComplex}(v)$ is the complex Lorentz tensor in the space $\mathbb{C}T(\text{colorToComplex} \circ c)$ defined by the expansion: $$\text{toComplex}(v) = \sum_i v_{\phi^{-1}(i)} E_i$$ where $\{E_i\}$ is the standard basis for the complex Lorentz tensor space, $v_j$ are the scalar components of $v$ with respect to the standard real Lorentz basis, and $\phi: \text{ComponentIdx}(c) \cong \text{ComponentIdx}(\text{colorToComplex} \circ c)$ is the canonical bijection identifying real multi-indices with complex multi-indices.

Let $c$ be a sequence of real Lorentz colors for tensors of rank $n$ and $v \in \mathbb{R}T(3, c)$ be a real Lorentz tensor. For any component multi-index $i$, let $v_i \in \mathbb{R}$ be the $i$-th scalar component of $v$ with respect to the canonical basis of the real Lorentz tensor space. Then the $\phi(i)$-th component of the complexified tensor $\text{toComplex}(v)$ with respect to the canonical complex basis is equal to $v_i$ coerced to a complex number, where $\phi$ is the canonical bijection between real and complex multi-indices.

Let $c$ be a sequence of real Lorentz colors for tensors of rank $n$. For any multi-index $i$ of the component indices, let $e_i$ be the $i$-th element of the canonical basis for the real Lorentz tensor space $\mathbb{R}T(3, c)$. The complexification map $\text{toComplex}$ maps $e_i$ to the corresponding basis element $E_{\text{complexify}(i)}$ of the complex Lorentz tensor space $\mathbb{C}T(\text{colorToComplex} \circ c)$: $$\text{toComplex}(e_i) = E_{\text{complexify}(i)}$$ where $\text{complexify}(i)$ is the multi-index $i$ viewed as an index for complex Lorentz tensors.

Let $c$ be a sequence of real Lorentz colors (specifying contravariant or covariant indices) for tensors of rank $n$. For any multi-index $b$ of the component indices, let $e_b = e_{b_0} \otimes \dots \otimes e_{b_{n-1}}$ be the associated pure basis vector—the tensor product of the basis vectors of the underlying real vector spaces—in the real Lorentz tensor space $\mathbb{R}T(3, c)$. The complexification map $\text{toComplex}$ maps this real pure basis vector to the corresponding complex pure basis vector $E_{\text{complexify}(b)}$ in the complex Lorentz tensor space $\mathbb{C}T(\text{colorToComplex} \circ c)$: $$\text{toComplex}(e_b) = E_{\text{complexify}(b)}$$ where $\text{complexify}(b)$ is the bijection identifying the real multi-index with its complex counterpart.

For any natural number $n$, any sequence of real Lorentz colors $c: \{0, \dots, n-1\} \to \text{Color}$, any real scalar $r \in \mathbb{R}$, and any real Lorentz tensor $t$ of type $c$, the complexification map $\text{toComplex}$ satisfies: $$\text{toComplex}(r \cdot t) = r \cdot \text{toComplex}(t)$$ where the scalar multiplication on the right-hand side treats $r$ as a complex number.

For any natural number $n$ and any sequence of real Lorentz colors $c: \{0, \dots, n-1\} \to \text{Color}$, let $v$ be a real Lorentz tensor in the space $\mathbb{R}T(3, c)$. The complexification of this tensor, denoted $\text{toComplex}(v)$, is equal to zero if and only if $v = 0$.

For any natural number $n$ and any sequence of real Lorentz colors $c: \{0, \dots, n-1\} \to \text{Color}$ (specifying contravariant or covariant indices), the complexification map $\text{toComplex}: \mathbb{R}T(3, c) \to \mathbb{C}T(\text{colorToComplex} \circ c)$, which embeds real Lorentz tensors into the corresponding complexified tensor space, is injective.

Let $c$ be a sequence of $n$ Lorentz colors. Suppose the $k$-th color is contravariant, i.e., $c_k = \text{up}$. For any $\Lambda \in SL(2, \mathbb{C})$, let $L$ be the corresponding real Lorentz transformation. Given a multi-index $b$ for a real tensor and a multi-index $i$ for a complexified tensor, the matrix element of the representation of $L$ acting on the $k$-th index slot (embedded in $\mathbb{C}$) is equal to the entry of the complexified Lorentz matrix $L$ at row $i_k$ and column $b_k$: $$\left( [L]^{i_k}_{\phantom{i_k}b_k} \right)_{\mathbb{C}} = (L_{\mathbb{C}})^{i_k}_{\phantom{i_k}b_k}$$ where $[L]^i_j$ denotes the matrix elements of the Lorentz transformation acting on the standard basis of the real vector representation.

This lemma isolates the matrix calculation for a covariant ("down") slot in the proof that the complexification map for Lorentz tensors is equivariant. For a real Lorentz tensor of rank $n$ with index colors $c_0, \dots, c_{n-1}$, suppose the $k$-th index is covariant ($c_k = \text{down}$). Let $\Lambda \in SL(2, \mathbb{C})$ and let $L(\Lambda)$ be the corresponding $4 \times 4$ real Lorentz transformation. Let $\{e_0, \dots, e_3\}$ be the standard basis for the covariant representation space. The theorem states that the $i_k$-th component of the transformed basis vector $\rho(L(\Lambda)) e_{b_k}$ (coerced to a complex number) is equal to the $(b_k, i_k)$ entry of the inverse Lorentz matrix $(L(\Lambda))^{-1}$: $$[\rho(L(\Lambda))]^{i_k}_{b_k} = (L(\Lambda)^{-1})_{b_k i_k}$$ where $b_k$ denotes the basis vector index and $i_k$ denotes the component index for the $k$-th slot.

Let $c: \{0, \dots, n-1\} \to \text{Color}$ be a sequence of real Lorentz colors specifying the index structure of a tensor. For any real Lorentz tensor $v \in \mathbb{R}T(3, c)$ and any element $\Lambda \in SL(2, \mathbb{C})$, the complexification map $\text{toComplex}$ satisfies: $$\Lambda \cdot \text{toComplex}(v) = \text{toComplex}(L(\Lambda) \cdot v)$$ where $L(\Lambda)$ is the real Lorentz transformation corresponding to $\Lambda$. This demonstrates that the map from real to complex Lorentz tensors is equivariant with respect to the action of the Lorentz group (viewed via its universal cover $SL(2, \mathbb{C})$).

Let $n, m \in \mathbb{N}$ be natural numbers. Let $c: \{0, \dots, n-1\} \to \text{Color}_{\mathbb{R}}$ and $c_1: \{0, \dots, m-1\} \to \text{Color}_{\mathbb{R}}$ be maps assigning colors to indices of real Lorentz tensors. Suppose a map $\sigma: \{0, \dots, m-1\} \to \{0, \dots, n-1\}$ satisfies the permutation condition for $c$ and $c_1$, meaning that $\sigma$ is a bijection and $c(\sigma(i)) = c_1(i)$ for all $i \in \{0, \dots, m-1\}$. Then $\sigma$ also satisfies the permutation condition for the complexified color maps $\text{colorToComplex} \circ c$ and $\text{colorToComplex} \circ c_1$.

Let $c: \{0, \dots, n-1\} \to \text{Color}$ and $c_1: \{0, \dots, m-1\} \to \text{Color}$ be sequences of colors (representing index types) for real Lorentz tensors. Let $\sigma: \{0, \dots, m-1\} \to \{0, \dots, n-1\}$ be a map satisfying the permutation condition $c \circ \sigma = c_1$, which implies $\sigma$ is a bijection. For any multi-index $b$ in the component index set of $c$, let $e_b$ denote the corresponding canonical basis element of the real Lorentz tensor space. The permutation operator $\text{permT}_\sigma$ maps the basis element $e_b$ to the basis element $e_{b \circ \sigma}$ in the target tensor space, where the components of the permuted multi-index are $(b \circ \sigma)_j = b_{\sigma(j)}$ for $j \in \{0, \dots, m-1\}$. $$\text{permT}_\sigma(e_b) = e_{b \circ \sigma}$$

Let $c: \{0, \dots, n-1\} \to \text{Color}$ and $c_1: \{0, \dots, m-1\} \to \text{Color}$ be sequences of colors representing representations of $SL(2, \mathbb{C})$. Let $\sigma: \{0, \dots, m-1\} \to \{0, \dots, n-1\}$ be a map satisfying the permutation condition $c_1 = c \circ \sigma$ (which implies $n = m$ and $\sigma$ is a bijection). For any multi-index $b$ in the component index set $\text{ComponentIdx}(c)$, let $e_b$ denote the canonical basis element of the tensor space associated with $c$. The action of the permutation operator $\text{permT}$ on the basis element $e_b$ is given by: $$\text{permT}(\sigma, h, e_b) = e_{b \circ \sigma}$$ where $e_{b \circ \sigma}$ is the basis element in the target tensor space associated with $c_1$, indexed by the permuted multi-index $(b \circ \sigma)_j = b_{\sigma(j)}$ for $j \in \{0, \dots, m-1\}$.

Let $c: \{0, \dots, n-1\} \to \text{Color}_{\mathbb{R}}$ and $c_1: \{0, \dots, m-1\} \to \text{Color}_{\mathbb{R}}$ be sequences of colors for real Lorentz tensors. Let $\sigma: \{0, \dots, m-1\} \to \{0, \dots, n-1\}$ be a map satisfying the permutation condition $c \circ \sigma = c_1$ (which implies $\sigma$ is a bijection and $n = m$). For any real Lorentz tensor $t \in \mathbb{R}T(3, c)$, the complexification map $\text{toComplex}$ commutes with the permutation operator $\text{permT}_{\sigma}$: $$\text{toComplex}(\text{permT}_{\sigma}(t)) = \text{permT}_{\sigma}(\text{toComplex}(t))$$ where the permutation on the right-hand side is performed on the complexified tensor space $\mathbb{C}T(\text{colorToComplex} \circ c)$.

For any natural numbers $n$ and $m$, and for any sequences of real Lorentz tensor colors $c: \text{Fin } n \to \text{realLorentzTensor.Color}$ and $c_1: \text{Fin } m \to \text{realLorentzTensor.Color}$, the map $\text{colorToComplex}$ commutes with the concatenation of color sequences. That is, the complexification of the concatenated sequence $\text{colorToComplex} \circ (\text{Fin.append } c \ c_1)$ is equal to the concatenation of the complexified sequences $\text{Fin.append } (\text{colorToComplex} \circ c) \ (\text{colorToComplex} \circ c_1)$.

For any natural numbers $n$ and $m$, and sequences of real Lorentz tensor colors $c: \text{Fin } n \to \text{realLorentzTensor.Color}$ and $c_1: \text{Fin } m \to \text{realLorentzTensor.Color}$, the identity map $\text{id} : \text{Fin}(n+m) \to \text{Fin}(n+m)$ satisfies the permutation condition $\text{PermCond}$ between the concatenation of the complexified color sequences, $\text{Fin.append } (\text{colorToComplex} \circ c) \ (\text{colorToComplex} \circ c_1)$, and the complexification of the concatenated real color sequences, $\text{colorToComplex} \circ (\text{Fin.append } c \ c_1)$.

Given sequences of real Lorentz colors $c: \text{Fin } n \to \text{Color}_{\mathbb{R}}$ and $c_1: \text{Fin } m \to \text{Color}_{\mathbb{R}}$, this function defines a bilinear map that takes a complex Lorentz tensor $x$ with color configuration $\text{colorToComplex} \circ c$ and a complex Lorentz tensor $y$ with color configuration $\text{colorToComplex} \circ c_1$, and returns their tensor product $x \otimes y$. The resulting tensor is associated with the color configuration $\text{colorToComplex} \circ (c \oplus c_1)$, where $c \oplus c_1$ denotes the concatenation of the color sequences. This is achieved by taking the standard tensor product of $x$ and $y$ and applying a permutation (specifically the identity map) to reconcile the representation of the indices.

Let $n, m \in \mathbb{N}$ be natural numbers, and let $c: \text{Fin } n \to \text{Color}_{\mathbb{R}}$ and $c_1: \text{Fin } m \to \text{Color}_{\mathbb{R}}$ be sequences of real Lorentz tensor colors. For any component multi-indices $b$ and $b_1$ corresponding to these colors, the complexification of their concatenated product $b \cdot b_1$ is equal to the concatenated product of their individual complexifications. That is, \[ \text{complexify}(b \cdot b_1) = \text{complexify}(b) \cdot \text{complexify}(b_1). \] Note that the equality holds because the complexification of the concatenated color sequence $\text{colorToComplex} \circ (c \mathbin{+\!+} c_1)$ is identical to the concatenation of the complexified color sequences $(\text{colorToComplex} \circ c) \mathbin{+\!+} (\text{colorToComplex} \circ c_1)$.

Let $n, m \in \mathbb{N}$ be natural numbers, and let $c: \text{Fin } n \to \text{Color}_{\mathbb{R}}$ and $c_1: \text{Fin } m \to \text{Color}_{\mathbb{R}}$ be sequences of real Lorentz tensor colors. For any real Lorentz tensors $t \in \mathbb{R}T(3, c)$ and $t_1 \in \mathbb{R}T(3, c_1)$, the complexification of their tensor product is equal to the tensor product of their individual complexifications. That is, \[ \text{toComplex}(t \otimes t_1) = \text{toComplex}(t) \otimes_{\mathbb{C}} \text{toComplex}(t_1) \] where $\otimes$ denotes the tensor product of real Lorentz tensors (`prodT`) and $\otimes_{\mathbb{C}}$ denotes the tensor product for complex Lorentz tensors (`prodTColorToComplex`).

For any color $x$ of a real Lorentz tensor (representing representation types such as contravariant "up" or covariant "down"), the duality involution $\tau$ commutes with the mapping $\text{colorToComplex}$ which transports real Lorentz colors to complex ones. Specifically, the identity holds: \[ \tau_{\mathbb{C}}(\text{colorToComplex}(x)) = \text{colorToComplex}(\tau_{\mathbb{R}}(x)), \] where $\tau_{\mathbb{R}}$ and $\tau_{\mathbb{C}}$ are the duality involutions for real and complex Lorentz tensors, respectively.

Let $n$ be a natural number, and let $c: \{0, \dots, n+1\} \to \text{Color}_{\mathbb{R}}$ be a sequence of real Lorentz colors (specifying the representation type of each tensor index). Let $\epsilon_{i,j}: \{0, \dots, n-1\} \to \{0, \dots, n+1\}$ be the `dropPairEmb` map that skips the indices $i$ and $j$. Given a multi-index $b$ for a tensor of rank $n+2$ with index structure $c$, the operation `complexify` (which maps real component indices to their complex equivalents) commutes with restricting the multi-index to the indices selected by $\epsilon_{i,j}$. Specifically, for any $m \in \{0, \dots, n-1\}$, the $m$-th index of the complexified restricted multi-index is equal to the index at position $\epsilon_{i,j}(m)$ of the complexified full multi-index: \[ (\text{complexify}(b \circ \epsilon_{i,j}))_m = (\text{complexify}(b))_{\epsilon_{i,j}(m)} \]

Let $n$ be a natural number and $c: \{0, \dots, n+1\} \to \text{Color}_{\mathbb{R}}$ be a sequence of $n+2$ real Lorentz colors. For any two distinct indices $i, j$ such that the color at $j$ is the dual of the color at $i$ (i.e., $c(j) = \tau(c(i))$), and for any multi-index $b$ of the component indices, let $e_b$ be the corresponding basis vector in the space of real pure Lorentz tensors. Then, the complexification of the contraction of $e_b$ at indices $i$ and $j$ is equal to the contraction of the complexified basis vector $E_{\text{complexify}(b)}$ at the same indices: $$\text{toComplex}(\text{contrP}_{i,j}(e_b)) = \text{contrP}_{i,j}(E_{\text{complexify}(b)})$$ where $\text{toComplex}$ is the semilinear map from real to complex Lorentz tensors, $\text{contrP}_{i,j}$ is the contraction operator for pure tensors, and $\text{complexify}(b)$ is the multi-index $b$ viewed as an index for complex Lorentz tensors.

Let $n$ be a natural number and $c: \{0, \dots, n+1\} \to \text{Color}_{\mathbb{R}}$ be a sequence of $n+2$ real Lorentz colors (specifying whether each index is contravariant or covariant). Given two distinct indices $i, j \in \{0, \dots, n+1\}$ such that the color at $j$ is the dual of the color at $i$ (i.e., $\tau(c(i)) = c(j)$), let $t$ be a real Lorentz tensor of rank $n+2$ with color sequence $c$. The complexification of the contraction of $t$ at indices $i$ and $j$ is equal to the contraction of the complexified tensor $\text{toComplex}(t)$ at the same indices: $$\text{toComplex}(\text{contrT}_{i,j}(t)) = \text{contrT}_{i,j}(\text{toComplex}(t))$$ where $\text{contrT}_{i,j}$ denotes the tensor contraction operator and $\text{toComplex}$ is the canonical semilinear map from real Lorentz tensors to complex Lorentz tensors.

Let $n$ be a natural number and $c : \{0, \dots, n\} \to \text{realLorentzTensor.Color}$ be a sequence of real Lorentz colors (such as "up" or "down"). Let $b$ be a multi-index (an element of type `ComponentIdx`) for a tensor with the index structure $c$. For any index $i \in \{0, \dots, n\}$, let $i.\text{succAbove} : \{0, \dots, n-1\} \to \{0, \dots, n\}$ be the function that skips the index $i$. Then for any $m \in \{0, \dots, n-1\}$, the $m$-th component of the complexified multi-index derived from the reduced index structure $b \circ i.\text{succAbove}$ is equal to the $(i.\text{succAbove}(m))$-th component of the complexified multi-index of the original $b$. That is: \[ (\text{complexify}(b \circ i.\text{succAbove}))_m = (\text{complexify } b)_{i.\text{succAbove}(m)} \] where $\text{complexify}$ is the canonical bijection between the sets of multi-indices for real and complex Lorentz tensors.

In the framework of complex Lorentz tensors, the dimension of the representation space corresponding to the contravariant Lorentz vector color (denoted by `Color.up`) is $4$.

In the framework of complex Lorentz tensors, the dimension of the representation space corresponding to the covariant Lorentz vector color (denoted by `Color.down`) is $4$.

Given a sequence of $n+1$ real Lorentz colors $c: \{0, \dots, n\} \to \text{Color}_{\mathbb{R}}$, a specific index position $i \in \{0, \dots, n\}$, and a component index $b \in \{0, \dots, \text{dim}(c_i) - 1\}$, the function maps $b$ to the corresponding index in the complexified representation space of dimension $\text{dim}(\text{colorToComplex}(c_i))$. This conversion is a canonical cast, as the dimensions for both real and complex Lorentz vector representations are equal to 4.

Let $c: \{0, \dots, n\} \to \text{Color}_{\mathbb{R}}$ be a sequence of colors for real Lorentz tensors. Given a specific index position $i \in \{0, \dots, n\}$ and a basis index $b$ for the representation space corresponding to the real color $c_i$, this function defines a linear map acting on complex Lorentz tensors of color sequence $\text{colorToComplex} \circ c$. It evaluates the $i$-th index of the tensor at the complexified basis index corresponding to $b$ (using `evalIdxToComplex`). The resulting tensor is a complex Lorentz tensor of rank $n$ with the color sequence $\text{colorToComplex} \circ (c \circ i.\text{succAbove})$, which represents the complexification of the original color sequence with the $i$-th entry removed.

Let $c: \{0, \dots, n\} \to \text{Color}_{\mathbb{R}}$ be a sequence of real Lorentz colors. For any index position $i \in \{0, \dots, n\}$, any basis index $b \in \{0, \dots, \text{dim}(c_i) - 1\}$, and any multi-index $b' \in \text{ComponentIdx}(c)$, let $e_{b'}$ be the corresponding real pure basis vector in the real Lorentz tensor space $\mathbb{R}T(3, c)$ and $E_{\text{complexify}(b')}$ be the corresponding complex pure basis vector in $\mathbb{C}T(\text{colorToComplex} \circ c)$. Then the complexification of the evaluation of the $i$-th index of $e_{b'}$ at $b$ is equal to the evaluation of the $i$-th index of $E_{\text{complexify}(b')}$ at the complexified index $\text{complexify}(b)$: $$\text{toComplex}(\text{evalP}_i(b, e_{b'})) = \text{evalP}_i(\text{complexify}(b), E_{\text{complexify}(b')})$$ (Note: On the right-hand side, a trivial permutation $\text{permT}$ with the identity map is applied to align the tensor index types).

Let $c: \{0, \dots, n\} \to \text{Color}_{\mathbb{R}}$ be a sequence of real Lorentz colors. For any index position $i \in \{0, \dots, n\}$, basis index $b$ for the representation associated with the color $c_i$, and real Lorentz tensor $t$ of type $c$, the complexification of the tensor resulting from evaluating the $i$-th index of $t$ at $b$ is equal to the evaluation of the complexified tensor $\text{toComplex}(t)$ at the same index: $$\text{toComplex}(\text{evalT}_i^b(t)) = \text{evalT}_{\mathbb{C}, i}^b(\text{toComplex}(t))$$ where $\text{evalT}_i^b$ is the $i$-th index evaluation operator for real tensors, and $\text{evalT}_{\mathbb{C}, i}^b$ is the corresponding evaluation operator acting on complexified Lorentz tensors.