Physlib.QFT.PerturbationTheory.WickContraction.Erase

Given a Wick contraction $c$ on $n+1$ elements and an index $i \in \{0, \dots, n\}$, the function `WickContraction.erase` produces a Wick contraction on $n$ elements. This is obtained by removing the index $i$ and any pair in $c$ that contains $i$. The remaining indices in $\{0, \dots, n\} \setminus \{i\}$ are relabeled to $\{0, \dots, n-1\}$ using the unique order-preserving bijection (the inverse of `i.succAbove`). If $i$ was originally contracted with some $j$ in $c$, then $j$ becomes uncontracted in the resulting contraction.

For a Wick contraction $c$ on $n+1$ indices and an index $i \in \{0, \dots, n\}$, let $c.\text{erase } i$ be the Wick contraction on $n$ indices obtained by removing $i$ and any pair containing $i$. For any index $j \in \{0, \dots, n-1\}$, $j$ is uncontracted in $c.\text{erase } i$ if and only if the corresponding index $i.\text{succAbove}(j)$ was either already uncontracted in $c$ or was paired with $i$ in $c$.

Let $c$ be a Wick contraction on $n+1$ indices $\{0, \dots, n\}$. Suppose an index $i$ is contracted in $c$, and let $j$ be the index it is paired with (so that $\{i, j\} \in c$). For any pair of indices $a$ that is a contraction in $c$, if $a$ is not the pair $\{i, j\}$, then there exists a pair $a'$ in the erased contraction $c.\text{erase } i$ such that $a$ is the image of $a'$ under the order-preserving embedding $i.\text{succAbove} : \{0, \dots, n-1\} \to \{0, \dots, n\} \setminus \{i\}$.

Let $c$ be a Wick contraction on $n+1$ indices $\{0, \dots, n\}$. Suppose an index $i \in \{0, \dots, n\}$ is uncontracted in $c$ (i.e., $\text{getDual? } i = \text{none}$). Then for any pair of indices $a$ that belongs to the contraction $c$, there exists a pair $a'$ in the erased contraction $c.\text{erase } i$ such that $a$ is the image of $a'$ under the order-preserving embedding $i.\text{succAbove} : \{0, \dots, n-1\} \to \{0, \dots, n\} \setminus \{i\}$.

Given a Wick contraction $c$ on $n+1$ elements and an index $i \in \{0, \dots, n\}$, this function returns the element that was paired with $i$ in $c$, now viewed as an uncontracted element in the reduced contraction $\text{erase}(c, i)$. Specifically, if $i$ is part of a contracted pair $\{i, j\}$ in $c$, the function returns the index $j$ (relabeled to the set $\{0, \dots, n-1\}$) as an element of the set of uncontracted indices of $\text{erase}(c, i)$. If $i$ is not contracted in $c$, the function returns $\text{none}$.

For any Wick contraction $c$ on $n+1$ elements and an index $i \in \{0, \dots, n\}$, the function $\text{getDualErase}(c, i)$—which identifies the partner of $i$ that becomes uncontracted in the reduced contraction—returns a value (is not `none`) if and only if the index $i$ is contracted in $c$ (i.e., $\text{getDual?}(c, i)$ is not `none`).

For any Wick contraction $c$ on a single element and any index $i$ in that set (i.e., $i \in \{0\}$), the function $\text{getDualErase}(c, i)$—which returns the partner of $i$ as an uncontracted element in the contraction resulting from erasing $i$—is equal to $\text{none}$.