Physlib.QFT.PerturbationTheory.WickContraction.Join

Given a list of field operators $\phi_s$, let $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ be the sub-list of field operators in $\phi_s$ that are not participating in any pairing in $\Lambda$. Given a second Wick contraction $\Lambda'$ defined on the indices of this uncontracted list $[\Lambda]^{uc}$, the function $\text{join}(\Lambda, \Lambda')$ defines a new Wick contraction on the original list $\phi_s$. This contraction is the union of the set of pairings in $\Lambda$ and the set of pairings in $\Lambda'$, where the indices in $\Lambda'$ are mapped back to their original positions in $\phi_s$ using the embedding $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ and $\Lambda'$ be two Wick contractions on the indices $\{0, \dots, |\phi_s|-1\}$. If $\Lambda = \Lambda'$, then for any Wick contraction $\Lambda_{uc}$ on the indices of the list of uncontracted operators $[\Lambda]^{uc}$, the joined contraction $\text{join}(\Lambda, \Lambda_{uc})$ is equal to $\text{join}(\Lambda', \Lambda'_{uc})$, where $\Lambda'_{uc}$ is the contraction $\Lambda_{uc}$ mapped to the indices of $[\Lambda']^{uc}$ via the equivalence induced by the equality of the base contractions.

Let $\phi_s$ be a list of field operators, $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and $\Lambda'$ be a Wick contraction on the list of uncontracted operators $[\Lambda]^{uc}$. This function maps any pairing $a$ belonging to the contraction $\Lambda$ to the corresponding pairing in the joined Wick contraction $\text{join}(\Lambda, \Lambda')$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the sub-list of operators $[\Lambda]^{uc}$ that are not contracted by $\Lambda$. The mapping $\text{joinLiftLeft}$, which sends each pairing in the original contraction $\Lambda$ to its corresponding pairing in the joined contraction $\text{join}(\Lambda, \Lambda')$, is injective.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the sub-list of field operators that remain uncontracted by $\Lambda$. Given a second Wick contraction $\Lambda'$ acting on the indices of $[\Lambda]^{uc}$, the function `joinLiftRight` maps a pairing (a set of two indices) $a \in \Lambda'$ to its corresponding pairing in the joined contraction $\text{join}(\Lambda, \Lambda')$. This mapping is defined by applying the embedding $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$ to the indices in $a$, which translates the local indices of the uncontracted list back to their original positions in $\phi_s$.

Let $\phi_s$ be a list of field operators and let $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the sub-list of field operators from $\phi_s$ that are not participating in any pairing in $\Lambda$. Given a second Wick contraction $\Lambda'$ defined on the indices of the list $[\Lambda]^{uc}$, the mapping $\text{joinLiftRight}$—which maps each pairing $a \in \Lambda'$ to its corresponding pairing in the joined contraction $\text{join}(\Lambda, \Lambda')$ by translating the indices of $[\Lambda]^{uc}$ back to their original positions in $\phi_s$—is injective.

Let $\phi_s$ be a list of field operators, $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and $\Lambda'$ be a Wick contraction on the sub-list of uncontracted operators $[\Lambda]^{uc}$. For any pairing $a \in \Lambda$ and any pairing $b \in \Lambda'$, the set of indices in the original list $\phi_s$ associated with $a$ and the set of indices in $\phi_s$ associated with $b$ (after mapping the indices of $b$ from the uncontracted list back to their original positions in $\phi_s$) are disjoint.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the list of operators $[\Lambda]^{uc}$ that remain uncontracted by $\Lambda$. For any pairing $a \in \Lambda$ and any pairing $b \in \Lambda'$, the pairings in the joined contraction $\text{join}(\Lambda, \Lambda')$ obtained by lifting $a$ (via $\text{joinLiftLeft}$) and $b$ (via $\text{joinLiftRight}$) are distinct; that is, $\text{joinLiftLeft}(a) \neq \text{joinLiftRight}(b)$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the indices of the sub-list of operators $[\Lambda]^{uc}$ that remain uncontracted by $\Lambda$. The function maps an element of the disjoint union of the pairings in $\Lambda$ and the pairings in $\Lambda'$ to its corresponding pairing in the joined contraction $\text{join}(\Lambda, \Lambda')$. Specifically, a pairing originating from $\Lambda$ is mapped using `joinLiftLeft`, and a pairing originating from $\Lambda'$ is mapped using `joinLiftRight`.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the sub-list of field operators in $\phi_s$ that are not participating in any pairing in $\Lambda$. Given a Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, the function $\text{joinLift}$ maps the pairings from the disjoint union of $\Lambda$ and $\Lambda'$ to the pairings in the joined contraction $\text{join}(\Lambda, \Lambda')$. This mapping $\text{joinLift}$ is injective.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the sub-list of field operators in $\phi_s$ that are not participating in any pairing in $\Lambda$. Given a Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, the function $\text{joinLift}$ maps the pairings from the disjoint union of $\Lambda$ and $\Lambda'$ to the pairings in the joined contraction $\text{join}(\Lambda, \Lambda')$. This mapping $\text{joinLift}$ is surjective.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the sub-list of field operators in $\phi_s$ that are not participating in any pairing in $\Lambda$. Given a Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, the function $\text{joinLift}$ maps the pairings from the disjoint union of the set of pairings in $\Lambda$ and the set of pairings in $\Lambda'$ to the pairings in the joined contraction $\text{join}(\Lambda, \Lambda')$. This mapping $\text{joinLift}$ is bijective.

Let $\phi_s$ be a list of field operators and $M$ be a commutative monoid. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the indices of the sub-list of operators $[\Lambda]^{uc}$ that remain uncontracted by $\Lambda$. For any function $f$ mapping the pairings of the joined contraction $\text{join}(\Lambda, \Lambda')$ to $M$, the product over all pairings in the joined contraction satisfies: $$\prod_{a \in \text{join}(\Lambda, \Lambda')} f(a) = \left( \prod_{a \in \Lambda} f(\text{joinLiftLeft}(a)) \right) \cdot \left( \prod_{a \in \Lambda'} f(\text{joinLiftRight}(a)) \right)$$ where $\text{joinLiftLeft}$ and $\text{joinLiftRight}$ are the natural inclusions of the pairings from $\Lambda$ and $\Lambda'$ into $\text{join}(\Lambda, \Lambda')$, respectively.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the sub-list of operators $[\Lambda]^{uc}$ that remain uncontracted by $\Lambda$. For any pairing $a$ (a set of two indices) in the joined Wick contraction $\text{join}(\Lambda, \Lambda')$, there exists either a pairing $b \in \Lambda$ such that $a = \text{joinLiftLeft}(b)$, or a pairing $b' \in \Lambda'$ such that $a = \text{joinLiftRight}(b')$.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the sub-list of field operators in $\phi_s$ that are not contracted by $\Lambda$. Let $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$ be the embedding (the function `uncontractedListEmd`) that maps the index of an operator in the uncontracted list to its original position in $\phi_s$. For any Wick contraction $\Lambda'$ defined on the indices of $[\Lambda]^{uc}$ and any contracted pair $a \in \Lambda'$, the first index (the smaller of the two indices) of the corresponding pair in the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the image under $f$ of the first index of the pair $a$ in $\Lambda'$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$, and let $[\Lambda]^{uc}$ denote the sub-list of field operators that remain uncontracted. Suppose $\Lambda'$ is a second Wick contraction acting on the indices of the uncontracted list $[\Lambda]^{uc}$. Let $f: \{0, 1, \dots, \text{length}([\Lambda]^{uc}) - 1\} \hookrightarrow \{0, 1, \dots, \text{length}(\phi_s) - 1\}$ be the embedding that maps the index of an operator in the uncontracted list back to its original index in $\phi_s$. For any contracted pair $a \in \Lambda'$, the larger index of the lifted pairing in the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the image under $f$ of the larger index of $a$ in $\Lambda'$: $$\text{snd}(\text{joinLiftRight}(a)) = f(\text{snd}(a))$$ where $\text{snd}$ denotes the function that returns the larger index of a contracted pair.

Let $\phi_s$ be a list of field operators belonging to a field specification $\mathcal{F}$. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the indices of the list of uncontracted operators $[\Lambda]^{uc}$. For any contracted pair $a \in \Lambda$, the first index (the smaller of the two indices) of the corresponding pair in the joined Wick contraction $\text{join}(\Lambda, \Lambda')$ is equal to the first index of $a$ in the original contraction $\Lambda$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the list of uncontracted operators $[\Lambda]^{uc}$. For any contracted pair $a \in \Lambda$, the larger index of the corresponding pair in the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the larger index of $a$ in the original contraction $\Lambda$.

Let $\phi_s$ be a list of field operators from a field specification $\mathcal{F}$. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the indices of the list of operators $[\Lambda]^{uc}$ that remain uncontracted by $\Lambda$. Let $f$ be the embedding (`uncontractedListEmd`) that maps an index of the uncontracted list to its corresponding original index in $\phi_s$. For any finite set of indices $a$ from the uncontracted list, $a$ is a contracted pair in $\Lambda'$ if and only if the image of $a$ under $f$, denoted $\{f(i) \mid i \in a\}$, is a contracted pair in the joined Wick contraction $\text{join}(\Lambda, \Lambda')$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ denote the list of operators remaining uncontracted by $\Lambda$. If $\Lambda'$ is a Wick contraction on the indices of $[\Lambda]^{uc}$, then the number of pairings in the joined Wick contraction $\text{join}(\Lambda, \Lambda')$ is equal to the sum of the number of pairings in $\Lambda$ and $\Lambda'$: $$|\text{join}(\Lambda, \Lambda')| = |\Lambda| + |\Lambda'|$$

For any list of field operators $\phi_s$, let $\Lambda_\emptyset$ be the empty Wick contraction on the indices of $\phi_s$. Let $[\Lambda_\emptyset]^{uc}$ be the list of field operators in $\phi_s$ that are not participating in any pairing in $\Lambda_\emptyset$ (which is identically $\phi_s$). For any Wick contraction $\Lambda$ defined on the indices of $[\Lambda_\emptyset]^{uc}$, the join of $\Lambda_\emptyset$ and $\Lambda$ is equal to the Wick contraction $\Lambda$ (under the canonical equivalence induced by the identification of indices).

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. The join of $\Lambda$ with the empty Wick contraction (defined on the set of indices uncontracted by $\Lambda$) is equal to $\Lambda$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ be the sub-list of field operators in $\phi_s$ that remain uncontracted by $\Lambda$. For any Wick contraction $\Lambda'$ defined on the indices of the uncontracted list $[\Lambda]^{uc}$, the time contraction of the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the product of the time contraction of $\Lambda$ and the time contraction of $\Lambda'$: $$\text{timeContract}(\text{join}(\Lambda, \Lambda')) = \text{timeContract}(\Lambda) \cdot \text{timeContract}(\Lambda')$$ where the time contraction of a Wick contraction $\Lambda$ is defined as the product of the pairwise contractions of the field operators corresponding to the indices paired in $\Lambda$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ be the list of field operators that remain uncontracted by $\Lambda$. Given a second Wick contraction $\Lambda'$ defined on the indices of $[\Lambda]^{uc}$, the static contraction of the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the product of the static contractions of $\Lambda$ and $\Lambda'$: $$\text{staticContract}(\text{join}(\Lambda, \Lambda')) = \text{staticContract}(\Lambda) \cdot \text{staticContract}(\Lambda')$$ where the static contraction of a contraction $c$ is defined as the product $\prod_{\{j, k\} \in c, j < k} [\phi_j^-, \phi_k]_s$ of super-commutators of the annihilation parts of the field operators.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ be the sub-list of field operators that are uncontracted by $\Lambda$. Given a second Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, let $\text{join}(\Lambda, \Lambda')$ be the union of these two contractions. If an index $i$ is uncontracted in $\Lambda'$, then the corresponding original index in $\phi_s$, given by the embedding $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$, is uncontracted in the combined contraction $\text{join}(\Lambda, \Lambda')$.

Let $\phi_s$ be a list of field operators and let $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ be the list of field operators remaining uncontracted by $\Lambda$, and let $\Lambda'$ be a Wick contraction on the indices of $[\Lambda]^{uc}$. For any index $i \in \{0, \dots, \text{length}(\phi_s)-1\}$, if $i$ is uncontracted in the joined contraction $\text{join}(\Lambda, \Lambda')$, then $i$ is necessarily uncontracted in the initial contraction $\Lambda$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ be the sub-list of field operators that are uncontracted by $\Lambda$. Let $\Lambda'$ be a Wick contraction on the indices of $[\Lambda]^{uc}$. If an index $i \in \{0, \dots, \text{length}(\phi_s)-1\}$ is uncontracted in the joined contraction $\text{join}(\Lambda, \Lambda')$, then there exists an index $a \in \{0, \dots, \text{length}([\Lambda]^{uc})-1\}$ such that $i$ is the original index corresponding to $a$ via the embedding $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$, and $a$ is uncontracted in the second contraction $\Lambda'$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the sub-list of operators $[\Lambda]^{uc}$ that are uncontracted by $\Lambda$. The sorted list of indices in $\phi_s$ that remain uncontracted under the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the list obtained by applying the embedding $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$ to each element of the sorted list of indices uncontracted by $\Lambda'$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $\Lambda'$ be a Wick contraction on the indices of the sub-list $[\Lambda]^{uc}$ consisting of operators uncontracted by $\Lambda$. Let $f: \{0, \dots, \text{length}([\Lambda]^{uc})-1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s)-1\}$ be the embedding (`uncontractedListEmd`) that maps an index in the uncontracted sub-list to its original position in $\phi_s$. Then, the $j$-th element of the sorted list of indices uncontracted by the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to $f(k)$, where $k$ is the $j$-th element of the sorted list of indices uncontracted by $\Lambda'$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ denote the sub-list of field operators in $\phi_s$ that are uncontracted by $\Lambda$. Given a second Wick contraction $\Lambda'$ defined on the indices of the list $[\Lambda]^{uc}$, the list of field operators that remain uncontracted under the joined contraction $\text{join}(\Lambda, \Lambda')$ is equal to the list of operators in $[\Lambda]^{uc}$ that are uncontracted by $\Lambda'$.

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ denote the sub-list of field operators that are uncontracted by $\Lambda$. Given a second Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, let $f_{\Lambda} : \{0, \dots, |[\Lambda]^{uc}|-1\} \hookrightarrow \{0, \dots, |\phi_s|-1\}$ be the embedding that maps an index in the uncontracted list of $\Lambda$ to its original position in $\phi_s$, and let $f_{\Lambda'} : \{0, \dots, |[\Lambda']^{uc}|-1\} \hookrightarrow \{0, \dots, |[\Lambda]^{uc}|-1\}$ be the embedding that maps an index in the uncontracted list of $\Lambda'$ to its position in $[\Lambda]^{uc}$. Then the embedding for the joined contraction $f_{\text{join}(\Lambda, \Lambda')} : \{0, \dots, |[\text{join}(\Lambda, \Lambda')]^{uc}|-1\} \hookrightarrow \{0, \dots, |\phi_s|-1\}$ is equal to the composition of these embeddings: $$f_{\text{join}(\Lambda, \Lambda')} = f_{\Lambda} \circ f_{\Lambda'}$$ (where the domains are identified using the fact that the uncontracted operators of the joined contraction are exactly those of $\Lambda'$).

Let $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$, and let $[\Lambda]^{uc}$ denote the sub-list of operators in $\phi_s$ that remain uncontracted by $\Lambda$. Let $\Lambda'$ be a second Wick contraction acting on the indices of $[\Lambda]^{uc}$, and let $\Lambda''$ be a third Wick contraction acting on the indices of the operators uncontracted by the joined contraction $\text{join}(\Lambda, \Lambda')$. Since the list of uncontracted operators of the joined contraction $\text{join}(\Lambda, \Lambda')$ is identical to the list of uncontracted operators of $\Lambda'$ (i.e., $[\text{join}(\Lambda, \Lambda')]^{uc} = [\Lambda']^{uc}$), the contraction $\Lambda''$ can be identified as a contraction acting on the uncontracted indices of $\Lambda'$. Under this identification, the operation of joining Wick contractions is associative: $$\text{join}(\text{join}(\Lambda, \Lambda'), \Lambda'') = \text{join}(\Lambda, \text{join}(\Lambda', \Lambda''))$$

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the list of field operators that are not contracted by $\Lambda$, and let $f: \{0, 1, \dots, \text{length}([\Lambda]^{uc}) - 1\} \hookrightarrow \{0, 1, \dots, \text{length}(\phi_s) - 1\}$ be the embedding that maps the index of an operator in the uncontracted list back to its original position in $\phi_s$. Given a second Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, the index $f(i)$ is uncontracted in the joined contraction $\text{join}(\Lambda, \Lambda')$ if and only if the index $i$ is uncontracted in $\Lambda'$.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ denote the list of field operators that are not contracted by $\Lambda$, and let $f: \{0, 1, \dots, \text{length}([\Lambda]^{uc}) - 1\} \hookrightarrow \{0, 1, \dots, \text{length}(\phi_s) - 1\}$ be the embedding that maps the index of an operator in the uncontracted list back to its original position in $\phi_s$. Given a second Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, the index $f(i)$ is contracted in the joined contraction $\text{join}(\Lambda, \Lambda')$ if and only if the index $i$ is contracted in $\Lambda'$.

Let $\phi_s$ be a list of field operators and let $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ be the list of operators in $\phi_s$ that are not contracted by $\Lambda$, and let $f: \{0, \dots, \text{length}([\Lambda]^{uc}) - 1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s) - 1\}$ be the embedding that maps an index in the uncontracted list to its original position in $\phi_s$. Suppose $\Lambda'$ is a second Wick contraction defined on the indices of $[\Lambda]^{uc}$. For any index $i$ in the uncontracted list, if the index $f(i)$ has a partner in the joined contraction $\text{join}(\Lambda, \Lambda')$, then this partner is given by $f(j)$, where $j$ is the partner of $i$ in the contraction $\Lambda'$.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Let $[\Lambda]^{uc}$ be the sub-list of operators in $\phi_s$ that are not contracted by $\Lambda$, and let $f: \{0, \dots, \text{length}([\Lambda]^{uc}) - 1\} \hookrightarrow \{0, \dots, \text{length}(\phi_s) - 1\}$ be the embedding that maps an index in the uncontracted list back to its original position in $\phi_s$. Given a second Wick contraction $\Lambda'$ on the indices of $[\Lambda]^{uc}$, the contraction partner of the index $f(i)$ in the joined contraction $\text{join}(\Lambda, \Lambda')$ is the image under $f$ of the contraction partner of $i$ in $\Lambda'$. That is, $$\text{getDual?}(\text{join}(\Lambda, \Lambda'), f(i)) = \text{Option.map}(f, \text{getDual?}(\Lambda', i))$$ where $\text{getDual?}(c, k)$ returns the index paired with $k$ in contraction $c$, or $\text{none}$ if $k$ is unpaired.

Let $\phi_s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\phi_s$. For any subset $S \subseteq \Lambda$ of the contracted pairs, let $\Lambda|_S$ be the sub-contraction containing only the pairs in $S$, and let $\Lambda/S$ be the quotient contraction containing the pairs in $\Lambda$ that are not in $S$. Then the join of the sub-contraction and the quotient contraction recovers the original contraction: $$\text{join}(\Lambda|_S, \Lambda/S) = \Lambda$$

Let $\phi_s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\phi_s$. For any subset of contracted pairs $S \subseteq \Lambda$, let $\Lambda|_S$ be the sub-contraction containing only the pairs in $S$, and let $\Lambda/S$ be the quotient contraction containing the pairs in $\Lambda$ that are not in $S$ (relabeled to correspond to the indices of the operators left uncontracted by $\Lambda|_S$). The sum of the number of pairings in the sub-contraction and the number of pairings in the quotient contraction is equal to the total number of pairings in the original contraction $\Lambda$: $$|\Lambda|_S| + |\Lambda/S| = |\Lambda|$$