Physlib.QFT.PerturbationTheory.WickContraction.SubContraction

Let $\varphi s$ be a sequence of field operators and $\varphi s\Lambda$ be a Wick contraction on $\varphi s$. Given a subset $S$ of the set of contracted pairs $\varphi s\Lambda.1$, the sub-contraction is the Wick contraction on the same sequence $\varphi s$ whose set of contracted pairs is exactly $S$.

Let $\varphi s$ be a sequence of field operators. Let $\Lambda$ be a Wick contraction on $\varphi s$, and let $\Lambda.1$ denote the set of its contracted pairs (where each pair is represented as a set of indices). For any subset of pairs $S \subseteq \Lambda.1$, let $\Lambda|_S$ be the sub-contraction formed by the pairs in $S$. If a set of indices $a$ is a contracted pair in the sub-contraction $\Lambda|_S$, then $a$ is also a contracted pair in the original contraction $\Lambda$.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Given a subset $S \subseteq \Lambda$ of the contracted pairs, the quotient contraction is the Wick contraction defined on the sequence of operators $[\Lambda|_S]^{uc}$ that remain after the contractions in $S$ have been performed. This contraction consists of all pairs in $\Lambda$ that are not in $S$, with their indices relabeled to correspond to their positions in the uncontracted list.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$, where $\Lambda.1$ denotes the set of contracted pairs. Let $S \subseteq \Lambda.1$ be a subset of these pairs. The quotient contraction of $\Lambda$ by $S$ is a Wick contraction defined on the sequence of operators that remain uncontracted after the pairs in $S$ are removed. If $a$ is a contracted pair in this quotient contraction, then the set of indices obtained by mapping $a$ back to the original sequence using the embedding $\text{uncontractedListEmd}$ is a contracted pair in the original contraction $\Lambda$. In mathematical notation, if $a \in (\text{quotContraction } S).1$, then $\{\text{uncontractedListEmd}(i) \mid i \in a\} \in \Lambda.1$.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Given a subset $S \subseteq \Lambda$ of the contracted pairs, any contracted pair $a \in \Lambda$ either belongs to the sub-contraction defined by $S$, or it corresponds to a pair in the quotient contraction of $\Lambda$ by $S$. Specifically, if $a \in \Lambda$, then: $$a \in S \lor \exists a', \text{map}(\text{uncontractedListEmd}, a') = a \text{ and } a' \in \text{quotContraction}(S)$$ where $\text{uncontractedListEmd}$ is the embedding that maps the indices of the operators remaining after the contractions in $S$ back to their original indices in the sequence $\varphi s$.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Let $S \subseteq \Lambda.1$ be a subset of the contracted pairs of $\Lambda$, which defines a sub-contraction. For any index $a$ into the list of operators remaining uncontracted after this sub-contraction (denoted as $[\text{subContraction}(S)]^{uc}$), the operator at that position is equal to the operator in the original sequence $\varphi s$ at the index $\text{uncontractedListEmd}(a)$, where $\text{uncontractedListEmd}$ is the embedding from the indices of the uncontracted list back to the indices of the original sequence. In formula terms: $$[\text{subContraction}(S)]^{uc}_a = \varphi s_{\text{uncontractedListEmd}(a)}$$

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Let $S \subseteq \Lambda.1$ be a subset of the contracted pairs of $\Lambda$, defining a sub-contraction $\Lambda|_S$. For any pair $a$ in the set of contracted pairs of the sub-contraction, the first field operator of the pair $a$ in $\Lambda|_S$ is equal to the first field operator of $a$ in the original contraction $\Lambda$. In formula terms: $$(\Lambda|_S).\text{fstFieldOfContract}(a) = \Lambda.\text{fstFieldOfContract}(a)$$ where $\text{fstFieldOfContract}$ identifies the first operator (typically the one with the smaller index) in a contracted pair.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Let $S \subseteq \Lambda.1$ be a subset of the contracted pairs of $\Lambda$, which defines a sub-contraction $\Lambda|_S$. For any pair $a$ in the set of contracted pairs of the sub-contraction, the second field operator of the pair $a$ in $\Lambda|_S$ is equal to the second field operator of the same pair in the original contraction $\Lambda$. In formula terms: $$(\Lambda|_S).\text{sndFieldOfContract}(a) = \Lambda.\text{sndFieldOfContract}(a)$$ where $\text{sndFieldOfContract}$ identifies the second field operator (typically the one with the larger index) in a contracted pair.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$, where $\Lambda.1$ denotes the set of contracted pairs. Let $S \subseteq \Lambda.1$ be a subset of these pairs, and let $\Lambda/S$ (denoted `quotContraction S hs`) be the quotient contraction on the sequence of operators that remain uncontracted after the pairs in $S$ are removed. Let $e$ be the embedding `uncontractedListEmd` which maps indices from the uncontracted sequence back to the original sequence. For any pair $a$ in the quotient contraction $\Lambda/S$, the image under $e$ of the first operator's index in $a$ is equal to the index of the first operator of the corresponding pair in the original contraction $\Lambda$. In mathematical notation: $$e\left((\Lambda/S).\text{fstFieldOfContract}(a)\right) = \Lambda.\text{fstFieldOfContract}\left(\{e(i) \mid i \in a\}\right)$$ where $\text{fstFieldOfContract}$ identifies the first operator (typically the one with the smaller index) in a contracted pair.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Let $S \subseteq \Lambda.1$ be a subset of the contracted pairs of $\Lambda$, and let $\text{quotContraction}(S)$ be the Wick contraction defined on the sequence of operators that remain after the contractions in $S$ have been performed. Let $\text{uncontractedListEmd}$ be the embedding function that maps the index of an operator in the uncontracted sequence back to its original index in $\varphi s$. For any pair $a$ in the quotient contraction, the image of its second field operator's index under $\text{uncontractedListEmd}$ is equal to the index of the second field operator of the corresponding mapped pair in the original contraction $\Lambda$. In mathematical notation: $$\text{uncontractedListEmd}\left( \text{sndFieldOfContract}_{\Lambda/S}(a) \right) = \text{sndFieldOfContract}_{\Lambda}\left( \{ \text{uncontractedListEmd}(i) \mid i \in a \} \right)$$ where $\text{sndFieldOfContract}$ identifies the index of the second field in a contracted pair (typically the one with the larger index).

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Suppose that $\Lambda$ is grading compliant, meaning its contracted pairs satisfy the required statistical properties according to the fields involved. For any subset $S \subseteq \Lambda.1$ of the contracted pairs, let $\Lambda/S$ (the quotient contraction) be the Wick contraction defined on the sequence of operators $[\text{subContraction}(S)]^{uc}$ that remain after the contractions in $S$ have been performed. Then the quotient contraction $\Lambda/S$ is also grading compliant.

Let $\varphi s$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\varphi s$. Let $S \subseteq \Lambda$ be a subset of the contracted pairs. The quotient contraction $\Lambda/S$ is the Wick contraction defined on the operators remaining uncontracted by $S$. Let $\text{uncontractedListEmd}$ be the embedding that maps the indices of these uncontracted operators back to their original positions in $\varphi s$. For any set of indices $a$ in the uncontracted list, $a$ is a contracted pair in the quotient contraction $\Lambda/S$ if and only if its image under $\text{uncontractedListEmd}$ is a contracted pair in $\Lambda$ that is not in $S$: $$a \in (\Lambda/S) \iff \text{map}(\text{uncontractedListEmd}, a) \in \Lambda \setminus S$$