Physlib.QFT.PerturbationTheory.WickContraction.InsertAndContract

Given a list of field operators $\phi_s$ of length $n$ and a Wick contraction $\Lambda$ associated with it, let $\phi$ be a field operator to be inserted at index $i \in \{0, \dots, n\}$. Let $k$ be an optional index (either `none` or an element from the set of uncontracted indices of $\Lambda$). The function returns a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, k$ for the list of length $n+1$ formed by inserting $\phi$ into $\phi_s$ at position $i$. The new contraction is defined by: 1. Shifting the indices of all existing contracted pairs in $\Lambda$ to account for the insertion of $\phi$ at index $i$. 2. If $k$ is some index $j$, adding a new contraction pair between the newly inserted operator at index $i$ and the operator originally at index $j$ (now shifted). 3. If $k$ is `none`, the new operator at index $i$ remains uncontracted.

The notation $\phi_s \hookleftarrow_\Lambda \phi, i, j$ represents the operation of inserting a field operator $\phi$ into a list of field operators $\phi_s$ at the index $i$, while simultaneously updating the Wick contraction $\Lambda$. The parameter $j$ specifies whether the newly inserted operator $\phi$ is to be contracted: if $j$ identifies an currently uncontracted index in $\Lambda$, $\phi$ is paired with that operator; if $j$ is empty (or indicates no contraction), $\phi$ remains uncontracted in the resulting Wick contraction.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$. Consider the Wick contraction $\Lambda' = \Lambda \hookleftarrow \phi, i, j$ obtained by inserting a field operator $\phi$ at index $i \in \{0, \dots, n\}$ and optionally contracting it with an uncontracted operator at index $j$. For any existing contraction pair $a$ in the original contraction $\Lambda$, let $a'$ be the corresponding "lifted" contraction pair in $\Lambda'$. Then the first field index of the pair $a'$ in $\Lambda'$ is given by $i.\text{succAbove}(l)$, where $l$ is the first field index of the original pair $a$ in $\Lambda$. The function $i.\text{succAbove}$ shifts indices to account for the insertion, mapping $k \mapsto k$ if $k < i$ and $k \mapsto k+1$ if $k \geq i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$. Consider the Wick contraction $\Lambda' = \Lambda \hookleftarrow \phi, i, j$ obtained by inserting a field operator $\phi$ at index $i \in \{0, \dots, n\}$ and optionally contracting it with an uncontracted operator at index $j$. For any existing contraction pair $a$ in the original contraction $\Lambda$, let $a'$ be the corresponding "lifted" contraction pair in $\Lambda'$. Then the second field index of the pair $a'$ in $\Lambda'$ is given by $i.\text{succAbove}(l)$, where $l$ is the second field index of the original pair $a$ in $\Lambda$. The function $i.\text{succAbove}$ shifts indices to account for the insertion, mapping $k \mapsto k$ if $k < i$ and $k \mapsto k+1$ if $k \geq i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we form a new Wick contraction $\Lambda'$ by inserting a field operator $\phi$ at index $i \in \{0, \dots, n\}$ and contracting it with an operator that was previously uncontracted at index $j$ in $\Lambda$. This operation creates a new contraction pair in $\Lambda'$ consisting of the index $i$ and the shifted index $i.\text{succAbove}(j)$ (where $i.\text{succAbove}(j)$ is $j$ if $j < i$ and $j+1$ if $j \geq i$). The first field index (the smaller of the two indices) of this new contraction pair is $i$ if $i < i.\text{succAbove}(j)$, and $i.\text{succAbove}(j)$ otherwise.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we form a new Wick contraction $\Lambda' = \Lambda \hookleftarrow \phi, i, \text{none}$ by inserting a field operator $\phi$ at index $i$ and choosing not to contract it with any existing operator. Then, the dual of the index $i$ in the resulting contraction $\Lambda'$ is $\text{none}$; that is, the newly inserted field at index $i$ remains uncontracted.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we form a new Wick contraction $\Lambda'$ by inserting a field operator $\phi$ at index $i$ and contracting it with a previously uncontracted index $j$ of $\Lambda$. Then, the dual of the index $i$ in the resulting contraction $\Lambda'$ exists; that is, the newly inserted field at index $i$ is successfully paired with another field.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we form a new Wick contraction $\Lambda' = \Lambda \hookleftarrow \phi, i, \text{some } j$ by inserting a field operator $\phi$ at index $i$ and contracting it with an existing uncontracted index $j$ of $\Lambda$. Then, the dual of the index $i$ in the resulting contraction $\Lambda'$ is not $\text{none}$; that is, the newly inserted field at index $i$ is successfully contracted with another field.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction of $\phi_s$. Suppose a field operator $\phi$ is inserted into the list at index $i$ and contracted with a previously uncontracted index $j$ of $\Lambda$, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$. Then, the contraction partner (dual) of the newly inserted operator at index $i$ is the operator originally at index $j$, whose index in the expanded list is given by $j' = \text{succAbove}_i(j)$. Mathematically, this is expressed as: $$\text{getDual?}_{\Lambda'}(i) = \text{some}(\text{succAbove}_i(j))$$ where $\text{getDual?}$ returns the index of the operator's contraction partner, and the shift function is defined as $\text{succAbove}_i(j) = j$ if $j < i$ and $\text{succAbove}_i(j) = j + 1$ if $j \geq i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction of $\phi_s$. Suppose a field operator $\phi$ is inserted into the list at index $i \in \{0, \dots, n\}$ and contracted with a previously uncontracted index $j$ of $\Lambda$, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ for the expanded list. Then, the contraction partner (dual) of the operator originally at index $j$ (whose index in the expanded list is $j' = \text{succAbove}_i(j)$) is the newly inserted operator at index $i$. Mathematically, this is expressed as: $$\text{getDual?}_{\Lambda'}(\text{succAbove}_i(j)) = \text{some}(i)$$ where $\text{getDual?}$ returns the index of an operator's contraction partner, and the shift function is defined as $\text{succAbove}_i(j) = j$ if $j < i$ and $\text{succAbove}_i(j) = j + 1$ if $j \geq i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction of $\phi_s$. Suppose a field operator $\phi$ is inserted into the list at index $i$ without forming a new contraction, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$. For any index $j$ in the original list, let $j' = \text{succAbove}_i(j)$ be the corresponding index in the expanded list (defined as $j' = j$ if $j < i$ and $j' = j+1$ if $j \ge i$). Then, the operator at index $j'$ is uncontracted in $\Lambda'$ if and only if the operator at index $j$ was uncontracted in $\Lambda$. Mathematically, this is expressed as: $$\text{getDual?}_{\Lambda'}(j') = \text{none} \iff \text{getDual?}_{\Lambda}(j) = \text{none}$$ where $\text{getDual?}$ returns the index of the operator's contraction partner or $\text{none}$ if it is uncontracted.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction of $\phi_s$. Suppose we insert a field operator $\phi$ at index $i \in \{0, \dots, n\}$ and contract it with an operator originally at index $k$ (which was uncontracted in $\Lambda$), resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, k$ for the list of length $n+1$. For any index $j$ in the original list such that $j \neq k$, let $j' = \text{succAbove}_i(j)$ be the shifted index in the new list (defined as $j' = j$ if $j < i$ and $j' = j+1$ if $j \ge i$). The contraction status of $j'$ in $\Lambda'$ is given by shifting the contraction status of $j$ in $\Lambda$. Specifically: $$\text{getDual?}_{\Lambda'}(j') = \text{Option.map}(\text{succAbove}_i, \text{getDual?}_{\Lambda}(j))$$ where $\text{getDual?}$ returns the index of an operator's contraction partner or $\text{none}$ if it is uncontracted. In other words, if $j$ was contracted with $d$ in $\Lambda$, then $j'$ is contracted with $\text{succAbove}_i(d)$ in $\Lambda'$; if $j$ was uncontracted in $\Lambda$, then $j'$ remains uncontracted in $\Lambda'$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction of $\phi_s$. Suppose a field operator $\phi$ is inserted into the list at index $i$ without forming a new contraction, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$. For any index $j$ in the original list, let $j' = \text{succAbove}_i(j)$ be the corresponding index in the expanded list (where $j' = j$ if $j < i$ and $j' = j+1$ if $j \ge i$). Then, the operator at index $j'$ is part of a contracted pair in $\Lambda'$ if and only if the operator at index $j$ was part of a contracted pair in $\Lambda$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction of $\phi_s$. Suppose we insert a field operator $\phi$ at index $i \in \{0, \dots, n\}$ without contracting it, resulting in a new contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ for the list of length $n+1$. For any index $j \in \{0, \dots, n-1\}$, let $j'$ be the corresponding index in the new list, defined by the shift $j' = \text{succAbove}_i(j)$ (where indices $\ge i$ are incremented by 1). If $j'$ is part of a contracted pair in $\Lambda'$, then the index of its partner in $\Lambda'$ is equal to the shifted index of the partner of $j$ in the original contraction $\Lambda$. Formally, if $\text{dual}_{\Lambda}(j)$ is the index paired with $j$ in $\Lambda$, then: $$\text{dual}_{\Lambda'}(j') = \text{succAbove}_i(\text{dual}_{\Lambda}(j))$$

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we insert a field operator $\phi$ at index $i$ and contract it with a previously uncontracted index $j$ of $\Lambda$. This results in a new contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ for the expanded list. The newly created contraction pair in $\Lambda'$ consists of the index $i$ and the shifted index $j' = \text{succAbove}_i(j)$. The second index of this contraction pair, denoted by $\text{sndFieldOfContract}$, is equal to $j'$ if $i < j'$, and is equal to $i$ otherwise.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we insert a field operator $\phi$ at index $i \in \{0, \dots, n\}$ without creating a new contraction pair, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ for the expanded list. For any function $f$ mapping contraction pairs of $\Lambda'$ to a commutative monoid $M$, the product of $f$ over all contraction pairs in $\Lambda'$ is given by: $$\prod_{a \in \Lambda'} f(a) = \prod_{a \in \Lambda} f(\text{lift}_i(a))$$ where $\text{lift}_i(a)$ denotes the contraction pair in $\Lambda'$ obtained by shifting the indices of the original pair $a \in \Lambda$ using the $\text{succAbove}_i$ mapping to account for the insertion of $\phi$ at position $i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we insert a field operator $\phi$ at index $i$ and contract it with a previously uncontracted index $j$ of $\Lambda$. This results in a new contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ for the expanded list. For any function $f$ from the set of contraction pairs of $\Lambda'$ to a commutative monoid $M$, the product of $f$ over all contraction pairs in $\Lambda'$ is given by: $$\prod_{a \in \Lambda'} f(a) = f(\{i, \text{succAbove}_i(j)\}) \cdot \prod_{a \in \Lambda} f(\text{lift}_i(j, a))$$ where $\{i, \text{succAbove}_i(j)\}$ is the newly formed contraction pair between the inserted field at index $i$ and the shifted uncontracted field, and $\text{lift}_i(j, a)$ denotes the original contraction pairs of $\Lambda$ shifted to account for the insertion of $\phi$ at index $i$.

For a field operator $\phi$, a list of field operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$, and an insertion index $i \in \{0, \dots, n\}$, let $\phi_s'$ be the list of length $n+1$ obtained by inserting $\phi$ into $\phi_s$ at position $i$. This function maps a finite set of indices $a \subseteq \{0, \dots, n-1\}$ to a finite set of indices $a' \subseteq \{0, \dots, n\}$ by applying the mapping $\text{succAbove}_i$ to each element of $a$. This mapping shifts indices $j \geq i$ by one to account for the insertion of the new operator at position $i$, effectively preserving the relative positions of the original operators in the new list.

Let $\mathcal{F}$ be a field specification, $\phi \in \text{FieldOp}(\mathcal{F})$ be a field operator, and $\phi_s$ be a list of field operators of length $n$. Let $i \in \{0, \dots, n\}$ be the index at which $\phi$ is inserted into the list $\phi_s$. For any finite set of indices $a \subseteq \{0, \dots, n-1\}$, let $a'$ be the lifted set of indices in the resulting list of length $n+1$, obtained by applying the mapping $\text{succAbove}_i$ to each element of $a$. Then, the insertion index $i$ is not an element of the lifted set $a'$.

Let $\mathcal{F}$ be a field specification and $\phi \in \text{FieldOp}(\mathcal{F})$ be a field operator. Let $\phi_s$ be a list of field operators of length $n$, and let $i \in \{0, \dots, n\}$ be the index at which $\phi$ is inserted into the list $\phi_s$. For any finite set of indices $a \subseteq \{0, \dots, n-1\}$, let $a'$ be the lifted set of indices in the resulting list of length $n+1$, obtained by applying the mapping $\text{succAbove}_i$ to each element of $a$. Then, for any index $j \in \{0, \dots, n-1\}$, the shifted index $i.\text{succAbove}(j)$ is an element of the lifted set $a'$ if and only if $j$ is an element of the original set $a$.

For a given field specification $\mathcal{F}$, let $\varphi \in \text{FieldOp}(\mathcal{F})$ be a field operator and $\varphi_s$ be a list of field operators. Let $i \in \{0, \dots, |\varphi_s|\}$ be an index. For any index $j$ in the list obtained by inserting $\varphi$ into $\varphi_s$ at position $i$ (denoted $\text{insertIdx}(\varphi_s, i, \varphi)$), it holds that either $j$ is the insertion index $i$, or there exists an index $k \in \{0, \dots, |\varphi_s| - 1\}$ such that $j$ corresponds to the shifted index $i.\text{succAbove}(k)$. Here, $i.\text{succAbove}(k)$ is defined as $k$ if $k < i$ and $k+1$ if $k \geq i$.

Let $\mathcal{F}$ be a field specification, $\phi_s$ be a list of field operators, and $\phi$ be a field operator to be inserted into $\phi_s$ at index $i \in \{0, \dots, |\phi_s|\}$. For any function $f$ mapping Wick contractions of the resulting augmented list to an additive commutative monoid $M$, the sum over all possible Wick contractions $c$ of the augmented list is equal to the nested sum over all Wick contractions $\Lambda$ of the original list $\phi_s$ and all choices $k$ of either leaving $\phi$ uncontracted or pairing it with one of the uncontracted operators in $\Lambda$: $$\sum_{c \in \text{WickContraction}(|\phi_s|+1)} f(c) = \sum_{\Lambda \in \text{WickContraction}(|\phi_s|)} \sum_{k \in \text{Option}(\Lambda.\text{uncontracted})} f(\Lambda \hookleftarrow_\Lambda \phi, i, k)$$ where $\Lambda \hookleftarrow_\Lambda \phi, i, k$ denotes the Wick contraction of the augmented list formed by inserting $\phi$ at position $i$ and applying the contraction specified by $k$.

Let $\mathcal{F}$ be a field specification, $\phi \in \text{FieldOp}(\mathcal{F})$ a field operator, and $\phi_s$ a list of field operators. For any Wick contraction $\Lambda$ of $\phi_s$ and any insertion index $i \in \{0, \dots, |\phi_s|\}$, let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ be the Wick contraction formed by inserting $\phi$ at index $i$ into $\phi_s$ such that $\phi$ remains uncontracted. Then the list of uncontracted operators of $\Lambda'$, denoted $[\Lambda']^{uc}$, is equal to the list of uncontracted operators of $\Lambda$, denoted $[\Lambda]^{uc}$, with $\phi$ inserted at the position determined by $\text{uncontractedListOrderPos}(\Lambda, i)$. That is, $$[\Lambda \hookleftarrow_\Lambda \phi, i, \text{none}]^{uc} = \text{insertIdx}([\Lambda]^{uc}, \text{uncontractedListOrderPos}(\Lambda, i), \phi)$$ where $\text{uncontractedListOrderPos}(\Lambda, i)$ is the index in the uncontracted list corresponding to the insertion index $i$ in the full list of operators.

Let $\mathcal{F}$ be a field specification, $\phi \in \text{FieldOp}(\mathcal{F})$ be a field operator, and $\phi_s$ be a list of field operators. For any Wick contraction $\Lambda$ of $\phi_s$, let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, 0, \text{none}$ be the Wick contraction formed by inserting $\phi$ at the beginning (index 0) of the list such that $\phi$ remains uncontracted. Then the list of uncontracted operators of $\Lambda'$, denoted $[\Lambda']^{uc}$, is equal to the list of uncontracted operators of $\Lambda$, denoted $[\Lambda]^{uc}$, with $\phi$ prepended to the front. That is, $$[\Lambda \hookleftarrow_\Lambda \phi, 0, \text{none}]^{uc} = \phi :: [\Lambda]^{uc}$$

For a field specification $\mathcal{F}$, let $\phi$ be a field operator, $\phi_s$ be a list of field operators, and $i$ be an insertion index. Let $a$ be a finite set of indices pointing to elements in $\phi_s$, and let $a' = \text{insertAndContractLiftFinset}(\phi, i, a)$ be the set of indices in the augmented list $\phi_s' = \text{insertIdx}(\phi_s, i, \phi)$ obtained by shifting the original indices in $a$ to account for the insertion of $\phi$ at position $i$. The collective field statistic of the operators in the list $\phi_s'$ indexed by $a'$ is equal to the collective field statistic of the operators in the original list $\phi_s$ indexed by $a$, i.e., $$\mathcal{F} \triangleright_s \langle \phi_s'.\text{get}, a' \rangle = \mathcal{F} \triangleright_s \langle \phi_s.\text{get}, a \rangle$$