Physlib.QFT.PerturbationTheory.WickContraction.Sign.InsertNone

Let $\mathcal{F}$ be a field specification. Let $\phi_s$ be a list of field operators of length $n$ and $\Lambda$ be a Wick contraction on $\phi_s$. Suppose a field operator $\phi$ is inserted into $\phi_s$ at index $i \in \{0, \dots, n\}$ without being contracted, resulting in a new contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ for the augmented list of length $n+1$. For any two original indices $i_1, i_2 \in \{0, \dots, n-1\}$, let $i_1' = \text{succAbove}_i(i_1)$ and $i_2' = \text{succAbove}_i(i_2)$ be their shifted positions in the new list. The set of indices contributing to the sign of a contraction between $i_1'$ and $i_2'$ in $\Lambda'$ (denoted $\text{signFinset}_{\Lambda'}(i_1', i_2')$) is related to the original set $\text{signFinset}_{\Lambda}(i_1, i_2)$ as follows: $$\text{signFinset}_{\Lambda'}(i_1', i_2') = \begin{cases} \{i\} \cup \text{lift}_i(\text{signFinset}_{\Lambda}(i_1, i_2)) & \text{if } i_1' < i < i_2' \\ \text{lift}_i(\text{signFinset}_{\Lambda}(i_1, i_2)) & \text{otherwise} \end{cases}$$ where $\text{lift}_i$ is the operator that shifts all indices in a set by one if they are greater than or equal to $i$ (formally `insertAndContractLiftFinset`).

Let $\mathcal{F}$ be a field specification. Given a list of field operators $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$, a Wick contraction $\Lambda$ on $\phi_s$, and a new field operator $\phi$ to be inserted into the list at index $i \in \{0, \dots, n\}$ without being contracted, the sign factor $\text{signInsertNone}$ is the product of the statistical signs $\mathcal{S}$ accumulated by $\phi$ as it "crosses" existing contracted pairs. For each contracted pair $\{a_1, a_2\} \in \Lambda$ (where $a_1 < a_2$), let $a_1'$ and $a_2'$ be the shifted indices of the pair in the new list of length $n+1$. The function picks up a sign contribution $\mathcal{S}(\phi, \phi_{a_2})$ if the insertion index $i$ lies strictly between the new positions of the pair: $$\text{signInsertNone}(\phi, \phi_s, \Lambda, i) = \prod_{\{a_1, a_2\} \in \Lambda} \begin{cases} \mathcal{S}(\phi, \phi_{a_2}) & \text{if } a_1' < i < a_2' \\ 1 & \text{otherwise} \end{cases}$$ Here, $\mathcal{S}(\phi, \psi)$ represents the statistical exchange factor (typically $-1$ if both fields are fermions and $+1$ otherwise) determined by the field statistics of $\phi$ and $\psi$ in $\mathcal{F}$.

Let $\mathcal{F}$ be a field specification. Let $\phi_s$ be a list of field operators of length $n$ and $\Lambda$ be a Wick contraction on $\phi_s$. Suppose a field operator $\phi$ is inserted into $\phi_s$ at index $i \in \{0, \dots, n\}$ without being contracted, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ for the augmented list. The sign of the resulting contraction $\Lambda'$ is equal to the product of the sign factor $\text{signInsertNone}(\phi, \phi_s, \Lambda, i)$ and the sign of the original contraction $\Lambda$: $$\text{sign}(\Lambda \hookleftarrow_\Lambda \phi, i, \text{none}) = \text{signInsertNone}(\phi, \phi_s, \Lambda, i) \cdot \text{sign}(\Lambda)$$ where $\text{signInsertNone}$ is the statistical sign factor accumulated by $\phi$ as it "crosses" the existing contracted pairs in $\Lambda$ to reach position $i$.

Let $\mathcal{F}$ be a field specification. Given a list of field operators $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$, a Wick contraction $\Lambda$ on $\phi_s$, and a new field operator $\phi$ to be inserted at index $i \in \{0, \dots, n\}$ without being contracted, the sign factor $\text{signInsertNone}(\phi, \phi_s, \Lambda, i)$ is given by the product over all contracted pairs $\{a_1, a_2\} \in \Lambda$ (where $a_1 < a_2$): $$\text{signInsertNone}(\phi, \phi_s, \Lambda, i) = \prod_{\{a_1, a_2\} \in \Lambda} \left( \sigma(a_1, i) \cdot \sigma(a_2, i) \right)$$ where the factor $\sigma(x, i)$ for an index $x$ is defined as: $$\sigma(x, i) = \begin{cases} \mathcal{S}(\phi, \phi_{a_2}) & \text{if } x < i \\ 1 & \text{otherwise} \end{cases}$$ In this expression, $\mathcal{S}(\phi, \psi)$ represents the statistical exchange factor determined by the field statistics of $\phi$ and $\psi$ in $\mathcal{F}$, and the condition $x < i$ identifies whether the original index $x$ precedes the insertion point $i$.

Let $\mathcal{F}$ be a field specification. Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\phi_s$. Suppose a new field operator $\phi$ is inserted into the sequence at index $i \in \{0, \dots, n\}$ without being contracted. If the contraction $\Lambda$ is grading compliant, then the sign factor $\text{signInsertNone}(\phi, \phi_s, \Lambda, i)$ is given by the nested product over all contracted pairs $a \in \Lambda$ and the individual indices $x$ within those pairs: $$\text{signInsertNone}(\phi, \phi_s, \Lambda, i) = \prod_{a \in \Lambda} \prod_{x \in a} \begin{cases} \mathcal{S}(\phi, \phi_x) & \text{if } x < i \\ 1 & \text{otherwise} \end{cases}$$ where $\mathcal{S}(\phi, \phi_x)$ denotes the statistical exchange factor between $\phi$ and the field operator at index $x$ in $\phi_s$, and the condition $x < i$ checks if the original index precedes the insertion point.

Let $\mathcal{F}$ be a field specification. Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\phi_s$. Suppose the contraction $\Lambda$ is grading compliant. If a new field operator $\phi$ is inserted into the sequence at index $i \in \{0, \dots, n\}$ without being contracted, then the sign factor $\text{signInsertNone}(\phi, \phi_s, \Lambda, i)$ is given by the product over all indices $x \in \{0, \dots, n-1\}$: $$\text{signInsertNone}(\phi, \phi_s, \Lambda, i) = \prod_{x=0}^{n-1} \begin{cases} \mathcal{S}(\phi, \phi_x) & \text{if } x \text{ is part of a contracted pair in } \Lambda \text{ and } x < i \\ 1 & \text{otherwise} \end{cases}$$ where $\mathcal{S}(\phi, \phi_x)$ denotes the statistical exchange factor between $\phi$ and the field operator at index $x$ in $\phi_s$.

Let $\mathcal{F}$ be a field specification. Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a sequence of field operators and $\Lambda$ be a grading-compliant Wick contraction on $\phi_s$. Suppose a new field operator $\phi$ is inserted into the sequence at index $i \in \{0, \dots, n\}$ without being contracted. The sign factor $\text{signInsertNone}(\phi, \phi_s, \Lambda, i)$ is equal to the statistical exchange factor between $\phi$ and the sub-list of $\phi_s$ containing only those field operators $\phi_x$ such that the index $x$ is part of a contracted pair in $\Lambda$ and $x < i$: $$\text{signInsertNone}(\phi, \phi_s, \Lambda, i) = \mathcal{S}\left(\phi, [\phi_x \in \phi_s \mid x < i \text{ and } x \text{ is part of a pair in } \Lambda]\right)$$ where $\mathcal{S}$ denotes the statistical sign factor determined by the field statistics (bosonic or fermionic) of the operators as specified in $\mathcal{F}$.

Let $\mathcal{F}$ be a field specification. Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\phi_s$ that is grading compliant. Suppose a new field operator $\phi$ is inserted into the sequence at index $i \in \{0, \dots, n\}$ without being contracted. Then the sign factor $\text{signInsertNone}(\phi, \phi_s, \Lambda, i)$ is equal to the statistical exchange factor $\mathcal{S}$ between the field $\phi$ and the collection of all field operators $\phi_x$ in $\phi_s$ such that the index $x$ is part of a contracted pair in $\Lambda$ and $x < i$. Mathematically, $$\text{signInsertNone}(\phi, \phi_s, \Lambda, i) = \mathcal{S}\left(\phi, \prod_{x \in I} \phi_x\right)$$ where $I = \{x \in \{0, \dots, n-1\} \mid x \text{ is contracted in } \Lambda \text{ and } x < i\}$.

Let $\mathcal{F}$ be a field specification. Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a sequence of field operators and $\Lambda$ be a Wick contraction on $\phi_s$ that is grading compliant. Suppose a new field operator $\phi$ is inserted into the sequence at index $i \in \{0, \dots, n\}$ without being contracted, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$. The sign of the resulting contraction $\Lambda'$ is equal to the product of the original sign $\text{sign}(\Lambda)$ and the statistical exchange factor $\mathcal{S}$ accumulated by $\phi$ as it moves past the field operators $\phi_x$ in $\phi_s$ that are already contracted in $\Lambda$ and have original indices $x < i$. Mathematically, $$\text{sign}(\Lambda \hookleftarrow_\Lambda \phi, i, \text{none}) = \mathcal{S}\left(\phi, \prod_{x \in I} \phi_x\right) \cdot \text{sign}(\Lambda)$$ where $I = \{x \in \{0, \dots, n-1\} \mid x \text{ is part of a contracted pair in } \Lambda \text{ and } x < i\}$, and $\mathcal{S}$ is the statistical sign factor determined by the field statistics of the operators in $\mathcal{F}$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$ and $\phi$ be a field operator. If $\phi$ is inserted at the beginning of the list (index 0) without being contracted, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, 0, \text{none}$, then the sign of the new contraction is equal to the sign of the original contraction: $$\text{sign}(\Lambda \hookleftarrow_\Lambda \phi, 0, \text{none}) = \text{sign}(\Lambda)$$