Physlib.QFT.PerturbationTheory.WickContraction.Sign.Join

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction acting on them. Let $[\Lambda]^{uc}$ be the sub-list of field operators remaining uncontracted, and let $f$ be the embedding that maps an index in $[\Lambda]^{uc}$ to its original index in $\phi_s$. Given a subsequent Wick contraction $\Lambda_{uc}$ acting on the uncontracted operators and two indices $i, j$ in the uncontracted list, let $S$ be the sign finset (the set of indices between $i$ and $j$ that are not contracted before $i$ in $\Lambda_{uc}$). The collective field statistic (which is fermionic if the set contains an odd number of fermionic operators and bosonic otherwise) of the indices $S$ relative to the list $[\Lambda]^{uc}$ is equal to the collective field statistic of the mapped indices $f(S)$ relative to the original list $\phi_s$: $$\text{statistic}([\Lambda]^{uc}, S) = \text{statistic}(\phi_s, f(S))$$

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on its indices. Let $[\Lambda]^{uc}$ be the list of operators uncontracted by $\Lambda$, and let $f$ be the embedding function `uncontractedListEmd` that maps indices from $[\Lambda]^{uc}$ back to their original positions in $\phi_s$. Given a subsequent Wick contraction $\Lambda'$ acting on the indices of the uncontracted list $[\Lambda]^{uc}$, let $C = \text{join}(\Lambda, \Lambda')$ be the Wick contraction on $\phi_s$ formed by their union. For any two indices $i, j$ in the uncontracted list, the image under $f$ of the sign finset of $\Lambda'$ between $i$ and $j$ is equal to the sign finset of the joined contraction $C$ between $f(i)$ and $f(j)$, restricted to the set of indices that were uncontracted by $\Lambda$. Mathematically: $$f(S(\Lambda', i, j)) = S(\text{join}(\Lambda, \Lambda'), f(i), f(j)) \cap \text{uncontracted}(\Lambda)$$ where $S(c, a, b)$ is the `signFinset` of contraction $c$ relative to indices $a$ and $b$, and $\text{uncontracted}(\Lambda)$ is the set of indices not paired in $\Lambda$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ 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 operators uncontracted by $\Lambda$. Let $f$ be the embedding that maps the index of an operator in the uncontracted list to its original index in $\phi_s$. For any Wick contraction $\Lambda'$ on the uncontracted list $[\Lambda]^{uc}$, let $C = \text{join}(\Lambda, \Lambda')$ be the Wick contraction on $\phi_s$ formed by the union of the pairings in $\Lambda$ and the mapped pairings from $\Lambda'$. The sign of the contraction $\Lambda'$ is equal to the product of two terms: $$\text{sign}(\Lambda') = \left( \prod_{\{i, j\} \in \Lambda', i < j} \mathcal{S}\left(\sigma(\phi'_j), \mathcal{F} \triangleright_s \langle \phi_s, S_C(f(i), f(j)) \setminus \text{uncontracted}(\Lambda) \rangle\right) \right) \cdot \left( \prod_{\{i, j\} \in \Lambda', i < j} \mathcal{S}\left(\sigma(\phi'_j), \mathcal{F} \triangleright_s \langle \phi_s, S_C(f(i), f(j)) \rangle\right) \right)$$ where: - $\{i, j\}$ are the indices of a contracted pair in $\Lambda'$ with $i < j$. - $\phi'_j$ is the field operator at index $j$ in the uncontracted list $[\Lambda]^{uc}$. - $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of the operator $\phi$. - $S_C(I, J)$ is the `signFinset` of the joined contraction $C$ between indices $I$ and $J$. - $\mathcal{F} \triangleright_s \langle \phi_s, A \rangle$ denotes the collective field statistic of the operators in $\phi_s$ at the set of indices $A$. - $\text{uncontracted}(\Lambda)$ is the set of indices in $\phi_s$ not paired by the initial contraction $\Lambda$. - $\mathcal{S}(s_1, s_2)$ is the sign factor, which is $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise.

Let $\phi_s$ be a list of field operators and let $i, j$ be indices such that $i < j$. Let $\Lambda_{\{i,j\}}$ be the Wick contraction consisting of the single pair $\{i, j\}$, and let $\Lambda'$ be a Wick contraction on the indices of $\phi_s$ that remain uncontracted by $\Lambda_{\{i,j\}}$ (i.e., all indices except $i$ and $j$). Let $C = \text{join}(\Lambda_{\{i,j\}}, \Lambda')$ be the Wick contraction on $\phi_s$ formed by the union of these pairings. The set of indices $S(C, i, j)$ between $i$ and $j$ that contribute to the sign of the contraction of the pair $\{i, j\}$ in $C$ is equal to the set of indices $k \in (i, j)$ such that the contraction partner of $k$ in $C$ is not strictly less than $i$. Mathematically, $$S(C, i, j) = \{ k \in \{i+1, \dots, j-1\} \mid \text{partner}_C(k) \not< i \}$$ where $\text{partner}_C(k)$ denotes the index paired with $k$ in $C$ (or is considered null if $k$ is uncontracted).

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Given two indices $i, j$ such that $i < j$, let $\Lambda_{\{i,j\}}$ be the Wick contraction consisting of the single pair $\{i, j\}$. Let $\Lambda'$ be a Wick contraction on the indices of $\phi_s$ that are not $i$ or $j$, and let $C = \text{join}(\Lambda_{\{i,j\}}, \Lambda')$ be the resulting Wick contraction on the full list of indices. The collective field statistic of all indices $k$ strictly between $i$ and $j$ is equal to the product (in the commutative group of field statistics $\mathbb{Z}_2$) of: 1. The collective field statistic of the set $S(C, i, j)$, which contains indices $k \in (i, j)$ whose contraction partner in $C$ is not strictly less than $i$. 2. The collective field statistic of the set of indices $k \in (i, j)$ whose contraction partner in $C$ is strictly less than $i$. Mathematically, this is expressed as: $$\text{stat}(\{k \mid i < k < j\}) = \text{stat}(\{k \mid i < k < j, \text{partner}_C(k) \not< i\}) \cdot \text{stat}(\{k \mid i < k < j, \text{partner}_C(k) < i\})$$ where $\text{stat}(A)$ denotes the collective field statistic $\mathcal{F} \triangleright_s \langle \phi_s, A \rangle$ for a set of indices $A$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Given indices $i, j$ such that $i < j$, let $\Lambda_{\{i,j\}}$ be the Wick contraction consisting only of the pair $\{i, j\}$. Let $\Lambda'$ be a Wick contraction on the indices of the field operators in $\phi_s$ that are not $i$ or $j$. Let $C = \text{join}(\Lambda_{\{i,j\}}, \Lambda')$ be the Wick contraction on the full list $\phi_s$ formed by the union of these pairings. This function calculates a complex phase factor (the "right extra" sign) defined as the product over all contracted pairs $\{u, v\}$ in $\Lambda'$ (where $u < v$ are the mapped indices in the original list $\phi_s$): \[ \prod_{\{u, v\} \in \Lambda'} \mathcal{S}\left(\text{stat}(\phi_v), \text{stat}(K_{u,v})\right) \] where $K_{u,v} = \{k \in \{i, j\} \mid k \in S(C, u, v)\}$ is the subset of the indices from the singleton $\{i, j\}$ that lie in the sign set $S(C, u, v)$ of the pair $\{u, v\}$. The sign set $S(C, u, v)$ consists of indices $k$ such that $u < k < v$ and the contraction partner of $k$ in $C$ is not strictly less than $u$. The function $\mathcal{S}$ returns the statistical phase ($-1$ if both arguments are fermionic, $1$ otherwise). This factor represents the sign difference between the intrinsic sign of $\Lambda'$ and its contribution to the sign of the joined contraction $C$.

Given a list of field operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$, a pair of indices $i < j$ representing a singleton Wick contraction $\{i, j\}$, and a Wick contraction $\Lambda'$ on the remaining uncontracted indices, let $\Lambda = \text{join}(\text{singleton}\{i, j\}, \Lambda')$ be the total Wick contraction on $n$ indices. The function returns the statistical sign factor $\sigma(s(\phi_j), s(S))$, where: 1. $s(\phi_j)$ is the field statistic (bosonic or fermionic) of the operator at position $j$. 2. $S$ is the subset of indices $c$ strictly between $i$ and $j$ ($i < c < j$) such that $c$ is paired in the total contraction $\Lambda$ with an index $k < i$. 3. $s(S)$ is the collective field statistic of the operators indexed by $S$. This value represents the sign contribution arising from the exchange of the field at position $j$ with fields that are already contracted to the left of $i$ within the context of the larger contraction $\Lambda$.

Let $\mathcal{F}$ be a field specification and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. For indices $i < j$, let $\Lambda_{\{i,j\}}$ be the singleton Wick contraction consisting of the pair $\{i, j\}$. Let $\Lambda'$ be a Wick contraction on the list of operators in $\phi_s$ that are not at positions $i$ or $j$. Let $C = \text{join}(\Lambda_{\{i,j\}}, \Lambda')$ be the total Wick contraction on the full list of indices. The sign of the singleton contraction $\Lambda_{\{i,j\}}$ can be decomposed as: \[ \text{sign}(\Lambda_{\{i,j\}}) = \mathcal{S}\left(\text{stat}(\phi_j), \text{stat}(S(C, i, j))\right) \cdot \text{joinSignLeftExtra}(i, j, \Lambda') \] where: 1. $\text{sign}(\Lambda_{\{i,j\}})$ is the permutation sign of the singleton contraction (calculated relative to all indices in $(i, j)$). 2. $\mathcal{S}(s_1, s_2)$ is the statistical phase factor (equal to $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise). 3. $\text{stat}(\phi_j)$ is the field statistic (bosonic or fermionic) of the operator at index $j$. 4. $S(C, i, j)$ is the "sign set" of the joined contraction $C$ for the pair $\{i, j\}$, containing indices $k \in (i, j)$ that are either uncontracted in $C$ or paired with an index $l > i$. 5. $\text{stat}(S(C, i, j))$ is the collective field statistic of the operators indexed by $S(C, i, j)$. 6. $\text{joinSignLeftExtra}(i, j, \Lambda')$ is the extra sign factor accounting for the exchange of $\phi_j$ with indices $k \in (i, j)$ that are contracted in $C$ with indices strictly less than $i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. For indices $i < j$ in $\phi_s$, let $\Lambda_{\{i, j\}}$ be the singleton Wick contraction consisting of the pair $\{i, j\}$. Let $\phi'_s = [\Lambda_{\{i,j\}}]^{uc}$ be the sub-list of operators remaining after removing $\phi_s[i]$ and $\phi_s[j]$. For any Wick contraction $\Lambda'$ defined on the sub-list $\phi'_s$, the sign of $\Lambda'$ is related to the joined contraction $C = \text{join}(\Lambda_{\{i, j\}}, \Lambda')$ on the original list $\phi_s$ by: $$\text{sign}(\Lambda') = \text{joinSignRightExtra}(i, j, \Lambda') \cdot \prod_{\{u, v\} \in \Lambda', u < v} \mathcal{S}\left(\sigma(\phi'_v), \sigma(S_C(f(u), f(v)))\right)$$ where: - $\text{sign}(\Lambda')$ is the permutation sign of the contraction $\Lambda'$ on the sub-list $\phi'_s$. - $\text{joinSignRightExtra}(i, j, \Lambda')$ is the right extra phase factor accounting for the exchange of indices in $\Lambda'$ with the pair $\{i, j\}$. - $u$ and $v$ (with $u < v$) are the indices of a contracted pair in $\Lambda'$. - $f: \text{Fin}(|\phi'_s|) \hookrightarrow \text{Fin}(|\phi_s|)$ is the embedding that maps indices of the sub-list to their original positions in $\phi_s$. - $\sigma(\phi'_v)$ is the field statistic (bosonic or fermionic) of the operator at index $v$ in the sub-list. - $S_C(f(u), f(v))$ is the `signFinset` of the joined contraction $C$ between the mapped indices $f(u)$ and $f(v)$. - $\sigma(S)$ is the collective field statistic of the operators at indices in the set $S$. - $\mathcal{S}(s_1, s_2)$ is the statistical phase factor ($-1$ if both $s_1$ and $s_2$ are fermionic, $1$ otherwise).

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. For any two indices $i, j$ such that $i < j$, let $\Lambda'$ be a Wick contraction on the list of operators formed by removing $\phi_s[i]$ and $\phi_s[j]$ from the original list. The phase factor $\text{joinSignRightExtra}$ is equal to the product over all contracted pairs $a = \{u, v\} \in \Lambda'$ (where $u < v$ are the indices mapped back to their original positions in $\phi_s$) of the statistical phase: \[ \prod_{\{u, v\} \in \Lambda'} \mathcal{S}\left(\text{stat}(\phi_v), \text{stat}(K_{u,v})\right) \] In this formula, $\text{stat}(\phi_v)$ is the field statistic of the operator at index $v$, $\mathcal{S}$ is the statistical phase factor (which is $-1$ if both arguments are fermionic and $1$ otherwise), and $\text{stat}(K_{u,v})$ is the collective field statistic of the indices in the set $K_{u,v} \subseteq \{i, j\}$ defined by: \[ K_{u,v} = \left( \text{if } u < j < v \text{ and } u < i \text{ then } \{j\} \text{ else } \emptyset \right) \cup \left( \text{if } u < i < v \text{ then } \{i\} \text{ else } \emptyset \right) \]

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $i$ and $j$ be indices in the list such that $i < j$. If the field statistics (bosonic or fermionic) of the operators at these indices are identical, i.e., $$\mathcal{F} \triangleright_s \phi_s[i] = \mathcal{F} \triangleright_s \phi_s[j]$$ then for any Wick contraction $\Lambda'$ defined on the list of operators remaining after $i$ and $j$ are removed, the left extra sign factor $\text{joinSignLeftExtra}$ and the right extra sign factor $\text{joinSignRightExtra}$ associated with joining the pair $\{i, j\}$ to $\Lambda'$ are equal: $$\text{joinSignLeftExtra}(i, j, \Lambda') = \text{joinSignRightExtra}(i, j, \Lambda')$$

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $i$ and $j$ be indices in the list such that $i < j$. If the field statistics (bosonic or fermionic) of the operators at these indices are identical, i.e., $$\mathcal{F} \triangleright_s \phi_s[i] = \mathcal{F} \triangleright_s \phi_s[j]$$ then for any Wick contraction $\Lambda'$ defined on the list of operators remaining after $\phi_s[i]$ and $\phi_s[j]$ are removed, the sign of the joined Wick contraction $\text{join}(\Lambda_{\{i,j\}}, \Lambda')$ is equal to the product of the sign of the singleton contraction $\Lambda_{\{i,j\}}$ and the sign of $\Lambda'$: $$\text{sign}(\text{join}(\Lambda_{\{i,j\}}, \Lambda')) = \text{sign}(\Lambda_{\{i,j\}}) \cdot \text{sign}(\Lambda')$$ where $\Lambda_{\{i,j\}}$ is the singleton contraction consisting of the pair $\{i, j\}$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$ that is grading compliant, and let $\Lambda'$ be a Wick contraction on the list of uncontracted operators $[\Lambda]^{uc}$. For any natural number $n$ such that the number of pairs in $\Lambda$ is $n$ (i.e., $|\Lambda| = n$), the sign of the joined Wick contraction is the product of the signs of the individual contractions: $$\text{sign}(\text{join}(\Lambda, \Lambda')) = \text{sign}(\Lambda) \cdot \text{sign}(\Lambda')$$

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$ that is grading compliant, and let $\Lambda'$ be a Wick contraction on the list of uncontracted operators $[\Lambda]^{uc}$ (the sub-list of operators in $\phi_s$ that are not participating in any pairing in $\Lambda$). The sign of the joined Wick contraction $\text{join}(\Lambda, \Lambda')$ is equal to the product of the signs of the individual contractions: $$\text{sign}(\text{join}(\Lambda, \Lambda')) = \text{sign}(\Lambda) \cdot \text{sign}(\Lambda')$$

Let $\mathcal{F}$ be a field specification and $\phi_s$ 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 $[\Lambda]^{uc}$. The product of the sign and the time contraction of the joined Wick contraction $\text{join}(\Lambda, \Lambda')$ is equal to the product of the corresponding terms for $\Lambda$ and $\Lambda'$: $$\text{sign}(\text{join}(\Lambda, \Lambda')) \cdot \text{timeContract}(\text{join}(\Lambda, \Lambda')) = (\text{sign}(\Lambda) \cdot \text{timeContract}(\Lambda)) \cdot (\text{sign}(\Lambda') \cdot \text{timeContract}(\Lambda'))$$ where the time contraction is viewed as an element of the Wick algebra, and the sign acts by scalar multiplication.