Physlib.QFT.PerturbationTheory.WickContraction.Sign.Basic

Given a Wick contraction $c$ of $n$ indices and two specific indices $i_1, i_2 \in \{0, 1, \dots, n-1\}$, this function returns the set of indices $i$ such that $i_1 < i < i_2$ and $i$ is not contracted with any index $j \le i_1$. Specifically, an index $i$ is an element of this set if: 1. It lies strictly between $i_1$ and $i_2$ ($i_1 < i < i_2$). 2. It is either uncontracted in $c$, or the index $j$ it is paired with satisfies $j > i_1$. If $i_1 \ge i_2$, the resulting set is empty. This set identifies the indices between $i_1$ and $i_2$ that contribute to the permutation sign when considering the contraction of fields at positions $i_1$ and $i_2$ in Wick's theorem.

Given a list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ and a Wick contraction $\Lambda$ (represented as a set of disjoint pairs $\{i, j\}$ of indices), the sign of the contraction is the complex number (specifically $1$ or $-1$) defined by: $$\prod_{\{i, j\} \in \Lambda, i < j} \mathcal{S}(\sigma(\phi_j), \sigma_{\text{mid}})$$ where $\sigma(\phi)$ denotes the statistic of an operator (bosonic or fermionic), $\mathcal{S}(s_1, s_2)$ is the sign factor which is $-1$ if both statistics $s_1$ and $s_2$ are fermionic and $1$ otherwise, and $\sigma_{\text{mid}}$ is the aggregate statistic of all operators $\phi_k$ such that $i < k < j$ and index $k$ has not been paired with any index $l < i$. This sign corresponds to the total number of swaps of adjacent fermionic operators required to bring all contracted pairs together.

For any list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ belonging to a field specification $\mathcal{F}$, the sign associated with the empty Wick contraction $\Lambda = \emptyset$ (the contraction that contains no pairs of indices) is equal to $1$.

Let $\Phi$ and $\Phi'$ be lists of field operators belonging to a field specification $\mathcal{F}$. If the two lists are equal ($\Phi = \Phi'$), then for any Wick contraction $\Lambda$ of the indices of $\Phi$, the sign of the contraction $\text{sign}(\Phi, \Lambda)$ is equal to the sign of the corresponding contraction $\Lambda'$ for the list $\Phi'$. Here, $\Lambda'$ is the contraction obtained by identifying the indices of $\Phi$ and $\Phi'$ via the equality of the lists.