Physlib.QFT.PerturbationTheory.WickContraction.StaticContract

Given a list of field operators $\phi_0, \phi_1, \dots, \phi_{n-1}$ from a field specification $\mathcal{F}$ and a Wick contraction $\Lambda$ acting on these operators, the static contraction is the element of the center of the Wick algebra $\mathcal{W}(\mathcal{F})$ defined by the product: $$\text{staticContract}(\Lambda) = \prod_{\{j, k\} \in \Lambda, j < k} [\phi_j^-, \phi_k]_s$$ where: - The product is taken over all pairs of indices $\{j, k\}$ contracted by $\Lambda$, with the condition $j < k$. - $\phi_j^-$ denotes the annihilation part (the `anPart`) of the $j$-th field operator. - $[\cdot, \cdot]_s$ denotes the super-commutator.

Let $\mathcal{F}$ be a field specification and $\phi_s = (\phi_0, \dots, \phi_{n-1})$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose a new field operator $\phi$ is inserted into the list $\phi_s$ at index $i \in \{0, \dots, n\}$ such that $\phi$ remains uncontracted, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ on the extended list of length $n+1$. Then, the static contraction of $\Lambda'$ is equal to the static contraction of $\Lambda$: $$\text{staticContract}(\Lambda \hookleftarrow_\Lambda \phi, i, \text{none}) = \text{staticContract}(\Lambda)$$ where the static contraction $\text{staticContract}(\Lambda)$ is the element of the center of the Wick algebra defined by the product of super-commutators $\prod_{\{j, k\} \in \Lambda, j < k} [\phi_j^-, \phi_k]_s$.

Let $\mathcal{F}$ be a field specification and $\phi_s = (\phi_0, \dots, \phi_{n-1})$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we insert a field operator $\phi$ at index $i \in \{0, \dots, n\}$ and contract it with the operator $\phi_j$ (where $j$ was previously uncontracted in $\Lambda$). Let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ be the resulting Wick contraction on the expanded list of length $n+1$. The static contraction of $\Lambda'$ is given by: $$\text{staticContract}(\Lambda') = \begin{cases} [\phi^-, \phi_j]_s \cdot \text{staticContract}(\Lambda) & \text{if } i \leq j \\ [\phi_j^-, \phi]_s \cdot \text{staticContract}(\Lambda) & \text{if } j < i \end{cases}$$ where $\phi^-$ denotes the annihilation part (`anPart`) of the field operator $\phi$, and $[\cdot, \cdot]_s$ denotes the super-commutator in the Wick algebra.

Let $\mathcal{F}$ be a field specification. Consider a list of field operators $\phi_s = (\phi_0, \dots, \phi_{n-1})$ and a Wick contraction $\Lambda$ on these operators. Let $\phi$ be a new field operator inserted into the list at index $i \in \{0, \dots, n\}$, and let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } k$ be the Wick contraction formed by contracting $\phi$ with the field operator originally at index $k$ (where $k$ was an uncontracted index in $\Lambda$). If the insertion index $i$ is such that $i < \text{succAbove}_i(k)$ (which implies the insertion point is at or to the left of the original index $k$, i.e., $i \le k$), then the static contraction of $\Lambda'$ is given by: $$\text{staticContract}(\Lambda') = \mathcal{S}(\phi, \Phi_{\mathcal{U}, <k}) \cdot (\text{contractStateAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k)) \cdot \text{staticContract}(\Lambda))$$ where: - $\mathcal{S}(\phi, \Phi_{\mathcal{U}, <k})$ is the statistic sign factor associated with the field $\phi$ and the collection of uncontracted fields in $\Lambda$ whose indices are strictly less than $k$. - $[\Lambda]^{uc}$ is the list of uncontracted field operators of $\phi_s$ under the contraction $\Lambda$. - $\text{pos}(k)$ is the position of the index $k$ within the sorted list of uncontracted indices, as determined by the equivalence `uncontractedFieldOpEquiv`. - $\text{contractStateAtIndex}$ computes the contraction contribution (typically the super-commutator $[\phi^-, \phi_k]_s$) between $\phi$ and the uncontracted field at the specified position. - $\text{staticContract}(\Lambda)$ is the product of super-commutators for all pairs already contracted in $\Lambda$.

Let $\mathcal{F}$ be a field specification and $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators in $\text{FieldOp}(\mathcal{F})$. For any Wick contraction $\Lambda$ acting on these operators, if $\Lambda$ is not grading-compliant with respect to the field operators $\phi_s$, then its associated static contraction is zero: $$\text{staticContract}(\Lambda) = 0$$