Physlib.QFT.PerturbationTheory.WickContraction.TimeContract

For a list of field operators $\Phi = (\phi_0, \phi_1, \dots, \phi_{n-1})$ and a Wick contraction $\Lambda$ (a set of disjoint pairs of indices from $\{0, \dots, n-1\}$), the time contraction of $\Lambda$ is defined as the product of the pairwise contractions of the operators corresponding to the indices in $\Lambda$: $$\prod_{\{j, k\} \in \Lambda, j < k} \text{timeContract}(\phi_j, \phi_k)$$ This product is an element of the center of the Wick algebra $\mathcal{W}(\mathcal{F})$ over $\mathbb{C}$. Each term $\text{timeContract}(\phi_j, \phi_k)$ represents the contraction of the $j$-th and $k$-th operators in the sequence.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. Consider the Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$ obtained by inserting a field operator $\phi$ at index $i$ into the list $\phi_s$ without creating any new contraction pairs. Then the time contraction of $\Lambda'$ is equal to the time contraction of $\Lambda$: $$(\Lambda \hookleftarrow_\Lambda \phi, i, \text{none}).\text{timeContract} = \Lambda.\text{timeContract}$$

Let $\mathcal{F}$ be a field specification and $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$ and $j$ be an index belonging to the set of uncontracted indices of $\Lambda$. For a field operator $\phi$ and an insertion index $i \in \{0, \dots, n\}$, let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ be the Wick contraction formed by inserting $\phi$ at index $i$ and contracting it with the operator originally at index $j$. The time contraction of $\Lambda'$ is given by the product of the time contraction of $\Lambda$ and a pairwise contraction: $$\text{timeContract}(\Lambda') = \begin{cases} \text{timeContract}(\phi, \phi_j) \cdot \text{timeContract}(\Lambda) & \text{if } i < j' \\ \text{timeContract}(\phi_j, \phi) \cdot \text{timeContract}(\Lambda) & \text{if } j' < i \end{cases}$$ where $j'$ is the shifted position of the original $j$-th field in the new list, defined as $j' = j$ if $j < i$ and $j' = j+1$ if $j \geq i$.

For any list of field operators $\Phi = (\phi_0, \phi_1, \dots, \phi_{n-1})$ associated with a field specification $\mathcal{F}$, the time contraction of the empty Wick contraction (the contraction containing no paired indices) is equal to $1$ in the center of the Wick algebra $\mathcal{W}(\mathcal{F})$.

Let $\mathcal{F}$ be a field specification and $\Phi = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\Phi$, and let $k$ be an index of a field that is uncontracted in $\Lambda$. Suppose a field operator $\phi$ is inserted into the list at index $i \le k$ and contracted with the original $k$-th field, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } k$ for the expanded list. If $\phi$ is chronologically later than or equal to $\phi_k$ (i.e., the relation $\text{timeOrderRel}(\phi, \phi_k)$ holds), then the time contraction of $\Lambda'$ is given by: $$\text{timeContract}(\Lambda') = \mathcal{S}(\phi, \{\phi_j \in [\Lambda]^{uc} \mid j < k\}) \cdot (\text{contractStateAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k)) \cdot \text{timeContract}(\Lambda))$$ where: - $\text{timeContract}(\Lambda)$ is the product of pairwise contractions for $\Lambda$. - $[\Lambda]^{uc}$ is the list of uncontracted field operators in $\Phi$. - $\text{pos}(k)$ is the position of the $k$-th field within the list of uncontracted operators. - $\text{contractStateAtIndex}$ represents the pairwise contraction of the annihilation part of $\phi$ with $\phi_k$. - $\mathcal{S}(\phi, \{\dots\})$ is the exchange sign between $\phi$ and the set of uncontracted fields in $\Phi$ whose original indices are strictly less than $k$.

Let $\mathcal{F}$ be a field specification and $\Phi = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\Phi$ and $k$ be an index in the set of uncontracted indices of $\Lambda$. Suppose a new field operator $\phi$ is inserted into the list at index $i$, where $k < i$, and $\phi$ is contracted with $\phi_k$ to form a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } k$. If $\phi_k$ is not chronologically later than or equal to $\phi$ (i.e., $\neg \text{timeOrderRel}(\phi_k, \phi)$), then the time contraction of $\Lambda'$ is given by: $$\text{timeContract}(\Lambda') = \mathcal{S}(\phi, \mathcal{U}_{\le k}) \cdot (\text{contractStateAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k)) \cdot \text{timeContract}(\Lambda))$$ where: - $\mathcal{U}_{\le k}$ is the sublist of field operators in $\Phi$ that are uncontracted in $\Lambda$ and have indices $j \le k$. - $\mathcal{S}(\phi, \mathcal{U}_{\le k})$ is the exchange sign (grading factor) obtained by commuting $\phi$ with the uncontracted operators in $\mathcal{U}_{\le k}$. - $[\Lambda]^{uc}$ is the list of all uncontracted field operators in $\Phi$ relative to $\Lambda$. - $\text{pos}(k)$ is the position of the operator $\phi_k$ within the list $[\Lambda]^{uc}$.

For a list of field operators $\Phi = (\phi_0, \phi_1, \dots, \phi_{n-1})$ of a field specification $\mathcal{F}$ and a Wick contraction $\Lambda$ on these operators, if $\Lambda$ is not grading compliant with respect to $\Phi$, then the time contraction of $\Lambda$ is zero: $$\text{timeContract}(\Lambda) = 0$$