Physlib.QFT.PerturbationTheory.WickContraction.TimeCond

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$. The property `EqTimeOnly` holds for $\Lambda$ if, for every pair of indices $\{i, j\}$ that form a contraction in $\Lambda$, the time-ordering relation $\text{timeOrderRel}$ holds in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. Physically, this condition signifies that the contraction only pairs operators that exist at the same time (for position-space operators) or belong to the same asymptotic temporal limit (both incoming or both outgoing).

For a list of field operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ and a Wick contraction $\Lambda$ of the indices $\{0, 1, \dots, n-1\}$, the property $\text{EqTimeOnly}(\phi_s, \Lambda)$ is decidable. This property holds if, for every pair of indices $\{i, j\}$ contracted in $\Lambda$, the time-ordering relation $\text{timeOrderRel}$ holds in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$. Suppose $\Lambda$ satisfies the property `EqTimeOnly`, which means that for every pair of indices $\{i, j\}$ contracted in $\Lambda$, the time-ordering relation holds in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. Then, for any specific pair of indices $\{i, j\}$ contained in $\Lambda$, the time-ordering relation $\text{timeOrderRel}(\phi_i, \phi_j)$ holds.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$. Suppose $\Lambda$ satisfies the property $\text{EqTimeOnly}$, meaning that for every pair of indices $\{i, j\}$ contracted in $\Lambda$, the time-ordering relation $\text{timeOrderRel}$ holds in both directions. Then, for any specific pair of indices $\{i, j\}$ contained in $\Lambda$, the time-ordering relation holds in both directions: $\text{timeOrderRel}(\phi_i, \phi_j) \land \text{timeOrderRel}(\phi_j, \phi_i)$. Physically, the $\text{EqTimeOnly}$ condition signifies that the contraction only pairs operators that exist at the same time (for position-space operators) or belong to the same asymptotic temporal limit (both incoming or both outgoing).

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$. The property $\text{EqTimeOnly}(\Lambda)$ holds if and only if for every contracted pair $a \in \Lambda$, the time-ordering relation $\text{timeOrderRel}$ holds in both directions for the operators associated with the indices of $a$. Specifically, if $i$ is the smaller index and $j$ is the larger index of the pair $a$, the condition is: $$\text{timeOrderRel}(\phi_i, \phi_j) \wedge \text{timeOrderRel}(\phi_j, \phi_i)$$ Physically, this means the contraction $\Lambda$ consists only of equal-time pairings.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. The empty Wick contraction on these $n$ indices satisfies the property $\text{EqTimeOnly}$, meaning it vacuously satisfies the condition that all its contracted pairs are at equal times.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$. If $\Lambda$ satisfies the property $\mathtt{EqTimeOnly}$—meaning that every contracted pair of indices $\{i, j\} \in \Lambda$ corresponds to operators $\phi_i$ and $\phi_j$ that occur at the same time (formally, the time-ordering relation $\text{timeOrderRel}$ holds in both directions)—then the static contraction of $\Lambda$ is equal to its time contraction: $$\text{staticContract}(\Lambda) = \text{timeContract}(\Lambda)$$ where $\text{staticContract}(\Lambda)$ is the product of super-commutators of the annihilation parts of the fields, and $\text{timeContract}(\Lambda)$ is the product of the pairwise time-ordered contractions.

Let $\phi_s$ and $\phi_s'$ be two lists of field operators in a field specification $\mathcal{F}$. If $\phi_s = \phi_s'$, then for any Wick contraction $\Lambda$ on the indices of $\phi_s$, the property `EqTimeOnly` holds for $\Lambda$ with respect to $\phi_s$ if and only if the congruent Wick contraction (induced by the equality of lengths) satisfies `EqTimeOnly` with respect to $\phi_s'$.

Let $\phi_s = [\phi_0, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on these operators. Suppose $\Lambda$ satisfies the property $\text{EqTimeOnly}$, meaning that for every pair of contracted indices $\{i, j\} \in \Lambda$, the associated field operators $\phi_i$ and $\phi_j$ satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. Given a subset of contracted pairs $S \subseteq \Lambda$, the quotient contraction $\Lambda/S$ (which consists of the pairs in $\Lambda \setminus S$ relabeled to act on the list of operators remaining after the contractions in $S$ are performed) also satisfies the $\text{EqTimeOnly}$ property.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$ that contains at least one pair (i.e., its cardinality $|\Lambda| > 0$) and satisfies the $\text{EqTimeOnly}$ property (meaning every contracted pair consists of fields at the same time or same asymptotic limit). Then there exist indices $i, j$ with $i < j$ and a Wick contraction $\Lambda'$ defined on the list of field operators remaining after the pair $\{i, j\}$ is removed (denoted $[\text{singleton}(i, j)]^{uc}$) such that: 1. $\Lambda$ can be decomposed as the join of the singleton contraction of $\{i, j\}$ and the remaining contraction $\Lambda'$. 2. The field operators at indices $i$ and $j$ satisfy the bidirectional time-ordering relation $\text{timeOrderRel}(\phi_i, \phi_j) \land \text{timeOrderRel}(\phi_j, \phi_i)$. 3. The remaining Wick contraction $\Lambda'$ also satisfies the $\text{EqTimeOnly}$ property. 4. The cardinality of the original contraction is one greater than the cardinality of the remaining contraction ($|\Lambda| = |\Lambda'| + 1$).

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Suppose $\Lambda$ satisfies the equal-time property ($\text{EqTimeOnly}$), which implies that for every pair of indices $\{i, j\}$ contracted in $\Lambda$, the corresponding field operators satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. Let $a$ and $b$ be elements of the Wick algebra $\mathcal{W}(\mathcal{F})$. For any natural number $n$, if the cardinality of the contraction $\Lambda$ is $n$ ($|\Lambda| = n$), then the time-ordering operator $\mathcal{T}$ satisfies: \[ \mathcal{T}(a \cdot \text{timeContract}(\Lambda) \cdot b) = \text{timeContract}(\Lambda) \cdot \mathcal{T}(a \cdot b) \] where $\text{timeContract}(\Lambda)$ is the central element of the algebra representing the product of the pairwise contractions in $\Lambda$.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Suppose $\Lambda$ satisfies the equal-time property ($\text{EqTimeOnly}$), which implies that for every pair of indices $\{i, j\}$ contracted in $\Lambda$, the corresponding field operators satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. Let $a$ and $b$ be elements of the Wick algebra $\mathcal{W}(\mathcal{F})$. Then the time-ordering operator $\mathcal{T}$ satisfies: \[ \mathcal{T}(a \cdot \text{timeContract}(\Lambda) \cdot b) = \text{timeContract}(\Lambda) \cdot \mathcal{T}(a \cdot b) \] where $\text{timeContract}(\Lambda)$ is the central element of the algebra representing the product of the pairwise contractions in $\Lambda$.

Let $\phi_s$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. Suppose $\Lambda$ satisfies the equal-time property ($\text{EqTimeOnly}$), which implies that for every pair of indices $\{i, j\}$ contracted in $\Lambda$, the corresponding field operators satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. For any element $b$ of the Wick algebra $\mathcal{W}(\mathcal{F})$, the time-ordering operator $\mathcal{T}$ satisfies: \[ \mathcal{T}(\text{timeContract}(\Lambda) \cdot b) = \text{timeContract}(\Lambda) \cdot \mathcal{T}(b) \] where $\text{timeContract}(\Lambda)$ is the central element of the algebra representing the product of the pairwise contractions in $\Lambda$.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on these operators. If $\Lambda$ is not an equal-time contraction (i.e., the property $\text{EqTimeOnly}(\Lambda)$ does not hold), then there exist indices $i$ and $j$ with $i < j$ and a Wick contraction $\Lambda_{uc}$ on the list of operators remaining after contracting the pair $\{i, j\}$, such that: 1. $\Lambda$ is the join of the singleton contraction $\{i, j\}$ and $\Lambda_{uc}$. 2. The pair $\{i, j\}$ violates the equal-time condition, meaning either $\neg \text{timeOrderRel}(\phi_i, \phi_j)$ or $\neg \text{timeOrderRel}(\phi_j, \phi_i)$.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of $\phi_s$. If $\Lambda$ is not an equal-time contraction (i.e., the property $\text{EqTimeOnly}(\Lambda)$ does not hold), then the time-ordering $\mathcal{T}$ of the time contraction of $\Lambda$ is zero: \[ \mathcal{T}(\Lambda.\text{timeContract}) = 0 \] This means that if $\Lambda$ contains at least one pair of indices $\{i, j\}$ such that the operators $\phi_i$ and $\phi_j$ do not satisfy the two-way time-ordering relation $\text{timeOrderRel}$, then the time-ordered product of its pairwise contractions vanishes.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on these operators. If $\Lambda$ is not an equal-time contraction—meaning there exists at least one contracted pair of indices $\{i, j\} \in \Lambda$ such that the corresponding operators $\phi_i$ and $\phi_j$ do not occur at the same time (i.e., the time-ordering relation $\text{timeOrderRel}$ does not hold in both directions)—then the time-ordering $\mathcal{T}$ of the static contraction of $\Lambda$ is zero: $$\mathcal{T}(\text{staticContract}(\Lambda)) = 0$$ where the static contraction is defined as the product of super-commutators $\prod_{\{j, k\} \in \Lambda, j < k} [\phi_j^-, \phi_k]_s$.

Given a list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ and a Wick contraction $\mathcal{C}$ (a collection of disjoint pairs of indices $\{i, j\}$), the property $\text{HaveEqTime}(\mathcal{C})$ holds if there exists a pair $\{i, j\} \in \mathcal{C}$ such that the corresponding field operators $\phi_i$ and $\phi_j$ satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. This condition signifies that the contraction contains at least one pair of fields that are evaluated at the same time.

Given a list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ and a Wick contraction $\mathcal{C}$ (a collection of disjoint pairs of indices), it is decidable whether there exists a pair $\{i, j\} \in \mathcal{C}$ such that the corresponding field operators $\phi_i$ and $\phi_j$ satisfy both $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. This condition signifies that the contraction $\mathcal{C}$ contains at least one pair of fields evaluated at the same time.

Let $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\mathcal{C}$ be a Wick contraction (a collection of disjoint pairs of indices from $\{0, 1, \dots, n-1\}$). The property $\text{HaveEqTime}(\mathcal{C})$ holds if and only if there exists a pair $\{i, j\} \in \mathcal{C}$ such that the corresponding operators $\phi_i$ and $\phi_j$ satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$.

For any list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$, the empty Wick contraction (the contraction containing no pairs of indices) does not satisfy the property $\text{HaveEqTime}$. The property $\text{HaveEqTime}(\mathcal{C})$ holds if there exists a pair of indices $\{i, j\}$ in the contraction $\mathcal{C}$ such that the corresponding field operators $\phi_i$ and $\phi_j$ are evaluated at the same time, meaning they satisfy the time-ordering relation in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$.

Given a list of field operators $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ and a Wick contraction $\Lambda$ (which is a collection of disjoint pairs of indices $\{i, j\}$), the function returns the subset of these pairs such that the corresponding field operators are at the same time. Specifically, a pair $\{i, j\} \in \Lambda$ belongs to this subset if the time-ordering relation $\text{timeOrderRel}$ holds in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. For position-space operators, this condition is equivalent to the time components of their spacetime coordinates being equal ($x_i^0 = x_j^0$).

Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators belonging to a field specification $\mathcal{F}$. For any Wick contraction $\Lambda$ of these indices (defined as a collection of disjoint pairs $\{i, j\} \subset \{0, 1, \dots, n-1\}$), the set of equal-time contracted pairs in $\Lambda$—consisting of those pairs $\{i, j\} \in \Lambda$ where the operators $\phi_i$ and $\phi_j$ satisfy the time-ordering relation in both directions—is a subset of the set of all pairs in $\Lambda$.

Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators and $\Lambda$ be a Wick contraction of $n$ indices. If a pair of indices $a = \{i, j\}$ is an element of the set of equal-time contracted pairs $\text{eqTimeContractSet}(\Lambda)$, then $a$ is a member of the set of pairs in the contraction $\Lambda$.

Let $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\Lambda$ be a Wick contraction of these indices, and let $[\Lambda]^{uc}$ be the list of field operators that remain uncontracted by $\Lambda$. Suppose $\Lambda_{uc}$ is a further Wick contraction performed on the indices of the uncontracted list $[\Lambda]^{uc}$. The set of equal-time contracted pairs in the joined contraction $\text{join}(\Lambda, \Lambda_{uc})$ is the union of: 1. The set of equal-time contracted pairs in the initial contraction $\Lambda$. 2. The set of equal-time contracted pairs in the secondary contraction $\Lambda_{uc}$, where the indices of these pairs are mapped back to their original positions in $\Phi$ via the embedding $f: \{0, \dots, \text{length}([\Lambda]^{uc}) - 1\} \hookrightarrow \{0, \dots, n-1\}$. In symbols: $$\text{eqTimeContractSet}(\text{join}(\Lambda, \Lambda_{uc})) = \text{eqTimeContractSet}(\Lambda) \cup f(\text{eqTimeContractSet}(\Lambda_{uc}))$$ where $f$ denotes the lifting of the index pairs from the uncontracted sub-list to the original list.

Let $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\mathcal{C}$ be a Wick contraction of these operators. If $\mathcal{C}$ does not contain any pairs $\{i, j\}$ such that the operators $\phi_i$ and $\phi_j$ are evaluated at the same time (i.e., the property $\text{HaveEqTime}(\mathcal{C})$ is false), then the set of equal-time contracted pairs $\text{eqTimeContractSet}(\mathcal{C})$ is empty.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices of these operators. If $\Lambda$ is an equal-time contraction (meaning the property $\text{EqTimeOnly}(\Lambda)$ holds, such that every contracted pair satisfies the bidirectional time-ordering relation), then the set of equal-time contracted pairs $\text{eqTimeContractSet}(\Lambda)$ is equal to the set of all contracted pairs in $\Lambda$.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$. Let $S$ be the set of equal-time contracted pairs in $\Lambda$ relative to $\phi_s$ (denoted as `eqTimeContractSet`), consisting of pairs $\{i, j\} \in \Lambda$ such that the bidirectional time-ordering relation $\text{timeOrderRel}(\phi_i, \phi_j) \wedge \text{timeOrderRel}(\phi_j, \phi_i)$ holds. Then, the sub-contraction formed by the pairs in $S$ satisfies the property $\text{EqTimeOnly}$, meaning all its contracted pairs consist of operators evaluated at the same time.

Let $\phi_s = (\phi_0, \phi_1, \dots, \phi_{n-1})$ be a list of field operators and let $\Lambda$ be a Wick contraction on $n$ indices. For any pair of indices $\{i, j\}$ that is a member of $\Lambda$, $\{i, j\}$ belongs to the set of equal-time contracted pairs of $\Lambda$ if and only if the time-ordering relation $\text{timeOrderRel}$ holds in both directions: $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$. This condition signifies that the two field operators are chronologically at the same time.

Let $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and let $\Lambda$ be a Wick contraction on these indices. If $\Lambda$ contains at least one equal-time pair (i.e., the property $\text{HaveEqTime}(\Lambda)$ holds, meaning there exists a pair $\{i, j\} \in \Lambda$ such that $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$ both hold), then the sub-contraction consisting of all equal-time contracted pairs in $\Lambda$ is not the empty contraction.

Let $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $\Lambda$ be a Wick contraction on $\Phi$. Let $S = \text{eqTimeContractSet}(\Lambda)$ be the subset of all pairs $\{i, j\} \in \Lambda$ such that the field operators $\phi_i$ and $\phi_j$ are evaluated at the same time (i.e., $\text{timeOrderRel}(\phi_i, \phi_j)$ and $\text{timeOrderRel}(\phi_j, \phi_i)$ both hold). Then the quotient contraction $\Lambda/S$, which consists of the remaining pairs in $\Lambda$ relabeled for the reduced list of field operators, contains no equal-time pairs (i.e., $\text{HaveEqTime}(\Lambda/S)$ is false).

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$ such that $\Lambda$ is non-empty and satisfies the `EqTimeOnly` property (meaning every pair of indices $\{i, j\} \in \Lambda$ corresponds to fields $\phi_i, \phi_j$ that exist at the same time). For any Wick contraction $\Lambda'$ acting on the uncontracted field operators $[\Lambda]^{uc}$, the combined contraction $\text{join}(\Lambda, \Lambda')$ contains at least one equal-time pair (i.e., it satisfies the property `HaveEqTime`).

Let $\phi_s$ be a list of field operators. Consider the set of dependent pairs $(\Lambda, \Lambda')$ where: 1. $\Lambda$ is a non-empty Wick contraction on the indices of $\phi_s$ that satisfies the equal-time property (i.e., it consists only of pairings between fields at the same time). 2. $\Lambda'$ is a Wick contraction on the indices of the list of field operators left uncontracted by $\Lambda$ (denoted $[\Lambda]^{uc}$) such that $\Lambda'$ contains no equal-time pairs. The theorem states that for any two such pairs $x_1 = (\Lambda_1, \Lambda'_1)$ and $x_2 = (\Lambda_2, \Lambda'_2)$, if the first components are equal ($\Lambda_1 = \Lambda_2$) and the second components are equal (where $\Lambda'_2$ is identified with a contraction on the same indices as $\Lambda'_1$ via the equality of the first components), then the pairs $x_1$ and $x_2$ are equal.

Let $\phi_s$ be a list of field operators $[\phi_0, \phi_1, \dots, \phi_{n-1}]$. There exists an equivalence (a bijection) between the set of Wick contractions $\mathcal{C}$ on the indices of $\phi_s$ that contain at least one equal-time pair ($\text{HaveEqTime } \mathcal{C}$) and the set of pairs $(\Lambda, \Lambda')$ such that: 1. $\Lambda$ is a non-empty Wick contraction on the indices of $\phi_s$ that consists exclusively of equal-time pairs ($\text{EqTimeOnly } \Lambda$). 2. $\Lambda'$ is a Wick contraction on the list of field operators left uncontracted by $\Lambda$ (denoted $[\Lambda]^{uc}$) such that $\Lambda'$ contains no equal-time pairs ($\neg\text{HaveEqTime } \Lambda'$). Physically, this states that any Wick contraction containing equal-time pairings can be uniquely decomposed into a part $\Lambda$ containing all its equal-time pairings and a part $\Lambda'$ containing all its remaining (non-equal-time) pairings.

Let $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators and $M$ be an additive commutative monoid. Let $f$ be a function that maps a Wick contraction of the indices of $\Phi$ to an element of $M$. The sum of $f(\mathcal{C})$ over all Wick contractions $\mathcal{C}$ that contain at least one equal-time pair is given by: $$\sum_{\substack{\mathcal{C} \in \text{Wick}(n) \\ \text{HaveEqTime}(\mathcal{C})}} f(\mathcal{C}) = \sum_{\substack{\Lambda \in \text{Wick}(n) \\ \Lambda \neq \emptyset \\ \text{EqTimeOnly}(\Lambda)}} \sum_{\substack{\Lambda' \in \text{Wick}([\Lambda]^{uc}) \\ \neg \text{HaveEqTime}(\Lambda')}} f(\Lambda \cup \Lambda')$$ where: - $\text{HaveEqTime}(\mathcal{C})$ indicates that $\mathcal{C}$ contains at least one pair $\{i, j\}$ such that the operators $\phi_i$ and $\phi_j$ are evaluated at the same time. - $\text{EqTimeOnly}(\Lambda)$ indicates that every pair in the contraction $\Lambda$ consists of operators evaluated at the same time. - $[\Lambda]^{uc}$ denotes the list of field operators in $\Phi$ whose indices are not contracted by $\Lambda$. - $\Lambda \cup \Lambda'$ (denoted formally as `join`) is the Wick contraction formed by the union of the pairings in $\Lambda$ and $\Lambda'$.