Physlib.QFT.PerturbationTheory.WickContraction.Perm

Let $\mathcal{F}$ be a field specification and $\phi s$ be a list of field operators in $\mathcal{F}.\text{FieldOp}$ of length $n$. Given two Wick contractions $\Lambda_1$ and $\Lambda_2$ of the indices $\{0, 1, \dots, n-1\}$, they are said to be permutations of each other if the Wick terms they produce from the list $\phi s$ are equal.

Let $\mathcal{F}$ be a field specification and $\phi s$ be a list of field operators in $\mathcal{F}.\text{FieldOp}$ of length $n$. For any Wick contraction $\Lambda$ of the indices $\{0, 1, \dots, n-1\}$, the relation $\text{Perm}(\Lambda, \Lambda)$ holds, meaning $\Lambda$ is a permutation of itself.

Let $\mathcal{F}$ be a field specification and $\phi s$ be a list of field operators in $\mathcal{F}.\text{FieldOp}$ of length $n$. For any two Wick contractions $\Lambda_1$ and $\Lambda_2$ of the indices $\{0, 1, \dots, n-1\}$, if $\Lambda_1$ is a permutation of $\Lambda_2$ (meaning they produce the same Wick term from the list $\phi s$), then $\Lambda_2$ is a permutation of $\Lambda_1$.

Let $\phi s$ be a list of field operators and let $n$ be its length. For any three Wick contractions $\Lambda_1, \Lambda_2$, and $\Lambda_3$ of the indices $\{0, 1, \dots, n-1\}$, if $\Lambda_1$ is a permutation of $\Lambda_2$ and $\Lambda_2$ is a permutation of $\Lambda_3$, then $\Lambda_1$ is a permutation of $\Lambda_3$. Two Wick contractions are defined to be permutations of each other if the Wick terms they produce from the list of field operators are equal.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Suppose $\Lambda_1$ and $\Lambda_2$ are two Wick contractions of the indices of $\phi_s$ that are permutations of each other (meaning they produce the same Wick term). If both $\Lambda_1$ and $\Lambda_2$ are grading-compliant and $\Lambda_1$ is a full contraction (meaning its set of uncontracted indices is empty), then $\Lambda_2$ is also a full contraction.

Let $\phi_s$ be a list of field operators. Suppose $\Lambda_1$ and $\Lambda_2$ are two Wick contractions of $\phi_s$ that are permutations of each other (meaning they produce the same Wick term). If both $\Lambda_1$ and $\Lambda_2$ are grading compliant, then the list of uncontracted field operators $[\Lambda_1]^{uc}$ is a permutation of the list of uncontracted field operators $[\Lambda_2]^{uc}$.