Physlib.QFT.PerturbationTheory.WickContraction.Involutions

Given a Wick contraction $c$ on $n$ indices, this definition constructs an involution $\sigma: \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$. For any index $i$, $\sigma(i)$ is defined as the dual index of $i$ if $i$ is contracted in $c$; otherwise, $\sigma(i) = i$.

Given an involutive function $f : \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$ (such that $f(f(i)) = i$ for all $i$), this definition constructs a Wick contraction on $n$ elements. The contraction is defined as the collection of all subsets $\{i, f(i)\}$ that have cardinality $2$. These sets correspond precisely to the orbits of $f$ that contain two distinct elements.

Let $f: \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$ be an involutive function (i.e., $f(f(i)) = i$ for all $i$). Let $c$ be the Wick contraction constructed from $f$, where $c$ consists of all pairs $\{j, f(j)\}$ such that $j \neq f(j)$. For any index $i \in \{0, 1, \dots, n-1\}$, the index $i$ is paired with a dual index in $c$ if and only if $i$ is not a fixed point of $f$ ($f(i) \neq i$).

Let $f: \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$ be an involutive function (i.e., $f(f(i)) = i$ for all $i$). Let $C$ be the Wick contraction constructed from $f$. For any index $i \in \{0, 1, \dots, n-1\}$, if $i$ is contracted in $C$ (meaning it belongs to a pair in the contraction), then the index it is paired with is equal to $f(i)$.

Let $f: \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$ be an involutive function (such that $f(f(i)) = i$ for all $i$). Let $C$ be the Wick contraction constructed from $f$, which consists of all pairs $\{j, f(j)\}$ where $j \neq f(j)$. For any index $i \in \{0, 1, \dots, n-1\}$, if $i$ is contracted in $C$ (meaning it belongs to one of these pairs), then the dual index it is paired with is equal to $f(i)$.

Let $c$ be a Wick contraction on the set of indices $\{0, 1, \dots, n-1\}$. Let $\sigma$ be the involution associated with $c$, defined such that for any index $i$, $\sigma(i)$ is the dual index of $i$ if $i$ is contracted in $c$, and $\sigma(i) = i$ otherwise. Then, the Wick contraction constructed from $\sigma$ (defined as the collection of all pairs $\{i, \sigma(i)\}$ where $i \neq \sigma(i)$) is equal to the original contraction $c$.

Let $f: \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$ be an involutive function, such that $f(f(i)) = i$ for all $i$. Let $c$ be the Wick contraction constructed from $f$, defined as the collection of all pairs $\{j, f(j)\}$ where $j \neq f(j)$. Then, the involution associated with $c$ is equal to $f$.

This definition establishes an equivalence (a bijection) between the set of Wick contractions on $n$ indices and the set of involutions $f: \{0, 1, \dots, n-1\} \to \{0, 1, \dots, n-1\}$ (functions such that $f(f(i)) = i$ for all $i$). - The map from a Wick contraction $c$ to an involution $\sigma$ is defined such that $\sigma(i) = j$ if the index $i$ is contracted with $j$ in $c$, and $\sigma(i) = i$ if $i$ is not contracted. - The inverse map from an involution $f$ to a Wick contraction $c$ is defined as the set of all pairs $\{i, f(i)\}$ where $i \neq f(i)$.