Physlib.QFT.PerturbationTheory.WickContraction.Singleton

Given two indices $i, j \in \{0, 1, \dots, n-1\}$ such that $i < j$, this definition constructs a Wick contraction on $n$ elements consisting of the single pair $\{i, j\}$.

For any indices $i, j \in \{0, 1, \dots, n-1\}$ such that $i < j$, the pair $\{i, j\}$ is an element of the Wick contraction consisting of that single pair.

Let $n$ be a natural number and let $i, j \in \{0, 1, \dots, n-1\}$ be indices such that $i < j$. Let $\mathcal{C}$ be the Wick contraction consisting of the single pair $\{i, j\}$. For any finite set of indices $a \subseteq \{0, 1, \dots, n-1\}$, $a$ is an element of the contraction $\mathcal{C}$ if and only if $a = \{i, j\}$.

Let $n$ be a natural number and $i, j \in \{0, 1, \dots, n-1\}$ be indices such that $i < j$. Let $\mathcal{C}$ be the Wick contraction consisting of the single pair $\{i, j\}$. For any element $a$ in the set of pairs of $\mathcal{C}$, it holds that $a = \{i, j\}$.

Let $\phi_s$ be a list of field operators and let $i, j$ be indices of these operators such that $i < j$. Let $\mathcal{C}$ be the Wick contraction consisting of the single pair $\{i, j\}$. For any function $f$ that maps the pairs in $\mathcal{C}$ to a commutative monoid $M$, the product over all pairs in the contraction is given by $\prod_{a \in \mathcal{C}} f(a) = f(\{i, j\})$.

Let $n$ be a natural number and let $i, j \in \{0, 1, \dots, n-1\}$ be indices such that $i < j$. For the Wick contraction consisting of the single pair $\{i, j\}$, the index of the first field in that pair is equal to $i$.

Let $n$ be a natural number and let $i, j \in \{0, 1, \dots, n-1\}$ be indices such that $i < j$. For the Wick contraction consisting of the single pair $\{i, j\}$, the index of the second field in that pair is equal to $j$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. For any indices $i, j$ such that $i < j$, let $\mathcal{C}$ be the Wick contraction consisting of the single pair $\{i, j\}$. The sign of this contraction, denoted $\text{sign}(\mathcal{C})$, is given by $$\text{sign}(\mathcal{C}) = 𝓢(\mathcal{F} \triangleright_s \phi_j, \mathcal{F} \triangleright_s \phi|_S)$$ where $\mathcal{F} \triangleright_s \phi_j$ is the field statistic (e.g., bosonic or fermionic) of the $j$-th operator, $S = \{k \mid i < k < j\}$ is the set of indices between $i$ and $j$, and $𝓢$ is the statistical swap function that calculates the phase factor obtained by permuting the operator $\phi_j$ past the operators with indices in $S$.

Let $n$ be a natural number and let $i, j \in \{0, 1, \dots, n-1\}$ be indices such that $i < j$. For the Wick contraction consisting of the single pair $\{i, j\}$, the contraction partner (dual) of an index $a \in \{0, 1, \dots, n-1\}$ is undefined (denoted by `none`) if and only if $a \neq i$ and $a \neq j$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ a list of field operators. For any indices $i, j$ of $\phi_s$ such that $i < j$, let $c$ be the Wick contraction consisting of the single pair $\{i, j\}$. For any index $a$ in the list of uncontracted indices of $c$, the $a$-th uncontracted index is not equal to $i$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. For any indices $i, j \in \{0, 1, \dots, |\phi_s|-1\}$ such that $i < j$, let $c$ be the Wick contraction consisting of the single pair $\{i, j\}$. For any index $a$ in the list of uncontracted indices of $c$, the value of the $a$-th uncontracted index is not equal to $j$.

Let $n$ be a natural number and let $i, j \in \{0, 1, \dots, n-1\}$ be indices such that $i < j$. Consider the Wick contraction consisting of the single pair $\{i, j\}$. An index $a \in \{0, 1, \dots, n-1\}$ is an element of the sign finset associated with the indices $i$ and $j$ if and only if $a$ lies strictly between $i$ and $j$, i.e., $i < a < j$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a sequence of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$. For any contracted pair $a = \{i, j\}$ in $\Lambda$ with $i < j$, the sub-contraction of $\Lambda$ consisting of only the set of pairs $\{a\}$ is equal to the singleton Wick contraction formed by the pair $\{i, j\}$.

Let $\mathcal{F}$ be a field specification and let $\phi_s$ be a list of field operators. For any indices $i$ and $j$ of $\phi_s$ such that $i < j$, the time-ordered contraction associated with the singleton Wick contraction $\{i, j\}$ is equal to the time-ordered contraction of the field operators $\phi_i$ and $\phi_j$.

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a sequence of field operators. For any indices $i, j$ such that $i < j$, the static contraction of the singleton Wick contraction consisting of the pair $\{i, j\}$ is equal to the super-commutator $[\phi_i^{(-)}, \phi_j]_s$, where $\phi_i^{(-)}$ denotes the annihilation part of the field operator at index $i$, and $\phi_j$ is the field operator at index $j$.