Physlib.QFT.PerturbationTheory.WickAlgebra.WickTerm

For a list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ and a Wick contraction $\Lambda$ of these operators, the Wick term is an element of the Wick algebra $\mathcal{W}(\mathcal{F})$ defined as: $$\text{wickTerm}(\Lambda, \Phi) = \epsilon(\Lambda, \Phi) \cdot \mathcal{C}(\Lambda, \Phi) \cdot \mathcal{N}([\Phi]_{\Lambda}^{uc})$$ where: - $\epsilon(\Lambda, \Phi)$ is the sign $(\pm 1)$ associated with the fermion permutations required for the contraction $\Lambda$. - $\mathcal{C}(\Lambda, \Phi)$ is the time contraction, defined as the product of pairwise contractions for all index pairs $\{i, j\} \in \Lambda$. - $\mathcal{N}(\cdot)$ is the normal ordering operator. - $[\Phi]_{\Lambda}^{uc}$ is the list of field operators from $\Phi$ that remain uncontracted by $\Lambda$. This term represents an individual summand in the expansion of a time-ordered product of field operators as provided by Wick's theorem.

For the empty list of field operators $\Phi = []$ associated with a field specification $\mathcal{F}$, the Wick term of the empty Wick contraction $\Lambda = \emptyset$ (which is the unique Wick contraction for a list of length zero) is equal to $1$ in 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 $\Phi$. Suppose a new field operator $\phi$ is inserted into the sequence $\Phi$ at index $i \in \{0, \dots, n\}$ without being contracted, resulting in a new list $\Phi'$ and a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{none}$. The Wick term of the resulting contraction $\Lambda'$ is given by: $$\text{wickTerm}(\Lambda', \Phi') = \mathcal{S}\left(\sigma(\phi), \sigma\left(\prod_{j < i} \phi_j\right)\right) \cdot \left( \epsilon(\Lambda, \Phi) \cdot \mathcal{C}(\Lambda, \Phi) \cdot \mathcal{N}(\phi :: [\Phi]_\Lambda^{uc}) \right)$$ where: - $\mathcal{S}(s_1, s_2)$ is the statistical sign factor (equal to $-1$ if both statistics $s_1, s_2$ are fermionic, and $1$ otherwise). - $\sigma(\cdot)$ denotes the collective field statistic of an operator or a product of operators. - $\epsilon(\Lambda, \Phi)$ is the sign $(\pm 1)$ associated with the original Wick contraction $\Lambda$. - $\mathcal{C}(\Lambda, \Phi)$ is the time contraction of $\Lambda$. - $\mathcal{N}(\cdot)$ is the normal ordering operator. - $[\Phi]_\Lambda^{uc}$ is the list of field operators in $\Phi$ that are not contracted by $\Lambda$.

Let $\mathcal{F}$ be a field specification. 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 $\phi$ be a field operator to be inserted into the list at index $i \in \{0, \dots, n\}$, and let $k$ be an index that is uncontracted in $\Lambda$. Suppose that $\phi$ is chronologically later than or equal to all operators in $\Phi$ (i.e., $\text{timeOrderRel}(\phi, \phi_j)$ for all $j$), and that all operators $\phi_j$ with $j < i$ are chronologically strictly earlier than $\phi$. The Wick term for the new contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } k$, formed by inserting $\phi$ at index $i$ and contracting it with the operator originally at index $k$, is given by: $$\text{wickTerm}(\Lambda', \Phi') = \mathcal{S}(\sigma(\phi), \sigma(\Phi_{<i})) \cdot \left( \epsilon(\Lambda) \cdot (\text{contractAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k)) \cdot \mathcal{C}(\Lambda)) \cdot \mathcal{N}([\Lambda]^{uc} \setminus \{\phi_k\}) \right)$$ where: - $\Phi'$ is the new list of operators after insertion. - $\mathcal{S}(s_1, s_2)$ is the statistical sign factor (equal to $-1$ if both statistics $s_1, s_2$ are fermionic, and $1$ otherwise). - $\sigma(\Phi_{<i})$ is the aggregate field statistic of the operators in the original list with indices strictly less than $i$. - $\epsilon(\Lambda)$ is the sign $(\pm 1)$ of the original Wick contraction. - $\mathcal{C}(\Lambda)$ is the time contraction of the original Wick contraction. - $\text{contractAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k))$ represents the pairwise contraction of the annihilation part of $\phi$ with $\phi_k$, including the sign associated with moving $\phi$ through uncontracted fields. - $[\Lambda]^{uc}$ is the list of uncontracted operators in $\Phi$, and $\text{pos}(k)$ is the position of $\phi_k$ within that list. - $\mathcal{N}(\cdot)$ is the normal ordering operator.

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 $\text{wickTerm}(\Lambda, \Phi)$ be its associated Wick term. Suppose a new field operator $\phi$ is inserted into the list at index $i \in \{0, \dots, n\}$, creating a new list $\Phi'$. Assume the following chronological conditions hold: 1. $\phi$ is chronologically later than or equal to all operators in $\Phi$ (i.e., $\text{timeOrderRel}(\phi, \phi_k)$ for all $k$). 2. All operators $\phi_k$ that appear before the insertion index $i$ in $\Phi'$ are chronologically strictly earlier than $\phi$. Then the product of the operator $\phi$ and the Wick term of $\Lambda$ is given by: $$\phi \cdot \text{wickTerm}(\Lambda, \Phi) = \mathcal{S}(\sigma(\phi), \sigma(\Phi_{<i})) \cdot \sum_{k \in \mathcal{U} \cup \{\text{none}\}} \text{wickTerm}(\Lambda \hookleftarrow_\Lambda \phi, i, k)$$ where: - $\mathcal{S}(s_1, s_2)$ is the statistical sign factor, which is $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise. - $\sigma(\phi)$ is the statistic of operator $\phi$, and $\sigma(\Phi_{<i})$ is the aggregate statistic of the operators in $\Phi$ that are placed before index $i$. - $\mathcal{U}$ is the set of indices in $\Phi$ that are uncontracted by $\Lambda$. - $\Lambda \hookleftarrow_\Lambda \phi, i, k$ is the augmented Wick contraction on $\Phi'$ formed by inserting $\phi$ at index $i$ and either leaving it uncontracted (if $k = \text{none}$) or contracting it with the operator at uncontracted index $k$.