Physlib.QFT.PerturbationTheory.WickAlgebra.StaticWickTerm

{φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.WickAlgebra

For the empty list of field operators $[]$ in a field specification $\mathcal{F}$, the static Wick term corresponding to the empty Wick contraction $\emptyset$ (which is the unique contraction for a list of length 0) is equal to $1$ in the Wick algebra $\mathcal{W}(\mathcal{F})$.

Let $\mathcal{F}$ be a field specification. For a list of field operators $\phi_s$, a Wick contraction $\Lambda$ of $\phi_s$, and a field operator $\phi$, let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, 0, \text{none}$ be the Wick contraction formed by inserting $\phi$ at the beginning of the list without contracting it. Then the static Wick term of $\Lambda'$ is given by: $$\text{staticWickTerm}(\Lambda \hookleftarrow_\Lambda \phi, 0, \text{none}) = \text{sign}(\Lambda) \cdot \text{staticContract}(\Lambda) * \mathcal{N}(\phi :: [\Lambda]^{uc})$$ where $\text{sign}(\Lambda)$ is the sign of the contraction, $\text{staticContract}(\Lambda)$ is the product of super-commutators of the contracted pairs, $[\Lambda]^{uc}$ is the list of uncontracted field operators of $\Lambda$, and $\mathcal{N}$ denotes the normal ordering operator in the Wick algebra.

Let $\mathcal{F}$ be a field specification. For a list of field operators $\phi_s$ and a Wick contraction $\Lambda$ acting on them, let $\phi$ be a field operator and $k$ be an uncontracted index in $\Lambda$. Let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, 0, \text{some } k$ be the Wick contraction formed by inserting $\phi$ at the beginning of the list and contracting it with the operator at index $k$. Then the static Wick term of $\Lambda'$ is equal to the product of: - The sign of the original contraction $\text{sign}(\Lambda)$. - The static contraction value $\text{staticContract}(\Lambda)$. - The contraction factor (super-commutator) between the annihilation part of $\phi$ and the uncontracted field operator originally at index $k$, given by $\text{contractStateAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k))$. - The normal ordering of the remaining uncontracted field operators, $\mathcal{N}([\Lambda]^{uc} \setminus \{\phi_k\})$. Mathematically, this is expressed as: $$\text{staticWickTerm}(\Lambda \hookleftarrow_\Lambda \phi, 0, \text{some } k) = \text{sign}(\Lambda) \cdot \text{staticContract}(\Lambda) \cdot \text{contractStateAtIndex}(\phi, [\Lambda]^{uc}, \text{pos}(k)) \cdot \mathcal{N}([\Lambda]^{uc} \setminus \{\phi_k\})$$ where $[\Lambda]^{uc}$ is the list of uncontracted operators of $\Lambda$.

For a field specification $\mathcal{F}$, let $\phi$ be a field operator and $\phi_s = [\phi_0, \dots, \phi_{n-1}]$ be a list of field operators. Let $\Lambda$ be a Wick contraction of the list $\phi_s$. The product of $\phi$ and the static Wick term associated with $\Lambda$ satisfies the identity: $$\phi \cdot \text{staticWickTerm}(\Lambda) = \sum_{k \in \operatorname{Option}(\Lambda^{uc})} \text{staticWickTerm}(\Lambda \hookleftarrow_\Lambda \phi, 0, k)$$ where $\Lambda^{uc}$ is the set of uncontracted indices of $\Lambda$, and $\Lambda \hookleftarrow_\Lambda \phi, 0, k$ denotes the Wick contraction formed by inserting $\phi$ at index $0$ and either keeping it uncontracted (if $k = \text{none}$) or contracting it with the operator corresponding to index $k$ (if $k \in \Lambda^{uc}$).