Physlib.QFT.PerturbationTheory.FieldSpecification.Filters

Given a field specification $\mathcal{F}$ and a list of field operator components $[\varphi_1, \varphi_2, \dots, \varphi_n]$ where each $\varphi_i \in \mathcal{F}.\text{CrAnFieldOp}$, this function returns a sublist containing only those elements classified as creation operators. An operator component $\varphi$ is included in the output list if and only if $\mathcal{F} \rhd^c \varphi = \text{create}$, where $\text{create} \in \text{CreateAnnihilate}$. For example, if the input list contains components representing $[a_1^\dagger, a_1, a_2^\dagger, a_2]$, the function returns the list $[a_1^\dagger, a_2^\dagger]$.

For a field specification $\mathcal{F}$, let $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ be a field operator component and $\varphi_s$ be a list of field operator components. If $\varphi$ is a creation operator, i.e., $\mathcal{F} \rhd^c \varphi = \text{create}$, then the sublist of creation operators extracted from the list formed by prepending $\varphi$ to $\varphi_s$ is equal to $\varphi$ prepended to the sublist of creation operators of $\varphi_s$. In symbols: $$\text{createFilter}(\varphi :: \varphi_s) = \varphi :: \text{createFilter}(\varphi_s)$$ where $::$ denotes the list construction (cons) operator.

Let $\mathcal{F}$ be a field specification. For any field operator component $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ and any list of field operator components $\varphi_s$, if $\varphi$ is classified as an annihilation operator (i.e., $\mathcal{F} \rhd^c \varphi = \text{annihilate}$), then the sublist of creation operators extracted from the list $\varphi :: \varphi_s$ is equal to the sublist of creation operators extracted from $\varphi_s$ alone: $$\text{createFilter}(\varphi :: \varphi_s) = \text{createFilter}(\varphi_s)$$

Given a field specification $\mathcal{F}$ and two lists $\varphi_s$ and $\varphi_s'$ of creation and annihilation operator components (elements of $\mathcal{F}.\text{CrAnFieldOp}$), the operation $\text{createFilter}$, which selects the sublist of components classified as creation operators, satisfies the property: $$\text{createFilter}(\varphi_s \mathbin{+\!+} \varphi_s') = \text{createFilter}(\varphi_s) \mathbin{+\!+} \text{createFilter}(\varphi_s')$$ where $\mathbin{+\!+}$ denotes the concatenation of lists.

Let $\mathcal{F}$ be a field specification and $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component. If the creation/annihilation label of $\varphi$ is $\text{create}$ (denoted as $\mathcal{F} \rhd^c \varphi = \text{create}$), then the sublist of creation operators obtained from the singleton list $[\varphi]$ is simply the list $[\varphi]$ itself.

For a field specification $\mathcal{F}$ and a field operator component $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$, if the creation/annihilation label of $\varphi$ is $\text{annihilate}$ (i.e., $\mathcal{F} \rhd^c \varphi = \text{annihilate}$), then filtering the singleton list $[\varphi]$ for creation operators results in the empty list $[]$.

For a given field specification $\mathcal{F}$, the function `annihilateFilter` takes a list $\varphi_s$ of creation and annihilation operator components (elements of $\mathcal{F}.\text{CrAnFieldOp}$) and returns a sublist containing only those elements $\varphi \in \varphi_s$ whose creation/annihilation label is $\text{annihilate}$.

For a field specification $\mathcal{F}$, let $\varphi$ be a component of a field operator (an element of $\mathcal{F}.\text{CrAnFieldOp}$) and let $\varphi_s$ be a list of such components. If $\varphi$ is classified as a creation operator, such that its label $\mathcal{F}|>^c \varphi$ is equal to $\text{create}$, then the result of applying the annihilation filter to the list formed by prepending $\varphi$ to $\varphi_s$ (denoted $\varphi :: \varphi_s$) is identical to the annihilation filter of $\varphi_s$ alone: $$\text{annihilateFilter}(\varphi :: \varphi_s) = \text{annihilateFilter}(\varphi_s)$$

For a given field specification $\mathcal{F}$, let $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ be a field operator component and let $\varphi_s$ be a list of such components. If the creation/annihilation label of $\varphi$ is $\text{annihilate}$ (denoted by the notation $\mathcal{F} \rhd^c \varphi = \text{annihilate}$), then the result of applying the `annihilateFilter` to the list formed by prepending $\varphi$ to $\varphi_s$ is equal to $\varphi$ prepended to the result of applying the `annihilateFilter` to $\varphi_s$. That is, $$\text{annihilateFilter}(\varphi :: \varphi_s) = \varphi :: \text{annihilateFilter}(\varphi_s)$$ where the operator `::` denotes the construction of a list by prepending an element to an existing list.

For any field specification $\mathcal{F}$ and any two lists $\varphi_s$ and $\varphi_s'$ of creation and annihilation operator components (elements of $\mathcal{F}.\text{CrAnFieldOp}$), the annihilation filter operation satisfies $$\text{annihilateFilter}(\varphi_s \mathbin{+\mkern-10mu+} \varphi_s') = \text{annihilateFilter}(\varphi_s) \mathbin{+\mkern-10mu+} \text{annihilateFilter}(\varphi_s')$$