Physlib.QFT.PerturbationTheory.WickAlgebra.Basic

For a given field specification $\mathcal{F}$, the set $\text{fieldOpIdealSet}$ is a subset of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ consisting of elements that are intended to vanish in the physical operator algebra. An element $x$ belongs to this set if it satisfies one of the following conditions: 1. $x = [\phi_1, [\phi_2, \phi_3]_s^F]_s^F$ for any creation or annihilation operators $\phi_1, \phi_2, \phi_3 \in \mathcal{F}.\text{CrAnFieldOp}$. This ensures that the super-commutator of any two operators belongs to the center of the algebra. 2. $x = [\phi_c, \phi_{c'}]_s^F$ where both $\phi_c$ and $\phi_{c'}$ are creation operators. 3. $x = [\phi_a, \phi_{a'}]_s^F$ where both $\phi_a$ and $\phi_{a'}$ are annihilation operators. 4. $x = [\phi, \phi']_s^F$ where the operators $\phi$ and $\phi'$ have different field statistics (i.e., one is bosonic and the other is fermionic). Here, $[a, b]_s^F$ denotes the super-commutator in the free algebra.

For a given field specification $\mathcal{F}$, the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is the quotient of the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ (generated over $\mathbb{C}$ by creation and annihilation operator components $\phi \in \mathcal{F}.\text{CrAnFieldOp}$) by the two-sided ideal generated by the following relations: 1. Two creation operators $\phi_c, \phi_{c'}$ always super-commute: $[\phi_c, \phi_{c'}]_s = 0$. 2. Two annihilation operators $\phi_a, \phi_{a'}$ always super-commute: $[\phi_a, \phi_{a'}]_s = 0$. 3. Operators with different statistics (one bosonic and one fermionic) always super-commute: $[\phi, \phi']_s = 0$. 4. The super-commutator of any two operators lies in the center of the algebra, expressed by the condition $[\phi_1, [\phi_2, \phi_3]_s]_s = 0$ for all $\phi_1, \phi_2, \phi_3$. Here, $[a, b]_s$ denotes the super-commutator. This algebra provides the minimal axiomatic structure required to prove Wick's theorem and satisfies the necessary universality conditions with respect to the operator algebra.

For a given field specification $\mathcal{F}$, the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equipped with a semiring structure. This structure is inherited from the quotient of the free algebra of field operators by the two-sided ideal generated by the relations defined in $\mathcal{F}.\text{fieldOpIdealSet}$.

For a given field specification $\mathcal{F}$, the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equipped with the structure of an associative algebra over the field of complex numbers $\mathbb{C}$. This structure is inherited from the $\mathbb{C}$-algebra structure of the free associative algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ through the quotient by the two-sided ideal generated by the field operator relations $\text{fieldOpIdealSet}$.

For a given field specification $\mathcal{F}$, the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equipped with a ring structure. This structure is inherited from the quotient of the free associative algebra of field operators by the two-sided ideal generated by the relations defined in $\mathcal{F}.\text{fieldOpIdealSet}$.

This definition provides a canonical coercion from the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. It maps an element $x$ in the free algebra to its corresponding equivalence class $[x]$ in the Wick algebra, which is defined as the quotient of the free algebra by the two-sided ideal generated by the field operator relations $\text{fieldOpIdealSet}$.

For a given field specification $\mathcal{F}$, let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free algebra of creation and annihilation operators. This definition provides a setoid structure on $\mathcal{F}.\text{FieldOpFreeAlgebra}$ where two elements $x$ and $y$ are equivalent, denoted $x \sim y$, if and only if their difference $x - y$ belongs to the two-sided ideal generated by the set $\text{fieldOpIdealSet}$. This equivalence relation is used to define the Wick algebra as a quotient of the free algebra.

For a given field specification $\mathcal{F}$, let $x$ and $y$ be elements of the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$. Then $x$ is equivalent to $y$ (denoted $x \approx y$) under the setoid relation defining the Wick algebra if and only if their difference $x - y$ belongs to the two-sided ideal generated by the set $\mathcal{F}.\text{fieldOpIdealSet}$.

For a given field specification $\mathcal{F}$, let $x$ and $y$ be elements of the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$. Then $x$ is equivalent to $y$ (denoted $x \approx y$) under the setoid relation defining the Wick algebra if and only if there exists an element $a$ in the two-sided ideal generated by the set $\mathcal{F}.\text{fieldOpIdealSet}$ such that $x = y + a$.

For a given field specification $\mathcal{F}$, the map $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is the canonical $\mathbb{C}$-algebra homomorphism that sends each element of the free algebra of creation and annihilation operators to its corresponding equivalence class in the Wick algebra. This projection is defined by the quotient of the free algebra by the two-sided ideal generated by the set of relations $\mathcal{F}.\text{fieldOpIdealSet}$.

For a given field specification $\mathcal{F}$, the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ from the free algebra of creation and annihilation operators to the Wick algebra is surjective.

For a given field specification $\mathcal{F}$ and an element $x$ in the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the image of $x$ under the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is equal to the equivalence class of $x$ in the Wick algebra.

Let $\mathcal{F}$ be a field specification. For any element $x$ in the free associative algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, if $x$ belongs to the generating set of the field operator ideal $\mathcal{F}.\text{fieldOpIdealSet}$, then its image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero.

For a given field specification $\mathcal{F}$, let $\phi_c$ and $\phi_{c'}$ be two creation/annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$. If both $\phi_c$ and $\phi_{c'}$ are classified as creation operators, then the image of their super-commutator $[\phi_c, \phi_{c'}]_s^F$ under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi_a, \phi_{a'} \in \mathcal{F}.\text{CrAnFieldOp}$ that are both classified as annihilation operators, the image of their super-commutator $[\phi_a, \phi_{a'}]_s^F$ under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ with different field statistics (where one is bosonic and the other is fermionic), the image of their super-commutator $[\phi, \psi]_s^F$ under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$, let $a_\phi, a_\psi \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be their corresponding generators in the free associative algebra. If their super-commutator $[a_\phi, a_\psi]_s^F$ is fermionic (i.e., it belongs to the fermionic statistic submodule), then its image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$\iota([a_\phi, a_\psi]_s^F) = 0.$$

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$, let $a_\phi, a_\psi \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be their corresponding generators in the free associative algebra. Then, the super-commutator $[a_\phi, a_\psi]_s^F$ in the free algebra is either bosonic (i.e., it belongs to the bosonic statistic submodule), or its image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$[a_\phi, a_\psi]_s^F \in \text{statisticSubmodule}(\text{bosonic}) \lor \iota([a_\phi, a_\psi]_s^F) = 0$$

Let $\mathcal{F}$ be a field specification. For any three creation or annihilation operator components $\phi_1, \phi_2, \phi_3 \in \mathcal{F}.\text{CrAnFieldOp}$, let $a_1, a_2, a_3$ be their corresponding generators in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The image of the nested super-commutator $[a_1, [a_2, a_3]_s^F]_s^F$ under the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$\iota([a_1, [a_2, a_3]_s^F]_s^F) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification. For any three creation or annihilation operator components $\phi_1, \phi_2, \phi_3 \in \mathcal{F}.\text{CrAnFieldOp}$, let $a_1, a_2, a_3$ be their corresponding generators in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The image of the nested super-commutator $[[a_1, a_2]_s^F, a_3]_s^F$ under the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$\iota([[a_1, a_2]_s^F, a_3]_s^F) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi_1, \phi_2 \in \mathcal{F}.\text{CrAnFieldOp}$, let $a_1, a_2$ be their corresponding generators in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. For any list of creation and annihilation operator components $\phi_s = [\psi_1, \dots, \psi_n]$, let $C = \text{ofCrAnListF}(\phi_s)$ be their product in the free algebra. The image of the nested super-commutator $[[a_1, a_2]_s^F, C]_s^F$ under the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$\iota([[a_1, a_2]_s^F, C]_s^F) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi_1, \phi_2 \in \mathcal{F}.\text{CrAnFieldOp}$, let $g_1, g_2$ be their corresponding generators in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. For any element $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, the image of the nested super-commutator $[[g_1, g_2]_s^F, a]_s^F$ under the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$\iota([[g_1, g_2]_s^F, a]_s^F) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi_1, \phi_2 \in \mathcal{F}.\text{CrAnFieldOp}$, let $g_1, g_2 \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be their corresponding generators in the free associative algebra. For any element $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, let $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical projection. Then the image of the super-commutator $[g_1, g_2]_s^F$ commutes with the image of $a$ in the Wick algebra: $$\iota(a) \cdot \iota([g_1, g_2]_s^F) - \iota([g_1, g_2]_s^F) \cdot \iota(a) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$, let $[g_\phi, g_\psi]_s^F$ be the super-commutator of their corresponding generators in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. Then the image of this super-commutator under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ lies in the center of the Wick algebra.

For a given field specification $\mathcal{F}$, let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$. Let $\iota : \mathcal{A}_{\mathcal{F}} \to \text{WickAlgebra}(\mathcal{F})$ be the canonical $\mathbb{C}$-algebra homomorphism to the Wick algebra. For any element $x \in \mathcal{A}_{\mathcal{F}}$, the image $\iota(x)$ is zero if and only if $x$ belongs to the two-sided ideal generated by the set of field operator relations $\mathcal{F}.\text{fieldOpIdealSet}$.

Let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$ for a given field specification $\mathcal{F}$. Let $I_{\text{set}} \subset \mathcal{A}_{\mathcal{F}}$ be the generating set for the field operator ideal (denoted `fieldOpIdealSet`), and let $P_{\text{bosonic}} : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the linear projection onto the bosonic submodule of the algebra. For any element $x \in I_{\text{set}}$, then either $P_{\text{bosonic}}(x) \in I_{\text{set}}$ or $P_{\text{bosonic}}(x) = 0$.

In the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$ associated with a field specification $\mathcal{F}$, let $\text{fieldOpIdealSet}$ be the generating set for the field operator ideal. For any element $x \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, if $x \in \text{fieldOpIdealSet}$, then its fermionic projection $\text{fermionicProjF}(x)$ is either an element of $\text{fieldOpIdealSet}$ or is equal to zero.

Let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators defined by a field specification $\mathcal{F}$. Let $\mathcal{I}$ be the two-sided ideal in $\mathcal{A}_{\mathcal{F}}$ generated by the set $I_{\text{set}}$ (the `fieldOpIdealSet`). Let $P_{\text{bosonic}} : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the $\mathbb{C}$-linear projection onto the bosonic submodule of $\mathcal{A}_{\mathcal{F}}$, which consists of elements with even parity. For any element $x \in \mathcal{A}_{\mathcal{F}}$, if $x \in \mathcal{I}$, then $P_{\text{bosonic}}(x) \in \mathcal{I}$.

Let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators defined by a field specification $\mathcal{F}$. Let $\mathcal{I}$ be the two-sided ideal in $\mathcal{A}_{\mathcal{F}}$ generated by the set $\text{fieldOpIdealSet}$. Let $P_{\text{fermionic}} : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the $\mathbb{C}$-linear projection onto the fermionic submodule of $\mathcal{A}_{\mathcal{F}}$, which consists of elements with odd parity (an odd number of fermionic operators). For any element $x \in \mathcal{A}_{\mathcal{F}}$, if $x \in \mathcal{I}$, then $P_{\text{fermionic}}(x) \in \mathcal{I}$.

Let $\mathcal{F}$ be a field specification and $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$. Let $\iota: \mathcal{A}_{\mathcal{F}} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical $\mathbb{C}$-algebra homomorphism. For any element $x \in \mathcal{A}_{\mathcal{F}}$, let $P_{\text{bosonic}}(x)$ and $P_{\text{fermionic}}(x)$ be its projections onto the bosonic and fermionic submodules of $\mathcal{A}_{\mathcal{F}}$, respectively. Then, the image of $x$ in the Wick algebra is zero if and only if the images of both its bosonic and fermionic projections are zero: $$\iota(x) = 0 \iff \iota(P_{\text{bosonic}}(x)) = 0 \land \iota(P_{\text{fermionic}}(x)) = 0$$

For a given field specification $\mathcal{F}$, the function `ofFieldOp` maps a field operator $\phi \in \mathcal{F}.\text{FieldOp}$ to its corresponding element in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. It is defined as the image of $\text{ofFieldOpF}(\phi)$ under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$, where $\text{ofFieldOpF}(\phi)$ is the sum of the creation and annihilation components of $\phi$ in the free algebra.

For a given field specification $\mathcal{F}$ and a field operator $\phi \in \mathcal{F}.\text{FieldOp}$, the element in the Wick algebra representing $\phi$ is equal to the image of its free-algebra representation under the canonical projection. Specifically, let $\text{ofFieldOp}(\phi)$ be the field operator as an element of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, and let $\text{ofFieldOpF}(\phi)$ be the sum of its creation and annihilation components in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. Then: $$\text{ofFieldOp}(\phi) = \iota(\text{ofFieldOpF}(\phi))$$ where $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is the canonical $\mathbb{C}$-algebra projection.

For a given field specification $\mathcal{F}$ and a list of field operators $[\phi_1, \phi_2, \dots, \phi_n]$ where each $\phi_i \in \mathcal{F}.\text{FieldOp}$, the function `ofFieldOpList` computes the product of these operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. It is defined as the image of the product of these operators from the free algebra under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$: $$\text{ofFieldOpList}([\phi_1, \dots, \phi_n]) = \iota \left( \prod_{i=1}^n (\phi_{i, \text{create}} + \phi_{i, \text{annihilate}}) \right)$$ where each operator $\phi_i$ is treated as the sum of its creation and annihilation components in the underlying free algebra. If the list is empty, the result is the identity element $1$ of the Wick algebra.

For a given field specification $\mathcal{F}$ and a list of field operators $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ (where $\phi_i \in \mathcal{F}.\text{FieldOp}$), the product of these operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, denoted as $\text{ofFieldOpList}(\phi_s)$, is equal to the image of their product in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, denoted as $\text{ofFieldOpListF}(\phi_s)$, under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$. That is: $$\text{ofFieldOpList}(\phi_s) = \iota(\text{ofFieldOpListF}(\phi_s))$$

For a given field specification $\mathcal{F}$ and two lists of field operators $\phi_s, \psi_s \in \text{List}(\mathcal{F}.\text{FieldOp})$, the representation of the concatenated list $\phi_s ++ \psi_s$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the product of the representations of the individual lists $\phi_s$ and $\psi_s$: $$\text{ofFieldOpList}(\phi_s \text{ ++ } \psi_s) = \text{ofFieldOpList}(\phi_s) \cdot \text{ofFieldOpList}(\psi_s)$$ where $\text{ofFieldOpList}$ is the map that takes a list of field operators $[\phi_1, \dots, \phi_n]$ to the ordered product $\prod_{i=1}^n \phi_i$ in the Wick algebra.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator and $\phi_s$ be a list of field operators. The representation of the list formed by prepending $\phi$ to $\phi_s$ (denoted $\phi :: \phi_s$) in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the product of the representation of $\phi$ and the representation of the list $\phi_s$. That is, $$\text{ofFieldOpList}(\phi :: \phi_s) = \text{ofFieldOp}(\phi) \cdot \text{ofFieldOpList}(\phi_s)$$

Given a field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator. The representation of the singleton list $[\phi]$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, denoted by $\text{ofFieldOpList}([\phi])$, is equal to the representation of the single field operator $\phi$ in the Wick algebra, denoted by $\text{ofFieldOp}(\phi)$.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component. The function `ofCrAnOp` maps $\phi$ to its corresponding representative in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. This is defined by taking the generator associated with $\phi$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ and applying the canonical projection $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component. The representation of $\phi$ in the Wick algebra, denoted by $\text{ofCrAnOp}(\phi) \in \mathcal{F}.\text{WickAlgebra}$, is equal to the image of its corresponding generator in the free algebra, $\text{ofCrAnOpF}(\phi) \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, under the canonical projection map $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator. Its representation in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, denoted by $\text{ofFieldOp}(\phi)$, is equal to the sum of its creation and annihilation components: $$\text{ofFieldOp}(\phi) = \sum_{i \in \mathcal{F}.\text{fieldOpToCrAnType}(\phi)} \text{ofCrAnOp}(\langle \phi, i \rangle)$$ where $\mathcal{F}.\text{fieldOpToCrAnType}(\phi)$ is the finite set of creation and annihilation modes associated with $\phi$, and $\text{ofCrAnOp}(\langle \phi, i \rangle)$ represents the corresponding component in the Wick algebra. In particular, for a position-space field operator, this sum expresses the decomposition $\phi = \phi_{\text{create}} + \phi_{\text{annihilate}}$.

For a given field specification $\mathcal{F}$ and a list of creation and annihilation operators $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$, the function $\text{ofCrAnList}(\phi_s)$ returns the corresponding product $\phi_1 \cdot \phi_2 \cdot \dots \cdot \phi_n$ as an element of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. This is formally computed by taking the product of the operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ and mapping it into the Wick algebra via the canonical projection $\iota$. If the list is empty, the function returns the identity element $1$.

For a given field specification $\mathcal{F}$, let $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$. The product of these operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the image of their product in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ under the canonical projection $\iota$. That is, $$\text{ofCrAnList}(\phi_s) = \iota(\text{ofCrAnListF}(\phi_s))$$ where $\text{ofCrAnListF}(\phi_s)$ is the product $\phi_1 \cdot \phi_2 \cdot \dots \cdot \phi_n$ in the free algebra and $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is the canonical $\mathbb{C}$-algebra homomorphism.

For a given field specification $\mathcal{F}$, the product of an empty list of creation and annihilation operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, denoted by $\text{ofCrAnList}([])$, is equal to the identity element $1$.

For a given field specification $\mathcal{F}$, let $\phi_s$ and $\psi_s$ be two lists of creation and annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$. Then the function $\text{ofCrAnList}$, which maps a list of operators to their product in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, satisfies: $$\text{ofCrAnList}(\phi_s \mathbin{+\!+} \psi_s) = \text{ofCrAnList}(\phi_s) \cdot \text{ofCrAnList}(\psi_s)$$ where $\mathbin{+\!+}$ denotes the concatenation of lists and $\cdot$ denotes the multiplication operation in the Wick algebra.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component. In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the product associated with the singleton list $[\phi]$, denoted by $\text{ofCrAnList}([\phi])$, is equal to the element representing the operator itself, $\text{ofCrAnOp}(\phi)$.

For a given field specification $\mathcal{F}$ and a list of field operators $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n]$ where each $\phi_i \in \mathcal{F}.\text{FieldOp}$, the product of these operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the sum over all possible creation and annihilation sections $s \in \text{CrAnSection}(\varphi_s)$ of their corresponding component products: $$\text{ofFieldOpList}(\varphi_s) = \sum_{s \in \text{CrAnSection}(\varphi_s)} \text{ofCrAnList}(s)$$ where $\text{ofFieldOpList}(\varphi_s)$ represents the product of the field operators $\prod_{i=1}^n \phi_i$, and $\text{ofCrAnList}(s)$ is the product of the specific creation or annihilation parts chosen by the section $s$.

For a given field specification $\mathcal{F}$, the function `anPart` maps a field operator $\phi \in \mathcal{F}.\text{FieldOp}$ to its corresponding annihilation component in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. Formally, it is defined as the canonical projection $\iota$ of the annihilation part in the free algebra $\text{anPartF}(\phi)$ into the Wick algebra. The mapping behaves as follows: - If $\phi$ is an incoming asymptotic state, $\text{anPart}(\phi) = 0$ (since these are treated as creation operators). - If $\phi$ is a position-space field operator, $\text{anPart}(\phi)$ is the image of the generator representing the annihilation mode of the field at that spacetime position. - If $\phi$ is an outgoing asymptotic state, $\text{anPart}(\phi)$ is the image of the operator itself, as outgoing asymptotic operators act as annihilation operators.

For a field specification $\mathcal{F}$ and a field operator $\phi \in \text{FieldOp}(\mathcal{F})$, the annihilation part of $\phi$ in the Wick algebra, denoted $\text{anPart}(\phi)$, is equal to the image of the annihilation part of $\phi$ in the free algebra, $\text{anPartF}(\phi)$, under the canonical projection map $\iota: \text{FieldOpFreeAlgebra}(\mathcal{F}) \to \text{WickAlgebra}(\mathcal{F})$.

For any field specification $\mathcal{F}$, the annihilation part $\text{anPart}(\phi)$ of an incoming asymptotic field operator $\phi$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to zero: $$\text{anPart}(\phi) = 0$$ where $\phi$ is an incoming asymptotic field operator defined by a field label and its momentum.

For a given field specification $\mathcal{F}$, let $\phi$ denote the parameters of a position-space field operator (consisting of the field type, position label, and spacetime coordinates). The annihilation part $\text{anPart}(\text{position}(\phi))$ of this operator in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the element $\text{ofCrAnOp}(\langle \text{position}(\phi), \text{annihilate} \rangle)$, which represents the generator corresponding to the annihilation mode of the field at that position.

For a given field specification $\mathcal{F}$, let $\phi$ be an outgoing asymptotic state consisting of a field $f$, an asymptotic label $e$, and a 3-momentum $\mathbf{p}$. In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the annihilation part $\text{anPart}(\text{outAsymp}(\phi))$ is equal to the generator representing the operator itself, denoted by $\text{ofCrAnOp} \langle \text{outAsymp}(\phi), () \rangle$.

For a given field specification $\mathcal{F}$, the function $\text{crPart}$ maps a field operator $\phi \in \mathcal{F}.\text{FieldOp}$ to its creation component within the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. It is defined as the image of the operator's creation part in the free algebra under the canonical projection $\iota$: \[ \text{crPart}(\phi) = \iota(\text{crPart}_F(\phi)) \] Specifically: - For an incoming asymptotic operator $\phi$, $\text{crPart}(\phi)$ is the corresponding creation generator in the Wick algebra. - For a position-space field operator $\phi(x)$, $\text{crPart}(\phi)$ is the component designated as the creation part of that field. - For an outgoing asymptotic operator $\phi$, $\text{crPart}(\phi) = 0$, as these operators are purely annihilating.

For any field operator $\phi$ in a field specification $\mathcal{F}$, the creation part of $\phi$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the image of its creation part in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ under the canonical projection $\iota$. That is, $\text{crPart}(\phi) = \iota(\text{crPart}_F(\phi))$.

For a given field specification $\mathcal{F}$, let $\phi$ denote the parameters of an asymptotic field state (consisting of a field $f$, an asymptotic label $e$, and a 3-momentum $\mathbf{p}$). The creation part $\text{crPart}$ of the incoming asymptotic field operator $\text{inAsymp}(\phi)$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the operator mapped as a creation/annihilation generator with its unique mode label $()$: $$\text{crPart}(\text{inAsymp}(\phi)) = \text{ofCrAnOp}(\langle \text{inAsymp}(\phi), () \rangle)$$ This indicates that within the Wick algebra, an incoming asymptotic field operator is considered to be its own creation part.

For a given field specification $\mathcal{F}$, let $\phi(x)$ be a position-space field operator at spacetime point $x$ (formally represented as `FieldOp.position φ`). The creation part of this operator in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, denoted by $\text{crPart}(\phi(x))$, is equal to the element in the Wick algebra corresponding to the creation mode of that field, denoted by $\text{ofCrAnOp} \langle \phi(x), \text{create} \rangle$.

For a given field specification $\mathcal{F}$, let $\varphi$ be an outgoing asymptotic state (comprising a field label and momentum). The creation part of the corresponding outgoing asymptotic field operator $\phi_{\text{out}}(\varphi)$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is zero: \[ \text{crPart}(\phi_{\text{out}}(\varphi)) = 0 \] This reflects the physical property that outgoing asymptotic operators are purely annihilating.

For a given field specification $\mathcal{F}$ and any field operator $\phi \in \mathcal{F}.\text{FieldOp}$, the representation of the operator in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is the sum of its creation part and its annihilation part: $$\text{ofFieldOp}(\phi) = \text{crPart}(\phi) + \text{anPart}(\phi)$$ where $\text{ofFieldOp}(\phi)$ is the field operator as an element of the Wick algebra, $\text{crPart}(\phi)$ is its creation component, and $\text{anPart}(\phi)$ is its annihilation component.