Physlib.QFT.PerturbationTheory.WickAlgebra.Grading

For a given field specification $\mathcal{F}$ and a field statistic $f \in \{\text{bosonic}, \text{fermionic}\}$, the submodule $\text{statSubmodule}(f)$ is the complex submodule of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ spanned by all products of creation and annihilation operators whose collective field statistic is equal to $f$. The collective statistic of a product is determined by the parity of the number of fermionic operators in the product: it is fermionic if there is an odd number of fermionic operators and bosonic otherwise.

For a given field specification $\mathcal{F}$, let $\varphi_s = [\phi_1, \dots, \phi_n]$ be a list of creation and annihilation operators. Let $s(\varphi_s)$ denote the collective statistic of the list, which is $\text{fermionic}$ if the list contains an odd number of fermionic operators and $\text{bosonic}$ otherwise. If $s(\varphi_s) = f$ for some statistic $f \in \{\text{bosonic}, \text{fermionic}\}$, then the product of these operators in the Wick algebra, denoted by $\text{ofCrAnList}(\varphi_s)$, belongs to the submodule $\text{statSubmodule}(f)$ consisting of elements with statistic $f$.

For a given field specification $\mathcal{F}$, let $\phi_s = [\phi_1, \dots, \phi_n]$ be a list of creation and annihilation operators (elements of $\mathcal{F}.\text{CrAnFieldOp}$). Let $\text{stat}(\phi_s)$ denote the collective statistic of this list, which is $\text{fermionic}$ if the list contains an odd number of fermionic operators and $\text{bosonic}$ otherwise. Then the product of these operators in the Wick algebra, denoted by $\text{ofCrAnList}(\phi_s)$, is an element of the submodule $\text{statSubmodule}(\text{stat}(\phi_s))$ corresponding to that statistic.

For a given field specification $\mathcal{F}$, let $a$ be an element of the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ generated by the creation and annihilation operators. If $a$ belongs to the submodule $\text{statisticSubmodule}(\text{bosonic})$—the $\mathbb{C}$-linear submodule spanned by products of operators with an even number of fermionic components—then its image under the canonical map $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ belongs to the bosonic submodule $\text{statSubmodule}(\text{bosonic})$ of the Wick algebra.

For a given field specification $\mathcal{F}$, let $a$ be an element of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ of creation and annihilation operators. If $a$ belongs to the submodule $\text{statisticSubmodule}(\text{fermionic})$ (the $\mathbb{C}$-linear submodule spanned by operator products with an odd number of fermionic components), then its image $\iota(a)$ under the canonical map into the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ belongs to the fermionic submodule $\text{statSubmodule}(\text{fermionic})$.

For a given field specification $\mathcal{F}$ and a field statistic $f \in \{\text{bosonic}, \text{fermionic}\}$, let $a$ be an element of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ generated by the creation and annihilation operators. If $a$ belongs to the submodule $\text{statisticSubmodule}(f)$—the $\mathbb{C}$-linear submodule of the free algebra spanned by operator products whose collective statistic is $f$—then its image $\iota(a)$ under the canonical map $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ belongs to the corresponding submodule $\text{statSubmodule}(f)$ in the Wick algebra.

For a given field specification $\mathcal{F}$ and a field statistic $f \in \{\text{bosonic}, \text{fermionic}\}$, this definition specifies the $\mathbb{C}$-linear map from the submodule of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ consisting of elements with statistic $f$ to the corresponding submodule $\text{statSubmodule}(f)$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. This map is essentially the restriction of the canonical quotient map $\iota$ to the submodules defined by the statistic $f$.

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 $\mathcal{F}.\text{CrAnFieldOp}$. The map $\text{bosonicProjFree}$ is a $\mathbb{C}$-linear map from $\mathcal{A}_{\mathcal{F}}$ to the bosonic submodule of the Wick algebra, denoted as $\text{statSubmodule}(\text{bosonic}) \subseteq \mathcal{F}.\text{WickAlgebra}$. It is defined as the composition of the projection map $\text{bosonicProjF}$ (which extracts the bosonic component of an element in the free algebra) and the canonical linear map $\iota$ that maps elements of the free algebra to the Wick algebra. Specifically, for any element $a \in \mathcal{A}_{\mathcal{F}}$, $\text{bosonicProjFree}(a)$ is the image under the quotient map of the part of $a$ consisting of products with an even number of fermionic operators.

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 $\iota : \mathcal{A}_{\mathcal{F}} \to \mathcal{W}_{\mathcal{F}}$ be the canonical quotient map from the free algebra to the Wick algebra $\mathcal{W}_{\mathcal{F}}$. For any element $a \in \mathcal{A}_{\mathcal{F}}$, the projection of $a$ into the bosonic submodule of the Wick algebra, denoted $\text{bosonicProjFree}(a)$, is equal to the image under $\iota$ of the bosonic projection of $a$ within the free algebra, $\text{bosonicProjF}(a)$. That is, $$\text{bosonicProjFree}(a) = \iota(\text{bosonicProjF}(a))$$ where the left-hand side is viewed as an element of the Wick algebra.

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 $\iota : \mathcal{A}_{\mathcal{F}} \to \mathcal{W}_{\mathcal{F}}$ be the canonical quotient map from the free algebra to the Wick algebra $\mathcal{W}_{\mathcal{F}}$. Let $\text{bosonicProjFree}: \mathcal{A}_{\mathcal{F}} \to \text{statSubmodule}(\text{bosonic})$ be the linear map that projects an element of the free algebra onto the bosonic submodule of the Wick algebra (defined as the image under $\iota$ of the component of the element consisting of products with an even number of fermionic operators). For any element $a \in \mathcal{A}_{\mathcal{F}}$, if $\iota(a) = 0$, then $\text{bosonicProjFree}(a) = 0$.

Let $\mathcal{F}$ be a field specification and $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the set of creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$. For any two elements $a, b \in \mathcal{A}_{\mathcal{F}}$, if $a \approx b$ (which implies that their images under the canonical quotient map $\iota: \mathcal{A}_{\mathcal{F}} \to \mathcal{W}_{\mathcal{F}}$ to the Wick algebra are equal), then their projections onto the bosonic submodule of the Wick algebra are equal: $$\text{bosonicProjFree}(a) = \text{bosonicProjFree}(b)$$ where $\text{bosonicProjFree}$ is the $\mathbb{C}$-linear map that extracts the part of an element in the free algebra consisting of products with an even number of fermionic operators and maps it into the Wick algebra.

For a given field specification $\mathcal{F}$, let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra, which is the quotient of the free algebra $\mathcal{A}_{\mathcal{F}}$ generated by creation and annihilation operators. The map $\text{bosonicProj}$ is a $\mathbb{C}$-linear projection from the Wick algebra $\mathcal{W}_{\mathcal{F}}$ to its bosonic submodule $\text{statSubmodule}(\text{bosonic})$. This submodule is spanned by elements that consist of products with an even number of fermionic operators. The map is formally defined by lifting the projection $\text{bosonicProjFree}$ from the free algebra to the Wick algebra quotient.

Let $\mathcal{F}$ be a field specification. Let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators, and let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra. Let $\iota: \mathcal{A}_{\mathcal{F}} \to \mathcal{W}_{\mathcal{F}}$ be the canonical quotient map. For any element $a \in \mathcal{A}_{\mathcal{F}}$, the bosonic projection of its image in the Wick algebra is equal to the value of the bosonic projection defined on the free algebra: $$\text{bosonicProj}(\iota(a)) = \text{bosonicProjFree}(a)$$ where $\text{bosonicProj}$ is the linear projection from the Wick algebra to its bosonic submodule (the space spanned by products with an even number of fermionic operators), and $\text{bosonicProjFree}$ is the map that extracts the bosonic component of an element in the free algebra and maps it into the Wick algebra.

For a given field specification $\mathcal{F}$, let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators. The map $\text{fermionicProjFree}$ is the $\mathbb{C}$-linear map from the free algebra to the fermionic submodule $\text{statSubmodule}(\text{fermionic})$ of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. It is defined as the composition $\iota_{\text{stat}} \circ \text{fermionicProjF}$, where $\text{fermionicProjF}$ projects an element of the free algebra onto its fermionic component (the part consisting of products with an odd number of fermionic operators), and $\iota_{\text{stat}}$ is the canonical linear map from the free algebra's fermionic submodule to the Wick algebra's fermionic submodule.

Let $\mathcal{F}$ be a field specification. Let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra generated by creation and annihilation operators and $\mathcal{F}.\text{WickAlgebra}$ be the corresponding Wick algebra. Let $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical linear map. For any element $a$ in the free algebra, the value of its fermionic projection into the Wick algebra, $\text{fermionicProjFree}(a)$, is equal to the image of its internal free-algebra fermionic projection, $\text{fermionicProjF}(a)$, under the map $\iota$. That is, $$(\text{fermionicProjFree}(a)) = \iota(\text{fermionicProjF}(a))$$ where the fermionic projection filters for terms consisting of products with an odd number of fermionic operators.

Let $\mathcal{F}$ be a field specification. Let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators, and let $\mathcal{F}.\text{WickAlgebra}$ be the corresponding Wick algebra. Let $\iota : \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical linear map. For any element $a$ in the free algebra, if $\iota(a) = 0$, then its fermionic projection into the Wick algebra, $\text{fermionicProjFree}(a)$, is also zero.

Let $\mathcal{F}$ be a field specification and let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators of $\mathcal{F}$. Let $\approx$ be the equivalence relation on the free algebra such that the quotient is the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. For any two elements $a, b$ in the free algebra, if $a \approx b$, then their fermionic projections into the Wick algebra are equal: $$\text{fermionicProjFree}(a) = \text{fermionicProjFree}(b)$$ where $\text{fermionicProjFree}$ is the $\mathbb{C}$-linear map that projects an element of the free algebra onto the fermionic submodule of the Wick algebra.

For a given field specification $\mathcal{F}$, the map $\text{fermionicProj}$ is a $\mathbb{C}$-linear projection from the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ to its fermionic submodule $\text{statSubmodule}(\text{fermionic})$. This submodule is spanned by products of creation and annihilation operators that contain an odd number of fermionic operators. The map is formally defined by lifting the projection from the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to the Wick algebra (the quotient of the free algebra by the Wick relations).

Let $\mathcal{F}$ be a field specification. Let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators, and let $\iota$ be the canonical quotient map from the free algebra to the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. For any element $a$ in the free algebra, the fermionic projection of its image in the Wick algebra is equal to the fermionic projection of $a$ defined from the free algebra: $$\text{fermionicProj}(\iota(a)) = \text{fermionicProjFree}(a)$$ where $\text{fermionicProj}$ is the projection from the Wick algebra onto its fermionic submodule, and $\text{fermionicProjFree}$ is the projection from the free algebra onto the same fermionic submodule.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra, which is the algebra generated by creation and annihilation operators. Let $P_{\text{bosonic}}$ and $P_{\text{fermionic}}$ denote the $\mathbb{C}$-linear projections from $\mathcal{W}_{\mathcal{F}}$ onto the submodules of elements with bosonic and fermionic statistics, respectively. For any element $a \in \mathcal{W}_{\mathcal{F}}$, the sum of its bosonic and fermionic projections is equal to the element itself: $$P_{\text{bosonic}}(a) + P_{\text{fermionic}}(a) = a$$ In this context, an element has a bosonic statistic if it consists of products containing an even number of fermionic operators, and a fermionic statistic if it contains an odd number of such operators.

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$, and let $S_{\text{bosonic}}$ be the $\mathbb{C}$-linear submodule of $\mathcal{W}_{\mathcal{F}}$ consisting of elements with a bosonic statistic (those spanned by products containing an even number of fermionic operators). Let $P_{\text{bosonic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ be the bosonic projection map. For any element $a \in \mathcal{W}_{\mathcal{F}}$, if $a$ is contained in $S_{\text{bosonic}}$, then $P_{\text{bosonic}}(a) = a$.

Let $\mathcal{F}$ be a field specification and $a$ be an element of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. If $a$ belongs to the submodule of elements with a fermionic statistic, $\text{statSubmodule}(\text{fermionic})$, then the fermionic projection $\text{fermionicProj}$ applied to $a$ is equal to $a$ itself (formally considered as an element of the fermionic submodule).

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$, and let $S_{\text{fermionic}}$ be the $\mathbb{C}$-linear submodule of $\mathcal{W}_{\mathcal{F}}$ consisting of elements with a fermionic statistic (those spanned by products containing an odd number of fermionic operators). Let $P_{\text{bosonic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ be the bosonic projection map onto the bosonic submodule. For any element $a \in \mathcal{W}_{\mathcal{F}}$, if $a \in S_{\text{fermionic}}$, then $P_{\text{bosonic}}(a) = 0$.

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$. Let $S_{\text{bosonic}}$ be the complex submodule of $\mathcal{W}_{\mathcal{F}}$ consisting of elements with bosonic statistics (spanned by products of creation and annihilation operators containing an even number of fermionic operators). Let $P_{\text{fermionic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{fermionic}}$ be the $\mathbb{C}$-linear fermionic projection map onto the submodule of elements with fermionic statistics. For any element $a \in \mathcal{W}_{\mathcal{F}}$, if $a \in S_{\text{bosonic}}$, then $P_{\text{fermionic}}(a) = 0$.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra. Let $S_{\text{bosonic}}$ be the complex submodule of the Wick algebra consisting of elements with bosonic statistics (spanned by products containing an even number of fermionic operators). Let $P_{\text{fermionic}}$ be the $\mathbb{C}$-linear fermionic projection map from the Wick algebra onto the fermionic submodule. For any element $a \in \mathcal{W}_{\mathcal{F}}$, $a$ belongs to $S_{\text{bosonic}}$ if and only if its fermionic projection is zero: $$a \in S_{\text{bosonic}} \iff P_{\text{fermionic}}(a) = 0$$

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$. Let $S_{\text{fermionic}}$ be the $\mathbb{C}$-linear submodule of $\mathcal{W}_{\mathcal{F}}$ consisting of elements with fermionic statistics, which are spanned by products of creation and annihilation operators containing an odd number of fermionic operators. Let $P_{\text{bosonic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ be the $\mathbb{C}$-linear bosonic projection map onto the bosonic submodule. For any element $a \in \mathcal{W}_{\mathcal{F}}$, $a$ belongs to the fermionic submodule $S_{\text{fermionic}}$ if and only if its bosonic projection is zero: $$a \in S_{\text{fermionic}} \iff P_{\text{bosonic}}(a) = 0$$

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$. Let $P_{\text{bosonic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ and $P_{\text{fermionic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{fermionic}}$ be the $\mathbb{C}$-linear projection maps onto the bosonic submodule $S_{\text{bosonic}}$ and fermionic submodule $S_{\text{fermionic}}$, respectively. For any element $a \in \mathcal{W}_{\mathcal{F}}$, applying the bosonic projection to the result of the fermionic projection yields zero: $$P_{\text{bosonic}}(P_{\text{fermionic}}(a)) = 0$$

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$. Let $P_{\text{bosonic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ and $P_{\text{fermionic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{fermionic}}$ be the $\mathbb{C}$-linear projection maps onto the bosonic and fermionic submodules, respectively. For any element $a \in \mathcal{W}_{\mathcal{F}}$, the fermionic projection of its bosonic part is zero: $$P_{\text{fermionic}}(P_{\text{bosonic}}(a)) = 0$$

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$. Let $P_{\text{bosonic}} : \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ be the $\mathbb{C}$-linear projection map onto the bosonic submodule $S_{\text{bosonic}}$, which consists of elements spanned by products of creation and annihilation operators with an even number of fermionic operators. For any element $a \in \mathcal{W}_{\mathcal{F}}$, applying the bosonic projection twice is equal to applying it once: $$P_{\text{bosonic}}(P_{\text{bosonic}}(a)) = P_{\text{bosonic}}(a)$$

Let $\mathcal{F}$ be a field specification. For any element $a$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the fermionic projection of its fermionic component is equal to the original fermionic projection: $$\text{fermionicProj}(\text{fermionicProj}(a)) = \text{fermionicProj}(a)$$ where $\text{fermionicProj}$ is the $\mathbb{C}$-linear projection onto the submodule $\text{statSubmodule}(\text{fermionic})$ spanned by products of creation and annihilation operators containing an odd number of fermionic operators.

In the Wick algebra $\mathcal{W}_{\mathcal{F}}$ associated with a field specification $\mathcal{F}$, let $S_{\text{bosonic}}$ and $S_{\text{fermionic}}$ be the submodules consisting of elements with bosonic and fermionic statistics, respectively. For any element $a$ in the direct sum $\bigoplus_{i \in \{\text{bosonic, fermionic}\}} S_i$, let $a_{\text{bosonic}}$ denote its component in $S_{\text{bosonic}}$. The bosonic projection $P_{\text{bosonic}}: \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ satisfies $P_{\text{bosonic}}(a_{\text{bosonic}}) = a_{\text{bosonic}}$.

Let $\mathcal{W}_{\mathcal{F}}$ be the Wick algebra associated with a field specification $\mathcal{F}$, which is graded by the field statistics $i \in \{\text{bosonic, fermionic}\}$. Let $S_{\text{bosonic}}$ and $S_{\text{fermionic}}$ be the submodules of $\mathcal{W}_{\mathcal{F}}$ consisting of elements with bosonic and fermionic statistics, respectively. For any element $a$ in the direct sum $\bigoplus_{i} S_i$, let $a_{\text{fermionic}}$ denote its component in $S_{\text{fermionic}}$. The bosonic projection $P_{\text{bosonic}}: \mathcal{W}_{\mathcal{F}} \to S_{\text{bosonic}}$ satisfies: $$P_{\text{bosonic}}(a_{\text{fermionic}}) = 0$$ where $a_{\text{fermionic}}$ is treated as an element of the Wick algebra.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}_{\mathcal{F}}$ be its associated Wick algebra. Let $S_{\text{bosonic}}$ and $S_{\text{fermionic}}$ be the submodules of $\mathcal{W}_{\mathcal{F}}$ containing elements with bosonic and fermionic statistics, respectively. For any element $a$ in the direct sum $S_{\text{bosonic}} \oplus S_{\text{fermionic}}$, let $a_{\text{bosonic}}$ denote its component in the bosonic submodule. The fermionic projection map $P_{\text{fermionic}}: \mathcal{W}_{\mathcal{F}} \to S_{\text{fermionic}}$ satisfies: $$P_{\text{fermionic}}(a_{\text{bosonic}}) = 0$$ where $a_{\text{bosonic}}$ is coerced into the Wick algebra $\mathcal{W}_{\mathcal{F}}$.

Let $\mathcal{F}$ be a field specification, and let $V_{\text{bosonic}}$ and $V_{\text{fermionic}}$ be the submodules of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ corresponding to the bosonic and fermionic statistics, respectively. For any element $a$ in the direct sum $V_{\text{bosonic}} \oplus V_{\text{fermionic}}$, let $a_{\text{fermionic}}$ be its component in the fermionic submodule. The fermionic projection map $\text{fermionicProj}$ satisfies: $$\text{fermionicProj}(a_{\text{fermionic}}) = a_{\text{fermionic}}$$ where the component $a_{\text{fermionic}}$ on the left is coerced into the Wick algebra and the result on the right is the element within the submodule.

For a field specification $\mathcal{F}$, let $V_{\text{bosonic}}$ and $V_{\text{fermionic}}$ be the submodules of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ corresponding to the bosonic and fermionic statistics, respectively. For any element $a$ in the direct sum $V_{\text{bosonic}} \oplus V_{\text{fermionic}}$, the image of $a$ under the canonical homomorphism to the Wick algebra is equal to the sum of its bosonic and fermionic components: $$\text{coe}(a) = a_{\text{bosonic}} + a_{\text{fermionic}}$$ where $a_{\text{bosonic}}$ and $a_{\text{fermionic}}$ are the components of $a$ in the respective submodules.

Let $\mathcal{F}$ be a field specification and $V_s$ be the submodule of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ corresponding to the field statistic $s \in \{\text{bosonic}, \text{fermionic}\}$. For any element $a$ in the direct sum $\bigoplus_{s \in \{\text{bosonic}, \text{fermionic}\}} V_s$, $a$ is equal to the sum of its components in the bosonic and fermionic submodules: $$a = \iota_{\text{bosonic}}(a_{\text{bosonic}}) + \iota_{\text{fermionic}}(a_{\text{fermionic}})$$ where $a_s$ denotes the component of $a$ corresponding to statistic $s$, and $\iota_s$ is the canonical injection into the direct sum.

For a given field specification $\mathcal{F}$, the Wick algebra $\mathcal{W}_{\mathcal{F}}$ is a graded algebra indexed by $\text{FieldStatistic} = \{\text{bosonic}, \text{fermionic}\}$. The grading is defined by the submodules $V_s = \text{statSubmodule}(s)$, where $V_{\text{bosonic}}$ is spanned by products of operators with an overall bosonic statistic (containing an even number of fermionic operators) and $V_{\text{fermionic}}$ is spanned by products with an overall fermionic statistic (containing an odd number of fermionic operators). The structure satisfies the following: 1. The identity element $1$ is in $V_{\text{bosonic}}$. 2. For any elements $a \in V_{s_1}$ and $b \in V_{s_2}$, the product $ab$ is in $V_{s_1 + s_2}$, where the addition of statistics follows parity rules ($\text{bosonic} + s = s$ and $\text{fermionic} + \text{fermionic} = \text{bosonic}$). 3. Any element $x \in \mathcal{W}_{\mathcal{F}}$ can be uniquely decomposed into its bosonic and fermionic parts using the projections $\text{bosonicProj}(x)$ and $\text{fermionicProj}(x)$.