Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.Grading

For a given field specification $\mathcal{F}$ and a statistic $f \in \{\text{bosonic}, \text{fermionic}\}$, the `statisticSubmodule f` is the $\mathbb{C}$-linear submodule of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ spanned by the set of all operator products $\phi_1 \phi_2 \dots \phi_n$ whose collective statistic is equal to $f$. The collective statistic of a product is determined by the parity of fermionic operators in the sequence: it is $\text{fermionic}$ if the number of fermionic operators is odd, and $\text{bosonic}$ otherwise.

For a given field specification $\mathcal{F}$, let $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$. If the collective statistic of the list $\varphi_s$ (which is fermionic if there are an odd number of fermionic operators in the list and bosonic otherwise) is equal to $f \in \{\text{bosonic}, \text{fermionic}\}$, then the product of these operators $\prod_{i=1}^n \phi_i$ (represented as $\text{ofCrAnListF}(\varphi_s)$) is an element of the $\mathbb{C}$-linear submodule $\text{statisticSubmodule}(f)$ of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$.

For a given field specification $\mathcal{F}$, let $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$. The product of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, denoted as $\text{ofCrAnListF}(\varphi_s) = \phi_1 \phi_2 \dots \phi_n$, is an element of either the bosonic submodule $\text{statisticSubmodule}(\text{bosonic})$ or the fermionic submodule $\text{statisticSubmodule}(\text{fermionic})$.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component. In the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$, the element $\text{ofCrAnOpF}(\phi)$ corresponding to $\phi$ is an element of either the bosonic submodule $\text{statisticSubmodule}(\text{bosonic})$ or the fermionic submodule $\text{statisticSubmodule}(\text{fermionic})$.

For a given field specification $\mathcal{F}$, let $\mathcal{A}_{\mathcal{F}}$ denote the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$. The map $\text{bosonicProjF}$ is a $\mathbb{C}$-linear projection from $\mathcal{A}_{\mathcal{F}}$ onto its bosonic submodule. It is defined on the basis of operator products such that for any monomial $b = \phi_1 \phi_2 \dots \phi_n$ (where each $\phi_i \in \mathcal{F}.\text{CrAnFieldOp}$): $$\text{bosonicProjF}(b) = \begin{cases} b & \text{if the collective statistic of } [\phi_1, \dots, \phi_n] \text{ is bosonic} \\ 0 & \text{otherwise} \end{cases}$$ The collective statistic is bosonic if the number of operators in the product with fermionic statistics is even, and fermionic otherwise.

For a given field specification $\mathcal{F}$, let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra generated by the creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$. Let $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of these operators, and let $\Phi = \phi_1 \phi_2 \dots \phi_n$ be their corresponding product in $\mathcal{A}_{\mathcal{F}}$. Let $P_{\text{bosonic}}$ be the linear projection onto the bosonic submodule of $\mathcal{A}_{\mathcal{F}}$. The theorem states that: $$P_{\text{bosonic}}(\Phi) = \begin{cases} \Phi & \text{if the collective statistic of } \phi_s \text{ is bosonic} \\ 0 & \text{if the collective statistic of } \phi_s \text{ is fermionic} \end{cases}$$ The collective statistic of the list is bosonic if the number of operators in $\phi_s$ with fermionic statistics is even, and fermionic if that number is odd.

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 $P_{\text{bosonic}}$ be the $\mathbb{C}$-linear projection from $\mathcal{A}_{\mathcal{F}}$ onto its bosonic submodule. For any element $a \in \mathcal{A}_{\mathcal{F}}$, if $a$ is contained in the bosonic submodule, then its projection $P_{\text{bosonic}}(a)$ is equal to $a$.

For a given field specification $\mathcal{F}$, let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators. Let $S_{\text{fermionic}}$ be the submodule of $\mathcal{A}_{\mathcal{F}}$ consisting of operators with fermionic statistics (those spanned by products of operators containing an odd number of fermionic components). For any element $a \in S_{\text{fermionic}}$, the linear projection onto the bosonic submodule, denoted $\text{bosonicProjF}$, satisfies $\text{bosonicProjF}(a) = 0$.

Let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators for a field specification $\mathcal{F}$. Let $S_{\text{bosonic}}$ and $S_{\text{fermionic}}$ be the submodules of $\mathcal{A}_{\mathcal{F}}$ containing operators with bosonic and fermionic statistics, respectively. For an element $a$ in the direct sum $\bigoplus_{i \in \{\text{bosonic, fermionic}\}} S_i$, let $a_{\text{bosonic}}$ denote its component in the bosonic submodule. The $\mathbb{C}$-linear projection onto the bosonic submodule, $P_{\text{bosonic}}$, satisfies $P_{\text{bosonic}}(a_{\text{bosonic}}) = a_{\text{bosonic}}$.

Let $\mathcal{A}_{\mathcal{F}}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators, which is decomposed into a direct sum of submodules $\mathcal{A}_{\mathcal{F}} \cong S_{\text{bosonic}} \oplus S_{\text{fermionic}}$ according to quantum field statistics. For any element $a$ represented in this direct sum, let $a_{\text{fermionic}} \in S_{\text{fermionic}}$ be its component in the fermionic submodule. The $\mathbb{C}$-linear projection onto the bosonic submodule, $\text{bosonicProjF}$, satisfies: $$\text{bosonicProjF}(a_{\text{fermionic}}) = 0$$

For a given field specification $\mathcal{F}$, let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by the set of creation and annihilation operators $\mathcal{F}.\text{CrAnFieldOp}$. The function $\text{fermionicProjF}$ is the $\mathbb{C}$-linear projection from this algebra onto the submodule of elements with a $\text{fermionic}$ statistic. Specifically, for any basis element representing a product of operators $\varphi_1 \varphi_2 \dots \varphi_n$ corresponding to a list $\varphi s = [\varphi_1, \dots, \varphi_n]$, the map is defined as: $$\text{fermionicProjF}(\varphi_1 \varphi_2 \dots \varphi_n) = \begin{cases} \varphi_1 \varphi_2 \dots \varphi_n & \text{if the statistic of } \varphi s \text{ is } \text{fermionic} \\ 0 & \text{if the statistic of } \varphi s \text{ is } \text{bosonic} \end{cases}$$ where the statistic of the list is $\text{fermionic}$ if it contains an odd number of fermionic components and $\text{bosonic}$ otherwise.

For a given field specification $\mathcal{F}$, let $\varphi s = [\varphi_1, \dots, \varphi_n]$ be a list of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$, and let $\text{ofCrAnListF}(\varphi s)$ denote their product $\varphi_1 \varphi_2 \dots \varphi_n$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The fermionic projection $\text{fermionicProjF}$ applied to this product satisfies: $$\text{fermionicProjF}(\varphi_1 \varphi_2 \dots \varphi_n) = \begin{cases} \varphi_1 \varphi_2 \dots \varphi_n & \text{if the collective statistic of } \varphi s \text{ is } \text{fermionic} \\ 0 & \text{if the collective statistic of } \varphi s \text{ is } \text{bosonic} \end{cases}$$ where the collective statistic is $\text{fermionic}$ if the list contains an odd number of fermionic operators and $\text{bosonic}$ otherwise. (Note: in the case where the result is non-zero, it is formally treated as an element of the fermionic submodule).

For a given field specification $\mathcal{F}$, let $\varphi s = [\varphi_1, \dots, \varphi_n]$ be a list of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$, and let $\varphi_1 \dots \varphi_n$ denote their product in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. Let $\text{fermionicProjF}$ be the linear projection onto the submodule of elements with fermionic statistics. The projection of the product satisfies: $$\text{fermionicProjF}(\varphi_1 \dots \varphi_n) = \begin{cases} 0 & \text{if the collective statistic of } \varphi s \text{ is } \text{bosonic} \\ \varphi_1 \dots \varphi_n & \text{otherwise} \end{cases}$$ where the collective statistic is $\text{bosonic}$ if the list contains an even number of operators with fermionic statistics, and $\text{fermionic}$ otherwise. (Note: in the case where the result is non-zero, it is formally treated as an element of the fermionic submodule).

In the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$ associated with a field specification $\mathcal{F}$, let $a$ be an element that belongs to the submodule of operators with a $\text{fermionic}$ statistic. Then the fermionic projection $\text{fermionicProjF}$ applied to $a$ is equal to $a$ itself (formally treated as an element of the fermionic submodule).

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators of $\mathcal{F}$. For any element $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, if $a$ belongs to the $\mathbb{C}$-linear submodule $\text{statisticSubmodule}(\text{bosonic})$ (the submodule spanned by operator products with an even number of fermionic components), then its fermionic projection satisfies $\text{fermionicProjF}(a) = 0$.

Let $\mathcal{F}$ be a field specification and let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by its creation and annihilation operators. This algebra is graded by $\text{FieldStatistic} = \{\text{bosonic}, \text{fermionic}\}$. For any element $a$ in the direct sum of the statistic submodules $\bigoplus_{i \in \{\text{bosonic}, \text{fermionic}\}} \text{statisticSubmodule}(i)$, let $a_{\text{bosonic}}$ denote its component in the bosonic submodule. Then the fermionic projection of this component is zero, i.e., $\text{fermionicProjF}(a_{\text{bosonic}}) = 0$.

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by creation and annihilation operators. This algebra is graded by field statistics, where $V_{\text{bosonic}}$ and $V_{\text{fermionic}}$ are the submodules of elements with bosonic and fermionic statistics, respectively. For any element $a$ in the direct sum $V_{\text{bosonic}} \oplus V_{\text{fermionic}}$, let $a_{\text{fermionic}}$ denote its component in the fermionic submodule. Then the fermionic projection $\text{fermionicProjF}$ applied to $a_{\text{fermionic}}$ is equal to $a_{\text{fermionic}}$ itself: $$\text{fermionicProjF}(a_{\text{fermionic}}) = a_{\text{fermionic}}$$

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 $P_{\text{bosonic}}$ and $P_{\text{fermionic}}$ denote the $\mathbb{C}$-linear projections from $\mathcal{A}_{\mathcal{F}}$ onto the submodules of elements with bosonic and fermionic statistics, respectively. For any element $a \in \mathcal{A}_{\mathcal{F}}$, the sum of its bosonic and fermionic projections is equal to the element itself: $$P_{\text{bosonic}}(a) + P_{\text{fermionic}}(a) = a$$ Here, an element has a bosonic statistic if it is a linear combination of products containing an even number of fermionic operators, and a fermionic statistic if it contains an odd number.

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operators. This algebra is graded by the field statistics $\{\text{bosonic}, \text{fermionic}\}$, with $V_{\text{bosonic}}$ and $V_{\text{fermionic}}$ denoting the submodules of operators with the respective statistics. For any element $a$ in the direct sum $V_{\text{bosonic}} \oplus V_{\text{fermionic}}$, its image under the canonical additive monoid homomorphism into the algebra is the sum of its components: $$\text{coe}(a) = a_{\text{bosonic}} + a_{\text{fermionic}}$$ where $a_{\text{bosonic}}$ and $a_{\text{fermionic}}$ are the projections of $a$ onto the bosonic and fermionic submodules, respectively.

For a given field specification $\mathcal{F}$, let $M_s$ be the $\mathbb{C}$-linear submodule of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ corresponding to the field statistic $s \in \{\text{bosonic}, \text{fermionic}\}$. For any element $a$ in the direct sum $\bigoplus_{s \in \{\text{bosonic}, \text{fermionic}\}} M_s$, $a$ is equal to the sum of its bosonic and fermionic components: $$a = \iota_{\text{bosonic}}(a_{\text{bosonic}}) + \iota_{\text{fermionic}}(a_{\text{fermionic}})$$ where $a_s$ denotes the component of $a$ at index $s$, and $\iota_s$ is the canonical inclusion from the submodule $M_s$ into the direct sum.

For a given field specification $\mathcal{F}$, the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ (the algebra generated by creation and annihilation operators over $\mathbb{C}$) is a graded algebra indexed by the field statistics $\{\text{bosonic}, \text{fermionic}\}$. The grading is defined by the submodules $V_{\text{bosonic}}$ and $V_{\text{fermionic}}$, where $V_s$ is the submodule spanned by operator products whose total statistic is $s$. Specifically, this grading structure satisfies: - The identity element $1$ is contained in the bosonic submodule $V_{\text{bosonic}}$. - For any statistics $s_1, s_2$, the product of elements from $V_{s_1}$ and $V_{s_2}$ lies in $V_{s_1 + s_2}$, where the addition of statistics follows $\mathbb{Z}_2$ parity rules (e.g., $\text{fermionic} + \text{fermionic} = \text{bosonic}$). - The algebra is the direct sum of its bosonic and fermionic components: $\mathcal{F}.\text{FieldOpFreeAlgebra} \cong V_{\text{bosonic}} \oplus V_{\text{fermionic}}$.

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 $\mathcal{F}$. This algebra is $\mathbb{Z}_2$-graded by field statistics, with submodules $V_{\text{bosonic}}$ (even parity) and $V_{\text{fermionic}}$ (odd parity). Let $P_{\text{bosonic}}$ and $P_{\text{fermionic}}$ denote the $\mathbb{C}$-linear projections from $\mathcal{A}_{\mathcal{F}}$ onto these respective submodules. For any two operators $a, b \in \mathcal{A}_{\mathcal{F}}$, the bosonic component of their product is given by the sum of the products of their corresponding statistics: $$P_{\text{bosonic}}(a \cdot b) = P_{\text{bosonic}}(a) \cdot P_{\text{bosonic}}(b) + P_{\text{fermionic}}(a) \cdot P_{\text{fermionic}}(b)$$ This identity reflects the parity rule that a product of two operators is bosonic if both operators are bosonic or if both operators are fermionic.

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 $P_{\text{bosonic}}$ and $P_{\text{fermionic}}$ denote the $\mathbb{C}$-linear projections from $\mathcal{A}_{\mathcal{F}}$ onto the submodules of elements with bosonic and fermionic statistics, respectively. For any elements $a, b \in \mathcal{A}_{\mathcal{F}}$, the fermionic projection of their product is given by: $$P_{\text{fermionic}}(a \cdot b) = P_{\text{bosonic}}(a) \cdot P_{\text{fermionic}}(b) + P_{\text{fermionic}}(a) \cdot P_{\text{bosonic}}(b)$$