Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.NormalOrder

For a field specification $\mathcal{F}$, the normal ordering $\mathcal{N}^f$ is the $\mathbb{C}$-linear map from the free algebra of field operators $\text{FieldOpFreeAlgebra } \mathcal{F}$ to itself: $$\mathcal{N}^f : \text{FieldOpFreeAlgebra } \mathcal{F} \to \text{FieldOpFreeAlgebra } \mathcal{F}$$ The map is uniquely defined by its action on the basis consisting of products of creation and annihilation operators. For any list of operators $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$, the normal ordering of their product is given by: $$\mathcal{N}^f(\phi_1 \phi_2 \dots \phi_n) = \eta(\phi_s) \cdot (\phi_{\sigma(1)} \phi_{\sigma(2)} \dots \phi_{\sigma(n)})$$ where the resulting product $(\phi_{\sigma(1)} \dots \phi_{\sigma(n)})$ is the normal-ordered arrangement of the original list (where all creation operators are placed to the left of all annihilation operators), and $\eta(\phi_s) \in \{1, -1\}$ is the phase factor (sign) associated with the permutations of fermionic operators required to achieve this ordering.

The notation $\mathcal{N}^f(a)$ represents the normal ordering of an element $a$ in the free algebra of field operators $\text{FieldOpFreeAlgebra } \mathcal{F}$. It is defined as the application of the linear map $\text{normalOrderF}$ to the element $a$.

Let $\mathcal{F}$ be a field specification and $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operators (elements of $\mathcal{F}.\text{CrAnFieldOp}$). In the free algebra of field operators $\text{FieldOpFreeAlgebra } \mathcal{F}$, the normal ordering operator $\mathcal{N}^f$ applied to the product of these operators is given by: $$\mathcal{N}^f(\phi_1 \phi_2 \dots \phi_n) = \eta(\phi_s) \cdot (\phi_{\sigma(1)} \phi_{\sigma(2)} \dots \phi_{\sigma(n)})$$ where $(\phi_{\sigma(1)} \phi_{\sigma(2)} \dots \phi_{\sigma(n)})$ is the product of the operators rearranged according to the normal ordering relation (where all creation operators are positioned to the left of all annihilation operators), and $\eta(\phi_s) \in \{1, -1\}$ is the phase factor associated with the permutations of fermionic operators required to achieve this ordering.

For a given field specification $\mathcal{F}$ and a list of creation and annihilation operators $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$, let $\text{normalOrderList}(\phi_s)$ be the list rearranged into normal order (where all creation operators precede all annihilation operators). Let $\eta(\phi_s) \in \{1, -1\}$ be the normal ordering sign (the phase factor arising from permutations of fermionic operators) and $\mathcal{N}^f$ be the linear normal ordering operator on the free algebra of field operators. The product of the operators in their normal-ordered arrangement is equal to the product of the phase factor and the normal ordering operator applied to the original product: $$\text{ofCrAnListF}(\text{normalOrderList}(\phi_s)) = \eta(\phi_s) \cdot \mathcal{N}^f(\phi_1 \phi_2 \dots \phi_n)$$

Let $\mathcal{F}$ be a field specification. In the free algebra of field operators $\text{FieldOpFreeAlgebra } \mathcal{F}$, the normal ordering operator $\mathcal{N}^f$ maps the identity element $1$ to itself: $$\mathcal{N}^f(1) = 1$$

In the free algebra of creation and annihilation operators $\text{FieldOpFreeAlgebra}(\mathcal{F})$, let $\mathcal{N}^f$ denote the normal ordering operator. For any three elements $a, b, c$ in the algebra, the normal ordering of their product $abc$ is equal to the normal ordering of the product $a \mathcal{N}^f(b) c$. That is: $$\mathcal{N}^f(a b c) = \mathcal{N}^f(a \mathcal{N}^f(b) c)$$

Let $\mathcal{F}$ be a field specification. In the free algebra of creation and annihilation operators $\text{FieldOpFreeAlgebra}(\mathcal{F})$, the normal ordering operator $\mathcal{N}^f$ satisfies the following identity for any elements $a, b \in \text{FieldOpFreeAlgebra}(\mathcal{F})$: $$\mathcal{N}^f(ab) = \mathcal{N}^f(a \mathcal{N}^f(b))$$

In the free algebra of creation and annihilation operators $\text{FieldOpFreeAlgebra}(\mathcal{F})$, let $\mathcal{N}^f$ denote the normal ordering operator. For any two elements $a$ and $b$ in the algebra, the normal ordering of their product $ab$ is equal to the normal ordering of the product of $\mathcal{N}^f(a)$ and $b$. That is: $$\mathcal{N}^f(a b) = \mathcal{N}^f(\mathcal{N}^f(a) b)$$

Let $\mathcal{F}$ be a field specification. Suppose $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ is a creation operator (i.e., its label is $\text{create}$) and $\phi_s = [\phi_1, \dots, \phi_n]$ is a list of creation and annihilation operators. In the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the normal ordering operator $\mathcal{N}^f$ satisfies: $$\mathcal{N}^f(\phi \cdot \phi_1 \cdot \dots \cdot \phi_n) = \phi \cdot \mathcal{N}^f(\phi_1 \cdot \dots \cdot \phi_n)$$ where the product of the list is defined by the map $\text{ofCrAnListF}$.

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 operator components $\mathcal{F}.\text{CrAnFieldOp}$. Let $\mathcal{N}^f : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the linear normal ordering map. For any operator component $\phi$ that is a creation operator (i.e., its label is $\text{create}$) and any element $a \in \mathcal{A}_{\mathcal{F}}$, the normal ordering of their product satisfies: $$\mathcal{N}^f(\phi \cdot a) = \phi \cdot \mathcal{N}^f(a)$$ where $\phi$ is treated as the corresponding generator in $\mathcal{A}_{\mathcal{F}}$.

Let $\mathcal{F}$ be a field specification and $\mathcal{A}_{\mathcal{F}}$ be the free algebra generated by the creation and annihilation operator components $\mathcal{O}_{\mathcal{F}}$. Let $\mathcal{N}^f: \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the normal ordering linear map. For any list of operators $\Phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ and any operator $\phi \in \mathcal{O}_{\mathcal{F}}$ that is an annihilation operator, the normal ordering of their product satisfies: $$\mathcal{N}^f(\phi_1 \dots \phi_n \phi) = \mathcal{N}^f(\phi_1 \dots \phi_n) \cdot \phi$$ where the product is taken in the algebra $\mathcal{A}_{\mathcal{F}}$.

Let $\mathcal{F}$ be a field specification and $\mathcal{A}_{\mathcal{F}}$ be the free algebra generated by the creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$. Let $\mathcal{N}^f: \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the normal ordering linear map. For any element $a \in \mathcal{A}_{\mathcal{F}}$ and any operator component $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ that is classified as an annihilation operator, the following identity holds: $$\mathcal{N}^f(a \cdot \phi) = \mathcal{N}^f(a) \cdot \phi$$ where $\cdot$ denotes the multiplication in the algebra $\mathcal{A}_{\mathcal{F}}$.

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 operator components $\mathcal{F}.\text{CrAnFieldOp}$. Let $\mathcal{N}^f : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the linear normal ordering map. For any field operator $\phi \in \text{FieldOp}(\mathcal{F})$ and any element $a \in \mathcal{A}_{\mathcal{F}}$, the normal ordering of the product of the creation part of $\phi$ with $a$ satisfies: $$\mathcal{N}^f(\text{crPartF}(\phi) \cdot a) = \text{crPartF}(\phi) \cdot \mathcal{N}^f(a)$$ where $\text{crPartF}(\phi)$ is the component of $\phi$ consisting of creation operators in the algebra $\mathcal{A}_{\mathcal{F}}$.

Let $\mathcal{F}$ be a field specification and $\mathcal{A}_{\mathcal{F}}$ be the free algebra generated by the creation and annihilation operator components of the fields. Let $\mathcal{N}^f : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the $\mathbb{C}$-linear normal ordering operator. For any element $a \in \mathcal{A}_{\mathcal{F}}$ and any field operator $\phi \in \mathcal{F}.\text{FieldOp}$, the following identity holds: $$\mathcal{N}^f(a \cdot \text{anPartF}(\phi)) = \mathcal{N}^f(a) \cdot \text{anPartF}(\phi)$$ where $\text{anPartF}(\phi)$ denotes the annihilation component of the field operator $\phi$ within the algebra.

Let $\mathcal{F}$ be a field specification. Suppose $\phi_c$ and $\phi_a$ are creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$ respectively. Let $A$ and $A'$ be elements of the free algebra $\text{FieldOpFreeAlgebra } \mathcal{F}$ formed by the products of two lists of operators $\Phi_s$ and $\Phi_s'$. The normal ordering operator $\mathcal{N}^f$ satisfies the following identity: $$\mathcal{N}^f(A' \cdot \phi_c \cdot \phi_a \cdot A) = \epsilon(\phi_c, \phi_a) \cdot \mathcal{N}^f(A' \cdot \phi_a \cdot \phi_c \cdot A)$$ where $\epsilon(\phi_c, \phi_a)$ is the exchange sign (phase factor) determined by the quantum statistics (bosonic or fermionic) of $\phi_c$ and $\phi_a$.

Let $\mathcal{F}$ be a field specification. Suppose $\phi_c \in \mathcal{F}.\text{CrAnFieldOp}$ is a creation operator and $\phi_a \in \mathcal{F}.\text{CrAnFieldOp}$ is an annihilation operator. For any list of operators $\Phi_s$ and any element $A$ in the free algebra $\text{FieldOpFreeAlgebra } \mathcal{F}$, the normal ordering operator $\mathcal{N}^f$ satisfies: $$\mathcal{N}^f(\text{ofCrAnListF}(\Phi_s) \cdot \phi_c \cdot \phi_a \cdot A) = \epsilon(\phi_c, \phi_a) \cdot \mathcal{N}^f(\text{ofCrAnListF}(\Phi_s) \cdot \phi_a \cdot \phi_c \cdot A)$$ where $\epsilon(\phi_c, \phi_a)$ is the exchange sign (phase factor) determined by the quantum statistics of $\phi_c$ and $\phi_a$, and $\text{ofCrAnListF}(\Phi_s)$ denotes the product of the operators in the list $\Phi_s$.

Let $\mathcal{F}$ be a field specification and let $\text{FieldOpFreeAlgebra}(\mathcal{F})$ be the free algebra generated by creation and annihilation operators. Suppose $\phi_c$ is a creation operator and $\phi_a$ is an annihilation operator. For any elements $a, b$ in the algebra, the normal ordering operator $\mathcal{N}^f$ satisfies the following identity: $$\mathcal{N}^f(a \cdot \phi_c \cdot \phi_a \cdot b) = \epsilon(\phi_c, \phi_a) \cdot \mathcal{N}^f(a \cdot \phi_a \cdot \phi_c \cdot b)$$ where $\epsilon(\phi_c, \phi_a)$ is the exchange sign (phase factor) determined by the quantum statistics (bosonic or fermionic) of the operators $\phi_c$ and $\phi_a$.

Let $\mathcal{F}$ be a field specification and $\text{FieldOpFreeAlgebra}(\mathcal{F})$ be the free associative algebra generated by the set of creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$. Let $\phi_c$ be a creation operator and $\phi_a$ be an annihilation operator in $\mathcal{F}.\text{CrAnFieldOp}$. For any elements $a, b \in \text{FieldOpFreeAlgebra}(\mathcal{F})$, the normal ordering operator $\mathcal{N}^f$ satisfies the identity: $$\mathcal{N}^f(a \cdot [\phi_c, \phi_a]_s^F \cdot b) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification and $\text{FieldOpFreeAlgebra}(\mathcal{F})$ be the free associative algebra over $\mathbb{C}$ generated by the set of creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$. Let $\phi_a$ be an annihilation operator and $\phi_c$ be a creation operator in $\mathcal{F}.\text{CrAnFieldOp}$. For any elements $a, b \in \text{FieldOpFreeAlgebra}(\mathcal{F})$, the normal ordering operator $\mathcal{N}^f$ satisfies the identity: $$\mathcal{N}^f(a \cdot [\phi_a, \phi_c]_s^F \cdot b) = 0$$ where $[\cdot, \cdot]_s^F$ denotes the super-commutator in the free algebra.

Let $\mathcal{F}$ be a field specification. For any field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ and any elements $a, b$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the normal ordering operator $\mathcal{N}^f$ satisfies the following identity when swapping the creation part of $\phi$ and the annihilation part of $\phi'$: $$\mathcal{N}^f(a \cdot \text{crPartF}(\phi) \cdot \text{anPartF}(\phi') \cdot b) = \epsilon(\phi, \phi') \cdot \mathcal{N}^f(a \cdot \text{anPartF}(\phi') \cdot \text{crPartF}(\phi) \cdot b)$$ where $\text{crPartF}(\phi)$ is the creation component of $\phi$, $\text{anPartF}(\phi')$ is the annihilation component of $\phi'$, and $\epsilon(\phi, \phi')$ is the exchange sign (phase factor) determined by the quantum statistics (bosonic or fermionic) of the fields $\phi$ and $\phi'$.

Let $\mathcal{F}$ be a field specification. For any field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ and any elements $a, b$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the normal ordering operator $\mathcal{N}^f$ satisfies the following identity when swapping the annihilation part of $\phi$ and the creation part of $\phi'$: $$\mathcal{N}^f(a \cdot \text{anPartF}(\phi) \cdot \text{crPartF}(\phi') \cdot b) = \epsilon(\phi, \phi') \cdot \mathcal{N}^f(a \cdot \text{crPartF}(\phi') \cdot \text{anPartF}(\phi) \cdot b)$$ where $\text{anPartF}(\phi)$ is the annihilation component of $\phi$, $\text{crPartF}(\phi')$ is the creation component of $\phi'$, and $\epsilon(\phi, \phi')$ is the exchange sign (phase factor) determined by the quantum statistics (bosonic or fermionic) of the fields $\phi$ and $\phi'$.

Let $\mathcal{F}$ be a field specification. For any field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ and any elements $a, b$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the normal ordering operator $\mathcal{N}^f$ satisfies: $$\mathcal{N}^f(a \cdot [\text{crPartF}(\phi), \text{anPartF}(\phi')]_s^F \cdot b) = 0$$ where $\text{crPartF}(\phi)$ is the creation part of the field operator $\phi$, $\text{anPartF}(\phi')$ is the annihilation part of the field operator $\phi'$, and $[\cdot, \cdot]_s^F$ denotes the super-commutator.

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 components of its field operators. For any field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ and any elements $a, b$ in the free algebra, the normal ordering operator $\mathcal{N}^f$ satisfies: $$\mathcal{N}^f(a \cdot [\text{anPartF}(\phi), \text{crPartF}(\phi')]_s^F \cdot b) = 0$$ where $\text{anPartF}(\phi)$ is the annihilation component of $\phi$, $\text{crPartF}(\phi')$ is the creation component of $\phi'$, and $[\cdot, \cdot]_s^F$ denotes the super-commutator.

Let $\mathcal{F}$ be a field specification. For any field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$, let $\text{crPartF}(\phi)$ and $\text{crPartF}(\phi')$ denote their respective creation components in the free algebra of field operators. The normal ordering operator $\mathcal{N}^f$ satisfies: $$\mathcal{N}^f(\text{crPartF}(\phi) \cdot \text{crPartF}(\phi')) = \text{crPartF}(\phi) \cdot \text{crPartF}(\phi')$$

Let $\mathcal{F}$ be a field specification and $\text{FieldOpFreeAlgebra } \mathcal{F}$ be the free algebra generated by the creation and annihilation components of the fields. Let $\mathcal{N}^f$ be the $\mathbb{C}$-linear normal ordering operator on this algebra. For any two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$, the normal ordering of the product of their annihilation components is equal to the product itself: $$\mathcal{N}^f(\text{anPartF}(\phi) \cdot \text{anPartF}(\phi')) = \text{anPartF}(\phi) \cdot \text{anPartF}(\phi')$$ where $\text{anPartF}(\phi)$ denotes the annihilation component of the field operator $\phi$.

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 operator components of the fields. Let $\mathcal{N}^f : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the linear normal ordering operator. For any field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$, the normal ordering of the product of the creation part of $\phi$ and the annihilation part of $\phi'$ is equal to the product itself: $$\mathcal{N}^f(\text{crPartF}(\phi) \cdot \text{anPartF}(\phi')) = \text{crPartF}(\phi) \cdot \text{anPartF}(\phi')$$ where $\text{crPartF}(\phi)$ and $\text{anPartF}(\phi')$ denote the creation and annihilation components of the field operators in the algebra $\mathcal{A}_{\mathcal{F}}$, respectively.

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 operator components of the fields. Let $\mathcal{N}^f : \mathcal{A}_{\mathcal{F}} \to \mathcal{A}_{\mathcal{F}}$ be the linear normal ordering operator. For any field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$, the normal ordering of the product of the annihilation part of $\phi$ and the creation part of $\phi'$ is given by: $$\mathcal{N}^f(\text{anPartF}(\phi) \cdot \text{crPartF}(\phi')) = \epsilon(\phi, \phi') \cdot (\text{crPartF}(\phi') \cdot \text{anPartF}(\phi))$$ where $\text{anPartF}(\phi)$ and $\text{crPartF}(\phi')$ denote the annihilation and creation components of the field operators in the algebra, respectively, and $\epsilon(\phi, \phi')$ is the exchange sign (phase factor) determined by the quantum statistics (bosonic or fermionic) of the fields $\phi$ and $\phi'$.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$, the normal ordering $\mathcal{N}^f$ of their product in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is given by: $$\mathcal{N}^f(\phi \cdot \phi') = \phi_{\text{cr}} \phi'_{\text{cr}} + \epsilon(\phi, \phi') \cdot (\phi'_{\text{cr}} \phi_{\text{an}}) + \phi_{\text{cr}} \phi'_{\text{an}} + \phi_{\text{an}} \phi'_{\text{an}}$$ where $\phi_{\text{cr}}$ and $\phi_{\text{an}}$ denote the creation and annihilation components of the field operator $\phi$, respectively, and $\epsilon(\phi, \phi') \in \{1, -1\}$ is the exchange sign (or phase factor) determined by the quantum statistics (bosonic or fermionic) of the fields $\phi$ and $\phi'$.

Let $\mathcal{F}$ be a field specification. Let $\phi_c, \phi_{c'} \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation operators (i.e., $\mathcal{F} \rhd^c \phi_c = \text{create}$ and $\mathcal{F} \rhd^c \phi_{c'} = \text{create}$), and let $\phi_s$ and $\phi_{s'}$ be lists of creation and annihilation operators. In the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the normal ordering $\mathcal{N}^f$ of a product containing the super-commutator $[V(\phi_c), V(\phi_{c'})]_s^F$ is given by: $$\mathcal{N}^f(V(\phi_s) \cdot [V(\phi_c), V(\phi_{c'})]_s^F \cdot V(\phi_{s'})) = \eta \cdot \left( V(\text{createFilter}(\phi_s)) \cdot [V(\phi_c), V(\phi_{c'})]_s^F \cdot V(\text{createFilter}(\phi_{s'})) \cdot V(\text{annihilateFilter}(\phi_s \mathbin{+\!+} \phi_{s'})) \right)$$ where $V(\phi_s)$ denotes the product of operators in list $\phi_s$, $[ \cdot, \cdot ]_s^F$ is the super-commutator, and $\eta = \text{normalOrderSign}(\phi_s \mathbin{+\!+} [\phi_{c'}, \phi_c] \mathbin{+\!+} \phi_{s'})$ is the phase factor associated with the permutation required to normal-order the list of operators where $\phi_c$ and $\phi_{c'}$ are adjacent and in reversed order. The functions $\text{createFilter}$ and $\text{annihilateFilter}$ extract sublists containing only creation or annihilation operators, respectively.

Let $\mathcal{F}$ be a field specification. Let $\phi_a, \phi_{a'} \in \mathcal{F}.\text{CrAnFieldOp}$ be two operators whose labels are both $\text{annihilate}$, and let $\Phi_s, \Phi_s'$ be two lists of creation and annihilation operator components. In the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the normal ordering of the product of $V(\Phi_s)$, the super-commutator of $\phi_a$ and $\phi_{a'}$, and $V(\Phi_s')$ is given by: $$\mathcal{N}^f(V(\Phi_s) \cdot [\phi_a, \phi_{a'}]_s^F \cdot V(\Phi_s')) = \eta \cdot \left( V(\text{createFilter}(\Phi_s \mathbin{+\!+} \Phi_s')) \cdot V(\text{annihilateFilter}(\Phi_s)) \cdot [\phi_a, \phi_{a'}]_s^F \cdot V(\text{annihilateFilter}(\Phi_s')) \right)$$ where $\mathcal{N}^f$ is the normal ordering operator, $V(\Phi)$ denotes the product of operators in the list $\Phi$, $[\cdot, \cdot]_s^F$ is the super-commutator, and $\eta = \text{normalOrderSign}(\Phi_s \mathbin{+\!+} [\phi_{a'}, \phi_a] \mathbin{+\!+} \Phi_s')$ is the phase factor associated with the permutation required to reach a normal-ordered arrangement.

For a given field specification $\mathcal{F}$, let $\phi_s$ and $\phi_s'$ be lists of creation and annihilation operators. Let $V(\phi_s)$ denote the product of operators in $\phi_s$, and let $\mathcal{N}^f$ be the normal ordering operator. The super-commutator of $V(\phi_s)$ and the normal-ordered product $\mathcal{N}^f(V(\phi_s'))$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is given by: $$[V(\phi_s), \mathcal{N}^f(V(\phi_s'))]_s^F = V(\phi_s) \cdot \mathcal{N}^f(V(\phi_s')) - \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot \mathcal{N}^f(V(\phi_s')) \cdot V(\phi_s)$$ where $\sigma(\phi_s)$ and $\sigma(\phi_s')$ are the collective statistics (bosonic or fermionic) of the lists $\phi_s$ and $\phi_s'$ respectively, and $\mathcal{S}(\sigma_1, \sigma_2)$ is the sign factor equal to $-1$ if both $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

For a given field specification $\mathcal{F}$, let $\phi_s$ be a list of creation and annihilation operators and let $\phi_s'$ be a list of field operators. Let $V(\phi_s)$ denote the product of operators in $\phi_s$ (via `ofCrAnListF`), and let $\Phi(\phi_s')$ denote the product of field operators in $\phi_s'$ (via `ofFieldOpListF`). Let $\mathcal{N}^f$ be the normal ordering operator. The super-commutator of $V(\phi_s)$ and the normal-ordered product $\mathcal{N}^f(\Phi(\phi_s'))$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is given by: $$[V(\phi_s), \mathcal{N}^f(\Phi(\phi_s'))]_s^F = V(\phi_s) \cdot \mathcal{N}^f(\Phi(\phi_s')) - \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot \mathcal{N}^f(\Phi(\phi_s')) \cdot V(\phi_s)$$ where $\sigma(\phi_s)$ and $\sigma(\phi_s')$ are the collective statistics (bosonic or fermionic) of the lists $\phi_s$ and $\phi_s'$ respectively, and $\mathcal{S}(\sigma_1, \sigma_2)$ is the sign factor equal to $-1$ if both $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

For a given field specification $\mathcal{F}$, let $\phi_s$ be a list of creation and annihilation operators and $\phi_s'$ be a list of field operators. Let $V(\phi_s)$ denote the product of operators in $\phi_s$ (via `ofCrAnListF`), $\Phi(\phi_s')$ denote the product of field operators in $\phi_s'$ (via `ofFieldOpListF`), and $\mathcal{N}^f$ be the normal ordering operator. In the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the product of $V(\phi_s)$ and the normal-ordered product $\mathcal{N}^f(\Phi(\phi_s'))$ satisfies the identity: $$V(\phi_s) \cdot \mathcal{N}^f(\Phi(\phi_s')) = \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot \mathcal{N}^f(\Phi(\phi_s')) \cdot V(\phi_s) + [V(\phi_s), \mathcal{N}^f(\Phi(\phi_s'))]_s^F$$ where $\sigma(\phi_s)$ and $\sigma(\phi_s')$ are the collective statistics (bosonic or fermionic) of the respective lists, $\mathcal{S}$ is the statistical sign factor ($1$ unless both arguments are fermionic), and $[ \cdot, \cdot ]_s^F$ denotes the super-commutator.

For a given field specification $\mathcal{F}$, let $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator and $\phi_s'$ be a list of field operators. Let $\mathcal{N}^f$ be the normal ordering operator and $\Phi(\phi_s')$ be the product of field operators in the list $\phi_s'$. In the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the product of $\varphi$ and the normal-ordered product $\mathcal{N}^f(\Phi(\phi_s'))$ satisfies the identity: $$\varphi \cdot \mathcal{N}^f(\Phi(\phi_s')) = \mathcal{S}(\sigma(\varphi), \sigma(\phi_s')) \cdot \mathcal{N}^f(\Phi(\phi_s')) \cdot \varphi + [\varphi, \mathcal{N}^f(\Phi(\phi_s'))]_s^F$$ where $\sigma(\varphi)$ and $\sigma(\phi_s')$ are the statistics (bosonic or fermionic) of $\varphi$ and the list $\phi_s'$ respectively, $\mathcal{S}(\sigma_1, \sigma_2)$ is the sign factor ($1$ unless both arguments are fermionic), and $[ \cdot, \cdot ]_s^F$ denotes the super-commutator.

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. Let $\text{anPartF}(\phi)$ be the annihilation part of $\phi$ and $\mathcal{N}^f$ be the normal ordering operator. In the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the product of $\text{anPartF}(\phi)$ and the normal-ordered product of the fields in $\phi_s'$ satisfies the identity: $$\text{anPartF}(\phi) \cdot \mathcal{N}^f(\Phi(\phi_s')) = \mathcal{S}(\sigma(\phi), \sigma(\phi_s')) \cdot \mathcal{N}^f(\Phi(\phi_s') \cdot \text{anPartF}(\phi)) + [\text{anPartF}(\phi), \mathcal{N}^f(\Phi(\phi_s'))]_s^F$$ where $\Phi(\phi_s')$ is the algebraic product of the operators in $\phi_s'$, $\sigma(\cdot)$ denotes the field statistic (bosonic or fermionic), $\mathcal{S}$ is the statistical sign factor ($1$ unless both arguments are fermionic), and $[ \cdot, \cdot ]_s^F$ is the super-commutator.