Physlib.QFT.PerturbationTheory.WickAlgebra.NormalOrder.Lemmas

Let $\mathcal{F}$ be a field specification. Let $\text{FieldOpFreeAlgebra } \mathcal{F}$ be the free algebra generated by creation and annihilation operators, and let $\mathcal{W}$ be the corresponding Wick algebra. Let $\iota : \text{FieldOpFreeAlgebra } \mathcal{F} \to \mathcal{W}$ be the canonical map from the free algebra to the Wick algebra. For any element $a$ in the free algebra, the normal ordering operator $\mathcal{N}$ on the Wick algebra and the normal ordering operator $\mathcal{N}^f$ on the free algebra satisfy: $$\mathcal{N}(\iota(a)) = \iota(\mathcal{N}^f(a))$$

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 (elements of $\mathcal{F}.\text{CrAnFieldOp}$). In the Wick algebra associated with $\mathcal{F}$, the normal ordering operator $\mathcal{N}$ applied to the product of these operators is given by: $$\mathcal{N}(\phi_1 \phi_2 \dots \phi_n) = \eta(\phi_s) \cdot (\phi_{\sigma(1)} \phi_{\sigma(2)} \dots \phi_{\sigma(n)})$$ 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 $\mathcal{F}$, the normal ordering operator $\mathcal{N}$ applied to the identity element $1$ of the associated Wick algebra results in the identity element itself: $$\mathcal{N}(1) = 1$$

For a given field specification $\mathcal{F}$, let $\mathcal{N}$ be the normal ordering operator on the associated Wick algebra. The normal ordering of the product of an empty list of field operators (which is the identity element $1$ of the algebra) is equal to $1$. That is, $$\mathcal{N}(\text{ofFieldOpList}([])) = 1$$

For a given field specification $\mathcal{F}$, the normal ordering operator $\mathcal{N}$ applied to the product of an empty list of creation and annihilation operators in the Wick algebra is equal to the identity element $1$. That is, $$\mathcal{N}(\text{ofCrAnList}([])) = 1$$

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. Let $\text{normalOrderList}(\phi_s)$ be the list reordered such that all creation operators are positioned to the left of all annihilation operators. The product of the operators in this reordered list is equal to the phase factor $\eta(\phi_s)$ (the `normalOrderSign`) multiplied by the normal ordering operator $\mathcal{N}$ applied to the product of the original list: $$\text{ofCrAnList}(\text{normalOrderList}(\phi_s)) = \eta(\phi_s) \cdot \mathcal{N}(\phi_1 \phi_2 \dots \phi_n)$$ where $\eta(\phi_s) \in \{1, -1\}$ is the sign associated with the permutations of fermionic operators required to reach the normal-ordered arrangement.

Let $\mathcal{W}$ be the Wick algebra associated with a field specification $\mathcal{F}$, and let $\mathcal{N} : \mathcal{W} \to \mathcal{W}$ denote the normal ordering operator. For any three elements $a, b, c \in \mathcal{W}$, the normal ordering of their product $abc$ is equal to the normal ordering of the product where the middle element $b$ is already normal ordered: $$\mathcal{N}(abc) = \mathcal{N}(a \mathcal{N}(b) c)$$

Let $\mathcal{W}$ be the Wick algebra associated with a field specification $\mathcal{F}$, and let $\mathcal{N} : \mathcal{W} \to \mathcal{W}$ denote the normal ordering operator. For any two elements $a, b \in \mathcal{W}$, the normal ordering of their product $ab$ is equal to the normal ordering of the product where the left factor $a$ is already normal ordered: $$\mathcal{N}(ab) = \mathcal{N}(\mathcal{N}(a)b)$$

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be the associated Wick algebra. Let $\mathcal{N} : \mathcal{W} \to \mathcal{W}$ denote the normal ordering operator. For any two elements $a, b \in \mathcal{W}$, the normal ordering of their product is equal to the normal ordering of the product where the right-hand factor $b$ is already normal ordered: $$\mathcal{N}(ab) = \mathcal{N}(a \mathcal{N}(b))$$

Let $\mathcal{W}$ be the Wick algebra associated with a field specification $\mathcal{F}$, and let $\mathcal{N} : \mathcal{W} \to \mathcal{W}$ denote the normal ordering operator. For any element $a \in \mathcal{W}$, applying the normal ordering operator twice is equivalent to applying it once: $$\mathcal{N}(\mathcal{N}(a)) = \mathcal{N}(a)$$

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be the associated Wick algebra. Let $\mathcal{N} : \mathcal{W} \to \mathcal{W}$ denote the normal ordering operator. For any element $a \in \mathcal{W}$ and any field operator $\phi \in \mathcal{F}.\text{FieldOp}$, the normal ordering of the product of $a$ and the annihilation part of $\phi$ is given by: $$\mathcal{N}(a \cdot \text{anPart}(\phi)) = \mathcal{N}(a) \cdot \text{anPart}(\phi)$$ where $\text{anPart}(\phi)$ denotes the annihilation component of the field operator $\phi$.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be the associated Wick algebra. For any field operator $\phi$ and any element $a \in \mathcal{W}$, the normal ordering $\mathcal{N}$ of the product of the creation part of $\phi$ with $a$ satisfies: $$\mathcal{N}(\text{crPart}(\phi) \cdot a) = \text{crPart}(\phi) \cdot \mathcal{N}(a)$$ where $\text{crPart}(\phi)$ denotes the component of the field operator $\phi$ consisting of creation operators.

Let $\mathcal{F}$ be a field specification. For any two elements $a$ and $b$ in the Wick algebra associated with $\mathcal{F}$, the normal ordering of their super-commutator $[a, b]_s$ vanishes, that is: $$\mathcal{N}([a, b]_s) = 0$$ where $\mathcal{N}$ denotes the normal ordering operator.

Let $\mathcal{W}$ be the Wick algebra associated with a field specification $\mathcal{F}$. For any elements $a, b, c \in \mathcal{W}$, the normal ordering of the product of the super-commutator $[a, b]_s$ and the element $c$ is zero: $$\mathcal{N}([a, b]_s \cdot c) = 0$$ where $[a, b]_s$ denotes the super-commutator and $\mathcal{N}$ denotes the normal ordering operator on the Wick algebra.

Let $\mathcal{W}$ be the Wick algebra associated with a field specification $\mathcal{F}$. For any three elements $a, b, c \in \mathcal{W}$, the normal ordering $\mathcal{N}$ of the product of $c$ and the super-commutator $[a, b]_s$ is zero: $$\mathcal{N}(c \cdot [a, b]_s) = 0$$ Here, $[a, b]_s$ denotes the super-commutator in the Wick algebra, and $\mathcal{N}$ is the normal ordering operator.

Let $\mathcal{F}$ be a field specification. For any elements $a, b, c, d$ in the Wick algebra of $\mathcal{F}$, the normal ordering $\mathcal{N}$ of the product $a [c, d]_s b$ is zero, where $[c, d]_s$ denotes the super-commutator of $c$ and $d$: $$\mathcal{N}(a [c, d]_s b) = 0$$

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$, the normal ordering $\mathcal{N}$ of their product in the Wick algebra satisfies the following symmetry relation: $$\mathcal{N}(\phi \phi') = \mathcal{S}(s(\phi), s(\phi')) \cdot \mathcal{N}(\phi' \phi)$$ where $s(\phi)$ and $s(\phi')$ are the field statistics (bosonic or fermionic) of the operators, and $\mathcal{S}$ is the exchange sign, which is $-1$ if both operators are fermionic and $1$ otherwise.

Let $\mathcal{F}$ be a field specification. For any creation or annihilation operator $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ and any finite list of creation or annihilation operators $\Phi = [\varphi_1, \dots, \varphi_n]$, the normal ordering $\mathcal{N}$ of their product satisfies: $$\mathcal{N}(\varphi \cdot \prod_{i=1}^n \varphi_i) = \mathcal{S}(\text{stat}(\varphi), \text{stat}(\Phi)) \cdot \mathcal{N}((\prod_{i=1}^n \varphi_i) \cdot \varphi)$$ where $\text{stat}(\varphi)$ is the field statistic of $\varphi$, $\text{stat}(\Phi)$ is the collective statistic of the list $\Phi$ (the product of the statistics of its elements in $\mathbb{Z}_2$), and $\mathcal{S}$ is the exchange sign.

Let $\mathcal{F}$ be a field specification. For any creation or annihilation component of a field operator $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ and any list of field operators $\phi' = [\phi_1, \dots, \phi_n]$, the normal ordering of their product in the Wick algebra satisfies: $$\mathcal{N}(\varphi \cdot \phi_1 \dots \phi_n) = \mathcal{S}(\sigma(\varphi), \sigma(\phi')) \mathcal{N}(\phi_1 \dots \phi_n \cdot \varphi)$$ where $\mathcal{N}$ is the normal ordering operator, $\mathcal{S}$ is the exchange sign determined by the statistics of the operators, $\sigma(\varphi)$ is the statistic (bosonic or fermionic) of $\varphi$, and $\sigma(\phi')$ is the collective statistic of the list of field operators $\phi'$.

Let $\mathcal{F}$ be a field specification. For any field operator $\phi \in \mathcal{F}.\text{FieldOp}$ and any finite list of field operators $\phi' = [\phi_1, \dots, \phi_n]$, the normal ordering $\mathcal{N}$ of the product of the annihilation part of $\phi$ (denoted $\text{annihilate}(\phi)$) and the product of the operators in $\phi'$ satisfies: $$\mathcal{N}(\text{annihilate}(\phi) \cdot \phi_1 \dots \phi_n) = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \mathcal{N}(\phi_1 \dots \phi_n \cdot \text{annihilate}(\phi))$$ where $\sigma(\phi)$ is the field statistic of $\phi$, $\sigma(\phi')$ is the collective statistic of the list $\phi'$, and $\mathcal{S}$ is the exchange sign determined by these statistics.

Let $\mathcal{F}$ be a field specification. For any field operator $\phi \in \mathcal{F}.\text{FieldOp}$ and any finite list of field operators $\phi' = [\phi_1, \dots, \phi_n]$, the normal ordering $\mathcal{N}$ of the product of the operators in $\phi'$ and the annihilation part of $\phi$ (denoted $\text{annihilate}(\phi)$) satisfies: $$\mathcal{N}(\phi_1 \dots \phi_n \cdot \text{annihilate}(\phi)) = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \mathcal{N}(\text{annihilate}(\phi) \cdot \phi_1 \dots \phi_n)$$ where $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$, $\sigma(\phi')$ is the collective statistic of the list $\phi'$, and $\mathcal{S}$ is the exchange sign determined by these statistics.

Let $\mathcal{F}$ be a field specification. For any field operator $\phi \in \mathcal{F}.\text{FieldOp}$ and any finite list of field operators $\phi_s = [\phi_1, \dots, \phi_n]$, the product of the normal-ordered operators in $\phi_s$ and the annihilation part of $\phi$ (denoted $\text{anPart}(\phi)$) satisfies: $$\mathcal{N}(\phi_1 \dots \phi_n) \cdot \text{anPart}(\phi) = \mathcal{S}(\sigma(\phi), \sigma(\phi_s)) \cdot \mathcal{N}(\text{anPart}(\phi) \cdot \phi_1 \dots \phi_n)$$ where $\mathcal{N}$ denotes the normal ordering operator, $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$, $\sigma(\phi_s)$ is the collective statistic of the list $\phi_s$, and $\mathcal{S}$ is the exchange sign determined by these statistics.

Let $\mathcal{F}$ be a field specification. For any field operator $\phi \in \mathcal{F}.\text{FieldOp}$ and any finite list of field operators $\phi'_s = [\phi_1, \dots, \phi_n]$, the following identity holds in the Wick algebra: $$\text{anPart}(\phi) \cdot \mathcal{N}(\phi_1 \dots \phi_n) = \mathcal{S}(\sigma(\phi), \sigma(\phi'_s)) \cdot \mathcal{N}(\phi_1 \dots \phi_n \cdot \text{anPart}(\phi)) + [\text{anPart}(\phi), \mathcal{N}(\phi_1 \dots \phi_n)]_s$$ where: - $\text{anPart}(\phi)$ is the annihilation part of the field operator $\phi$. - $\mathcal{N}$ denotes the normal ordering operator on the Wick algebra. - $\phi_1 \dots \phi_n$ is the algebraic product of the operators in the list $\phi'_s$. - $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$, and $\sigma(\phi'_s)$ is the collective field statistic of the list $\phi'_s$. - $\mathcal{S}$ is the exchange sign factor, defined as $-1$ if both arguments are fermionic and $1$ otherwise. - $[\cdot, \cdot]_s$ denotes the super-commutator.

For a given field specification $\mathcal{F}$, let $\phi$ be a creation or annihilation operator and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of creation or annihilation operators. The supercommutator of $\phi$ with the normal-ordered product of the operators in $\phi_s$ is given by the sum: $$[\phi, \mathcal{N}(\phi_0 \phi_1 \dots \phi_{n-1})]_s = \sum_{i=0}^{n-1} \mathcal{S}(\phi, \{\phi_0, \dots, \phi_{i-1}\}) \cdot [\phi, \phi_i]_s \cdot \mathcal{N}(\phi_0 \dots \phi_{i-1} \phi_{i+1} \dots \phi_{n-1})$$ where $\mathcal{N}$ denotes the normal ordering operator, $[ \cdot, \cdot ]_s$ denotes the supercommutator, and $\mathcal{S}(\phi, \{\phi_0, \dots, \phi_{i-1}\})$ is the phase factor $(\pm 1)$ obtained from the exchange statistics of $\phi$ and the preceding operators in the list.

For a given field specification $\mathcal{F}$, let $\phi$ be a creation or annihilation operator and $\Phi_s = [\Phi_0, \Phi_1, \dots, \Phi_{n-1}]$ be a list of field operators. The supercommutator of $\phi$ with the normal-ordered product of the field operators in $\Phi_s$ is given by the sum: $$[\phi, \mathcal{N}(\Phi_0 \Phi_1 \dots \Phi_{n-1})]_s = \sum_{i=0}^{n-1} \mathcal{S}(\phi, \{\Phi_0, \dots, \Phi_{i-1}\}) \cdot [\phi, \Phi_i]_s \cdot \mathcal{N}(\Phi_0 \dots \Phi_{i-1} \Phi_{i+1} \dots \Phi_{n-1})$$ where $\mathcal{N}$ denotes the normal ordering operator, $[\cdot, \cdot]_s$ denotes the supercommutator, and $\mathcal{S}(\phi, \{\Phi_0, \dots, \Phi_{i-1}\})$ is the phase factor $(\pm 1)$ obtained from the exchange statistics of $\phi$ and the preceding field operators in the list.

For a given field specification $\mathcal{F}$, let $\phi$ be a field operator and $\Phi_s = [\Phi_0, \Phi_1, \dots, \Phi_{n-1}]$ be a list of field operators. The supercommutator of the annihilation part of $\phi$, denoted $\text{anPart } \phi$, with the normal-ordered product of the list $\Phi_s$ is given by: $$[\text{anPart } \phi, \mathcal{N}(\Phi_0 \Phi_1 \dots \Phi_{n-1})]_s = \sum_{i=0}^{n-1} \mathcal{S}(\phi, \{\Phi_0, \dots, \Phi_{i-1}\}) \cdot [\text{anPart } \phi, \Phi_i]_s \cdot \mathcal{N}(\Phi_0 \dots \Phi_{i-1} \Phi_{i+1} \dots \Phi_{n-1})$$ where: - $\text{anPart } \phi$ is the annihilation component of the field operator $\phi$. - $\mathcal{N}$ denotes the normal ordering operator. - $[\cdot, \cdot]_s$ denotes the supercommutator. - $\mathcal{S}(\phi, \{\Phi_0, \dots, \Phi_{i-1}\})$ is the phase factor $(\pm 1)$ determined by the exchange statistics between $\phi$ and the sub-list of preceding field operators. - Each $\Phi_i$ in the supercommutator on the right-hand side represents the $i$-th field operator in the algebra.

Let $\mathcal{F}$ be a field specification. For any field operator $\phi \in \text{FieldOp}(\mathcal{F})$ and any finite list of field operators $\Phi = [\phi_0, \dots, \phi_{n-1}]$, the product of the annihilation part of $\phi$ and the normal-ordered product of the list $\Phi$ satisfies the identity: $$\text{anPart}(\phi) \cdot \mathcal{N}(\phi_0 \dots \phi_{n-1}) = \mathcal{N}(\text{anPart}(\phi) \cdot \phi_0 \dots \phi_{n-1}) + [\text{anPart}(\phi), \mathcal{N}(\phi_0 \dots \phi_{n-1})]_s$$ where: - $\text{anPart}(\phi)$ is the annihilation component of the field operator $\phi$. - $\mathcal{N}$ denotes the normal ordering operator. - $[\cdot, \cdot]_s$ denotes the super-commutator.

Let $\mathcal{F}$ be a field specification. For any field operator $\phi \in \text{FieldOp}(\mathcal{F})$ and any finite list of field operators $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$, the product of $\phi$ and the normal-ordered product of the list $\Phi$ satisfies the identity: $$\phi \cdot \mathcal{N}(\phi_0 \phi_1 \dots \phi_{n-1}) = \mathcal{N}(\phi \phi_0 \phi_1 \dots \phi_{n-1}) + [\text{anPart}(\phi), \mathcal{N}(\phi_0 \phi_1 \dots \phi_{n-1})]_s$$ where: - $\phi$ represents the field operator as an element of the Wick algebra. - $\mathcal{N}$ denotes the normal ordering operator. - $\text{anPart}(\phi)$ is the annihilation component of the field operator $\phi$. - $[\cdot, \cdot]_s$ denotes the super-commutator.

The function `contractStateAtIndex` computes the coefficient or operator contribution arising from contracting a field operator $\phi$ with an element in a list of field operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{k-1}]$ at a specified optional index $n$. - If $n = \text{none}$, the function returns $1$, representing the term where no contraction occurs. - If $n$ is an index $i \in \{0, \dots, k-1\}$, the function returns the supercommutator of the annihilation part of $\phi$ (denoted $\phi^{(-)}$) and the $i$-th operator $\phi_i$, weighted by an exchange sign: $$\mathcal{S}(\text{stat}(\phi), \text{stat}([\phi_0, \dots, \phi_{i-1}])) \cdot [\phi^{(-)}, \phi_i]_s$$ where $\text{stat}(\phi)$ is the field statistic (bosonic or fermionic) of the operator $\phi$, and $\mathcal{S}$ is the exchange sign $\pm 1$ resulting from permuting $\phi$ past the first $i$ operators in the list.

For a field specification $\mathcal{F}$, let $\phi$ be a field operator and $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. The product of $\phi$ and the normal-ordered product of the list $\Phi$ satisfies the following identity in the Wick algebra: $$\phi \cdot \mathcal{N}(\phi_0 \phi_1 \dots \phi_{n-1}) = \mathcal{N}(\phi \phi_0 \phi_1 \dots \phi_{n-1}) + \sum_{i=0}^{n-1} \mathcal{S}(\phi, \{\phi_0, \dots, \phi_{i-1}\}) \cdot [\text{anPart}(\phi), \phi_i]_s \cdot \mathcal{N}(\phi_0 \dots \hat{\phi}_i \dots \phi_{n-1})$$ where: - $\mathcal{N}$ is the normal ordering operator. - $\text{anPart}(\phi)$ is the annihilation component of the field operator $\phi$. - $[\cdot, \cdot]_s$ denotes the supercommutator. - $\mathcal{S}(\phi, \{\phi_0, \dots, \phi_{i-1}\})$ is the exchange sign ($\pm 1$) determined by the statistics (bosonic or fermionic) of $\phi$ and the operators preceding $\phi_i$ in the list. - The notation $\hat{\phi}_i$ indicates that the $i$-th field operator is omitted from the product. In the formal sum over $n \in \text{Option}(\text{Fin } n)$, the case $n = \text{none}$ corresponds to the first term $\mathcal{N}(\phi \phi_0 \dots \phi_{n-1})$, while the cases $n = i$ correspond to the contractions in the summation.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be its corresponding Wick algebra. For any field operator $\phi$, a list of field operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$, and an insertion index $k \in \{0, \dots, n\}$, the normal ordering of the product $\phi \cdot \phi_0 \cdot \dots \cdot \phi_{n-1}$ is equal to the normal ordering of the product where $\phi$ is inserted at position $k$, multiplied by an exchange sign: $$\mathcal{N}(\phi \cdot \phi_0 \cdot \dots \cdot \phi_{n-1}) = \mathcal{S}(\text{stat}(\phi), \text{stat}([\phi_0, \dots, \phi_{k-1}])) \cdot \mathcal{N}(\phi_0 \cdot \dots \cdot \phi_{k-1} \cdot \phi \cdot \phi_k \cdot \dots \cdot \phi_{n-1})$$ where: - $\mathcal{N}$ denotes the normal ordering operator. - $\text{stat}(\phi)$ is the field statistic (bosonic or fermionic) of operator $\phi$. - $\text{stat}([\phi_0, \dots, \phi_{k-1}])$ is the collective statistic of the first $k$ operators in the list. - $\mathcal{S}(a, b)$ is the exchange sign, returning $-1$ if both statistics $a$ and $b$ are fermionic, and $1$ otherwise.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be its corresponding Wick algebra. For any field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$, let $\text{crPart}(\phi)$ and $\text{crPart}(\phi')$ denote their respective creation components in the Wick algebra. The normal ordering operator $\mathcal{N}$ on the Wick algebra satisfies: $$\mathcal{N}(\text{crPart}(\phi) \cdot \text{crPart}(\phi')) = \text{crPart}(\phi) \cdot \text{crPart}(\phi')$$

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be its corresponding Wick algebra. For any field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$, let $\text{anPart}(\phi)$ and $\text{anPart}(\phi')$ denote their respective annihilation components in the Wick algebra. The normal ordering operator $\mathcal{N}$ on the Wick algebra satisfies: $$\mathcal{N}(\text{anPart}(\phi) \cdot \text{anPart}(\phi')) = \text{anPart}(\phi) \cdot \text{anPart}(\phi')$$

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be its corresponding Wick algebra. For any field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$, let $\text{crPart}(\phi)$ and $\text{anPart}(\phi')$ denote their respective creation and annihilation components in the Wick algebra. The normal ordering operator $\mathcal{N}$ on the Wick algebra satisfies: $$\mathcal{N}(\text{crPart}(\phi) \cdot \text{anPart}(\phi')) = \text{crPart}(\phi) \cdot \text{anPart}(\phi')$$

Let $\mathcal{F}$ be a field specification and $\mathcal{W}$ be its corresponding Wick algebra. For any field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$, let $\text{anPart}(\phi)$ and $\text{crPart}(\phi')$ denote their respective annihilation and creation components in the Wick algebra. The normal ordering operator $\mathcal{N}$ on the Wick algebra satisfies: $$\mathcal{N}(\text{anPart}(\phi) \cdot \text{crPart}(\phi')) = \mathcal{S}(\phi, \phi') \cdot \text{crPart}(\phi') \cdot \text{anPart}(\phi)$$ where $\mathcal{S}(\phi, \phi')$ is the exchange sign (phase factor) determined by the quantum statistics (bosonic or fermionic) of the field operators $\phi$ and $\phi'$.

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