Physlib.QFT.PerturbationTheory.WickAlgebra.NormalOrder.Basic

Let $\mathcal{F}$ be a field specification. Let $\phi_a, \phi_{a'} \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components, and let $\Phi_s, \Phi_s'$ be lists of such components. Let $V(\Phi)$ denote the product of operators in a list $\Phi$ within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering $\mathcal{N}^f$ of the product of $V(\Phi_s)$, the super-commutator $[\phi_a, \phi_{a'}]_s^F$, and $V(\Phi_s')$ is zero: $$\iota\left( \mathcal{N}^f\left( V(\Phi_s) \cdot [\phi_a, \phi_{a'}]_s^F \cdot V(\Phi_s') \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. Let $\phi_a, \phi_{a'} \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components, $\Phi_s$ be a list of such components, and $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be an arbitrary element of the free algebra of field operators. Let $V(\Phi_s)$ denote the product of operators in the list $\Phi_s$ within the free algebra, and let $[\cdot, \cdot]_s^F$ denote the super-commutator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering $\mathcal{N}^f$ of the product $V(\Phi_s) \cdot [\phi_a, \phi_{a'}]_s^F \cdot a$ is zero: $$\iota\left( \mathcal{N}^f\left( V(\Phi_s) \cdot [\phi_a, \phi_{a'}]_s^F \cdot a \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. Let $\phi_a, \phi_{a'} \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components, and let $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be arbitrary elements of the free algebra of field operators. Let $[\cdot, \cdot]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [\phi_a, \phi_{a'}]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [\phi_a, \phi_{a'}]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. Let $\phi_a \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component, let $\Phi_s$ be a list of such components, and let $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be arbitrary elements of the free algebra of field operators. Let $V(\Phi_s)$ denote the product of operators in the list $\Phi_s$ in the free algebra, $[\cdot, \cdot]_s^F$ denote the super-commutator, and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [\phi_a, V(\Phi_s)]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [\phi_a, V(\Phi_s)]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. Let $\phi_a \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component, let $\Phi_s$ be a list of such components, and let $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be arbitrary elements of the free algebra of field operators. Let $V(\Phi_s)$ denote the product of operators in the list $\Phi_s$ in the free algebra, $[\cdot, \cdot]_s^F$ denote the super-commutator, and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [V(\Phi_s), \phi_a]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [V(\Phi_s), \phi_a]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. Let $\phi_s$ and $\phi_s'$ be lists of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$. Let $V(\phi_s)$ and $V(\phi_s')$ denote their corresponding products in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, and let $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be arbitrary elements. Let $[\cdot, \cdot]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [V(\phi_s), V(\phi_s')]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [V(\phi_s), V(\phi_s')]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. Let $\phi_s$ be a list of creation and annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$, and let $V(\phi_s)$ be its corresponding product in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. For any elements $a, b, c \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[\cdot, \cdot]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [V(\phi_s), c]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [V(\phi_s), c]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. For any elements $a, b, c, d$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[d, c]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [d, c]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [d, c]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. For any elements $b, c, d$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[d, c]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $[d, c]_s^F \cdot b$ is zero: $$\iota\left( \mathcal{N}^f\left( [d, c]_s^F \cdot b \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. For any elements $a, c, d$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[d, c]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [d, c]_s^F$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [d, c]_s^F \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. For any elements $a, b_1, b_2, c, d$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[d, c]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the product $a \cdot [d, c]_s^F \cdot b_1 \cdot b_2$ is zero: $$\iota\left( \mathcal{N}^f\left( a \cdot [d, c]_s^F \cdot b_1 \cdot b_2 \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification. For any elements $c$ and $d$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[d, c]_s^F$ denote the super-commutator and $\mathcal{N}^f$ denote the normal ordering operator. The image under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ of the normal ordering of the super-commutator $[d, c]_s^F$ is zero: $$\iota\left( \mathcal{N}^f\left( [d, c]_s^F \right) \right) = 0$$

Let $\mathcal{F}$ be a field specification, and let $\text{fieldOpIdealSet}$ be the generating set for the field operator ideal of $\mathcal{F}$. For any element $a$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, if $a$ belongs to the two-sided ideal spanned by $\text{fieldOpIdealSet}$, then the image of its normal ordering $\mathcal{N}^f(a)$ under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is zero: $$\iota(\mathcal{N}^f(a)) = 0$$

Let $\mathcal{F}$ be a field specification. For any elements $a$ and $b$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, if $a$ and $b$ are equivalent ($a \approx b$) under the setoid relation defining the Wick algebra, then the image of their normal orderings under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ are equal: $$\iota(\mathcal{N}^f(a)) = \iota(\mathcal{N}^f(b))$$

For a given field specification $\mathcal{F}$, the normal ordering operator $\mathcal{N}$ is the $\mathbb{C}$-linear map $$\mathcal{N} : \mathcal{F}.\text{WickAlgebra} \to \mathcal{F}.\text{WickAlgebra}$$ defined as the descent of the composition $\iota \circ \mathcal{N}^f$ from the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, where $\mathcal{N}^f$ is the normal ordering on the free algebra and $\iota$ is the canonical projection. This map is well-defined because $\iota(\mathcal{N}^f(a)) = \iota(\mathcal{N}^f(b))$ whenever $a$ and $b$ represent the same element in the Wick algebra. For any $a \in \mathcal{F}.\text{WickAlgebra}$, the result of this operation is denoted by $\mathcal{N}(a)$.

The notation $\mathcal{N}(a)$ represents the normal ordering operation applied to an element $a$ within the Wick algebra associated with a field specification $\mathcal{F}$.