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theorem

$\iota(b) = 0 \implies \iota([a, b]_s^F) = 0$

#ι_superCommuteF_eq_zero_of_ι_right_zero

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 $\mathcal{F}.\text{CrAnFieldOp}$ . Let $\iota: \mathcal{A}_{\mathcal{F}} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical $\mathbb{C}$ -algebra homomorphism. For any elements $a, b \in \mathcal{A}_{\mathcal{F}}$ , if the image of $b$ in the Wick algebra is zero ( $\iota(b) = 0$ ), then the image of the super-commutator $[a, b]_s^F$ in the Wick algebra is also zero: $\iota(b) = 0 \implies \iota([a, b]_s^F) = 0$

theorem

$\iota(a) = 0 \implies \iota([a, b]_s^F) = 0$

#ι_superCommuteF_eq_zero_of_ι_left_zero

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 $\iota : \mathcal{A}_{\mathcal{F}} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical $\mathbb{C}$ -algebra homomorphism. For any elements $a, b \in \mathcal{A}_{\mathcal{F}}$ , if $\iota(a) = 0$ , then the image of the super-commutator $[a, b]_s^F$ under $\iota$ is also zero: $\iota(a) = 0 \implies \iota([a, b]_s^F) = 0$

theorem

$b \in \text{span}(\text{fieldOpIdealSet}) \implies \iota([a, b]_s^F) = 0$

#ι_superCommuteF_right_zero_of_mem_ideal

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 $\iota : \mathcal{A}_{\mathcal{F}} \to \text{WickAlgebra}(\mathcal{F})$ be the canonical $\mathbb{C}$ -algebra homomorphism, and let $I$ be the two-sided ideal generated by the set of field operator relations $\mathcal{F}.\text{fieldOpIdealSet}$ . For any elements $a, b \in \mathcal{A}_{\mathcal{F}}$ , if $b$ belongs to the ideal $I$ , then the image $\iota([a, b]_s^F)$ of the super-commutator in the free algebra is zero.

theorem

$b_1 \approx b_2 \implies \iota([a, b_1]_s^F) = \iota([a, b_2]_s^F)$

#ι_superCommuteF_eq_of_equiv_right

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. Let $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical projection onto the Wick algebra. For any elements $a, b_1, b_2$ in the free algebra, if $b_1 \approx b_2$ (meaning $b_1 - b_2$ belongs to the ideal defining the Wick algebra), then the images of their super-commutators under $\iota$ are equal: $\iota([a, b_1]_s^F) = \iota([a, b_2]_s^F)$ where $[ \cdot, \cdot ]_s^F$ denotes the super-commutator in the free algebra.

definition

Right super-commutator map $[a, \cdot]_s$ on the Wick algebra

#superCommuteRight

For a given field specification $\mathcal{F}$ and an element $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ , the map $\text{superCommuteRight}_a$ is a $\mathbb{C}$ -linear map from the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ to itself. It is defined by lifting the super-commutator in the free algebra to the quotient space. Specifically, for any element $b$ in the free algebra, the map acts on its equivalence class $\llbracket b \rrbracket$ in the Wick algebra as: $\text{superCommuteRight}_a(\llbracket b \rrbracket) = \iota([a, b]_s^F)$ where $[a, b]_s^F$ is the super-commutator in the free algebra and $\iota$ is the canonical projection onto the Wick algebra.

theorem

$\text{superCommuteRight}_a(\iota(b)) = \iota([a, b]_s^F)$

#superCommuteRight_apply_ι

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. Let $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ be the canonical projection onto the Wick algebra. For any elements $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ , the right super-commutator map $\text{superCommuteRight}_a$ applied to the image $\iota(b)$ is equal to the image of the super-commutator in the free algebra, specifically $\text{superCommuteRight}_a(\iota(b)) = \iota([a, b]_s^F)$ .

theorem

$\text{superCommuteRight}_a(\llbracket b \rrbracket) = \iota([a, b]_s^F)$

#superCommuteRight_apply_quot

For a given field specification $\mathcal{F}$ and any two elements $a, b$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , the right super-commutator map $\text{superCommuteRight}_a$ applied to the equivalence class $\llbracket b \rrbracket$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is equal to the canonical projection $\iota$ of the super-commutator $[a, b]_s^F$ calculated in the free algebra.

theorem

$a_1 \approx a_2 \implies \text{superCommuteRight}_{a_1} = \text{superCommuteRight}_{a_2}$

#superCommuteRight_eq_of_equiv

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. For any two elements $a_1, a_2 \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ , if $a_1$ is equivalent to $a_2$ ( $a_1 \approx a_2$ ) in the sense that their difference belongs to the field operator ideal, then the right super-commutator maps associated with $a_1$ and $a_2$ are equal: $\text{superCommuteRight}_{a_1} = \text{superCommuteRight}_{a_2}$ .

definition

Super-commutator $[a, b]_s$ on the Wick algebra

#superCommute

For a given field specification $\mathcal{F}$ , the super-commutator $\text{superCommute}$ is a $\mathbb{C}$ -bilinear map on the Wick algebra, denoted as $[\cdot, \cdot]_s : \text{WickAlgebra } \mathcal{F} \to_\ell[\mathbb{C}] \text{WickAlgebra } \mathcal{F} \to_\ell[\mathbb{C}] \text{WickAlgebra } \mathcal{F}$ defined by descending the super-commutator from the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to the quotient. For any two elements $a, b \in \text{WickAlgebra } \mathcal{F}$ , the map satisfies the relation $[a, b]_s = a * b - \mathcal{S}(a, b) \bullet b * a$ where $\mathcal{S}(a, b)$ is the statistical factor determined by the grades of $a$ and $b$ . In particular, for lists of operators $\phi_s$ and $\phi_s'$ , the super-commutator captures the difference between their product and their graded-permuted product.

definition

Super-commutator notation $[a, b]_s$

#term[_,_]ₛ

This definition introduces the notation $[a, b]_s$ to represent the super-commutator of two elements $a$ and $b$ in the Wick algebra $\text{WickAlgebra}(\mathcal{F})$ . It is defined as the application of the bilinear map $\text{superCommute}$ to $a$ and $b$ .

theorem

$[\iota(a), \iota(b)]_s = \iota([a, b]_s^F)$

#superCommute_eq_ι_superCommuteF

For a given field specification $\mathcal{F}$ and any two elements $a, b$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , the super-commutator of their images under the canonical projection $\iota: \mathcal{F}.\text{FieldOpFreeAlgebra} \to \mathcal{F}.\text{WickAlgebra}$ is equal to the image of their super-commutator in the free algebra. That is, $[\iota(a), \iota(b)]_s = \iota([a, b]_s^F)$ where $[\cdot, \cdot]_s$ denotes the super-commutator on the Wick algebra and $[\cdot, \cdot]_s^F$ denotes the super-commutator on the free algebra.

theorem

The super-commutator of two creation operators in the Wick algebra is 0

#superCommute_create_create

For a given field specification $\mathcal{F}$ , let $\varphi$ and $\varphi'$ be two creation or annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$ . If both $\varphi$ and $\varphi'$ are creation operators (i.e., their creation/annihilation labels are $\text{create}$ ), then their super-commutated product in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is zero: $[\text{ofCrAnOp } \varphi, \text{ofCrAnOp } \varphi']_s = 0$

theorem

The super-commutator of two annihilation operators in the Wick algebra is zero.

#superCommute_annihilate_annihilate

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\varphi, \varphi' \in \mathcal{F}.\text{CrAnFieldOp}$ such that both are classified as annihilation operators (i.e., $\text{crAnFieldOpToCreateAnnihilate}(\varphi) = \text{annihilate}$ and $\text{crAnFieldOpToCreateAnnihilate}(\varphi') = \text{annihilate}$ ), their super-commutator $[\varphi, \varphi']_s$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is zero.

theorem

The super-commutator of operators with different statistics vanishes in the Wick algebra

#superCommute_diff_statistic

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \phi' \in \mathcal{F}.\text{CrAnFieldOp}$ , if their field statistics are different (i.e., one is bosonic and the other is fermionic), then their super-commutated product in the Wick algebra is zero: $[\text{ofCrAnOp } \phi, \text{ofCrAnOp } \phi']_s = 0$ where $[\cdot, \cdot]_s$ denotes the super-commutator on $\mathcal{F}.\text{WickAlgebra}$ .

theorem

The super-commutator of a creation/annihilation operator and a field operator with different statistics is zero

#superCommute_ofCrAnOp_ofFieldOp_diff_stat_zero

Let $\mathcal{F}$ be a field specification. For any creation or annihilation operator component $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$ and any field operator $\psi \in \mathcal{F}.\text{FieldOp}$ , if their field statistics are different (i.e., one is bosonic and the other is fermionic), then their super-commutator in the Wick algebra is zero: $[\text{ofCrAnOp}(\varphi), \text{ofFieldOp}(\psi)]_s = 0$ where $[\cdot, \cdot]_s$ denotes the super-commutator on the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ .

theorem

The super-commutator of $\text{anPart}(\phi)$ and $\text{ofFieldOp}(\psi)$ is $0$ if $\text{stat}(\phi) \neq \text{stat}(\psi)$

#superCommute_anPart_ofFieldOpF_diff_grade_zero

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$ , if their field statistics are different (i.e., one is bosonic and the other is fermionic), then the super-commutator of the annihilation part of $\phi$ and the field operator $\psi$ in the Wick algebra is zero: $[\text{anPart}(\phi), \text{ofFieldOp}(\psi)]_s = 0$ where $[\cdot, \cdot]_s$ denotes the super-commutator in $\mathcal{F}.\text{WickAlgebra}$ .

theorem

$[\phi, \phi']_s \in \text{center}(\text{WickAlgebra } \mathcal{F})$

#superCommute_ofCrAnOp_ofCrAnOp_mem_center

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \phi' \in \mathcal{F}.\text{CrAnFieldOp}$ , the super-commutator $[\phi, \phi']_s$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ lies in the center of the algebra. That is, for any element $a \in \mathcal{F}.\text{WickAlgebra}$ , it holds that $a \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot a$ .

theorem

$a \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot a$ for $\phi, \phi' \in \text{CrAnFieldOp}$

#superCommute_ofCrAnOp_ofCrAnOp_commute

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \phi' \in \mathcal{F}.\text{CrAnFieldOp}$ , their super-commutator $[\phi, \phi']_s$ commutes with any element $a$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . That is, $a \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot a$ .

theorem

$[\phi, \phi']_s$ lies in the center of the Wick algebra for $\phi \in \text{CrAnFieldOp}$ and $\phi' \in \text{FieldOp}$

#superCommute_ofCrAnOp_ofFieldOp_mem_center

Let $\mathcal{F}$ be a field specification. For any creation or annihilation operator component $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ and any field operator $\phi' \in \mathcal{F}.\text{FieldOp}$ , the super-commutator $[\text{ofCrAnOp}(\phi), \text{ofFieldOp}(\phi')]_s$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ belongs to the center of the algebra. That is, it commutes with all elements of the Wick algebra.

theorem

$a \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot a$ for $\phi \in \text{CrAnFieldOp}$ and $\phi' \in \text{FieldOp}$

#superCommute_ofCrAnOp_ofFieldOp_commute

Let $\mathcal{F}$ be a field specification. For any creation or annihilation operator component $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ , any field operator $\phi' \in \mathcal{F}.\text{FieldOp}$ , and any element $a$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ , the super-commutator $[\text{ofCrAnOp}(\phi), \text{ofFieldOp}(\phi')]_s$ commutes with $a$ , such that: $a \cdot [\text{ofCrAnOp}(\phi), \text{ofFieldOp}(\phi')]_s = [\text{ofCrAnOp}(\phi), \text{ofFieldOp}(\phi')]_s \cdot a$

theorem

$[\text{anPart}(\phi), \text{ofFieldOp}(\phi')]_s$ lies in the center of the Wick algebra

#superCommute_anPart_ofFieldOp_mem_center

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the super-commutator of the annihilation part of the first and the image of the second in the Wick algebra, denoted $[\text{anPart}(\phi), \text{ofFieldOp}(\phi')]_s$ , belongs to the center of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . That is, it commutes with all elements of the algebra.

theorem

The super-commutator of two operator products in the Wick algebra equals the difference of their concatenated products weighted by the exchange sign.

#superCommute_ofCrAnList_ofCrAnList

For a given field specification $\mathcal{F}$ , let $\phi_s$ and $\phi_s'$ be lists of creation and annihilation operators. The super-commutator of their corresponding products in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is given by: $[\text{ofCrAnList}(\phi_s), \text{ofCrAnList}(\phi_s')]_s = \text{ofCrAnList}(\phi_s \mathbin{+\mkern-10mu+} \phi_s') - \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot \text{ofCrAnList}(\phi_s' \mathbin{+\mkern-10mu+} \phi_s)$ where $\phi_s \mathbin{+\mkern-10mu+} \phi_s'$ denotes the concatenation of the operator lists, and $\mathcal{S}(\sigma(\phi_s), \sigma(\phi_s'))$ is the exchange sign factor determined by the collective statistics $\sigma(\phi_s)$ and $\sigma(\phi_s')$ of the two lists.

theorem

Super-commutator of creation/annihilation operators $\phi, \phi'$ in the Wick algebra

#superCommute_ofCrAnOp_ofCrAnOp

For a given field specification $\mathcal{F}$ , let $\phi$ and $\phi'$ be two creation or annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$ . Let $a = \text{ofCrAnOp}(\phi)$ and $a' = \text{ofCrAnOp}(\phi')$ denote their corresponding elements in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . Their super-commutator in the algebra is given by: $[a, a']_s = a \cdot a' - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot a' \cdot a$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of the operator component $\phi$ , and $\mathcal{S}$ is the exchange sign factor, which equals $-1$ if both $\phi$ and $\phi'$ are fermionic and $1$ otherwise.

theorem

$[V(\phi_{\text{cas}}), V(\phi_s)]_s = V(\phi_{\text{cas}}) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi_{\text{cas}})$ in the Wick algebra

#superCommute_ofCrAnList_ofFieldOpList

In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ associated with a field specification $\mathcal{F}$ , let $\phi_{\text{cas}}$ be a list of creation and annihilation operator components and $\phi_s$ be a list of field operators. The super-commutator of their corresponding products in the algebra is given by: $[\text{ofCrAnList}(\phi_{\text{cas}}), \text{ofFieldOpList}(\phi_s)]_s = \text{ofCrAnList}(\phi_{\text{cas}}) \cdot \text{ofFieldOpList}(\phi_s) - \mathcal{S}(\sigma(\phi_{\text{cas}}), \sigma(\phi_s)) \cdot \text{ofFieldOpList}(\phi_s) \cdot \text{ofCrAnList}(\phi_{\text{cas}})$ where $\text{ofCrAnList}(\phi_{\text{cas}})$ is the product of the component operators, $\text{ofFieldOpList}(\phi_s)$ is the product of the field operators, $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of each list, and $\mathcal{S}$ is the exchange sign factor.

theorem

$[V(\varphi_s), V(\varphi_{s'})]_s = V(\varphi_s) V(\varphi_{s'}) - \mathcal{S} \cdot V(\varphi_{s'}) V(\varphi_s)$ for products of field operators in the Wick algebra

#superCommute_ofFieldOpList_ofFieldOpList

For a given field specification $\mathcal{F}$ , let $\varphi_s$ and $\varphi_{s'}$ be two lists of field operators. Let $V(\varphi_s)$ and $V(\varphi_{s'})$ denote the corresponding products of these field operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator of these two elements is given by: $[V(\varphi_s), V(\varphi_{s'})]_s = V(\varphi_s) \cdot V(\varphi_{s'}) - \mathcal{S}(\sigma(\varphi_s), \sigma(\varphi_{s'})) \cdot V(\varphi_{s'}) \cdot V(\varphi_s)$ where $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of a list, and the exchange sign $\mathcal{S}(\sigma_1, \sigma_2)$ is $-1$ if both collective statistics are fermionic, and $1$ otherwise.

theorem

$[V(\phi), V(\phi_s)]_s = V(\phi) V(\phi_s) - \mathcal{S} \cdot V(\phi_s) V(\phi)$ in the Wick algebra

#superCommute_ofFieldOp_ofFieldOpList

For a given field specification $\mathcal{F}$ , let $\phi$ be a field operator and $\phi_s$ be a list of field operators. Let $V(\phi)$ denote the image of $\phi$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ , and let $V(\phi_s)$ denote the product of the field operators in the list $\phi_s$ within that algebra. The super-commutator of $V(\phi)$ and $V(\phi_s)$ is given by: $[V(\phi), V(\phi_s)]_s = V(\phi) \cdot V(\phi_s) - \mathcal{S}(\sigma(\phi), \sigma(\phi_s)) \cdot V(\phi_s) \cdot V(\phi)$ where $\sigma(\phi)$ is the field statistic of $\phi$ , $\sigma(\phi_s)$ is the collective field statistic of the list $\phi_s$ , and the exchange sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

Super-commutator of a product of field operators and a single field operator in the Wick algebra

#superCommute_ofFieldOpList_ofFieldOp

For a given field specification $\mathcal{F}$ , let $\varphi_s$ be a list of field operators and $\phi$ be a single field operator. Let $V(\varphi_s)$ be the product of the field operators in the list within the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ , and let $V(\phi)$ be the representation of $\phi$ in the same algebra. The super-commutator $[V(\varphi_s), V(\phi)]_s$ is given by: $[V(\varphi_s), V(\phi)]_s = V(\varphi_s) \cdot V(\phi) - \mathcal{S}(\sigma(\varphi_s), \sigma(\phi)) \cdot V(\phi) \cdot V(\varphi_s)$ where $\sigma(\varphi_s)$ is the collective statistic (bosonic or fermionic) of the list $\varphi_s$ , $\sigma(\phi)$ is the statistic of the field operator $\phi$ , and $\mathcal{S}$ is the exchange sign factor.

theorem

Super-commutator of $\text{anPart}(\phi)$ and $\text{crPart}(\phi')$ in the Wick algebra

#superCommute_anPart_crPart

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$ , let $\text{anPart}(\phi)$ and $\text{crPart}(\phi')$ be their respective annihilation and creation parts in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator of these elements satisfies the relation: $[\text{anPart}(\phi), \text{crPart}(\phi')]_s = \text{anPart}(\phi) \cdot \text{crPart}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPart}(\phi') \cdot \text{anPart}(\phi)$ where $\sigma(\phi)$ and $\sigma(\phi')$ denote the 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.

theorem

Super-commutator of creation and annihilation parts in the Wick algebra of $\mathcal{F}$

#superCommute_crPart_anPart

For a given field specification $\mathcal{F}$ and two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , let $\text{crPart}(\phi)$ and $\text{anPart}(\phi')$ denote their respective creation and annihilation components in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator of these components is given by: $[\text{crPart}(\phi), \text{anPart}(\phi')]_s = \text{crPart}(\phi) \cdot \text{anPart}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{anPart}(\phi') \cdot \text{crPart}(\phi)$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of the operator $\phi$ , and $\mathcal{S}(\sigma_1, \sigma_2)$ is the exchange sign factor which is $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

The super-commutator of creation parts $[\text{crPart}(\phi), \text{crPart}(\phi')]_s$ is zero in the Wick algebra.

#superCommute_crPart_crPart

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$ , the super-commutator of their creation parts in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is zero: $[\text{crPart}(\phi), \text{crPart}(\phi')]_s = 0$

theorem

$[\text{anPart}(\phi), \text{anPart}(\phi')]_s = 0$ in the Wick algebra

#superCommute_anPart_anPart

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the super-commutator of their respective annihilation parts in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ is zero: $[\text{anPart}(\phi), \text{anPart}(\phi')]_s = 0$

theorem

Super-commutator $[\text{crPart}(\phi), \prod \phi_i]_s$ in the Wick algebra.

#superCommute_crPart_ofFieldOpList

For a given field specification $\mathcal{F}$ , let $\phi \in \text{FieldOp}(\mathcal{F})$ be a field operator and $\Phi = [\phi_1, \dots, \phi_n]$ be a list of field operators. Let $\text{crPart}(\phi)$ be the creation part of $\phi$ and $\text{ofFieldOpList}(\Phi)$ be the product of the field operators in the list within the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator of these two elements is given by: $[\text{crPart}(\phi), \text{ofFieldOpList}(\Phi)]_s = \text{crPart}(\phi) \cdot \text{ofFieldOpList}(\Phi) - \mathcal{S}(\sigma(\phi), \sigma(\Phi)) \cdot \text{ofFieldOpList}(\Phi) \cdot \text{crPart}(\phi)$ where $\sigma(\phi)$ is the field statistic of $\phi$ , $\sigma(\Phi)$ is the collective field statistic of the list $\Phi$ , and $\mathcal{S}$ is the exchange sign factor which is $-1$ if both arguments are fermionic and $1$ otherwise.

theorem

Super-commutator of an annihilation part and a product of field operators in the Wick algebra $(\mathcal{F}.\text{WickAlgebra})$

#superCommute_anPart_ofFieldOpList

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator and $\phi_s = [\phi_1, \dots, \phi_n]$ be a list of field operators. Let $\text{anPart}(\phi)$ denote the annihilation part of $\phi$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ , and let $\text{ofFieldOpList}(\phi_s)$ denote the product $\prod_{i=1}^n \phi_i$ in the Wick algebra. The super-commutator of these two elements satisfies: $[\text{anPart}(\phi), \text{ofFieldOpList}(\phi_s)]_s = \text{anPart}(\phi) \cdot \text{ofFieldOpList}(\phi_s) - \mathcal{S}(\sigma(\phi), \sigma(\phi_s)) \cdot \text{ofFieldOpList}(\phi_s) \cdot \text{anPart}(\phi)$ where $\sigma(\phi)$ is the field statistic of $\phi$ , $\sigma(\phi_s)$ is the collective statistic of the list $\phi_s$ , and $\mathcal{S}$ is the exchange sign factor defined as $-1$ if both arguments are fermionic and $1$ otherwise.

theorem

Super-commutator $[\text{crPart}(\phi), \text{ofFieldOp}(\phi')]_s$ in the Wick Algebra

#superCommute_crPart_ofFieldOp

For a given field specification $\mathcal{F}$ and two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the super-commutator of the creation part of $\phi$ , denoted $\text{crPart}(\phi)$ , and the representation of $\phi'$ in the Wick algebra, denoted $\text{ofFieldOp}(\phi')$ , satisfies the identity: $[\text{crPart}(\phi), \text{ofFieldOp}(\phi')]_s = \text{crPart}(\phi) \cdot \text{ofFieldOp}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{ofFieldOp}(\phi') \cdot \text{crPart}(\phi)$ where $\sigma(\phi)$ and $\sigma(\phi')$ are the field statistics (bosonic or fermionic) of $\phi$ and $\phi'$ , and $\mathcal{S}$ is the exchange sign factor defined by these statistics.

theorem

Super-commutator of an annihilation part $A(\phi)$ and a field operator $\phi'$ in the Wick algebra

#superCommute_anPart_ofFieldOp

For a given field specification $\mathcal{F}$ , let $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ be field operators. Let $A(\phi)$ denote the annihilation part of $\phi$ and $V(\phi')$ denote the representation of $\phi'$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator $[A(\phi), V(\phi')]_s$ satisfies the relation: $[A(\phi), V(\phi')]_s = A(\phi) \cdot V(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot V(\phi') \cdot A(\phi)$ where $\sigma(\phi)$ and $\sigma(\phi')$ are the field statistics (bosonic or fermionic) of $\phi$ and $\phi'$ respectively, and $\mathcal{S}$ is the exchange sign factor.

theorem

Expansion of the product of two operator lists via the super-commutator

#ofCrAnList_mul_ofCrAnList_eq_superCommute

For a given field specification $\mathcal{F}$ , let $\phi_s$ and $\phi_s'$ be lists of creation and annihilation operators in $\mathcal{F}.\text{CrAnFieldOp}$ . Let $\text{ofCrAnList}(\phi_s)$ and $\text{ofCrAnList}(\phi_s')$ be their corresponding products in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . Then the product of these two elements satisfies the relation: $\text{ofCrAnList}(\phi_s) \cdot \text{ofCrAnList}(\phi_s') = \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot \text{ofCrAnList}(\phi_s') \cdot \text{ofCrAnList}(\phi_s) + [\text{ofCrAnList}(\phi_s), \text{ofCrAnList}(\phi_s')]_s$ where $[\cdot, \cdot]_s$ denotes the super-commutator, $\sigma(\phi_s)$ and $\sigma(\phi_s')$ are the collective field statistics of the operator lists, and $\mathcal{S}$ is the exchange sign factor.

theorem

$\text{ofCrAnOp}(\phi) \cdot \text{ofCrAnList}(\phi_s') = \mathcal{S} \cdot \text{ofCrAnList}(\phi_s') \cdot \text{ofCrAnOp}(\phi) + [\text{ofCrAnOp}(\phi), \text{ofCrAnList}(\phi_s')]_s$ in the Wick algebra

#ofCrAnOp_mul_ofCrAnList_eq_superCommute

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator and let $\phi_{s}' = [\phi_1, \dots, \phi_n]$ be a list of such operators. Let $\text{ofCrAnOp}(\phi)$ and $\text{ofCrAnList}(\phi_{s}')$ be their corresponding representations in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The product of these elements satisfies: $\text{ofCrAnOp}(\phi) \cdot \text{ofCrAnList}(\phi_{s}') = \mathcal{S}(\sigma(\phi), \sigma(\phi_{s}')) \cdot \text{ofCrAnList}(\phi_{s}') \cdot \text{ofCrAnOp}(\phi) + [\text{ofCrAnOp}(\phi), \text{ofCrAnList}(\phi_{s}')]_s$ where $\sigma(\phi)$ is the field statistic of $\phi$ , $\sigma(\phi_{s}')$ is the collective field statistic of the list $\phi_{s}'$ , $\mathcal{S}$ is the exchange sign, and $[\cdot, \cdot]_s$ denotes the super-commutator.

theorem

$V(\varphi_s) V(\varphi_{s'}) = \mathcal{S} \cdot V(\varphi_{s'}) V(\varphi_s) + [V(\varphi_s), V(\varphi_{s'})]_s$ for products of field operators in the Wick algebra

#ofFieldOpList_mul_ofFieldOpList_eq_superCommute

For a given field specification $\mathcal{F}$ , let $\varphi_s$ and $\varphi_{s'}$ be two lists of field operators. Let $V(\varphi_s)$ and $V(\varphi_{s'})$ denote the products of these lists of field operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The product of these two elements satisfies: $V(\varphi_s) \cdot V(\varphi_{s'}) = \mathcal{S}(\sigma(\varphi_s), \sigma(\varphi_{s'})) \cdot V(\varphi_{s'}) \cdot V(\varphi_s) + [V(\varphi_s), V(\varphi_{s'})]_s$ where $\sigma(\cdot)$ denotes the collective field statistic of a list, $\mathcal{S}$ is the exchange sign, and $[\cdot, \cdot]_s$ denotes the super-commutator.

theorem

$V(\phi) V(\phi_s') = \mathcal{S} \cdot V(\phi_s') V(\phi) + [V(\phi), V(\phi_s')]_s$ in the Wick algebra

#ofFieldOp_mul_ofFieldOpList_eq_superCommute

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator and $\phi_s' = [\phi_1, \dots, \phi_n]$ be a list of field operators. Let $V(\phi)$ denote the operator $\phi$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ and $V(\phi_s')$ denote the product of the operators in the list $\phi_s'$ . Their multiplication satisfies the relation: $V(\phi) \cdot V(\phi_s') = \mathcal{S}(\sigma(\phi), \sigma(\phi_s')) \cdot V(\phi_s') \cdot V(\phi) + [V(\phi), V(\phi_s')]_s$ where $\sigma(\phi)$ is the field statistic of $\phi$ , $\sigma(\phi_s')$ is the collective field statistic of the list $\phi_s'$ , $\mathcal{S}$ is the exchange sign, and $[ \cdot, \cdot ]_s$ denotes the super-commutator in the Wick algebra.

theorem

$V(\phi) V(\phi') = \mathcal{S} \cdot V(\phi') V(\phi) + [V(\phi), V(\phi')]_s$ for field operators in the Wick algebra

#ofFieldOp_mul_ofFieldOp_eq_superCommute

In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ associated with a field specification $\mathcal{F}$ , let $\phi$ and $\phi'$ be two field operators. Let $V(\phi)$ and $V(\phi')$ denote their corresponding elements in the Wick algebra. Their product satisfies the following relation: $V(\phi) \cdot V(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot V(\phi') \cdot V(\phi) + [V(\phi), V(\phi')]_s$ where $\sigma(\phi)$ and $\sigma(\phi')$ are the statistics of the field operators $\phi$ and $\phi'$ respectively, $\mathcal{S}$ is the exchange sign factor, and $[\cdot, \cdot]_s$ denotes the super-commutator in the algebra.

theorem

Expansion of the product $V(\varphi_s) \cdot V(\varphi)$ via the super-commutator

#ofFieldOpList_mul_ofFieldOp_eq_superCommute

In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ of a field specification $\mathcal{F}$ , for any list of field operators $\varphi_s$ and any single field operator $\varphi$ , the product of their corresponding algebra elements $V(\varphi_s)$ and $V(\varphi)$ satisfies: $V(\varphi_s) \cdot V(\varphi) = \mathcal{S}(\sigma(\varphi_s), \sigma(\varphi)) \cdot V(\varphi) \cdot V(\varphi_s) + [V(\varphi_s), V(\varphi)]_s$ where $\sigma(\varphi_s)$ is the collective statistic of the list $\varphi_s$ (determined by the number of fermionic operators therein), $\sigma(\varphi)$ is the statistic of the field operator $\varphi$ , $\mathcal{S}$ is the exchange sign factor, and $[ \cdot, \cdot ]_s$ denotes the super-commutator.

theorem

Decomposition of $V(\phi_s) \cdot V(\phi_s')$ into graded-permuted product and super-commutator in the Wick algebra

#ofCrAnList_mul_ofFieldOpList_eq_superCommute

In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ associated with a field specification $\mathcal{F}$ , let $\phi_s$ be a list of creation and annihilation operator components and $\phi_s'$ be a list of field operators. The product of their corresponding representations in the algebra satisfies: $\text{ofCrAnList}(\phi_s) \cdot \text{ofFieldOpList}(\phi_s') = \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot \text{ofFieldOpList}(\phi_s') \cdot \text{ofCrAnList}(\phi_s) + [\text{ofCrAnList}(\phi_s), \text{ofFieldOpList}(\phi_s')]_s$ where $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of each list, $\mathcal{S}$ is the exchange sign factor, and $[\cdot, \cdot]_s$ is the super-commutator.

theorem

Decomposition of $\text{crPart}(\phi) \cdot \text{anPart}(\phi')$ into graded product and super-commutator

#crPart_mul_anPart_eq_superCommute

For a given field specification $\mathcal{F}$ and two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the product of the creation part of $\phi$ and the annihilation part of $\phi'$ in the Wick algebra is given by $\text{crPart}(\phi) \cdot \text{anPart}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{anPart}(\phi') \cdot \text{crPart}(\phi) + [\text{crPart}(\phi), \text{anPart}(\phi')]_s$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of $\phi$ , $\mathcal{S}$ is the exchange sign, and $[\cdot, \cdot]_s$ denotes the super-commutator.

theorem

Expansion of $\text{anPart}(\phi) \cdot \text{crPart}(\phi')$ via super-commutator in the Wick algebra

#anPart_mul_crPart_eq_superCommute

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$ , let $\text{anPart}(\phi)$ be the annihilation part of $\phi$ and $\text{crPart}(\phi')$ be the creation part of $\phi'$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The product of these operators satisfies the relation: $\text{anPart}(\phi) \cdot \text{crPart}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPart}(\phi') \cdot \text{anPart}(\phi) + [\text{anPart}(\phi), \text{crPart}(\phi')]_s$ where $\sigma(\phi)$ and $\sigma(\phi')$ denote the field statistics of the operators, $\mathcal{S}$ is the exchange sign ( $\mathcal{S}=-1$ if both operators are fermionic, $1$ otherwise), and $[\cdot, \cdot]_s$ denotes the super-commutator.

theorem

$\text{crPart}(\phi) \cdot \text{crPart}(\phi') = \mathcal{S}(\sigma, \sigma') \cdot \text{crPart}(\phi') \cdot \text{crPart}(\phi)$ in the Wick Algebra

#crPart_mul_crPart_swap

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the product of their creation parts in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ satisfies the graded exchange relation: $\text{crPart}(\phi) \cdot \text{crPart}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPart}(\phi') \cdot \text{crPart}(\phi)$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of the operator $\phi$ , and $\mathcal{S}(\sigma, \sigma')$ is the exchange sign, which is $-1$ if both operators are fermionic and $1$ otherwise.

theorem

Exchange Symmetry of Annihilation Parts in the Wick Algebra

#anPart_mul_anPart_swap

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$ , let $\text{anPart}(\phi)$ and $\text{anPart}(\phi')$ be their corresponding annihilation parts in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The product of these annihilation parts satisfies the commutation relation: $\text{anPart}(\phi) \cdot \text{anPart}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{anPart}(\phi') \cdot \text{anPart}(\phi)$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of the operator $\phi$ , and $\mathcal{S}$ is the exchange sign, which is $-1$ if both operators are fermionic and $1$ otherwise.

theorem

$[V(\phi_s), V(\phi_s')]_s = -\mathcal{S} \cdot [V(\phi_s'), V(\phi_s)]_s$ in the Wick Algebra

#superCommute_ofCrAnList_ofCrAnList_symm

For a given field specification $\mathcal{F}$ , let $\phi_s$ and $\phi_s'$ be lists of creation and annihilation operators. Let $V(\phi_s)$ and $V(\phi_s')$ denote the products of these operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator $[\cdot, \cdot]_s$ satisfies the following symmetry relation: $[V(\phi_s), V(\phi_s')]_s = -\mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot [V(\phi_s'), V(\phi_s)]_s$ where $\sigma(\phi_s)$ is the collective statistic (bosonic or fermionic) of the operators in the list $\phi_s$ , and the exchange sign $\mathcal{S}$ is $-1$ if both arguments are fermionic and $1$ otherwise.

theorem

Symmetry of the Super-commutator for Creation and Annihilation Operators in the Wick Algebra

#superCommute_ofCrAnOp_ofCrAnOp_symm

Let $\mathcal{F}$ be a field specification. For any two creation or annihilation operator components $\phi, \phi' \in \mathcal{F}.\text{CrAnFieldOp}$ , let $a = \text{ofCrAnOp}(\phi)$ and $a' = \text{ofCrAnOp}(\phi')$ be their corresponding elements in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator $[ \cdot, \cdot ]_s$ on the Wick algebra satisfies the symmetry relation: $[a, a']_s = -\mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot [a', a]_s$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) associated with the operator $\phi$ , and $\mathcal{S}(\sigma_1, \sigma_2)$ is the exchange sign, which equals $-1$ if both $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

Expansion of $[V(\phi_s), V(\phi_s')]_s$ as a sum over elements of $\phi_s'$ in the Wick Algebra

#superCommute_ofCrAnList_ofCrAnList_eq_sum

For a given field specification $\mathcal{F}$ , 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 the corresponding products of these operators in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator $[ \cdot, \cdot ]_s$ of these two products satisfies the following identity: $[V(\phi_s), V(\phi_s')]_s = \sum_{i=0}^{|\phi_s'|-1} \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s'[0 \dots i-1])) \cdot V(\phi_s'[0 \dots i-1]) \cdot [V(\phi_s), \phi_i']_s \cdot V(\phi_s'[i+1 \dots |\phi_s'|-1])$ where: - $|\phi_s'|$ is the length of the list $\phi_s'$ . - $\phi_s'[0 \dots i-1]$ is the prefix list consisting of the first $i$ elements of $\phi_s'$ . - $\phi_i'$ is the element at index $i$ of the list $\phi_s'$ . - $\phi_s'[i+1 \dots |\phi_s'|-1]$ is the suffix list consisting of the elements in $\phi_s'$ after the $i$ -th index. - $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of a list of operators. - $\mathcal{S}(\cdot, \cdot)$ is the exchange sign factor determined by the statistics.

theorem

Expansion of $[V(\phi), V(\phi_s')]_s$ as a sum over operators in the Wick algebra

#superCommute_ofCrAnOp_ofCrAnList_eq_sum

Let $\mathcal{F}$ be a field specification. For any creation or annihilation operator component $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ and any list of such operator components $\phi_s' = [\phi_0', \phi_1', \dots, \phi_{m-1}']$ in $\mathcal{F}.\text{CrAnFieldOp}$ , let $V(\phi)$ and $V(\phi_s')$ denote their respective representatives in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator $[\cdot, \cdot]_s$ satisfies the following expansion: $[V(\phi), V(\phi_s')]_s = \sum_{n=0}^{m-1} \mathcal{S}(\sigma(\phi), \sigma(\phi_s'[0 \dots n-1])) \cdot [V(\phi), V(\phi_n')]_s * V(\phi_s' \setminus \{\phi_n'\})$ where: - $m$ is the length of the list $\phi_s'$ . - $\phi_s'[0 \dots n-1]$ is the prefix list of the first $n$ elements of $\phi_s'$ . - $\phi_n'$ is the element at index $n$ of the list $\phi_s'$ . - $\phi_s' \setminus \{\phi_n'\}$ is the list $\phi_s'$ with the element at index $n$ removed. - $\sigma(\cdot)$ denotes the field statistic (bosonic or fermionic) of an operator or the collective statistic of a list. - $\mathcal{S}(\cdot, \cdot)$ is the exchange sign factor, which is $-1$ if both arguments are fermionic and $1$ otherwise. - $\cdot$ denotes scalar multiplication and $*$ denotes multiplication in the Wick algebra.

theorem

Expansion of $[V(\phi_{s, \text{CrAn}}), V(\phi_{s, \text{Field}})]_s$ as a sum over field operators in the Wick algebra

#superCommute_ofCrAnList_ofFieldOpList_eq_sum

For a given field specification $\mathcal{F}$ , let $\phi_s$ be a list of creation and annihilation operator components and $\phi_s'$ be a list of field operators. Let $V(\phi_s)$ be the product $\prod \phi_i$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ and $V(\phi_s')$ be the product $\prod \phi'_j$ in the same algebra. The super-commutator $[\cdot, \cdot]_s$ satisfies the following expansion: $[V(\phi_s), V(\phi_s')]_s = \sum_{n=0}^{|\phi_s'|-1} \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s'[0 \dots n-1])) \cdot V(\phi_s'[0 \dots n-1]) \cdot [V(\phi_s), \phi_n']_s \cdot V(\phi_s'[n+1 \dots |\phi_s'|-1])$ where: - $|\phi_s'|$ is the length of the list of field operators $\phi_s'$ . - $\phi_s'[0 \dots n-1]$ denotes the prefix list of the first $n$ field operators in $\phi_s'$ . - $\phi_n'$ is the $n$ -th field operator in the list $\phi_s'$ . - $\phi_s'[n+1 \dots |\phi_s'|-1]$ denotes the suffix list of field operators in $\phi_s'$ following the $n$ -th operator. - $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of a list of operators. - $\mathcal{S}(\cdot, \cdot)$ is the exchange sign factor, equal to $-1$ if both arguments are fermionic and $1$ otherwise. - $\cdot$ denotes scalar multiplication by the exchange sign, and $*$ denotes the multiplication in the Wick algebra.

theorem

Expansion of $[V(\phi_{\text{CrAn}}), \prod \phi'_{\text{Field}}]_s$ as a sum over operators in the Wick algebra

#superCommute_ofCrAnOp_ofFieldOpList_eq_sum

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component and $\phi_s' = [\phi_0', \phi_1', \dots, \phi_{m-1}']$ be a list of field operators in $\mathcal{F}.\text{FieldOp}$ . Let $V(\phi)$ and $V(\phi_s')$ denote their respective representatives in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ . The super-commutator $[\cdot, \cdot]_s$ satisfies the following expansion: $[V(\phi), V(\phi_s')]_s = \sum_{n=0}^{m-1} \mathcal{S}(\sigma(\phi), \sigma(\phi_s'[0 \dots n-1])) \cdot [V(\phi), V(\phi_n')]_s * V(\phi_s' \setminus \{\phi_n'\})$ where: - $m$ is the length of the list $\phi_s'$ . - $\phi_s'[0 \dots n-1]$ denotes the prefix list of the first $n$ field operators in $\phi_s'$ . - $V(\phi_n')$ is the $n$ -th field operator in the list interpreted in the Wick algebra. - $V(\phi_s' \setminus \{\phi_n'\})$ is the product of the field operators in the list $\phi_s'$ with the $n$ -th element removed. - $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of an operator or a list of operators. - $\mathcal{S}(\cdot, \cdot)$ is the exchange sign factor, equal to $-1$ if both arguments are fermionic and $1$ otherwise. - $\cdot$ denotes scalar multiplication by the exchange sign, and $*$ denotes multiplication in the Wick algebra.

Physlib.QFT.PerturbationTheory.WickAlgebra.SuperCommute

SuperCommute on Field operator algebra

Defining normal order for `FiedOpAlgebra`.

Properties of `superCommute`.

Properties from the definition of WickAlgebra

`superCommute` on different constructors.

Mul equal superCommute

Symmetry of the super commutator.

splitting the super commute into sums

ι(b)=0 ⟹ ι([a,b]sF)=0\iota(b) = 0 \implies \iota([a, b]_s^F) = 0ι(b)=0⟹ι([a,b]sF​)=0

ι(a)=0 ⟹ ι([a,b]sF)=0\iota(a) = 0 \implies \iota([a, b]_s^F) = 0ι(a)=0⟹ι([a,b]sF​)=0

b∈span(fieldOpIdealSet) ⟹ ι([a,b]sF)=0b \in \text{span}(\text{fieldOpIdealSet}) \implies \iota([a, b]_s^F) = 0b∈span(fieldOpIdealSet)⟹ι([a,b]sF​)=0

b1≈b2 ⟹ ι([a,b1]sF)=ι([a,b2]sF)b_1 \approx b_2 \implies \iota([a, b_1]_s^F) = \iota([a, b_2]_s^F)b1​≈b2​⟹ι([a,b1​]sF​)=ι([a,b2​]sF​)

Right super-commutator map [a,⋅]s[a, \cdot]_s[a,⋅]s​ on the Wick algebra

superCommuteRighta(ι(b))=ι([a,b]sF)\text{superCommuteRight}_a(\iota(b)) = \iota([a, b]_s^F)superCommuteRighta​(ι(b))=ι([a,b]sF​)

superCommuteRighta(⟦b⟧)=ι([a,b]sF)\text{superCommuteRight}_a(\llbracket b \rrbracket) = \iota([a, b]_s^F)superCommuteRighta​([[b]])=ι([a,b]sF​)

a1≈a2 ⟹ superCommuteRighta1=superCommuteRighta2a_1 \approx a_2 \implies \text{superCommuteRight}_{a_1} = \text{superCommuteRight}_{a_2}a1​≈a2​⟹superCommuteRighta1​​=superCommuteRighta2​​

Super-commutator [a,b]s[a, b]_s[a,b]s​ on the Wick algebra

Super-commutator notation [a,b]s[a, b]_s[a,b]s​

[ι(a),ι(b)]s=ι([a,b]sF)[\iota(a), \iota(b)]_s = \iota([a, b]_s^F)[ι(a),ι(b)]s​=ι([a,b]sF​)

The super-commutator of two creation operators in the Wick algebra is 0

The super-commutator of two annihilation operators in the Wick algebra is zero.

The super-commutator of operators with different statistics vanishes in the Wick algebra

The super-commutator of a creation/annihilation operator and a field operator with different statistics is zero

The super-commutator of anPart(ϕ)\text{anPart}(\phi)anPart(ϕ) and ofFieldOp(ψ)\text{ofFieldOp}(\psi)ofFieldOp(ψ) is 000 if stat(ϕ)≠stat(ψ)\text{stat}(\phi) \neq \text{stat}(\psi)stat(ϕ)=stat(ψ)

[ϕ,ϕ′]s∈center(WickAlgebra F)[\phi, \phi']_s \in \text{center}(\text{WickAlgebra } \mathcal{F})[ϕ,ϕ′]s​∈center(WickAlgebra F)

a⋅[ϕ,ϕ′]s=[ϕ,ϕ′]s⋅aa \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot aa⋅[ϕ,ϕ′]s​=[ϕ,ϕ′]s​⋅a for ϕ,ϕ′∈CrAnFieldOp\phi, \phi' \in \text{CrAnFieldOp}ϕ,ϕ′∈CrAnFieldOp

[ϕ,ϕ′]s[\phi, \phi']_s[ϕ,ϕ′]s​ lies in the center of the Wick algebra for ϕ∈CrAnFieldOp\phi \in \text{CrAnFieldOp}ϕ∈CrAnFieldOp and ϕ′∈FieldOp\phi' \in \text{FieldOp}ϕ′∈FieldOp

a⋅[ϕ,ϕ′]s=[ϕ,ϕ′]s⋅aa \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot aa⋅[ϕ,ϕ′]s​=[ϕ,ϕ′]s​⋅a for ϕ∈CrAnFieldOp\phi \in \text{CrAnFieldOp}ϕ∈CrAnFieldOp and ϕ′∈FieldOp\phi' \in \text{FieldOp}ϕ′∈FieldOp

[anPart(ϕ),ofFieldOp(ϕ′)]s[\text{anPart}(\phi), \text{ofFieldOp}(\phi')]_s[anPart(ϕ),ofFieldOp(ϕ′)]s​ lies in the center of the Wick algebra

The super-commutator of two operator products in the Wick algebra equals the difference of their concatenated products weighted by the exchange sign.

Super-commutator of creation/annihilation operators ϕ,ϕ′\phi, \phi'ϕ,ϕ′ in the Wick algebra

[V(ϕcas),V(ϕs)]s=V(ϕcas)⋅V(ϕs)−S⋅V(ϕs)⋅V(ϕcas)[V(\phi_{\text{cas}}), V(\phi_s)]_s = V(\phi_{\text{cas}}) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi_{\text{cas}})[V(ϕcas​),V(ϕs​)]s​=V(ϕcas​)⋅V(ϕs​)−S⋅V(ϕs​)⋅V(ϕcas​) in the Wick algebra

[V(ϕ),V(ϕs)]s=V(ϕ)V(ϕs)−S⋅V(ϕs)V(ϕ)[V(\phi), V(\phi_s)]_s = V(\phi) V(\phi_s) - \mathcal{S} \cdot V(\phi_s) V(\phi)[V(ϕ),V(ϕs​)]s​=V(ϕ)V(ϕs​)−S⋅V(ϕs​)V(ϕ) in the Wick algebra

Super-commutator of a product of field operators and a single field operator in the Wick algebra

Super-commutator of anPart(ϕ)\text{anPart}(\phi)anPart(ϕ) and crPart(ϕ′)\text{crPart}(\phi')crPart(ϕ′) in the Wick algebra

Super-commutator of creation and annihilation parts in the Wick algebra of F\mathcal{F}F

The super-commutator of creation parts [crPart(ϕ),crPart(ϕ′)]s[\text{crPart}(\phi), \text{crPart}(\phi')]_s[crPart(ϕ),crPart(ϕ′)]s​ is zero in the Wick algebra.

[anPart(ϕ),anPart(ϕ′)]s=0[\text{anPart}(\phi), \text{anPart}(\phi')]_s = 0[anPart(ϕ),anPart(ϕ′)]s​=0 in the Wick algebra

Super-commutator [crPart(ϕ),∏ϕi]s[\text{crPart}(\phi), \prod \phi_i]_s[crPart(ϕ),∏ϕi​]s​ in the Wick algebra.

Super-commutator of an annihilation part and a product of field operators in the Wick algebra (F.WickAlgebra)(\mathcal{F}.\text{WickAlgebra})(F.WickAlgebra)

Super-commutator [crPart(ϕ),ofFieldOp(ϕ′)]s[\text{crPart}(\phi), \text{ofFieldOp}(\phi')]_s[crPart(ϕ),ofFieldOp(ϕ′)]s​ in the Wick Algebra

Super-commutator of an annihilation part A(ϕ)A(\phi)A(ϕ) and a field operator ϕ′\phi'ϕ′ in the Wick algebra

Expansion of the product of two operator lists via the super-commutator

V(φs)V(φs′)=S⋅V(φs′)V(φs)+[V(φs),V(φs′)]sV(\varphi_s) V(\varphi_{s'}) = \mathcal{S} \cdot V(\varphi_{s'}) V(\varphi_s) + [V(\varphi_s), V(\varphi_{s'})]_sV(φs​)V(φs′​)=S⋅V(φs′​)V(φs​)+[V(φs​),V(φs′​)]s​ for products of field operators in the Wick algebra

V(ϕ)V(ϕs′)=S⋅V(ϕs′)V(ϕ)+[V(ϕ),V(ϕs′)]sV(\phi) V(\phi_s') = \mathcal{S} \cdot V(\phi_s') V(\phi) + [V(\phi), V(\phi_s')]_sV(ϕ)V(ϕs′​)=S⋅V(ϕs′​)V(ϕ)+[V(ϕ),V(ϕs′​)]s​ in the Wick algebra

V(ϕ)V(ϕ′)=S⋅V(ϕ′)V(ϕ)+[V(ϕ),V(ϕ′)]sV(\phi) V(\phi') = \mathcal{S} \cdot V(\phi') V(\phi) + [V(\phi), V(\phi')]_sV(ϕ)V(ϕ′)=S⋅V(ϕ′)V(ϕ)+[V(ϕ),V(ϕ′)]s​ for field operators in the Wick algebra

Expansion of the product V(φs)⋅V(φ)V(\varphi_s) \cdot V(\varphi)V(φs​)⋅V(φ) via the super-commutator

Decomposition of V(ϕs)⋅V(ϕs′)V(\phi_s) \cdot V(\phi_s')V(ϕs​)⋅V(ϕs′​) into graded-permuted product and super-commutator in the Wick algebra

Decomposition of crPart(ϕ)⋅anPart(ϕ′)\text{crPart}(\phi) \cdot \text{anPart}(\phi')crPart(ϕ)⋅anPart(ϕ′) into graded product and super-commutator

Expansion of anPart(ϕ)⋅crPart(ϕ′)\text{anPart}(\phi) \cdot \text{crPart}(\phi')anPart(ϕ)⋅crPart(ϕ′) via super-commutator in the Wick algebra

crPart(ϕ)⋅crPart(ϕ′)=S(σ,σ′)⋅crPart(ϕ′)⋅crPart(ϕ)\text{crPart}(\phi) \cdot \text{crPart}(\phi') = \mathcal{S}(\sigma, \sigma') \cdot \text{crPart}(\phi') \cdot \text{crPart}(\phi)crPart(ϕ)⋅crPart(ϕ′)=S(σ,σ′)⋅crPart(ϕ′)⋅crPart(ϕ) in the Wick Algebra

Exchange Symmetry of Annihilation Parts in the Wick Algebra

[V(ϕs),V(ϕs′)]s=−S⋅[V(ϕs′),V(ϕs)]s[V(\phi_s), V(\phi_s')]_s = -\mathcal{S} \cdot [V(\phi_s'), V(\phi_s)]_s[V(ϕs​),V(ϕs′​)]s​=−S⋅[V(ϕs′​),V(ϕs​)]s​ in the Wick Algebra

Symmetry of the Super-commutator for Creation and Annihilation Operators in the Wick Algebra

Expansion of [V(ϕs),V(ϕs′)]s[V(\phi_s), V(\phi_s')]_s[V(ϕs​),V(ϕs′​)]s​ as a sum over elements of ϕs′\phi_s'ϕs′​ in the Wick Algebra

Expansion of [V(ϕ),V(ϕs′)]s[V(\phi), V(\phi_s')]_s[V(ϕ),V(ϕs′​)]s​ as a sum over operators in the Wick algebra

Expansion of [V(ϕs,CrAn),V(ϕs,Field)]s[V(\phi_{s, \text{CrAn}}), V(\phi_{s, \text{Field}})]_s[V(ϕs,CrAn​),V(ϕs,Field​)]s​ as a sum over field operators in the Wick algebra

Expansion of [V(ϕCrAn),∏ϕField′]s[V(\phi_{\text{CrAn}}), \prod \phi'_{\text{Field}}]_s[V(ϕCrAn​),∏ϕField′​]s​ as a sum over operators in the Wick algebra

$\iota(b) = 0 \implies \iota([a, b]_s^F) = 0$

$\iota(a) = 0 \implies \iota([a, b]_s^F) = 0$

$b \in \text{span}(\text{fieldOpIdealSet}) \implies \iota([a, b]_s^F) = 0$

$b_1 \approx b_2 \implies \iota([a, b_1]_s^F) = \iota([a, b_2]_s^F)$

Right super-commutator map $[a, \cdot]_s$ on the Wick algebra

$\text{superCommuteRight}_a(\iota(b)) = \iota([a, b]_s^F)$

$\text{superCommuteRight}_a(\llbracket b \rrbracket) = \iota([a, b]_s^F)$

$a_1 \approx a_2 \implies \text{superCommuteRight}_{a_1} = \text{superCommuteRight}_{a_2}$

Super-commutator $[a, b]_s$ on the Wick algebra

Super-commutator notation $[a, b]_s$

$[\iota(a), \iota(b)]_s = \iota([a, b]_s^F)$

The super-commutator of $\text{anPart}(\phi)$ and $\text{ofFieldOp}(\psi)$ is $0$ if $\text{stat}(\phi) \neq \text{stat}(\psi)$

$[\phi, \phi']_s \in \text{center}(\text{WickAlgebra } \mathcal{F})$

$a \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot a$ for $\phi, \phi' \in \text{CrAnFieldOp}$

$[\phi, \phi']_s$ lies in the center of the Wick algebra for $\phi \in \text{CrAnFieldOp}$ and $\phi' \in \text{FieldOp}$

$a \cdot [\phi, \phi']_s = [\phi, \phi']_s \cdot a$ for $\phi \in \text{CrAnFieldOp}$ and $\phi' \in \text{FieldOp}$

$[\text{anPart}(\phi), \text{ofFieldOp}(\phi')]_s$ lies in the center of the Wick algebra

Super-commutator of creation/annihilation operators $\phi, \phi'$ in the Wick algebra

$[V(\phi_{\text{cas}}), V(\phi_s)]_s = V(\phi_{\text{cas}}) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi_{\text{cas}})$ in the Wick algebra

$[V(\phi), V(\phi_s)]_s = V(\phi) V(\phi_s) - \mathcal{S} \cdot V(\phi_s) V(\phi)$ in the Wick algebra

Super-commutator of $\text{anPart}(\phi)$ and $\text{crPart}(\phi')$ in the Wick algebra

Super-commutator of creation and annihilation parts in the Wick algebra of $\mathcal{F}$

The super-commutator of creation parts $[\text{crPart}(\phi), \text{crPart}(\phi')]_s$ is zero in the Wick algebra.

$[\text{anPart}(\phi), \text{anPart}(\phi')]_s = 0$ in the Wick algebra

Super-commutator $[\text{crPart}(\phi), \prod \phi_i]_s$ in the Wick algebra.

Super-commutator of an annihilation part and a product of field operators in the Wick algebra $(\mathcal{F}.\text{WickAlgebra})$

Super-commutator $[\text{crPart}(\phi), \text{ofFieldOp}(\phi')]_s$ in the Wick Algebra

Super-commutator of an annihilation part $A(\phi)$ and a field operator $\phi'$ in the Wick algebra

$V(\varphi_s) V(\varphi_{s'}) = \mathcal{S} \cdot V(\varphi_{s'}) V(\varphi_s) + [V(\varphi_s), V(\varphi_{s'})]_s$ for products of field operators in the Wick algebra

$V(\phi) V(\phi_s') = \mathcal{S} \cdot V(\phi_s') V(\phi) + [V(\phi), V(\phi_s')]_s$ in the Wick algebra

$V(\phi) V(\phi') = \mathcal{S} \cdot V(\phi') V(\phi) + [V(\phi), V(\phi')]_s$ for field operators in the Wick algebra

Expansion of the product $V(\varphi_s) \cdot V(\varphi)$ via the super-commutator

Decomposition of $V(\phi_s) \cdot V(\phi_s')$ into graded-permuted product and super-commutator in the Wick algebra

Decomposition of $\text{crPart}(\phi) \cdot \text{anPart}(\phi')$ into graded product and super-commutator

Expansion of $\text{anPart}(\phi) \cdot \text{crPart}(\phi')$ via super-commutator in the Wick algebra

$\text{crPart}(\phi) \cdot \text{crPart}(\phi') = \mathcal{S}(\sigma, \sigma') \cdot \text{crPart}(\phi') \cdot \text{crPart}(\phi)$ in the Wick Algebra

$[V(\phi_s), V(\phi_s')]_s = -\mathcal{S} \cdot [V(\phi_s'), V(\phi_s)]_s$ in the Wick Algebra

Expansion of $[V(\phi_s), V(\phi_s')]_s$ as a sum over elements of $\phi_s'$ in the Wick Algebra

Expansion of $[V(\phi), V(\phi_s')]_s$ as a sum over operators in the Wick algebra

Expansion of $[V(\phi_{s, \text{CrAn}}), V(\phi_{s, \text{Field}})]_s$ as a sum over field operators in the Wick algebra

Expansion of $[V(\phi_{\text{CrAn}}), \prod \phi'_{\text{Field}}]_s$ as a sum over operators in the Wick algebra