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definition

Super-commutator $[a, b]_s^F$ on the field operator free algebra

For a given field specification $\mathcal{F}$ , the super-commutator $\text{superCommuteF}$ , denoted by the notation $[a, b]_s^F$ , is a $\mathbb{C}$ -bilinear map on the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . For two basis elements $a$ and $b$ , which are products of lists of creation and annihilation operators $\phi_s$ and $\phi_s'$ , the super-commutator is defined as: $[a, b]_s^F = a \cdot b - \mathcal{S}(a, b) \cdot b \cdot a$ where $a \cdot b$ and $b \cdot a$ are the products in the free algebra (corresponding to the concatenation of the operator lists), and $\mathcal{S}(a, b)$ is a sign factor determined by the statistics of the elements. The sign $\mathcal{S}(a, b)$ is $-1$ if both $a$ and $b$ are fermionic (meaning both contain an odd number of fermionic components) and $1$ otherwise.

definition

Notation for the super-commutator $[\phi, \psi]_s^F$

#term[_,_]ₛF

The notation $[\phi, \psi]_s^F$ represents the super-commutator of two elements $\phi$ and $\psi$ within the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . It is defined as a shorthand for the bilinear map $\text{superCommuteF}(\phi, \psi)$ over the complex numbers $\mathbb{C}$ .

theorem

$[V(\phi_s), V(\phi_s')]_s^F = V(\phi_s \mathbin{+\mkern-10mu+} \phi_s') - \mathcal{S} \cdot V(\phi_s' \mathbin{+\mkern-10mu+} \phi_s)$ for operator lists $\phi_s, \phi_s'$

#superCommuteF_ofCrAnListF_ofCrAnListF

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 corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these two elements is given by: $[V(\phi_s), V(\phi_s')]_s^F = V(\phi_s \mathbin{+\mkern-10mu+} \phi_s') - \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot V(\phi_s' \mathbin{+\mkern-10mu+} \phi_s)$ 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 a sign factor equal to $-1$ if the collective statistics $\sigma(\phi_s)$ and $\sigma(\phi_s')$ of both lists are fermionic, and $1$ otherwise.

theorem

$[V(\phi), V(\phi')]_s^F = V(\phi) \cdot V(\phi') - \mathcal{S} \cdot V(\phi') \cdot V(\phi)$ for creation/annihilation operators $\phi, \phi'$

#superCommuteF_ofCrAnOpF_ofCrAnOpF

For a given field specification $\mathcal{F}$ , let $\phi$ and $\phi'$ be two creation or annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$ . Let $V(\phi)$ and $V(\phi')$ denote their corresponding generators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these two operators is given by: $[V(\phi), V(\phi')]_s^F = V(\phi) \cdot V(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot V(\phi') \cdot V(\phi)$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) associated with the operator component $\phi$ , and the sign factor $\mathcal{S}(\sigma(\phi), \sigma(\phi'))$ is equal to $-1$ if both $\phi$ and $\phi'$ are fermionic, and $1$ otherwise.

theorem

$[V(\phi_{\text{cas}}), V(\phi_s)]_s^F = V(\phi_{\text{cas}}) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi_{\text{cas}})$ for products of components and field operators

#superCommuteF_ofCrAnListF_ofFieldOpFsList

For a given 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. Let $V(\phi_{\text{cas}})$ and $V(\phi_s)$ denote the corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these two elements is given by: $[V(\phi_{\text{cas}}), V(\phi_s)]_s^F = V(\phi_{\text{cas}}) \cdot V(\phi_s) - \mathcal{S}(\sigma(\phi_{\text{cas}}), \sigma(\phi_s)) \cdot V(\phi_s) \cdot V(\phi_{\text{cas}})$ where $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of the list, and the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both collective statistics are fermionic, and $1$ otherwise.

theorem

$[V(\varphi), V(\varphi_s)]_s^F = V(\varphi) \cdot V(\varphi_s) - \mathcal{S} \cdot V(\varphi_s) \cdot V(\varphi)$ for products of field operator lists

#superCommuteF_ofFieldOpListF_ofFieldOpFsList

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

theorem

$[V(\phi), V(\phi_s)]_s^F = V(\phi) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi)$ for a single field operator and a product of field operators

#superCommuteF_ofFieldOpF_ofFieldOpFsList

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 representation of the operator $\phi$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , 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: $[V(\phi), V(\phi_s)]_s^F = V(\phi) \cdot V(\phi_s) - \mathcal{S}(\sigma(\phi), \sigma(\phi_s)) \cdot V(\phi_s) \cdot V(\phi)$ where $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$ , $\sigma(\phi_s)$ is the collective field statistic of the list $\phi_s$ , and the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

$[V(\varphi_s), V(\phi)]_s^F = V(\varphi_s) \cdot V(\phi) - \mathcal{S} \cdot V(\phi) \cdot V(\varphi_s)$ for a product of field operators and a single field operator

#superCommuteF_ofFieldOpListF_ofFieldOpF

For a given field specification $\mathcal{F}$ , let $\varphi_s$ be a list of field operators and $\phi$ be a single field operator. Let $V(\varphi_s)$ denote the product of the operators in $\varphi_s$ within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , and let $V(\phi)$ denote the representation of $\phi$ in the same algebra. The super-commutator of these elements is given by: $[V(\varphi_s), V(\phi)]_s^F = V(\varphi_s) \cdot V(\phi) - \mathcal{S}(\sigma(\varphi_s), \sigma(\phi)) \cdot V(\phi) \cdot V(\varphi_s)$ 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 the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

Super-commutator of $\text{anPartF}(\phi)$ and $\text{crPartF}(\phi')$

#superCommuteF_anPartF_crPartF

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \text{FieldOp}(\mathcal{F})$ , let $\text{anPartF}(\phi)$ and $\text{crPartF}(\phi')$ be their respective annihilation and creation parts in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these elements is given by: $[\text{anPartF}(\phi), \text{crPartF}(\phi')]_s^F = \text{anPartF}(\phi) \cdot \text{crPartF}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPartF}(\phi') \cdot \text{anPartF}(\phi)$ where $\sigma(\phi)$ and $\sigma(\phi')$ denote the statistics (bosonic or fermionic) of the operators $\phi$ and $\phi'$ , and $\mathcal{S}(\sigma, \sigma')$ is the sign factor equal to $-1$ if both operators are fermionic and $1$ otherwise.

theorem

$[\text{crPartF}(\phi), \text{anPartF}(\phi')]_s^F = \text{crPartF}(\phi)\text{anPartF}(\phi') - \mathcal{S} \cdot \text{anPartF}(\phi')\text{crPartF}(\phi)$

#superCommuteF_crPartF_anPartF

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

theorem

Super-commutator of creation parts $[\text{crPartF}(\phi), \text{crPartF}(\phi')]_s^F$

#superCommuteF_crPartF_crPartF

For a given field specification $\mathcal{F}$ and two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the super-commutator of their creation parts in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is given by: $[\text{crPartF}(\phi), \text{crPartF}(\phi')]_s^F = \text{crPartF}(\phi) \cdot \text{crPartF}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPartF}(\phi') \cdot \text{crPartF}(\phi)$ where $\text{crPartF}(\phi)$ denotes the creation component of the operator $\phi$ in the algebra, $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$ , and $\mathcal{S}(\sigma, \sigma')$ is a sign factor equal to $-1$ if both $\sigma$ and $\sigma'$ are fermionic and $1$ otherwise.

theorem

Super-commutator of two annihilation parts $[\text{anPartF}(\phi), \text{anPartF}(\phi')]_s^F$

#superCommuteF_anPartF_anPartF

For any two field operators $\phi, \phi'$ in a field specification $\mathcal{F}$ , the super-commutator of their annihilation parts in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is given by: $[\text{anPartF}(\phi), \text{anPartF}(\phi')]_s^F = \text{anPartF}(\phi) \cdot \text{anPartF}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{anPartF}(\phi') \cdot \text{anPartF}(\phi)$ where $\text{anPartF}(\phi)$ is the annihilation component of the field operator $\phi$ in the algebra, $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$ , and $\mathcal{S}(\sigma_1, \sigma_2)$ is the statistical sign factor, which is $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

Super-commutator $[\text{crPartF}(\phi), \prod \phi_i]_s^F$

#superCommuteF_crPartF_ofFieldOpListF

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator and $\Phi = [\phi_1, \dots, \phi_n]$ be a list of field operators. Let $\text{crPartF}(\phi)$ denote the creation part of $\phi$ and $\text{ofFieldOpListF}(\Phi)$ denote the product of the field operators in the list within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these two elements is given by: $[\text{crPartF}(\phi), \text{ofFieldOpListF}(\Phi)]_s^F = \text{crPartF}(\phi) \cdot \text{ofFieldOpListF}(\Phi) - \mathcal{S}(\sigma(\phi), \sigma(\Phi)) \cdot \text{ofFieldOpListF}(\Phi) \cdot \text{crPartF}(\phi)$ where $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of $\phi$ , $\sigma(\Phi)$ is the collective field statistic of the list $\Phi$ , and the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

Super-commutator $[A(\phi), \prod \phi_i]_s^F$ of an annihilation part and a product of field operators

#superCommuteF_anPartF_ofFieldOpListF

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator and $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of field operators. Let $\text{anPartF}(\phi)$ denote the annihilation part of $\phi$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , and let $\text{ofFieldOpListF}(\phi_s)$ denote the algebraic product $\phi_1 \cdot \phi_2 \cdot \dots \cdot \phi_n$ in the same algebra. The super-commutator of these two elements is given by: $[\text{anPartF}(\phi), \text{ofFieldOpListF}(\phi_s)]_s^F = \text{anPartF}(\phi) \cdot \text{ofFieldOpListF}(\phi_s) - \mathcal{S}(\sigma(\phi), \sigma(\phi_s)) \cdot \text{ofFieldOpListF}(\phi_s) \cdot \text{anPartF}(\phi)$ where $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of the operator $\phi$ , $\sigma(\phi_s)$ is the collective statistic of the list of operators $\phi_s$ , and the statistical sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is $-1$ if both statistics are fermionic, and $1$ otherwise.

theorem

Super-commutator $[\text{crPartF}(\phi), \text{ofFieldOpF}(\phi')]_s^F$ of a creation part and a field operator

#superCommuteF_crPartF_ofFieldOpF

For a given field specification $\mathcal{F}$ , let $\phi$ and $\phi'$ be field operators in $\mathcal{F}.\text{FieldOp}$ . Let $\text{crPartF}(\phi)$ be the creation component of $\phi$ and $\text{ofFieldOpF}(\phi')$ be the representation of the operator $\phi'$ (the sum of its creation and annihilation components) within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these two elements is given by: $[\text{crPartF}(\phi), \text{ofFieldOpF}(\phi')]_s^F = \text{crPartF}(\phi) \cdot \text{ofFieldOpF}(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{ofFieldOpF}(\phi') \cdot \text{crPartF}(\phi)$ where $\sigma(\phi)$ and $\sigma(\phi')$ are the field statistics (bosonic or fermionic) of $\phi$ and $\phi'$ respectively, and the statistical sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

$[A(\phi), \phi']_s^F = A(\phi) \cdot \phi' - \mathcal{S} \cdot \phi' \cdot A(\phi)$ for field operators $\phi, \phi'$

#superCommuteF_anPartF_ofFieldOpF

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 $\phi$ (represented by `anPartF φ`) and $V(\phi')$ denote the representation of $\phi'$ as a sum of its creation and annihilation components (represented by `ofFieldOpF φ'`) in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator of these two elements is defined by the relation: $[A(\phi), V(\phi')]_s^F = A(\phi) \cdot V(\phi') - \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot V(\phi') \cdot A(\phi)$ where $\sigma(\phi)$ and $\sigma(\phi')$ are the field statistics (bosonic or fermionic) of $\phi$ and $\phi'$ respectively, and the statistical sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

$V(\phi_s) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi_s) + [V(\phi_s), V(\phi_s')]_s^F$ for operator lists $\phi_s, \phi_s'$

#ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF

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 corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The product of these two elements satisfies the relation: $V(\phi_s) \cdot V(\phi_s') = \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot V(\phi_s') \cdot V(\phi_s) + [V(\phi_s), V(\phi_s')]_s^F$ where $\sigma(\phi_s)$ and $\sigma(\phi_s')$ represent the collective statistics (bosonic or fermionic) of the respective operator lists, $[ \cdot, \cdot ]_s^F$ is the super-commutator, and $\mathcal{S}(\sigma(\phi_s), \sigma(\phi_s'))$ is a sign factor equal to $-1$ if both lists have fermionic statistics and $1$ otherwise.

theorem

$a_\phi \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot a_\phi + [a_\phi, V(\phi_s')]_s^F$ for operator $\phi$ and operator list $\phi_s'$

#ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component and $\phi_s'$ be a list of such components. Let $a_\phi$ denote the representation of $\phi$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ and $V(\phi_s')$ denote the product of the components in $\phi_s'$ . The product of these two elements satisfies the relation: $a_\phi \cdot V(\phi_s') = \mathcal{S}(\sigma(\phi), \sigma(\phi_s')) \cdot V(\phi_s') \cdot a_\phi + [a_\phi, V(\phi_s')]_s^F$ where $\sigma(\phi)$ and $\sigma(\phi_s')$ represent the statistics (bosonic or fermionic) of the operator and the operator list respectively, $[\cdot, \cdot]_s^F$ is the super-commutator, and $\mathcal{S}$ is a sign factor equal to $-1$ if both $\sigma(\phi)$ and $\sigma(\phi_s')$ are fermionic and $1$ otherwise.

theorem

$V(\varphi_s) \cdot V(\varphi_s') = \mathcal{S} \cdot V(\varphi_s') \cdot V(\varphi_s) + [V(\varphi_s), V(\varphi_s')]_s^F$ for field operator lists $\varphi_s, \varphi_s'$

#ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF

For a given field specification $\mathcal{F}$ , let $\varphi_s$ and $\varphi_s'$ be lists of field operators. Let $V(\varphi_s)$ and $V(\varphi_s')$ denote the corresponding products of these field operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The product of these two elements satisfies the relation: $V(\varphi_s) \cdot V(\varphi_s') = \mathcal{S}(\sigma(\varphi_s), \sigma(\varphi_s')) \cdot V(\varphi_s') \cdot V(\varphi_s) + [V(\varphi_s), V(\varphi_s')]_s^F$ where $\sigma(\varphi_s)$ and $\sigma(\varphi_s')$ represent the collective statistics (bosonic or fermionic) of the respective field operator lists, $[\cdot, \cdot]_s^F$ is the super-commutator, and $\mathcal{S}(\sigma_1, \sigma_2)$ is a sign factor equal to $-1$ if both collective statistics are fermionic and $1$ otherwise.

theorem

$V(\phi) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi) + [V(\phi), V(\phi_s')]_s^F$ for a field operator and a product of field operators

#ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF

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 $V(\phi)$ represent the field operator $\phi$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ (defined as the sum of its creation and annihilation components), and let $V(\phi_s')$ denote the product of the field operators in the list $\phi_s'$ within that algebra. The product of these two elements satisfies the relation: $V(\phi) \cdot V(\phi_s') = \mathcal{S}(\sigma(\phi), \sigma(\phi_s')) \cdot V(\phi_s') \cdot V(\phi) + [V(\phi), V(\phi_s')]_s^F$ where $\sigma(\phi)$ is the field statistic of $\phi$ , $\sigma(\phi_s')$ is the collective field statistic of the list $\phi_s'$ , and the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

$V(\varphi_s) \cdot V(\phi) = \mathcal{S} \cdot V(\phi) \cdot V(\varphi_s) + [V(\varphi_s), V(\phi)]_s^F$ for a product of field operators and a single field operator

#ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF

For a given field specification $\mathcal{F}$ , let $\varphi_s$ be a list of field operators and $\phi$ be a single field operator. Let $V(\varphi_s)$ denote the product of the operators in $\varphi_s$ within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , and let $V(\phi)$ denote the representation of $\phi$ in the same algebra. The product of these elements satisfies the relation: $V(\varphi_s) \cdot V(\phi) = \mathcal{S}(\sigma(\varphi_s), \sigma(\phi)) \cdot V(\phi) \cdot V(\varphi_s) + [V(\varphi_s), V(\phi)]_s^F$ 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 $\phi$ , $[\cdot, \cdot]_s^F$ is the super-commutator, and $\mathcal{S}(\sigma_1, \sigma_2)$ is a sign factor equal to $-1$ if both statistics are fermionic and $1$ otherwise.

theorem

$\text{crPartF}(\phi) \cdot \text{anPartF}(\phi') = \mathcal{S} \cdot \text{anPartF}(\phi') \cdot \text{crPartF}(\phi) + [\text{crPartF}(\phi), \text{anPartF}(\phi')]_s^F$

#crPartF_mul_anPartF_eq_superCommuteF

For a given field specification $\mathcal{F}$ and two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , let $\text{crPartF}(\phi)$ and $\text{anPartF}(\phi')$ denote their respective creation and annihilation components in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The product of these components satisfies the relation: $\text{crPartF}(\phi) \cdot \text{anPartF}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{anPartF}(\phi') \cdot \text{crPartF}(\phi) + [\text{crPartF}(\phi), \text{anPartF}(\phi')]_s^F$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of the operator $\phi$ , $\mathcal{S}(\sigma_1, \sigma_2)$ is the sign factor (which is $-1$ if both operators are fermionic and $1$ otherwise), and $[\cdot, \cdot]_s^F$ is the super-commutator on the free algebra.

theorem

$\text{anPartF}(\phi) \cdot \text{crPartF}(\phi') = \mathcal{S} \cdot \text{crPartF}(\phi') \cdot \text{anPartF}(\phi) + [\text{anPartF}(\phi), \text{crPartF}(\phi')]_s^F$

#anPartF_mul_crPartF_eq_superCommuteF

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , let $\text{anPartF}(\phi)$ be the annihilation part of $\phi$ and $\text{crPartF}(\phi')$ be the creation part of $\phi'$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The product of these elements satisfies the identity: $\text{anPartF}(\phi) \cdot \text{crPartF}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPartF}(\phi') \cdot \text{anPartF}(\phi) + [\text{anPartF}(\phi), \text{crPartF}(\phi')]_s^F$ where $\sigma(\phi)$ and $\sigma(\phi')$ are the field statistics (bosonic or fermionic) of the operators, and $\mathcal{S}(\sigma, \sigma')$ is the statistical sign factor which is $-1$ if both operators are fermionic and $1$ otherwise.

theorem

$\text{crPartF}(\phi) \cdot \text{crPartF}(\phi') = \mathcal{S}(\sigma, \sigma') \cdot \text{crPartF}(\phi') \cdot \text{crPartF}(\phi) + [\text{crPartF}(\phi), \text{crPartF}(\phi')]_s^F$

#crPartF_mul_crPartF_eq_superCommuteF

For a given field specification $\mathcal{F}$ and two field operators $\phi, \phi' \in \mathcal{F}.\text{FieldOp}$ , the product of their creation parts $\text{crPartF}(\phi)$ and $\text{crPartF}(\phi')$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is given by: $\text{crPartF}(\phi) \cdot \text{crPartF}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{crPartF}(\phi') \cdot \text{crPartF}(\phi) + [\text{crPartF}(\phi), \text{crPartF}(\phi')]_s^F$ where $\sigma(\phi)$ is the field statistic (bosonic or fermionic) of the operator $\phi$ , $\mathcal{S}(\sigma, \sigma')$ is the statistical sign factor (equal to $-1$ if both statistics are fermionic and $1$ otherwise), and $[ \cdot, \cdot ]_s^F$ denotes the super-commutator in the algebra.

theorem

$\text{anPartF}(\phi) \cdot \text{anPartF}(\phi') = \mathcal{S} \cdot \text{anPartF}(\phi') \cdot \text{anPartF}(\phi) + [\text{anPartF}(\phi), \text{anPartF}(\phi')]_s^F$

#anPartF_mul_anPartF_eq_superCommuteF

For any two field operators $\phi$ and $\phi'$ in a field specification $\mathcal{F}$ , the product of their annihilation parts $\text{anPartF}(\phi)$ and $\text{anPartF}(\phi')$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ can be expressed as: $\text{anPartF}(\phi) \cdot \text{anPartF}(\phi') = \mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot \text{anPartF}(\phi') \cdot \text{anPartF}(\phi) + [\text{anPartF}(\phi), \text{anPartF}(\phi')]_s^F$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) of $\phi$ , and $\mathcal{S}(\sigma_1, \sigma_2)$ is the statistical sign factor, which is $-1$ if both operators are fermionic and $1$ otherwise. Here, $[a, b]_s^F$ denotes the super-commutator in the algebra.

theorem

$V(\phi_s) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi_s) + [V(\phi_s), V(\phi_s')]_s^F$ for products of components and field operators

#ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF

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 $V(\phi_s)$ and $V(\phi_s')$ denote the corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The product of these two elements satisfies: $V(\phi_s) \cdot V(\phi_s') = \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s')) \cdot V(\phi_s') \cdot V(\phi_s) + [V(\phi_s), V(\phi_s')]_s^F$ where $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of the list, $\mathcal{S}(\sigma_1, \sigma_2)$ is the statistical sign factor (equal to $-1$ if both statistics are fermionic and $1$ otherwise), and $[ \cdot, \cdot ]_s^F$ denotes the super-commutator in the algebra.

theorem

$[V(\phi_s), V(\phi_s')]_s^F = -\mathcal{S} \cdot [V(\phi_s'), V(\phi_s)]_s^F$

#superCommuteF_ofCrAnListF_ofCrAnListF_symm

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

theorem

$[V(\phi), V(\phi')]_s^F = -\mathcal{S} \cdot [V(\phi'), V(\phi)]_s^F$ for creation and annihilation operators $\phi, \phi'$

#superCommuteF_ofCrAnOpF_ofCrAnOpF_symm

For a given field specification $\mathcal{F}$ , let $\phi$ and $\phi'$ be two creation or annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$ , and let $V(\phi)$ and $V(\phi')$ denote their corresponding generators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator $[ \cdot, \cdot ]_s^F$ satisfies the following symmetry relation: $[V(\phi), V(\phi')]_s^F = -\mathcal{S}(\sigma(\phi), \sigma(\phi')) \cdot [V(\phi'), V(\phi)]_s^F$ where $\sigma(\phi)$ denotes the field statistic (bosonic or fermionic) associated with the operator component $\phi$ , and the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

$[V(\phi_s), V(\phi :: \phi_s')]_s^F = [V(\phi_s), V(\phi)]_s^F V(\phi_s') + \mathcal{S} V(\phi) [V(\phi_s), V(\phi_s')]_s^F$

#superCommuteF_ofCrAnListF_ofCrAnListF_cons

For a given field specification $\mathcal{F}$ , let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator and let $\phi_s, \phi_s'$ be lists of such operators. Let $V(\phi_s)$ denote the product of the operators in the list $\phi_s$ within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator $[ \cdot, \cdot ]_s^F$ satisfies the following recursive relation when an operator $\phi$ is prepended to the list $\phi_s'$ (denoted $\phi :: \phi_s'$ ): $[V(\phi_s), V(\phi :: \phi_s')]_s^F = [V(\phi_s), V(\phi)]_s^F \cdot V(\phi_s') + \mathcal{S}(\sigma(\phi_s), \sigma(\phi)) \cdot V(\phi) \cdot [V(\phi_s), V(\phi_s')]_s^F$ where $\sigma(\cdot)$ denotes the collective statistic (bosonic or fermionic) of the elements, and $\mathcal{S}(\sigma_1, \sigma_2)$ is a sign factor equal to $-1$ if both statistics $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

$[V(\phi_s), V(\phi :: \phi_s')]_s^F = [V(\phi_s), V(\phi)]_s^F V(\phi_s') + \mathcal{S} V(\phi) [V(\phi_s), V(\phi_s')]_s^F$ for products of field operators

#superCommuteF_ofCrAnListF_ofFieldOpListF_cons

For a given field specification $\mathcal{F}$ , let $\phi_s$ be a list of creation and annihilation operator components, $\phi$ be a field operator, and $\phi_s'$ be a list of field operators. Let $V(\phi_s)$ and $V(\phi_s')$ denote the corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator $[\cdot, \cdot]_s^F$ satisfies the following identity when $\phi$ is prepended to the list $\phi_s'$ (denoted $\phi :: \phi_s'$ ): $[V(\phi_s), V(\phi :: \phi_s')]_s^F = [V(\phi_s), V(\phi)]_s^F \cdot V(\phi_s') + \mathcal{S}(\sigma(\phi_s), \sigma(\phi)) \cdot V(\phi) \cdot [V(\phi_s), V(\phi_s')]_s^F$ where $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of the elements, and the sign factor $\mathcal{S}(\sigma_1, \sigma_2)$ is equal to $-1$ if both statistics $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

Expansion of $[V(\phi_s), V(\phi_s')]_s^F$ as a sum over the elements of $\phi_s'$

#superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum

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 corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator $[ \cdot, \cdot ]_s^F$ of these two products satisfies the following identity: $[V(\phi_s), V(\phi_s')]_s^F = \sum_{i=0}^{|\phi_s'|-1} \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s'[0 \dots i-1])) \cdot V(\phi_s'[0 \dots i-1]) \cdot [V(\phi_s), \phi_i']_s^F \cdot V(\phi_s'[i+1 \dots |\phi_s'|-1])$ 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 $i$ -th element 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. - $\mathcal{S}(\sigma_1, \sigma_2)$ is a sign factor equal to $-1$ if both statistics $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

Expansion of $[V(\phi_s), V(\phi_s')]_s^F$ as a sum over the field operators in $\phi_s'$

#superCommuteF_ofCrAnListF_ofFieldOpListF_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 $V(\phi_s)$ and $V(\phi_s')$ denote the corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . The super-commutator $[\cdot, \cdot]_s^F$ of these two products satisfies the following identity: $[V(\phi_s), V(\phi_s')]_s^F = \sum_{i=0}^{|\phi_s'|-1} \mathcal{S}(\sigma(\phi_s), \sigma(\phi_s'[0 \dots i-1])) \cdot V(\phi_s'[0 \dots i-1]) \cdot [V(\phi_s), \phi_i']_s^F \cdot V(\phi_s'[i+1 \dots |\phi_s'|-1])$ 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 $i$ field operators of $\phi_s'$ . - $\phi_i'$ is the $i$ -th field operator of the list $\phi_s'$ . - $\phi_s'[i+1 \dots |\phi_s'|-1]$ is the suffix list consisting of the field operators in $\phi_s'$ after the $i$ -th index. - $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of a list of operators. - $\mathcal{S}(\sigma_1, \sigma_2)$ is a sign factor equal to $-1$ if both statistics $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

Super Jacobi Identity for Products of Operator Lists

#summerCommute_jacobi_ofCrAnListF

For a given field specification $\mathcal{F}$ , let $\phi_{s1}, \phi_{s2}, \phi_{s3}$ be three lists of creation and annihilation operators. Let $V(\phi_{si})$ denote the product of the operators in the list $\phi_{si}$ within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , and let $\sigma(\phi_{si})$ denote the collective field statistic of the list. The super-commutator $[a, b]_s^F$ satisfies the super Jacobi identity: $[V(\phi_{s1}), [V(\phi_{s2}), V(\phi_{s3})]_s^F]_s^F = \mathcal{S}(\sigma(\phi_{s1}), \sigma(\phi_{s3})) \cdot \left( -\mathcal{S}(\sigma(\phi_{s2}), \sigma(\phi_{s3})) \cdot [V(\phi_{s3}), [V(\phi_{s1}), V(\phi_{s2})]_s^F]_s^F - \mathcal{S}(\sigma(\phi_{s1}), \sigma(\phi_{s2})) \cdot [V(\phi_{s2}), [V(\phi_{s3}), V(\phi_{s1})]_s^F]_s^F \right)$ where $\mathcal{S}(s_a, s_b)$ is the sign factor equal to $-1$ if both statistics $s_a$ and $s_b$ are fermionic, and $1$ otherwise. This is equivalent to the cyclic form: $\mathcal{S}(\sigma(\phi_{s3}), \sigma(\phi_{s1})) [V(\phi_{s1}), [V(\phi_{s2}), V(\phi_{s3})]_s^F]_s^F + \mathcal{S}(\sigma(\phi_{s1}), \sigma(\phi_{s2})) [V(\phi_{s2}), [V(\phi_{s3}), V(\phi_{s1})]_s^F]_s^F + \mathcal{S}(\sigma(\phi_{s2}), \sigma(\phi_{s3})) [V(\phi_{s3}), [V(\phi_{s1}), V(\phi_{s2})]_s^F]_s^F = 0$

theorem

The super-commutator $[a, b]_s^F$ has statistic $f_1 + f_2$ for elements with statistics $f_1$ and $f_2$

#superCommuteF_grade

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the corresponding free algebra of field operators. Suppose $a$ and $b$ are elements of this algebra such that $a$ has the quantum statistic $f_1$ (i.e., $a$ belongs to the submodule $\text{statisticSubmodule}(f_1)$ ) and $b$ has the quantum statistic $f_2$ (i.e., $b$ belongs to the submodule $\text{statisticSubmodule}(f_2)$ ), where $f_1, f_2 \in \text{FieldStatistic}$ . Then their super-commutator $[a, b]_s^F$ has the statistic $f_1 + f_2$ . The addition of statistics follows the rules of $\mathbb{Z}_2$ parity, where $\text{bosonic}$ acts as $0$ and $\text{fermionic}$ acts as $1$ .

theorem

$[a, b]_s^F = a \cdot b - b \cdot a$ for bosonic elements $a$ and $b$

#superCommuteF_bosonic_bosonic

For any field specification $\mathcal{F}$ and any elements $a, b$ in the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , if $a$ and $b$ both belong to the bosonic submodule (meaning they have bosonic statistics), then their super-commutator $[a, b]_s^F$ is equal to the standard commutator $a \cdot b - b \cdot a$ .

theorem

$[a, b]_s^F = a \cdot b - b \cdot a$ for bosonic $a$ and fermionic $b$

#superCommuteF_bosonic_fermionic

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the associated free algebra of creation and annihilation operators over $\mathbb{C}$ . For any two elements $a, b$ in this algebra, if $a$ has bosonic statistics (i.e., $a$ belongs to the bosonic statistic submodule) and $b$ has fermionic statistics (i.e., $b$ belongs to the fermionic statistic submodule), then their super-commutator $[a, b]_s^F$ is equal to their standard commutator: $[a, b]_s^F = a \cdot b - b \cdot a$

theorem

$[a, b]_s^F = a \cdot b - b \cdot a$ for fermionic $a$ and bosonic $b$

#superCommuteF_fermionic_bonsonic

In the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ associated with a field specification $\mathcal{F}$ , let $a$ be an element with fermionic statistics ( $a \in \text{statisticSubmodule fermionic}$ ) and $b$ be an element with bosonic statistics ( $b \in \text{statisticSubmodule bosonic}$ ). Then their super-commutator $[a, b]_s^F$ is equal to their standard commutator: $[a, b]_s^F = a \cdot b - b \cdot a$

theorem

$[a, b]_s^F = a \cdot b - b \cdot a$ if $b$ is bosonic

#superCommuteF_bonsonic

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the associated free algebra of creation and annihilation operators. For any two elements $a, b$ in this algebra, if $b$ has bosonic statistics (i.e., $b$ belongs to the bosonic statistic submodule), then their super-commutator $[a, b]_s^F$ is equal to the standard commutator: $[a, b]_s^F = a \cdot b - b \cdot a$

theorem

$[a, b]_s^F = a \cdot b - b \cdot a$ for bosonic $a$

#bosonic_superCommuteF

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the associated free algebra of creation and annihilation operators. For any two elements $a, b$ in this algebra, if $a$ has bosonic statistics (i.e., $a$ belongs to the bosonic statistic submodule), then their super-commutator $[a, b]_s^F$ is equal to their standard commutator: $[a, b]_s^F = a \cdot b - b \cdot a$

theorem

$[a, b]_s^F = -[b, a]_s^F$ for bosonic $b$

#superCommuteF_bonsonic_symm

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the associated free algebra of creation and annihilation operators. For any two elements $a, b$ in this algebra, if $b$ has bosonic statistics (i.e., $b$ belongs to the bosonic statistic submodule), then the super-commutator $[a, b]_s^F$ satisfies the following symmetry relation: $[a, b]_s^F = -[b, a]_s^F$

theorem

$[a, b]_s^F = -[b, a]_s^F$ if $a$ is bosonic

#bonsonic_superCommuteF_symm

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the associated free algebra of creation and annihilation operators. For any two elements $a, b$ in this algebra, if $a$ has bosonic statistics (i.e., $a$ belongs to the bosonic statistic submodule), then the super-commutator $[a, b]_s^F$ is anti-symmetric: $[a, b]_s^F = -[b, a]_s^F$

theorem

$[a, b]_s^F = a \cdot b + b \cdot a$ for fermionic $a$ and $b$

#superCommuteF_fermionic_fermionic

For any two elements $a, b$ in the field operator free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ , if both $a$ and $b$ are fermionic (meaning they belong to the fermionic statistic submodule), then their super-commutator $[a, b]_s^F$ is equal to their anti-commutator $a \cdot b + b \cdot a$ .

theorem

$[a, b]_s^F = [b, a]_s^F$ for fermionic $a$ and $b$

#superCommuteF_fermionic_fermionic_symm

For a given field specification $\mathcal{F}$ , let $a$ and $b$ be elements of the free algebra of field operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . If both $a$ and $b$ are fermionic (meaning they belong to the fermionic statistic submodule), then their super-commutator $[a, b]_s^F$ is symmetric, satisfying: $[a, b]_s^F = [b, a]_s^F$

theorem

Expansion of $[a, b]_s^F$ via bosonic and fermionic projections

#superCommuteF_expand_bosonicProjF_fermionicProjF

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the associated free algebra of creation and annihilation operators. For any two elements $a, b$ in this algebra, let $P_B$ and $P_F$ denote the projections of an element onto the bosonic and fermionic statistic submodules, respectively. Then the super-commutator $[a, b]_s^F$ can be expanded as: $[a, b]_s^F = P_B(a) P_B(b) - P_B(b) P_B(a) + P_B(a) P_F(b) - P_F(b) P_B(a) + P_F(a) P_B(b) - P_B(b) P_F(a) + P_F(a) P_F(b) + P_F(b) P_F(a)$

theorem

The super-commutator $[A, B]_s^F$ of products of creation and annihilation operators is bosonic or fermionic

#superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic

Given a field specification $\mathcal{F}$ , let $\varphi_s$ and $\varphi_s'$ be two lists of creation and annihilation operators. Let $A = \text{ofCrAnListF}(\varphi_s)$ and $B = \text{ofCrAnListF}(\varphi_s')$ be the corresponding products of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . Then their super-commutator $[A, B]_s^F$ is either bosonic or fermionic; that is, it belongs to either $\text{statisticSubmodule}(\text{bosonic})$ or $\text{statisticSubmodule}(\text{fermionic})$ .

theorem

The super-commutator $[A, A']_s^F$ of two creation or annihilation operators is bosonic or fermionic

#superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic

Given a field specification $\mathcal{F}$ , let $\phi$ and $\phi'$ be two creation or annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$ . Let $A = \text{ofCrAnOpF}(\phi)$ and $A' = \text{ofCrAnOpF}(\phi')$ be their corresponding generators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . Then their super-commutator $[A, A']_s^F$ is either bosonic or fermionic; that is, it belongs to either the submodule $\text{statisticSubmodule}(\text{bosonic})$ or the submodule $\text{statisticSubmodule}(\text{fermionic})$ .

theorem

The double super-commutator $[A_1, [A_2, A_3]_s^F]_s^F$ of creation/annihilation operators is bosonic or fermionic

#superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic

Let $\mathcal{F}$ be a field specification and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the corresponding free algebra of field operators. Let $\phi_1, \phi_2, \phi_3 \in \mathcal{F}.\text{CrAnFieldOp}$ be three creation or annihilation operator components, and let $A_i = \text{ofCrAnOpF}(\phi_i)$ be their corresponding generators in the algebra for $i=1, 2, 3$ . Then the double super-commutator $[A_1, [A_2, A_3]_s^F]_s^F$ is either bosonic or fermionic; that is, it belongs to either the submodule $\text{statisticSubmodule}(\text{bosonic})$ or the submodule $\text{statisticSubmodule}(\text{fermionic})$ .

theorem

Expansion of $[a, \prod \phi_i]_s^F$ as a sum for bosonic $a$

#superCommuteF_bosonic_ofCrAnListF_eq_sum

Let $\mathcal{F}$ be a field specification. Let $a$ be an element of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ that is bosonic (i.e., $a$ belongs to the bosonic statistic submodule). For any list of creation and annihilation operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{m-1}]$ , the super-commutator of $a$ with the product of these operators satisfies the following identity: $[a, \text{ofCrAnListF}(\phi_s)]_s^F = \sum_{i=0}^{m-1} \text{ofCrAnListF}(\text{take } i \ \phi_s) \cdot [a, \phi_i]_s^F \cdot \text{ofCrAnListF}(\text{drop } (i+1) \ \phi_s)$ where $\text{ofCrAnListF}(\text{take } i \ \phi_s)$ is the product of the first $i$ operators in the list, $\phi_i$ is the $i$ -th operator, and $\text{ofCrAnListF}(\text{drop } (i+1) \ \phi_s)$ is the product of the remaining operators in the list after the $i$ -th one.

theorem

Expansion of $[a, \prod \phi_i]_s^F$ as a sum for fermionic $a$

#superCommuteF_fermionic_ofCrAnListF_eq_sum

Let $\mathcal{F}$ be a field specification. Let $a$ be an element of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ that is fermionic (i.e., $a$ belongs to the fermionic statistic submodule). For any list of creation and annihilation operators $\phi_s = [\phi_0, \phi_1, \dots, \phi_{m-1}]$ , the super-commutator of $a$ with the product of these operators satisfies the following identity: $[a, \text{ofCrAnListF}(\phi_s)]_s^F = \sum_{i=0}^{m-1} \mathcal{S}(\text{fermionic}, \sigma(\text{take } i \ \phi_s)) \cdot \text{ofCrAnListF}(\text{take } i \ \phi_s) \cdot [a, \phi_i]_s^F \cdot \text{ofCrAnListF}(\text{drop } (i+1) \ \phi_s)$ where $\text{ofCrAnListF}(\text{take } i \ \phi_s)$ is the product of the first $i$ operators in the list, $\phi_i$ is the $i$ -th operator, $\text{ofCrAnListF}(\text{drop } (i+1) \ \phi_s)$ is the product of the remaining operators in the list after the $i$ -th one, $\sigma(\cdot)$ denotes the collective field statistic (bosonic or fermionic) of a list of operators, and $\mathcal{S}(\sigma_1, \sigma_2)$ is a sign factor equal to $-1$ if both statistics $\sigma_1$ and $\sigma_2$ are fermionic, and $1$ otherwise.

theorem

If $[A, B]_s^F$ is fermionic, then $s(\varphi_s) \neq s(\varphi_s')$ or $[A, B]_s^F = 0$

#statistic_ne_of_superCommuteF_fermionic

Let $\mathcal{F}$ be a field specification. Let $\varphi_s$ and $\varphi_s'$ be lists of creation and annihilation operators, and let $A = \text{ofCrAnListF}(\varphi_s)$ and $B = \text{ofCrAnListF}(\varphi_s')$ be their corresponding products in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ . If the super-commutator $[A, B]_s^F$ is fermionic (i.e., it belongs to the fermionic statistic submodule), then either the collective quantum statistic of the list $\varphi_s$ is not equal to the collective quantum statistic of the list $\varphi_s'$ , or the super-commutator is zero.

Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.SuperCommute

Super-commutator [a,b]sF[a, b]_s^F[a,b]sF​ on the field operator free algebra

Notation for the super-commutator [ϕ,ψ]sF[\phi, \psi]_s^F[ϕ,ψ]sF​

[V(ϕ),V(ϕ′)]sF=V(ϕ)⋅V(ϕ′)−S⋅V(ϕ′)⋅V(ϕ)[V(\phi), V(\phi')]_s^F = V(\phi) \cdot V(\phi') - \mathcal{S} \cdot V(\phi') \cdot V(\phi)[V(ϕ),V(ϕ′)]sF​=V(ϕ)⋅V(ϕ′)−S⋅V(ϕ′)⋅V(ϕ) for creation/annihilation operators ϕ,ϕ′\phi, \phi'ϕ,ϕ′

[V(φ),V(φs)]sF=V(φ)⋅V(φs)−S⋅V(φs)⋅V(φ)[V(\varphi), V(\varphi_s)]_s^F = V(\varphi) \cdot V(\varphi_s) - \mathcal{S} \cdot V(\varphi_s) \cdot V(\varphi)[V(φ),V(φs​)]sF​=V(φ)⋅V(φs​)−S⋅V(φs​)⋅V(φ) for products of field operator lists

[V(ϕ),V(ϕs)]sF=V(ϕ)⋅V(ϕs)−S⋅V(ϕs)⋅V(ϕ)[V(\phi), V(\phi_s)]_s^F = V(\phi) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi)[V(ϕ),V(ϕs​)]sF​=V(ϕ)⋅V(ϕs​)−S⋅V(ϕs​)⋅V(ϕ) for a single field operator and a product of field operators

[V(φs),V(ϕ)]sF=V(φs)⋅V(ϕ)−S⋅V(ϕ)⋅V(φs)[V(\varphi_s), V(\phi)]_s^F = V(\varphi_s) \cdot V(\phi) - \mathcal{S} \cdot V(\phi) \cdot V(\varphi_s)[V(φs​),V(ϕ)]sF​=V(φs​)⋅V(ϕ)−S⋅V(ϕ)⋅V(φs​) for a product of field operators and a single field operator

Super-commutator of anPartF(ϕ)\text{anPartF}(\phi)anPartF(ϕ) and crPartF(ϕ′)\text{crPartF}(\phi')crPartF(ϕ′)

Super-commutator of creation parts [crPartF(ϕ),crPartF(ϕ′)]sF[\text{crPartF}(\phi), \text{crPartF}(\phi')]_s^F[crPartF(ϕ),crPartF(ϕ′)]sF​

Super-commutator of two annihilation parts [anPartF(ϕ),anPartF(ϕ′)]sF[\text{anPartF}(\phi), \text{anPartF}(\phi')]_s^F[anPartF(ϕ),anPartF(ϕ′)]sF​

Super-commutator [crPartF(ϕ),∏ϕi]sF[\text{crPartF}(\phi), \prod \phi_i]_s^F[crPartF(ϕ),∏ϕi​]sF​

Super-commutator [A(ϕ),∏ϕi]sF[A(\phi), \prod \phi_i]_s^F[A(ϕ),∏ϕi​]sF​ of an annihilation part and a product of field operators

Super-commutator [crPartF(ϕ),ofFieldOpF(ϕ′)]sF[\text{crPartF}(\phi), \text{ofFieldOpF}(\phi')]_s^F[crPartF(ϕ),ofFieldOpF(ϕ′)]sF​ of a creation part and a field operator

[A(ϕ),ϕ′]sF=A(ϕ)⋅ϕ′−S⋅ϕ′⋅A(ϕ)[A(\phi), \phi']_s^F = A(\phi) \cdot \phi' - \mathcal{S} \cdot \phi' \cdot A(\phi)[A(ϕ),ϕ′]sF​=A(ϕ)⋅ϕ′−S⋅ϕ′⋅A(ϕ) for field operators ϕ,ϕ′\phi, \phi'ϕ,ϕ′

aϕ⋅V(ϕs′)=S⋅V(ϕs′)⋅aϕ+[aϕ,V(ϕs′)]sFa_\phi \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot a_\phi + [a_\phi, V(\phi_s')]_s^Faϕ​⋅V(ϕs′​)=S⋅V(ϕs′​)⋅aϕ​+[aϕ​,V(ϕs′​)]sF​ for operator ϕ\phiϕ and operator list ϕs′\phi_s'ϕs′​

V(ϕ)⋅V(ϕs′)=S⋅V(ϕs′)⋅V(ϕ)+[V(ϕ),V(ϕs′)]sFV(\phi) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi) + [V(\phi), V(\phi_s')]_s^FV(ϕ)⋅V(ϕs′​)=S⋅V(ϕs′​)⋅V(ϕ)+[V(ϕ),V(ϕs′​)]sF​ for a field operator and a product of field operators

V(φs)⋅V(ϕ)=S⋅V(ϕ)⋅V(φs)+[V(φs),V(ϕ)]sFV(\varphi_s) \cdot V(\phi) = \mathcal{S} \cdot V(\phi) \cdot V(\varphi_s) + [V(\varphi_s), V(\phi)]_s^FV(φs​)⋅V(ϕ)=S⋅V(ϕ)⋅V(φs​)+[V(φs​),V(ϕ)]sF​ for a product of field operators and a single field operator

V(ϕs)⋅V(ϕs′)=S⋅V(ϕs′)⋅V(ϕs)+[V(ϕs),V(ϕs′)]sFV(\phi_s) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi_s) + [V(\phi_s), V(\phi_s')]_s^FV(ϕs​)⋅V(ϕs′​)=S⋅V(ϕs′​)⋅V(ϕs​)+[V(ϕs​),V(ϕs′​)]sF​ for products of components and field operators

[V(ϕs),V(ϕs′)]sF=−S⋅[V(ϕs′),V(ϕs)]sF[V(\phi_s), V(\phi_s')]_s^F = -\mathcal{S} \cdot [V(\phi_s'), V(\phi_s)]_s^F[V(ϕs​),V(ϕs′​)]sF​=−S⋅[V(ϕs′​),V(ϕs​)]sF​

[V(ϕ),V(ϕ′)]sF=−S⋅[V(ϕ′),V(ϕ)]sF[V(\phi), V(\phi')]_s^F = -\mathcal{S} \cdot [V(\phi'), V(\phi)]_s^F[V(ϕ),V(ϕ′)]sF​=−S⋅[V(ϕ′),V(ϕ)]sF​ for creation and annihilation operators ϕ,ϕ′\phi, \phi'ϕ,ϕ′

Expansion of [V(ϕs),V(ϕs′)]sF[V(\phi_s), V(\phi_s')]_s^F[V(ϕs​),V(ϕs′​)]sF​ as a sum over the elements of ϕs′\phi_s'ϕs′​

Expansion of [V(ϕs),V(ϕs′)]sF[V(\phi_s), V(\phi_s')]_s^F[V(ϕs​),V(ϕs′​)]sF​ as a sum over the field operators in ϕs′\phi_s'ϕs′​

Super Jacobi Identity for Products of Operator Lists

The super-commutator [a,b]sF[a, b]_s^F[a,b]sF​ has statistic f1+f2f_1 + f_2f1​+f2​ for elements with statistics f1f_1f1​ and f2f_2f2​

[a,b]sF=a⋅b−b⋅a[a, b]_s^F = a \cdot b - b \cdot a[a,b]sF​=a⋅b−b⋅a for bosonic elements aaa and bbb

[a,b]sF=a⋅b−b⋅a[a, b]_s^F = a \cdot b - b \cdot a[a,b]sF​=a⋅b−b⋅a for bosonic aaa and fermionic bbb

[a,b]sF=a⋅b−b⋅a[a, b]_s^F = a \cdot b - b \cdot a[a,b]sF​=a⋅b−b⋅a for fermionic aaa and bosonic bbb

[a,b]sF=a⋅b−b⋅a[a, b]_s^F = a \cdot b - b \cdot a[a,b]sF​=a⋅b−b⋅a if bbb is bosonic

[a,b]sF=a⋅b−b⋅a[a, b]_s^F = a \cdot b - b \cdot a[a,b]sF​=a⋅b−b⋅a for bosonic aaa

[a,b]sF=−[b,a]sF[a, b]_s^F = -[b, a]_s^F[a,b]sF​=−[b,a]sF​ for bosonic bbb

[a,b]sF=−[b,a]sF[a, b]_s^F = -[b, a]_s^F[a,b]sF​=−[b,a]sF​ if aaa is bosonic

[a,b]sF=a⋅b+b⋅a[a, b]_s^F = a \cdot b + b \cdot a[a,b]sF​=a⋅b+b⋅a for fermionic aaa and bbb

[a,b]sF=[b,a]sF[a, b]_s^F = [b, a]_s^F[a,b]sF​=[b,a]sF​ for fermionic aaa and bbb

Expansion of [a,b]sF[a, b]_s^F[a,b]sF​ via bosonic and fermionic projections

The super-commutator [A,B]sF[A, B]_s^F[A,B]sF​ of products of creation and annihilation operators is bosonic or fermionic

The super-commutator [A,A′]sF[A, A']_s^F[A,A′]sF​ of two creation or annihilation operators is bosonic or fermionic

The double super-commutator [A1,[A2,A3]sF]sF[A_1, [A_2, A_3]_s^F]_s^F[A1​,[A2​,A3​]sF​]sF​ of creation/annihilation operators is bosonic or fermionic

Expansion of [a,∏ϕi]sF[a, \prod \phi_i]_s^F[a,∏ϕi​]sF​ as a sum for bosonic aaa

Expansion of [a,∏ϕi]sF[a, \prod \phi_i]_s^F[a,∏ϕi​]sF​ as a sum for fermionic aaa

If [A,B]sF[A, B]_s^F[A,B]sF​ is fermionic, then s(φs)≠s(φs′)s(\varphi_s) \neq s(\varphi_s')s(φs​)=s(φs′​) or [A,B]sF=0[A, B]_s^F = 0[A,B]sF​=0

Super-commutator $[a, b]_s^F$ on the field operator free algebra

Notation for the super-commutator $[\phi, \psi]_s^F$

$[V(\phi), V(\phi')]_s^F = V(\phi) \cdot V(\phi') - \mathcal{S} \cdot V(\phi') \cdot V(\phi)$ for creation/annihilation operators $\phi, \phi'$

$[V(\varphi), V(\varphi_s)]_s^F = V(\varphi) \cdot V(\varphi_s) - \mathcal{S} \cdot V(\varphi_s) \cdot V(\varphi)$ for products of field operator lists

$[V(\phi), V(\phi_s)]_s^F = V(\phi) \cdot V(\phi_s) - \mathcal{S} \cdot V(\phi_s) \cdot V(\phi)$ for a single field operator and a product of field operators

$[V(\varphi_s), V(\phi)]_s^F = V(\varphi_s) \cdot V(\phi) - \mathcal{S} \cdot V(\phi) \cdot V(\varphi_s)$ for a product of field operators and a single field operator

Super-commutator of $\text{anPartF}(\phi)$ and $\text{crPartF}(\phi')$

Super-commutator of creation parts $[\text{crPartF}(\phi), \text{crPartF}(\phi')]_s^F$

Super-commutator of two annihilation parts $[\text{anPartF}(\phi), \text{anPartF}(\phi')]_s^F$

Super-commutator $[\text{crPartF}(\phi), \prod \phi_i]_s^F$

Super-commutator $[A(\phi), \prod \phi_i]_s^F$ of an annihilation part and a product of field operators

Super-commutator $[\text{crPartF}(\phi), \text{ofFieldOpF}(\phi')]_s^F$ of a creation part and a field operator

$[A(\phi), \phi']_s^F = A(\phi) \cdot \phi' - \mathcal{S} \cdot \phi' \cdot A(\phi)$ for field operators $\phi, \phi'$

$a_\phi \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot a_\phi + [a_\phi, V(\phi_s')]_s^F$ for operator $\phi$ and operator list $\phi_s'$

$V(\phi) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi) + [V(\phi), V(\phi_s')]_s^F$ for a field operator and a product of field operators

$V(\varphi_s) \cdot V(\phi) = \mathcal{S} \cdot V(\phi) \cdot V(\varphi_s) + [V(\varphi_s), V(\phi)]_s^F$ for a product of field operators and a single field operator

$V(\phi_s) \cdot V(\phi_s') = \mathcal{S} \cdot V(\phi_s') \cdot V(\phi_s) + [V(\phi_s), V(\phi_s')]_s^F$ for products of components and field operators

$[V(\phi_s), V(\phi_s')]_s^F = -\mathcal{S} \cdot [V(\phi_s'), V(\phi_s)]_s^F$

$[V(\phi), V(\phi')]_s^F = -\mathcal{S} \cdot [V(\phi'), V(\phi)]_s^F$ for creation and annihilation operators $\phi, \phi'$

Expansion of $[V(\phi_s), V(\phi_s')]_s^F$ as a sum over the elements of $\phi_s'$

Expansion of $[V(\phi_s), V(\phi_s')]_s^F$ as a sum over the field operators in $\phi_s'$

The super-commutator $[a, b]_s^F$ has statistic $f_1 + f_2$ for elements with statistics $f_1$ and $f_2$

$[a, b]_s^F = a \cdot b - b \cdot a$ for bosonic elements $a$ and $b$

$[a, b]_s^F = a \cdot b - b \cdot a$ for bosonic $a$ and fermionic $b$

$[a, b]_s^F = a \cdot b - b \cdot a$ for fermionic $a$ and bosonic $b$

$[a, b]_s^F = a \cdot b - b \cdot a$ if $b$ is bosonic

$[a, b]_s^F = a \cdot b - b \cdot a$ for bosonic $a$

$[a, b]_s^F = -[b, a]_s^F$ for bosonic $b$

$[a, b]_s^F = -[b, a]_s^F$ if $a$ is bosonic

$[a, b]_s^F = a \cdot b + b \cdot a$ for fermionic $a$ and $b$

$[a, b]_s^F = [b, a]_s^F$ for fermionic $a$ and $b$

Expansion of $[a, b]_s^F$ via bosonic and fermionic projections

The super-commutator $[A, B]_s^F$ of products of creation and annihilation operators is bosonic or fermionic

The super-commutator $[A, A']_s^F$ of two creation or annihilation operators is bosonic or fermionic

The double super-commutator $[A_1, [A_2, A_3]_s^F]_s^F$ of creation/annihilation operators is bosonic or fermionic

Expansion of $[a, \prod \phi_i]_s^F$ as a sum for bosonic $a$

Expansion of $[a, \prod \phi_i]_s^F$ as a sum for fermionic $a$

If $[A, B]_s^F$ is fermionic, then $s(\varphi_s) \neq s(\varphi_s')$ or $[A, B]_s^F = 0$