Physlib.QFT.PerturbationTheory.WickAlgebra.TimeOrder

For a given field specification $\mathcal{F}$, let $\varphi_1, \varphi_2, \varphi_3 \in \mathcal{F}.\text{CrAnFieldOp}$ be creation or annihilation field operators. Suppose these three operators are mutually simultaneous according to the time-ordering relation, such that $\text{crAnTimeOrderRel}(\varphi_i, \varphi_j)$ holds for all $i, j \in \{1, 2, 3\}$. Then, for any lists of operators $\varphi_{s1}$ and $\varphi_{s2}$, the image in the Wick algebra (via the canonical map $\iota$) of the time-ordered product containing the nested super-commutator $[\varphi_1, [\varphi_2, \varphi_3]_s^F]_s^F$ is zero: $$\iota \left( \mathcal{T}^f \left( \text{ofCrAnListF}(\varphi_{s1}) \cdot [\text{ofCrAnOpF}(\varphi_1), [\text{ofCrAnOpF}(\varphi_2), \text{ofCrAnOpF}(\varphi_3)]_s^F]_s^F \cdot \text{ofCrAnListF}(\varphi_{s2}) \right) \right) = 0$$ where $\mathcal{T}^f$ is the time-ordering linear map and $[a, b]_s^F$ denotes the super-commutator in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$.

For a given field specification $\mathcal{F}$, let $\varphi_1, \varphi_2, \varphi_3 \in \mathcal{F}.\text{CrAnFieldOp}$ be creation and annihilation field operators, and let $\varphi_{s1}, \varphi_{s2}$ be lists of such operators. Let $\text{ofCrAnOpF}(\varphi)$ denote the generator of the free algebra corresponding to $\varphi$, $[ \cdot, \cdot ]_s^F$ denote the super-commutator, and $\text{ofCrAnListF}(\varphi_s)$ denote the product of operators in a list. Then the time-ordering $\mathcal{T}^f$ of the product of $\varphi_{s1}$, the nested super-commutator of $\varphi_1, \varphi_2$, and $\varphi_3$, and the product of $\varphi_{s2}$ vanishes in the Wick algebra: $$\iota \left( \mathcal{T}^f \left( \text{ofCrAnListF}(\varphi_{s1}) \cdot [\text{ofCrAnOpF}(\varphi_1), [\text{ofCrAnOpF}(\varphi_2), \text{ofCrAnOpF}(\varphi_3)]_s^F]_s^F \cdot \text{ofCrAnListF}(\varphi_{s2}) \right) \right) = 0$$ where $\iota$ is the canonical map from the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to the Wick algebra.

For a given field specification $\mathcal{F}$, let $\phi_1, \phi_2, \phi_3 \in \mathcal{F}.\text{CrAnFieldOp}$ be creation or annihilation field operators, and let $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be arbitrary elements of the free algebra. The identity states that the image in the Wick algebra of the time-ordered product of $a$, the nested super-commutator of the three operators, and $b$ is zero: $$\iota \left( \mathcal{T}^f \left( a \cdot [\text{ofCrAnOpF}(\phi_1), [\text{ofCrAnOpF}(\phi_2), \text{ofCrAnOpF}(\phi_3)]_s^F]_s^F \cdot b \right) \right) = 0$$ where $\mathcal{T}^f$ is the time-ordering linear map, $[\cdot, \cdot]_s^F$ denotes the super-commutator, $\text{ofCrAnOpF}$ maps an operator to its generator in the free algebra, and $\iota$ is the canonical map from the free algebra to the Wick algebra.

Let $\mathcal{F}$ be a field specification. Let $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components that are contemporary, meaning the relations $\text{crAnTimeOrderRel}(\phi, \psi)$ and $\text{crAnTimeOrderRel}(\psi, \phi)$ both hold. For any elements $a, b \in \mathcal{F}.\text{FieldOpFreeAlgebra}$, the following identity holds for the time-ordering map $\mathcal{T}^f$ in the Wick algebra (represented by the map $\iota$): $$\iota(\mathcal{T}^f(a \cdot [V(\phi), V(\psi)]_s^F \cdot b)) = \iota([V(\phi), V(\psi)]_s^F \cdot \mathcal{T}^f(a \cdot b))$$ where $V(\phi)$ and $V(\psi)$ are the representations of the operators in the free algebra, $[ \cdot, \cdot ]_s^F$ denotes the super-commutator, and $\iota$ is the canonical map from the free algebra to the Wick algebra.

Let $\mathcal{F}$ be a field specification. Let $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components that are not contemporary, meaning the condition $\text{crAnTimeOrderRel}(\phi, \psi) \wedge \text{crAnTimeOrderRel}(\psi, \phi)$ is false (i.e., they do not occur at the same time). For any elements $a, b$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the action of the time-ordering map $\mathcal{T}^f$ followed by the canonical map $\iota$ to the Wick algebra on the product $a \cdot [V(\phi), V(\psi)]_s^F \cdot b$ is zero: $$\iota(\mathcal{T}^f(a \cdot [V(\phi), V(\psi)]_s^F \cdot b)) = 0$$ where $V(\cdot)$ maps an operator component to its generator in the free algebra and $[\cdot, \cdot]_s^F$ denotes the super-commutator.

Let $\mathcal{F}$ be a field specification. Let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by the creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$. For any element $a$ in the two-sided ideal generated by the set of field operator relations $\mathcal{F}.\text{fieldOpIdealSet}$, the image of its time-ordered product in the Wick algebra is zero: $$\iota(\mathcal{T}^f(a)) = 0$$ where $\mathcal{T}^f$ is the time-ordering linear map on the free algebra and $\iota$ is the canonical map from the free algebra to the Wick algebra.

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 operator components. For any two elements $a, b$ in the free algebra such that $a \approx b$ (meaning they are equivalent modulo the two-sided ideal generated by the field operator relations), the image of their time-ordered products in the Wick algebra are equal: $$\iota(\mathcal{T}^f(a)) = \iota(\mathcal{T}^f(b))$$ where $\mathcal{T}^f$ is the time-ordering linear map on the free algebra and $\iota$ is the canonical quotient map from the free algebra to the Wick algebra.

For a given field specification $\mathcal{F}$, the time-ordering operator $\mathcal{T}$ is the $\mathbb{C}$-linear map from the Wick algebra $\text{WickAlgebra } \mathcal{F}$ to itself. It is defined as the descent of the composition $\iota \circ \mathcal{T}^f$ to the quotient, where $\mathcal{T}^f$ is the time-ordering map on the free algebra $\text{FieldOpFreeAlgebra } \mathcal{F}$ and $\iota$ is the canonical quotient map into the Wick algebra. This map is well-defined because $\iota(\mathcal{T}^f(a)) = \iota(\mathcal{T}^f(b))$ whenever $a \approx b$ in the free algebra. For any element $a$ in the Wick algebra, the action is denoted by $\mathcal{T}(a)$.

The notation $\mathcal{T}(a)$ represents the time-ordering operator applied to an element $a$ in the Wick algebra of a field specification $\mathcal{F}$. It serves as a shorthand for the linear map $\text{timeOrder}(a)$.

For any element $a$ in the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$, the time-ordering operator $\mathcal{T}$ acting on the image of $a$ in the Wick algebra (via the canonical quotient map $\iota$) is equal to the image of the time-ordering operator $\mathcal{T}^f$ acting on $a$ in the free algebra. That is, $\mathcal{T}(\iota(a)) = \iota(\mathcal{T}^f(a))$.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$, if $\phi$ is chronologically later than or equal to $\psi$ (such that the relation $\text{timeOrderRel}(\phi, \psi)$ holds), then the time-ordering operator $\mathcal{T}$ acting on their product in the Wick algebra $\text{WickAlgebra } \mathcal{F}$ satisfies: $$\mathcal{T}(\phi \cdot \psi) = \phi \cdot \psi$$ where $\phi \cdot \psi$ denotes the product of the representations of the field operators in the Wick algebra.

For a given field specification $\mathcal{F}$ and two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$, if $\phi$ is not chronologically later than or equal to $\psi$ (i.e., $\neg \text{timeOrderRel}(\phi, \psi)$), then the action of the time-ordering operator $\mathcal{T}$ on their product in the Wick algebra is given by: $$\mathcal{T}(\text{ofFieldOp}(\phi) \cdot \text{ofFieldOp}(\psi)) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \psi) \cdot \text{ofFieldOp}(\psi) \cdot \text{ofFieldOp}(\phi)$$ where $\text{ofFieldOp}(\phi)$ and $\text{ofFieldOp}(\psi)$ are the images of the field operators in the Wick algebra, $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of the operator, and $\mathcal{S}$ is the exchange sign factor $\pm 1$ determined by the statistics of $\phi$ and $\psi$.

For a given field specification $\mathcal{F}$ and two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$, if $\phi$ is not chronologically later than or equal to $\psi$ (i.e., the relation $\text{timeOrderRel}(\phi, \psi)$ does not hold), then the action of the time-ordering linear map $\mathcal{T}$ on their product in the Wick algebra satisfies: $$\mathcal{T}(\text{ofFieldOp}(\phi) \cdot \text{ofFieldOp}(\psi)) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \psi) \cdot \mathcal{T}(\text{ofFieldOp}(\psi) \cdot \text{ofFieldOp}(\phi))$$ where $\text{ofFieldOp}(\phi)$ is the representation of the field operator in the Wick algebra, $\mathcal{F} \triangleright_s \phi$ denotes its field statistic (bosonic or fermionic), and $\mathcal{S}$ is the exchange sign $\pm 1$ determined by the statistics of $\phi$ and $\psi$.

For a given field specification $\mathcal{F}$, let $\mathcal{T}$ be the time-ordering linear map on the Wick algebra. The time-ordering operator applied to the identity element of the Wick algebra (represented by the algebraic product of an empty list of field operators) is equal to $1$. Mathematically, $\mathcal{T}(\text{ofFieldOpList}([])) = 1$.

For any field specification $\mathcal{F}$ and any field operator $\phi \in \text{FieldOp}(\mathcal{F})$, the time-ordering operator $\mathcal{T}$ acting on the product of a single field operator $\phi$ (represented in the Wick algebra) is equal to the field operator itself: $$\mathcal{T}(\phi) = \phi$$

For a given field specification $\mathcal{F}$, let $L = [\phi, \phi_1, \dots, \phi_n]$ be a list of field operators. Let $\phi_{\text{max}}$ be the chronologically latest operator in $L$ according to the time-ordering relation (taking the leftmost occurrence if multiple operators have the same maximum time), and let $L \setminus \{\phi_{\text{max}}\}$ denote the list after removing this first occurrence. The time-ordering operator $\mathcal{T}$ acting on the product of the operators in $L$ satisfies the recursive identity: $$\mathcal{T}(\phi \cdot \phi_1 \cdot \dots \cdot \phi_n) = \mathcal{S}(\mathcal{F} \triangleright_s \phi_{\text{max}}, \sigma_{<k}) \cdot \phi_{\text{max}} \cdot \mathcal{T}(\text{product of } L \setminus \{\phi_{\text{max}}\})$$ where: - $\mathcal{S}$ is the exchange sign factor reflecting the commutation or anti-commutation of the fields. - $\mathcal{F} \triangleright_s \phi_{\text{max}}$ is the field statistic (bosonic or fermionic) of the maximal operator. - $\sigma_{<k}$ is the collective statistic of the sub-sequence of operators that appeared to the left of $\phi_{\text{max}}$ in the original list $L$, represented here by a filtered finite set of indices.

Let $\mathcal{F}$ be a field specification. Let $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components that are contemporary, meaning that both the time-ordering relations $\text{crAnTimeOrderRel}(\phi, \psi)$ and $\text{crAnTimeOrderRel}(\psi, \phi)$ hold. For any elements $a, b$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the time-ordering operator $\mathcal{T}$ satisfies the following identity involving the super-commutator $[ \cdot, \cdot ]_s$: $$\mathcal{T}(a \cdot [\text{ofCrAnOp}(\phi), \text{ofCrAnOp}(\psi)]_s \cdot b) = [\text{ofCrAnOp}(\phi), \text{ofCrAnOp}(\psi)]_s \cdot \mathcal{T}(a \cdot b)$$ where $\text{ofCrAnOp}$ maps the components into the Wick algebra.

Let $\mathcal{F}$ be a field specification. Let $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be two creation or annihilation operator components that are contemporary, meaning that both the time-ordering relations $\text{crAnTimeOrderRel}(\phi, \psi)$ and $\text{crAnTimeOrderRel}(\psi, \phi)$ hold. For any element $b$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the time-ordering operator $\mathcal{T}$ satisfies the following identity: $$\mathcal{T}([\text{ofCrAnOp}(\phi), \text{ofCrAnOp}(\psi)]_s \cdot b) = [\text{ofCrAnOp}(\phi), \text{ofCrAnOp}(\psi)]_s \cdot \mathcal{T}(b)$$ where $[\cdot, \cdot]_s$ denotes the super-commutator and $\text{ofCrAnOp}$ maps the components into the Wick algebra.

Let $\mathcal{F}$ be a field specification. For any creation or annihilation operator components $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ that are not contemporary (i.e., the condition $\text{crAnTimeOrderRel}(\phi, \psi) \wedge \text{crAnTimeOrderRel}(\psi, \phi)$ does not hold), the time-ordering operator $\mathcal{T}$ applied to their super-commutator in the Wick algebra is zero: $$\mathcal{T}([\text{ofCrAnOp } \phi, \text{ofCrAnOp } \psi]_s) = 0$$ where $[\cdot, \cdot]_s$ denotes the super-commutator in the Wick algebra.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$ that are not contemporary (meaning the condition $\text{timeOrderRel}(\phi, \psi) \wedge \text{timeOrderRel}(\psi, \phi)$ is false), the time-ordering operator $\mathcal{T}$ applied to the super-commutator of the annihilation part of $\phi$ and the operator $\psi$ in the Wick algebra is zero: $$\mathcal{T}([\text{anPart } \phi, \text{ofFieldOp } \psi]_s) = 0$$ where $\text{anPart } \phi$ denotes the annihilation component of the field operator and $[\cdot, \cdot]_s$ denotes the super-commutator.

Let $\mathcal{F}$ be a field specification and $\mathcal{W}(\mathcal{F})$ be its associated Wick algebra. For any elements $a, b, c \in \mathcal{W}(\mathcal{F})$, let $\mathcal{T}$ denote the time-ordering linear map. The time-ordering of the product of these three elements is invariant under the time-ordering of the middle factor: $$\mathcal{T}(a \cdot b \cdot c) = \mathcal{T}(a \cdot \mathcal{T}(b) \cdot c)$$

Let $\mathcal{F}$ be a field specification and $\mathcal{W}(\mathcal{F})$ be its associated Wick algebra. For any elements $b, c \in \mathcal{W}(\mathcal{F})$, let $\mathcal{T}$ denote the time-ordering linear map. The time-ordering of the product of these two elements satisfies: $$\mathcal{T}(b \cdot c) = \mathcal{T}(\mathcal{T}(b) \cdot c)$$

Let $\mathcal{F}$ be a field specification and $\text{WickAlgebra } \mathcal{F}$ its corresponding Wick algebra. For any elements $a, b \in \text{WickAlgebra } \mathcal{F}$, let $\mathcal{T}$ denote the time-ordering linear map. The time-ordering of the product of two elements is equal to the time-ordering of the product of the first element and the time-ordered second element: $$\mathcal{T}(a \cdot b) = \mathcal{T}(a \cdot \mathcal{T}(b))$$

Let $\mathcal{F}$ be a field specification and $\text{WickAlgebra } \mathcal{F}$ be its associated Wick algebra. For any element $a$ in the Wick algebra, the time-ordering linear map $\mathcal{T}$ satisfies: $$\mathcal{T}(\mathcal{T}(a)) = \mathcal{T}(a)$$ This demonstrates that the time-ordering operator is idempotent and acts as a projection.