Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.TimeOrder

For a given field specification $\mathcal{F}$, the function $\text{timeOrderF}$ (denoted as $\mathcal{T}^f$) is the $\mathbb{C}$-linear map from the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ to itself defined by its action on the basis of operator products. For a basis element representing a product of creation and annihilation operators $\phi_1 \phi_2 \dots \phi_n$ associated with a list $\varphi_s$, the map is defined as: $$\mathcal{T}^f(\phi_1 \phi_2 \dots \phi_n) = \text{crAnTimeOrderSign}(\varphi_s) \cdot \text{ofCrAnListF}(\text{crAnTimeOrderList}(\varphi_s))$$ In other words, the operator rearranges the field operators into chronological order (where operators with later time coordinates appear first) and multiplies the result by the sign factor $(-1)^N$, where $N$ is the number of exchanges between fermionic operators required to perform the reordering.

The notation $\mathcal{T}^f(a)$ represents the time-ordering operator applied to an expression $a$ within the free algebra of field operators $\mathcal{F}$. It is defined as the application of the linear map $\text{timeOrderF}$ to the element $a$, where $\text{timeOrderF} : \text{FieldOpFreeAlgebra } \mathcal{F} \to_{\mathbb{C}} \text{FieldOpFreeAlgebra } \mathcal{F}$ is a $\mathbb{C}$-linear map that orders field operators according to their time coordinates.

For a given field specification $\mathcal{F}$, let $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$. Let $\text{ofCrAnListF}(\varphi_s)$ denote the product of these operators $\phi_1 \phi_2 \cdots \phi_n$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The action of the time-ordering linear map $\mathcal{T}^f$ on this product is given by: $$\mathcal{T}^f(\text{ofCrAnListF}(\varphi_s)) = \eta(\varphi_s) \cdot \text{ofCrAnListF}(\text{sort}(\varphi_s))$$ where $\text{sort}(\varphi_s)$ (formally `crAnTimeOrderList`) is the chronologically ordered permutation of the operators in $\varphi_s$ (arranged such that operators with later time coordinates appear first), and $\eta(\varphi_s)$ (formally `crAnTimeOrderSign`) is the sign factor $(-1)^N$ associated with the number of exchanges $N$ between fermionic operators required to reach that chronological order.

Let $\mathcal{F}$ be a field specification and let $\mathcal{T}^f$ be the time-ordering linear map on the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$. For any elements $a, b, c$ in this algebra, the time-ordering of their product is invariant under time-ordering the middle factor: $$\mathcal{T}^f(a \cdot b \cdot c) = \mathcal{T}^f(a \cdot \mathcal{T}^f(b) \cdot c)$$

Let $\mathcal{F}$ be a field specification and let $\mathcal{T}^f$ be the time-ordering linear map on the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$. For any elements $a$ and $b$ in this algebra, the time-ordering of their product is invariant under time-ordering the right factor: $$\mathcal{T}^f(a \cdot b) = \mathcal{T}^f(a \cdot \mathcal{T}^f(b))$$

Let $\mathcal{F}$ be a field specification and let $\mathcal{T}^f$ be the time-ordering linear map on the free algebra of creation and annihilation operators $\mathcal{F}.\text{FieldOpFreeAlgebra}$. For any elements $a, b$ in this algebra, the time-ordering of their product is invariant under time-ordering the left factor: $$\mathcal{T}^f(a \cdot b) = \mathcal{T}^f(\mathcal{T}^f(a) \cdot b)$$

For a given field specification $\mathcal{F}$ and a list of field operators $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n]$, the time-ordering map $\mathcal{T}^f$ acting on the product of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the product of the operators reordered chronologically, multiplied by the corresponding time-ordering sign: $$\mathcal{T}^f(\text{ofFieldOpListF}(\varphi_s)) = \text{timeOrderSign}(\varphi_s) \cdot \text{ofFieldOpListF}(\text{timeOrderList}(\varphi_s))$$ where $\text{ofFieldOpListF}(\varphi_s)$ denotes the algebraic product $\phi_1 \phi_2 \dots \phi_n$, $\text{timeOrderList}(\varphi_s)$ is the permutation of the operators sorted such that those with later time coordinates (or outgoing asymptotic states) appear first, and $\text{timeOrderSign}(\varphi_s)$ is the sign factor $\epsilon \in \{1, -1\}$ determined by the number of exchanges of fermionic operators required to achieve this chronological order.

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

For a given field specification $\mathcal{F}$ and a field operator $\phi \in \text{FieldOp}(\mathcal{F})$, the time-ordering map $\mathcal{T}^f$ acting on the representation of the operator $\phi$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the operator itself: $$\mathcal{T}^f(\phi) = \phi$$

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 $\psi$ (i.e., the relation $\text{timeOrderRel}(\phi, \psi)$ holds), then the time-ordering map $\mathcal{T}^f$ applied to their product in the field operator free algebra is simply the product itself: $$\mathcal{T}^f(\phi \cdot \psi) = \phi \cdot \psi$$ where $\phi \cdot \psi$ denotes the product of the representations of the field operators in the algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$.

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 $\mathcal{T}^f$ on the product of their representations in the free algebra is given by: $$\mathcal{T}^f(\text{ofFieldOpF}(\phi) \cdot \text{ofFieldOpF}(\psi)) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \psi) \cdot \text{ofFieldOpF}(\psi) \cdot \text{ofFieldOpF}(\phi)$$ where $\text{ofFieldOpF}(\phi)$ is the representation of the field operator $\phi$ in the algebra (the sum of its creation and annihilation components), $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of the operator, and $\mathcal{S}$ is the sign factor $\pm 1$ determined by the statistics of $\phi$ and $\psi$ according to the Koszul convention.

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 $\mathcal{T}^f$ on their product in the free algebra satisfies: $$\mathcal{T}^f(\text{ofFieldOpF}(\phi) \cdot \text{ofFieldOpF}(\psi)) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \psi) \cdot \mathcal{T}^f(\text{ofFieldOpF}(\psi) \cdot \text{ofFieldOpF}(\phi))$$ where $\text{ofFieldOpF}(\phi)$ is the representation of the field operator in the free algebra, $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of the operator, and $\mathcal{S}$ is the sign factor $\pm 1$ determined by the statistics of $\phi$ and $\psi$ according to the Koszul convention.

For a given field specification $\mathcal{F}$, let $\varphi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be creation and annihilation field operators. If $\varphi$ is not chronologically later than or equal to $\psi$ (meaning the relation $\text{crAnTimeOrderRel}(\varphi, \psi)$ does not hold), then the action of the time-ordering linear map $\mathcal{T}^f$ on the super-commutator $[V(\varphi), V(\psi)]_s^F$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is zero: $$\mathcal{T}^f([V(\varphi), V(\psi)]_s^F) = 0$$ where $V(\cdot)$ denotes the map from an operator component to its generator in the algebra.

For a given field specification $\mathcal{F}$, let $\varphi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be creation and annihilation field operators, and let $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be an arbitrary element of the free algebra. If $\varphi$ is not chronologically later than or equal to $\psi$ (meaning the relation $\text{crAnTimeOrderRel}(\varphi, \psi)$ does not hold), then the time-ordering linear map $\mathcal{T}^f$ applied to the product of $a$ and the super-commutator $[V(\varphi), V(\psi)]_s^F$ is zero: $$\mathcal{T}^f(a \cdot [V(\varphi), V(\psi)]_s^F) = 0$$ where $V(\cdot)$ denotes the mapping of an operator component to its corresponding generator in the algebra.

For a given field specification $\mathcal{F}$, let $\varphi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ be creation and annihilation field operators, and let $a \in \mathcal{F}.\text{FieldOpFreeAlgebra}$ be an arbitrary element of the free algebra. If $\varphi$ is not chronologically later than or equal to $\psi$ (meaning the relation $\text{crAnTimeOrderRel}(\varphi, \psi)$ does not hold), then the action of the time-ordering linear map $\mathcal{T}^f$ on the product of their super-commutator $[V(\varphi), V(\psi)]_s^F$ and the element $a$ is zero: $$\mathcal{T}^f([V(\varphi), V(\psi)]_s^F \cdot a) = 0$$ where $V(\cdot)$ denotes the map from an operator component to its generator in the algebra.

Let $\mathcal{F}$ be a field specification. For any creation and annihilation field operators $\varphi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$ and any elements $a, b$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, if $\varphi$ is not chronologically later than or equal to $\psi$ (i.e., the relation $\text{crAnTimeOrderRel}(\varphi, \psi)$ does not hold), then the time-ordering of the product $a \cdot [V(\varphi), V(\psi)]_s^F \cdot b$ is zero: $$\mathcal{T}^f(a \cdot [V(\varphi), V(\psi)]_s^F \cdot b) = 0$$ where $V(\cdot)$ denotes the map from an operator component to its generator in the free algebra, $[ \cdot, \cdot ]_s^F$ is the super-commutator, and $\mathcal{T}^f$ is the time-ordering linear map.

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 $V(\varphi)$ denote the generator in $\mathcal{A}_{\mathcal{F}}$ corresponding to an operator $\varphi$, and let $[\cdot, \cdot]_s^F$ denote the super-commutator. For any two creation and annihilation operators $\varphi_1, \varphi_2 \in \mathcal{F}.\text{CrAnFieldOp}$ and any element $a \in \mathcal{A}_{\mathcal{F}}$, if $\varphi_1$ is not chronologically later than or equal to $\varphi_2$ (i.e., the relation $\text{crAnTimeOrderRel}(\varphi_1, \varphi_2)$ does not hold), then the time-ordering linear map $\mathcal{T}^f$ applied to the nested super-commutator of $a$ with $[V(\varphi_1), V(\varphi_2)]_s^F$ is zero: $$\mathcal{T}^f([a, [V(\varphi_1), V(\varphi_2)]_s^F]_s^F) = 0$$

Let $\mathcal{F}$ be a field specification. For any creation and annihilation field operators $\varphi_1, \varphi_2, \varphi_3 \in \mathcal{F}.\text{CrAnFieldOp}$, let $V(\varphi_i)$ denote the corresponding generator in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, let $[\cdot, \cdot]_s^F$ denote the super-commutator, and let $\mathcal{T}^f$ be the time-ordering linear map. If $\varphi_1$ is not chronologically later than or equal to $\varphi_2$ and $\varphi_1$ is not chronologically later than or equal to $\varphi_3$ (i.e., $\neg \text{crAnTimeOrderRel}(\varphi_1, \varphi_2)$ and $\neg \text{crAnTimeOrderRel}(\varphi_1, \varphi_3)$), then the time-ordering of the nested super-commutator is zero: $$\mathcal{T}^f([V(\varphi_1), [V(\varphi_2), V(\varphi_3)]_s^F]_s^F) = 0$$

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 $V(\varphi)$ denote the generator in $\mathcal{A}_{\mathcal{F}}$ corresponding to an operator $\varphi \in \mathcal{F}.\text{CrAnFieldOp}$, and let $[\cdot, \cdot]_s^F$ denote the super-commutator. Let $\mathcal{T}^f$ denote the time-ordering linear map. For any three creation and annihilation operators $\varphi_1, \varphi_2, \varphi_3 \in \mathcal{F}.\text{CrAnFieldOp}$, if $\varphi_1$ is chronologically later than both $\varphi_2$ and $\varphi_3$ (specifically, if the relations $\text{crAnTimeOrderRel}(\varphi_2, \varphi_1)$ and $\text{crAnTimeOrderRel}(\varphi_3, \varphi_1)$ do not hold), then the time-ordering map applied to the nested super-commutator of $V(\varphi_1)$ with $[V(\varphi_2), V(\varphi_3)]_s^F$ is zero: $$\mathcal{T}^f([V(\varphi_1), [V(\varphi_2), V(\varphi_3)]_s^F]_s^F) = 0$$

Let $\mathcal{F}$ be a field specification. For any creation and annihilation field operators $\varphi_1, \varphi_2, \varphi_3 \in \mathcal{F}.\text{CrAnFieldOp}$, let $V(\varphi_i)$ denote the corresponding generator in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, $[\cdot, \cdot]_s^F$ denote the super-commutator, and $\mathcal{T}^f$ denote the time-ordering linear map. If it is not the case that $\varphi_1, \varphi_2,$ and $\varphi_3$ are all mutually simultaneous (meaning the conjunction of the time-ordering relations $\text{crAnTimeOrderRel}(\varphi_i, \varphi_j)$ for all $i, j \in \{1, 2, 3\}$ is false), then the time-ordering of the nested super-commutator is zero: $$\mathcal{T}^f([V(\varphi_1), [V(\varphi_2), V(\varphi_3)]_s^F]_s^F) = 0$$

For a given field specification $\mathcal{F}$ and creation or annihilation operators $\phi, \psi \in \mathcal{F}.\text{CrAnFieldOp}$, if $\phi$ and $\psi$ are simultaneous (meaning both $\text{crAnTimeOrderRel}(\phi, \psi)$ and $\text{crAnTimeOrderRel}(\psi, \phi)$ hold), then the action of the time-ordering map $\mathcal{T}^f$ on their super-commutator $[V(\phi), V(\psi)]_s^F$ is equal to the super-commutator itself: $$\mathcal{T}^f([V(\phi), V(\psi)]_s^F) = [V(\phi), V(\psi)]_s^F$$ where $V(\phi)$ and $V(\psi)$ denote the generators of the operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$.

For a given field specification $\mathcal{F}$, let $L = [\phi, \phi_1, \dots, \phi_n]$ be a sequence of field operators (where $\phi$ is the head and $\phi_s$ is the tail). Let $\phi_{\text{max}}$ be the chronologically latest operator in $L$ according to the relation $\text{timeOrderRel}$, and let $L_{<k}$ be the sub-sequence of operators appearing in $L$ before the first occurrence of $\phi_{\text{max}}$. The time-ordering linear map $\mathcal{T}^f$ acting on the algebraic product of these operators satisfies: $$\mathcal{T}^f(\phi \cdot \phi_1 \cdot \dots \cdot \phi_n) = \mathcal{S}(\mathcal{F} \triangleright_s \phi_{\text{max}}, \mathcal{F} \triangleright_s L_{<k}) \cdot \phi_{\text{max}} \cdot \mathcal{T}^f(\text{product of } L \setminus \{\phi_{\text{max}}\})$$ where $\mathcal{S}$ is the exchange sign factor ($\pm 1$) determined by the statistics (bosonic or fermionic) of $\phi_{\text{max}}$ and the preceding sub-sequence $L_{<k}$, and the product on the right-hand side is the time-ordering of the operators remaining after the first occurrence of $\phi_{\text{max}}$ is removed.

For a given field specification $\mathcal{F}$, let $L = [\phi, \phi_1, \dots, \phi_n]$ be a sequence of field operators in $\text{FieldOp}(\mathcal{F})$. Let $\phi_{\text{max}}$ be the chronologically latest operator in $L$ according to the relation $\text{timeOrderRel}$ (choosing the leftmost if there are ties), and let $L \setminus \{\phi_{\text{max}}\}$ be the sequence with its first occurrence removed. The time-ordering linear map $\mathcal{T}^f$ acting on the algebraic product of these operators satisfies: $$\mathcal{T}^f(\phi \cdot \phi_1 \cdot \dots \cdot \phi_n) = \mathcal{S}(\mathcal{F} \triangleright_s \phi_{\text{max}}, \sigma_{<k}) \cdot \text{ofFieldOpF}(\phi_{\text{max}}) \cdot \mathcal{T}^f(\text{product of } L \setminus \{\phi_{\text{max}}\})$$ where: - $\mathcal{S}$ is the exchange sign factor ($\pm 1$). - $\mathcal{F} \triangleright_s \phi_{\text{max}}$ is the statistic of the maximal operator. - $\sigma_{<k}$ is the collective statistic of the subset of operators that appeared before $\phi_{\text{max}}$ in the original list $L$, calculated using finite sets.