Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.Basic

For a given field specification $\mathcal{F}$, the algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is the free associative algebra over the complex numbers $\mathbb{C}$ generated by the set of creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$.

For a given field specification $\mathcal{F}$, this function maps a creation or annihilation operator component $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ to its corresponding generator in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$.

For a given field specification $\mathcal{F}$, let $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over the complex numbers $\mathbb{C}$ generated by the set of creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$, and let $i : \mathcal{F}.\text{CrAnFieldOp} \to \mathcal{F}.\text{FieldOpFreeAlgebra}$ be the canonical map (given by `ofCrAnOpF`) that identifies each component with its generator in the algebra. The algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ satisfies the following universal property: for any $\mathbb{C}$-algebra $A$ and any map $f : \mathcal{F}.\text{CrAnFieldOp} \to A$, there exists a unique $\mathbb{C}$-algebra homomorphism $g : \mathcal{F}.\text{FieldOpFreeAlgebra} \to A$ such that $g \circ i = f$.

For a given field specification $\mathcal{F}$, let $\mathcal{F}.\text{CrAnFieldOp}$ be the set of creation and annihilation operator components. The function $\text{ofCrAnListF}$ maps a list of these components $[\phi_1, \phi_2, \dots, \phi_n]$ to the corresponding product in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$. Specifically, the mapping is defined as: $$\text{ofCrAnListF}([\phi_1, \phi_2, \dots, \phi_n]) = \text{ofCrAnOpF}(\phi_1) \cdot \text{ofCrAnOpF}(\phi_2) \cdot \dots \cdot \text{ofCrAnOpF}(\phi_n)$$ where an empty list maps to the identity element $1$. The set of all such products for all possible lists of creation and annihilation operators forms a basis for the algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$.

For a given field specification $\mathcal{F}$, the map $\text{ofCrAnListF}$, which sends a list of creation and annihilation operator components to their product in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, maps the empty list $[]$ to the identity element $1$ of the algebra.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator and let $\phi s$ be a list of such operators. The function $\text{ofCrAnListF}$, which maps a list of operators to their product in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, satisfies the recursive relation: $$\text{ofCrAnListF}(\phi :: \phi s) = \text{ofCrAnOpF}(\phi) \cdot \text{ofCrAnListF}(\phi s)$$ where $\phi :: \phi s$ denotes the list formed by prepending the operator $\phi$ to the list $\phi s$.

For a given field specification $\mathcal{F}$, let $\phi_s$ and $\phi_s'$ be two lists of creation and annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$. Then the mapping $\text{ofCrAnListF}$ to the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ satisfies: $$\text{ofCrAnListF}(\phi_s \mathbin{+\!+} \phi_s') = \text{ofCrAnListF}(\phi_s) \cdot \text{ofCrAnListF}(\phi_s')$$ where $\mathbin{+\!+}$ denotes the concatenation of lists and $\cdot$ denotes the multiplication operation in the algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$.

For a given field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{CrAnFieldOp}$ be a creation or annihilation operator component. Then, in the free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$, the product of the singleton list containing $\phi$ is equal to the generator corresponding to $\phi$ itself. This is expressed as: $$\text{ofCrAnListF}([\phi]) = \text{ofCrAnOpF}(\phi)$$ where $\text{ofCrAnListF}$ is the map from a list of components to their product in the algebra and $\text{ofCrAnOpF}$ maps a single component to its representation in the algebra.

For a given field specification $\mathcal{F}$ and a field operator $\phi \in \mathcal{F}.\text{FieldOp}$, the function `ofFieldOpF` maps $\phi$ to an element in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ by summing its creation and annihilation components. The definition is given by: $$\text{ofFieldOpF}(\phi) = \sum_{i \in \mathcal{F}.\text{fieldOpToCrAnType}(\phi)} \text{ofCrAnOpF}(\langle \phi, i \rangle)$$ where $\mathcal{F}.\text{fieldOpToCrAnType}(\phi)$ denotes the set of available modes for $\phi$. - For a position-space field operator, this sum represents the decomposition into its creation and annihilation parts: $\phi = \phi_{\text{create}} + \phi_{\text{annihilate}}$. - For an incoming or outgoing asymptotic field operator, which possesses only a single mode (creation or annihilation respectively), the sum reduces to the single corresponding operator.

Given a field specification $\mathcal{F}$ and a list of field operators $[\phi_1, \phi_2, \dots, \phi_n]$ where $\phi_i \in \mathcal{F}.\text{FieldOp}$, the function `ofFieldOpListF` computes the product of these operators within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The result is defined by the ordered product: $$\text{ofFieldOpListF}([\phi_1, \phi_2, \dots, \phi_n]) = \prod_{i=1}^n \text{ofFieldOpF}(\phi_i) = \text{ofFieldOpF}(\phi_1) \cdot \text{ofFieldOpF}(\phi_2) \cdot \dots \cdot \text{ofFieldOpF}(\phi_n)$$ where $\text{ofFieldOpF}(\phi_i)$ is the representation of the field operator $\phi_i$ in the algebra (the sum of its creation and annihilation components). For an empty list, the function returns the identity element $1$ of the algebra.

This coercion maps a list of field operators $[\phi_1, \phi_2, \dots, \phi_n]$, where each $\phi_i \in \text{FieldOp}(\mathcal{F})$, to an element of the free algebra $\text{FieldOpFreeAlgebra}(\mathcal{F})$ by computing their ordered algebraic product $\prod_{i=1}^n \phi_i$. The conversion is performed using the function `ofFieldOpListF`, where each $\phi_i$ is treated as the sum of its creation and annihilation components.

For a given field specification $\mathcal{F}$, the algebraic product of an empty list of field operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is the identity element $1$. That is, $\text{ofFieldOpListF}([]) = 1$.

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. The representation of the concatenated list $\phi :: \phi_s$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the product of the representation of the single operator $\phi$ and the representation of the list $\phi_s$: $$\text{ofFieldOpListF}(\phi :: \phi_s) = \text{ofFieldOpF}(\phi) \cdot \text{ofFieldOpListF}(\phi_s)$$ where $\text{ofFieldOpListF}$ is the map from a list of field operators to their ordered product in the algebra, and $\text{ofFieldOpF}(\phi)$ is the sum of the creation and annihilation components of the operator $\phi$.

Given a field specification $\mathcal{F}$, let $\phi \in \mathcal{F}.\text{FieldOp}$ be a field operator. The representation of the singleton list $[\phi]$ in the creation and annihilation free algebra, denoted by $\text{ofFieldOpListF}([\phi])$, is equal to the representation of the single field operator $\phi$ in the algebra, denoted by $\text{ofFieldOpF}(\phi)$.

For a given field specification $\mathcal{F}$ and two lists of field operators $\phi_s, \phi'_s \in \text{List}(\mathcal{F}.\text{FieldOp})$, the representation of the concatenated list $\phi_s ++ \phi'_s$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the product of the representations of the individual lists $\phi_s$ and $\phi'_s$: $$\text{ofFieldOpListF}(\phi_s \text{ ++ } \phi'_s) = \text{ofFieldOpListF}(\phi_s) \cdot \text{ofFieldOpListF}(\phi'_s)$$ where $\text{ofFieldOpListF}$ is the map that takes a list of field operators $[\phi_1, \dots, \phi_n]$ to the ordered product $\prod_{i=1}^n \text{ofFieldOpF}(\phi_i)$ in the creation and annihilation algebra.

For a given field specification $\mathcal{F}$ and a list of field operators $\varphi_s = [\phi_1, \phi_2, \dots, \phi_n] \in \text{List}(\mathcal{F}.\text{FieldOp})$, the representation of the product of these operators in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the sum over all possible creation and annihilation sections $s \in \text{CrAnSection}(\varphi_s)$ of the products of their corresponding components: $$\text{ofFieldOpListF}(\varphi_s) = \sum_{s \in \text{CrAnSection}(\varphi_s)} \text{ofCrAnListF}(s)$$ where $\text{ofFieldOpListF}(\varphi_s)$ denotes the algebraic product $\prod_{i=1}^n \text{ofFieldOpF}(\phi_i)$, and each term in the sum is the product of a specific combination of creation and annihilation components of the operators in the list.

For a given field specification $\mathcal{F}$, the function $\text{crPartF}$ maps a field operator $\phi \in \text{FieldOp}(\mathcal{F})$ to its creation component within the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. - For an incoming asymptotic operator $\phi$, the function returns the operator as a creation generator in the algebra. - For a position-space operator $\phi$, the function returns the specific component designated as the creation part of the field. - For an outgoing asymptotic operator $\phi$, the function returns $0$, as these operators represent annihilation processes.

For a given field specification $\mathcal{F}$, let $\phi$ be an asymptotic field state consisting of a field $f$, an asymptotic label $e$, and a 3-momentum $\mathbf{p}$. The creation part $\text{crPartF}$ of the incoming asymptotic field operator $\text{inAsymp}(\phi)$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the operator mapped as a generator with its unique mode label $()$: $$\text{crPartF}(\text{inAsymp}(\phi)) = \text{ofCrAnOpF}(\langle \text{inAsymp}(\phi), () \rangle)$$ Since incoming asymptotic operators represent creation processes, their creation part is simply the operator itself in the algebra.

For a given field specification $\mathcal{F}$, let $\phi(x)$ be a position-space field operator at spacetime point $x$ (formally represented as `FieldOp.position φ`). The creation part of this operator in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$, denoted by $\text{crPartF}(\phi(x))$, is equal to the generator in the algebra corresponding to the creation mode of the field, denoted as $\text{ofCrAnOpF} \langle \phi(x), \text{create} \rangle$.

For a given field specification $\mathcal{F}$, let $\varphi$ be an outgoing asymptotic state (comprising a field label and momentum). The creation part of the corresponding outgoing asymptotic field operator $\text{outAsymp}(\varphi)$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is zero, i.e., $\text{crPartF}(\text{outAsymp}(\varphi)) = 0$.

For a given field specification $\mathcal{F}$, the function $\text{anPartF}$ maps a field operator $\phi \in \mathcal{F}.\text{FieldOp}$ to its annihilation component in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$. The mapping is defined based on the type of operator: - If $\phi$ is an incoming asymptotic field operator $\text{inAsymp}(f, e, \mathbf{p})$, its annihilation part is $0$ (as these are creation operators). - If $\phi$ is a position-space field operator $\text{position}(f, e, x)$, its annihilation part is the generator in the algebra corresponding to the annihilation mode of the field at that spacetime point. - If $\phi$ is an outgoing asymptotic field operator $\text{outAsymp}(f, e, \mathbf{p})$, the function maps it to its corresponding generator in the algebra, treating the entire operator as an annihilation operator.

For a field specification $\mathcal{F}$, the annihilation part of any incoming asymptotic field operator $\text{inAsymp}(\phi)$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is zero. That is, for any incoming state $\phi$ (parameterized by a field label and momentum), $\text{anPartF}(\text{inAsymp}(\phi)) = 0$.

For a given field specification $\mathcal{F}$, let $\phi$ represent the parameters (field type, position label, and spacetime coordinates) of a position-space field operator. The annihilation part of the position-space operator $\text{position}(\phi)$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the generator in the algebra corresponding to the annihilation mode of that operator, denoted by $\langle \text{position}(\phi), \text{annihilate} \rangle$.

For a given field specification $\mathcal{F}$, let $\phi$ be an outgoing asymptotic state consisting of a field $f$, an asymptotic label $e$, and a 3-momentum $\mathbf{p}$. The annihilation part $\text{anPartF}$ of the outgoing asymptotic field operator $\text{outAsymp}(\phi)$ in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is equal to the generator representing the operator itself, denoted by $\text{ofCrAnOpF} \langle \text{FieldOp.outAsymp } \phi, () \rangle$.

For a given field specification $\mathcal{F}$ and any field operator $\phi \in \mathcal{F}.\text{FieldOp}$, the representation of the field operator in the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ is the sum of its creation part and its annihilation part: $$\text{ofFieldOpF}(\phi) = \text{crPartF}(\phi) + \text{anPartF}(\phi)$$ where $\text{ofFieldOpF}(\phi)$ is the field operator as an element of the algebra, $\text{crPartF}(\phi)$ is its creation component, and $\text{anPartF}(\phi)$ is its annihilation component.

For a given field specification $\mathcal{F}$, let $\mathcal{S} = \mathcal{F}.\text{CrAnFieldOp}$ be the set of creation and annihilation operator components. The free associative algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ over $\mathbb{C}$ generated by $\mathcal{S}$ has a basis indexed by the set of all finite lists of elements from $\mathcal{S}$. Each element of this basis corresponds to a unique monomial $s_1 s_2 \cdots s_n$ in the algebra, where $[s_1, s_2, \dots, s_n]$ is a list of generators.

For a given field specification $\mathcal{F}$, let $\phi_s = [\phi_1, \phi_2, \dots, \phi_n]$ be a list of creation and annihilation operator components in $\mathcal{F}.\text{CrAnFieldOp}$. The basis element of the free algebra $\mathcal{F}.\text{FieldOpFreeAlgebra}$ indexed by the list $\phi_s$, denoted $\text{ofCrAnListFBasis}(\phi_s)$, is equal to the product of the generators corresponding to those components in the algebra, denoted $\text{ofCrAnListF}(\phi_s)$.

Let $\mathcal{F}$ be a field specification. Let $\mathcal{F}.\text{CrAnFieldOp}$ be the set of creation and annihilation operator components and $\mathcal{F}.\text{FieldOpFreeAlgebra}$ be the free associative algebra over $\mathbb{C}$ generated by these components. The function $\text{ofCrAnListF}$, which maps a finite list of components $[\phi_1, \phi_2, \dots, \phi_n]$ to their product $\phi_1 \phi_2 \cdots \phi_n$ in the algebra, is injective.

For a given field specification $\mathcal{F}$, let $\mathcal{A}$ denote the free algebra of creation and annihilation operators (the `FieldOpFreeAlgebra`). The mapping `mulLinearMap` is the $\mathbb{C}$-bilinear map from $\mathcal{A} \times \mathcal{A}$ to $\mathcal{A}$ that sends a pair of elements $(a, b)$ to their product $a \cdot b$. Formally, it is defined as a curried linear map $\mathcal{A} \to_{\ell} (\mathcal{A} \to_{\ell} \mathcal{A})$, where an element $a$ is mapped to the left-multiplication operator $L_a(b) = a \cdot b$.

Let $\mathcal{F}$ be a field specification and let $\mathcal{A}$ be the free algebra of creation and annihilation operators (the `FieldOpFreeAlgebra`) over $\mathbb{C}$ generated by the set of creation and annihilation operator components $\mathcal{F}.\text{CrAnFieldOp}$. For any two elements $a, b \in \mathcal{A}$, the value of the $\mathbb{C}$-bilinear multiplication map $\text{mulLinearMap}$ at $a$ and $b$ is equal to their product $a \cdot b$.

For a given field specification $\mathcal{F}$ and a complex scalar $c \in \mathbb{C}$, this definition represents the $\mathbb{C}$-linear operator on the free algebra of creation and annihilation operators, $\text{FieldOpFreeAlgebra}(\mathcal{F})$, which maps an element $a$ to its scalar multiple $c \cdot a$.