Physlib.QFT.PerturbationTheory.FieldSpecification.Basic

For a given field specification $\mathcal{F}$, the type $\mathcal{F}.\text{FieldOp}$ defines the fundamental operators associated with the fields in the theory. It is an inductive type consisting of three categories of operators: - $\text{inAsymp}(f, e, \mathbf{p})$: An incoming asymptotic field operator for a field $f \in \mathcal{F}$, where $e$ is an asymptotic label (such as spin $s$) and $\mathbf{p}$ is the 3-momentum. In the operator algebra, these correspond to creation operators $a(\mathbf{p}, s)$. - $\text{position}(f, e, x)$: A position-space field operator for a field $f \in \mathcal{F}$, where $e$ is a position label (such as a Lorentz index $\mu$) and $x$ is the spacetime position. These correspond to field operators $\phi(x)$ expanded as a sum of creation and annihilation operators. - $\text{outAsymp}(f, e, \mathbf{p})$: An outgoing asymptotic field operator for a field $f \in \mathcal{F}$, with asymptotic label $e$ and 3-momentum $\mathbf{p}$. In the operator algebra, these correspond to annihilation operators $a^\dagger(\mathbf{p}, s)$.

For a given field specification $\mathcal{F}$, this function maps a field operator $\omega \in \text{FieldOp}(\mathcal{F})$ to a Boolean value. It returns $\text{true}$ if $\omega$ is a position-space field operator $\text{position}(f, e, x)$, and $\text{false}$ if $\omega$ is an incoming or outgoing asymptotic field operator ($\text{inAsymp}$ or $\text{outAsymp}$).

For a given field specification $\mathcal{F}$, this function maps a field operator $\omega \in \text{FieldOp}(\mathcal{F})$ to its underlying field $f \in \text{Field}(\mathcal{F})$. Specifically, if $\omega$ is an incoming asymptotic operator $\text{inAsymp}(f, e, \mathbf{p})$, a position-space operator $\text{position}(f, e, x)$, or an outgoing asymptotic operator $\text{outAsymp}(f, e, \mathbf{p})$, the function returns the field $f$ from which the operator is derived.

For a field specification $\mathcal{F}$, the function `fieldOpStatistic` assigns a field statistic (such as bosonic or fermionic) to a field operator $\phi \in \text{FieldOp}(\mathcal{F})$. The statistic of the operator is defined as the statistic of its underlying field $f \in \text{Field}(\mathcal{F})$, effectively inheriting the commutation or anti-commutation properties of the field from which the operator is derived. This is denoted by the notation $\mathcal{F} \triangleright_s \phi$.

Given a field specification $\mathcal{F}$ and a field operator $\phi$, the notation $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (e.g., bosonic or fermionic) associated with the field operator $\phi$ within the specification $\mathcal{F}$.

Given a field specification $\mathcal{F}$ and a list of field operators $\varphi$, the notation $\mathcal{F} \rhd_s \varphi$ denotes the collective field statistic (such as Bose or Fermi) of the sequence $\varphi$. This value is determined by aggregating the statistics of the individual operators in $\varphi$ as defined by the specification $\mathcal{F}$.

For a given field specification $\mathcal{F}$, this notation calculates the aggregate field statistic (e.g., Bose or Fermi) for a finite collection of field operators. Given a finite set of fields $f$ and an index $a$, the expression $\mathcal{F} \rhd_s \langle f, a \rangle$ determines the total statistic by applying the aggregation function `FieldStatistic.ofFinset` to the individual statistics of the field operators as defined within the specification $\mathcal{F}$.