PhysLean/HepLean/PerturbationTheory/FieldSpecification/Basic.lean

/-
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.Lorentz.RealVector.Basic
import HepLean.PerturbationTheory.FieldStatistics.ExchangeSign
import HepLean.SpaceTime.Basic
import HepLean.PerturbationTheory.FieldStatistics.OfFinset
import HepLean.Meta.Remark.Basic
/-!

# Field specification

In this module is the definition of a field specification.
A field specification is a structure consisting of a type of fields and a
the field statistics of each field.

From each field we can create three different types of `FieldOp`.
- Negative asymptotic states.
- Position states.
- Positive asymptotic states.

These states carry the same field statistic as the field they are derived from.

## Some references

- https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf

-/

/--
The structure `FieldSpecification` is defined to have the following content:
  - A type `Field` whose elements are the constituent fields of the theory.
  - For every field `f` in `Field`, a type `PositionLabel f` whose elements label the different
    position operators associated with the field `f`. For example,
    - For `f` a *real-scalar field*, `PositionLabel f` will have a unique element.
    - For `f` a *complex-scalar field*, `PositionLabel f` will have two elements, one for the field
      operator and one for its conjugate.
    - For `f` a *Dirac fermion*, `PositionLabel f` will have eight elements, one for each Lorentz
      index of the field and its conjugate.
    - For `f` a *Weyl fermion*, `PositionLabel f` will have four elements, one for each Lorentz
      index of the field and its conjugate.
  - For every field `f` in `Field`, a type `AsymptoticLabel f` whose elements label the different
    types of incoming asymptotic field operators associated with the
    field `f` (this is also matches the types of outgoing asymptotic field operators).
    For example,
    - For `f` a *real-scalar field*, `AsymptoticLabel f` will have a unique element.
    - For `f` a *complex-scalar field*, `AsymptoticLabel f` will have two elements, one for the
      field operator and one for its conjugate.
    - For `f` a *Dirac fermion*, `AsymptoticLabel f` will have four elements, two for each spin.
    - For `f` a *Weyl fermion*, `AsymptoticLabel f` will have two elements, one for each spin.
  - For each field `f` in `Field`, a field statistic `statistic f` which classifies `f` as either
    `bosonic` or `fermionic`.
-/
structure FieldSpecification where
  /-- A type whose elements are the constituent fields of the theory. -/
  Field : Type
  /-- For every field `f` in `Field`, the type `PositionLabel f` has elements that label the
    different position operators associated with the field `f`. -/
  PositionLabel : Field → Type
  /-- For every field `f` in `Field`, the type `AsymptoticLabel f` has elements that label
    the different asymptotic based field operators associated with the field `f`. -/
  AsymptoticLabel : Field → Type
  /-- For every field `f` in `Field`, the field statistic `statistic f` classifies `f` as either
    `bosonic` or `fermionic`. -/
  statistic : Field → FieldStatistic

namespace FieldSpecification
variable (𝓕 : FieldSpecification)

/-- For a field specification `𝓕`, the inductive type `𝓕.FieldOp` is defined
  to contain the following elements:
- for every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
  element labelled `inAsymp f e p` corresponding to an incoming asymptotic field operator
  of the field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
- for every `f` in `𝓕.Field`, element of `e` of `PositionLabel f` and space-time position `x`, an
  element labelled `position f e x` corresponding to a position field operator of the field `f`,
  of label `e` (e.g. specifying the Lorentz index), and position `x`.
- for every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
  element labelled `outAsymp f e p` corresponding to an outgoing asymptotic field operator of the
  field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.

As some intuition, if `f` corresponds to a Weyl-fermion field, then
- For `inAsymp f e p`, `e` would correspond to a spin `s`, and `inAsymp f e p` would, once
  represented in the operator algebra,
  be proportional to the creation operator `a(p, s)`.
- `position f e x`, `e` would correspond to a Lorentz index `a`, and `position f e x` would,
  once represented in the operator algebra, be proportional to the operator

  `∑ s, ∫ d³p/(…) (xₐ(p,s)  a(p, s) e ^ (-i p x) + yₐ(p,s) a†(p, s) e ^ (-i p x))`.

- `outAsymp f e p`, `e` would correspond to a spin `s`, and `outAsymp f e p` would,
  once represented in the operator algebra, be proportional to the
  annihilation operator `a†(p, s)`.

This type contains all operators which are related to a field.
-/
inductive FieldOp (𝓕 : FieldSpecification) where
  | inAsymp : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ) → 𝓕.FieldOp
  | position : (Σ f, 𝓕.PositionLabel f) × SpaceTime → 𝓕.FieldOp
  | outAsymp : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ) → 𝓕.FieldOp

/-- The bool on `FieldOp` which is true only for position field operator. -/
def statesIsPosition : 𝓕.FieldOp → Bool
  | FieldOp.position _ => true
  | _ => false

/-- For a field specification `𝓕`, `𝓕.fieldOpToField` is defined to take field operators
  to their underlying field. -/
def fieldOpToField : 𝓕.FieldOp → 𝓕.Field
  | FieldOp.inAsymp (f, _) => f.1
  | FieldOp.position (f, _) => f.1
  | FieldOp.outAsymp (f, _) => f.1

/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`.
  The field statistic `fieldOpStatistic φ` is defined to be the statistic associated with
  the field underlying `φ`.

  The following notation is used in relation to `fieldOpStatistic`:
- For `φ` an element of `𝓕.FieldOp`, `𝓕 |>ₛ φ` is `fieldOpStatistic φ`.
- For `φs` a list of `𝓕.FieldOp`, `𝓕 |>ₛ φs` is the product of `fieldOpStatistic φ` over
  the list `φs`.
- For a function `f : Fin n → 𝓕.FieldOp` and a finset `a` of `Fin n`, `𝓕 |>ₛ ⟨f, a⟩` is the
  product of `fieldOpStatistic (f i)` for all `i ∈ a`. -/
def fieldOpStatistic : 𝓕.FieldOp → FieldStatistic := 𝓕.statistic ∘ 𝓕.fieldOpToField

@[inherit_doc fieldOpStatistic]
scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => fieldOpStatistic 𝓕 φ

@[inherit_doc fieldOpStatistic]
scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => FieldStatistic.ofList
    (fieldOpStatistic 𝓕) φ

@[inherit_doc fieldOpStatistic]
scoped[FieldSpecification] notation 𝓕 "|>ₛ" "⟨" f ","a "⟩"=> FieldStatistic.ofFinset
    (fieldOpStatistic 𝓕) f a

end FieldSpecification