88 lines
3.2 KiB
Text
88 lines
3.2 KiB
Text
/-
|
||
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 `States`.
|
||
- Negative asymptotic states.
|
||
- Position states.
|
||
- Positive asymptotic states.
|
||
|
||
These states carry the same field statistic as the field they are derived from.
|
||
|
||
-/
|
||
|
||
remark fieldSpecification_intro := "The raw ingredients of a field theory are:
|
||
- The specification of the fields.
|
||
- Whether each field is a boson or a fermion.
|
||
- Vertices present in the Lagrangian.
|
||
- The coefficent of each vertex.
|
||
|
||
We call the first two of these ingredients the `FieldSpecification` of the theory. "
|
||
|
||
/-- A field specification is a type, `Fields`, elements of which are fields
|
||
present in a theory, and a map `statistics` from `Fields` to `FieldStatistic` which
|
||
identifies each field as a boson or a fermion. -/
|
||
structure FieldSpecification where
|
||
/-- The type of fields. This also includes anti-states. -/
|
||
Fields : Type
|
||
/-- The specification if a field is bosonic or fermionic. -/
|
||
statistics : Fields → FieldStatistic
|
||
|
||
namespace FieldSpecification
|
||
variable (𝓕 : FieldSpecification)
|
||
|
||
/-- An incoming asymptotic state is a field and a momentum. -/
|
||
def IncomingAsymptotic : Type := 𝓕.Fields × Lorentz.Contr 4
|
||
|
||
/-- An outgoing asymptotic states is a field and a momentum. -/
|
||
def OutgoingAsymptotic : Type := 𝓕.Fields × Lorentz.Contr 4
|
||
|
||
/-- A position state is a field and a space-time position. -/
|
||
def PositionStates : Type := 𝓕.Fields × SpaceTime
|
||
|
||
/-- The type States is the inductive type combining the asymptotic states and position states. -/
|
||
inductive States (𝓕 : FieldSpecification) where
|
||
| inAsymp : 𝓕.IncomingAsymptotic → 𝓕.States
|
||
| position : 𝓕.PositionStates → 𝓕.States
|
||
| outAsymp : 𝓕.OutgoingAsymptotic → 𝓕.States
|
||
|
||
/-- Taking a state to its underlying field. -/
|
||
def statesToField : 𝓕.States → 𝓕.Fields
|
||
| States.inAsymp φ => φ.1
|
||
| States.position φ => φ.1
|
||
| States.outAsymp φ => φ.1
|
||
|
||
/-- The bool on `States` which is true only for position states. -/
|
||
def statesIsPosition : 𝓕.States → Bool
|
||
| States.position _ => true
|
||
| _ => false
|
||
|
||
/-- The statistics associated to a state. -/
|
||
def statesStatistic : 𝓕.States → FieldStatistic := 𝓕.statistics ∘ 𝓕.statesToField
|
||
|
||
/-- The field statistics associated with a state. -/
|
||
scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => statesStatistic 𝓕 φ
|
||
|
||
/-- The field statistics associated with a list states. -/
|
||
scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => FieldStatistic.ofList
|
||
(statesStatistic 𝓕) φ
|
||
|
||
/-- The field statistic associated with a finite set-/
|
||
scoped[FieldSpecification] notation 𝓕 "|>ₛ" "⟨" f ","a "⟩"=> FieldStatistic.ofFinset
|
||
(statesStatistic 𝓕) f a
|
||
|
||
end FieldSpecification
|