86 lines
2.6 KiB
Text
86 lines
2.6 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.FeynmanDiagrams.Basic
|
||
import Mathlib.Data.Real.Basic
|
||
import Mathlib.Algebra.Category.ModuleCat.Basic
|
||
/-!
|
||
# Momentum in Feynman diagrams
|
||
|
||
The aim of this file is to associate with each half-edge of a Feynman diagram a momentum,
|
||
and constrain the momentums based conservation at each vertex and edge.
|
||
|
||
The number of loops of a Feynman diagram is related to the dimension of the resulting
|
||
vector space.
|
||
|
||
-/
|
||
|
||
namespace FeynmanDiagram
|
||
|
||
open CategoryTheory
|
||
open PreFeynmanRule
|
||
|
||
variable {P : PreFeynmanRule} (F : FeynmanDiagram P)
|
||
variable (d : ℕ)
|
||
|
||
/-- The momentum space for a `d`-dimensional field theory for a single particle.
|
||
TODO: Move this definition, and define it as a four-vector. -/
|
||
def SingleMomentumSpace : Type := Fin d → ℝ
|
||
|
||
instance : AddCommGroup (SingleMomentumSpace d) := Pi.addCommGroup
|
||
|
||
instance : Module ℝ (SingleMomentumSpace d) := Pi.module _ _ _
|
||
|
||
|
||
/-- The type which asociates to each half-edge a `d`-dimensional vector.
|
||
This is to be interpreted as the momentum associated to that half-edge flowing from the
|
||
corresponding `edge` to the corresponding `vertex`. So all momentums flow into vertices. -/
|
||
def FullMomentumSpace : Type := F.𝓱𝓔 → Fin d → ℝ
|
||
|
||
instance : AddCommGroup (F.FullMomentumSpace d) := Pi.addCommGroup
|
||
|
||
instance : Module ℝ (F.FullMomentumSpace d) := Pi.module _ _ _
|
||
|
||
/-- The linear map taking a half-edge to its momentum.
|
||
(defined as flowing from the `edge` to the vertex.) -/
|
||
def toHalfEdgeMomentum (i : F.𝓱𝓔) : F.FullMomentumSpace d →ₗ[ℝ] SingleMomentumSpace d where
|
||
toFun x := x i
|
||
map_add' x y := by rfl
|
||
map_smul' c x := by rfl
|
||
|
||
namespace Hom
|
||
|
||
variable {F G : FeynmanDiagram P}
|
||
variable (f : F ⟶ G)
|
||
|
||
/-- The linear map induced by a morphism of Feynman diagrams. -/
|
||
def toLinearMap : G.FullMomentumSpace d →ₗ[ℝ] F.FullMomentumSpace d where
|
||
toFun x := x ∘ f.𝓱𝓔
|
||
map_add' x y := by rfl
|
||
map_smul' c x := by rfl
|
||
|
||
|
||
end Hom
|
||
|
||
/-- The contravariant functor from Feynman diagrams to Modules over `ℝ`. -/
|
||
noncomputable def funcFullMomentumSpace : FeynmanDiagram P ⥤ (ModuleCat ℝ)ᵒᵖ where
|
||
obj F := Opposite.op $ ModuleCat.of ℝ (F.FullMomentumSpace d)
|
||
map f := Opposite.op $ Hom.toLinearMap d f
|
||
|
||
|
||
/-!
|
||
## Edge constraints
|
||
|
||
There is a linear map from `F.FullMomentumSpace` to `F.EdgeMomentumSpace`, induced
|
||
by the constraints at each edge.
|
||
|
||
We impose the constraint that we live in the kernal of this linear map.
|
||
|
||
A similar result is true for the vertex constraints.
|
||
|
||
-/
|
||
|
||
|
||
end FeynmanDiagram
|