95 lines
2.5 KiB
Text
95 lines
2.5 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Logic.Equiv.Fin
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import Mathlib.Tactic.FinCases
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import Mathlib.Data.Finset.Card
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import Mathlib.CategoryTheory.IsomorphismClasses
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import LeanCopilot
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/-!
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# Feynman diagrams in Phi^4 theory
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The aim of this file is to start building up the theory of Feynman diagrams in the context of
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Phi^4 theory.
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-/
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namespace PhiFour
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/-- The type of vacuum Feynman diagrams for Phi-4 theory. -/
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structure VacuumFeynmanDiagram where
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HalfEdge : Type
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Edge : Type
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Vertex : Type
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𝓔 : HalfEdge → Edge
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𝓥 : HalfEdge → Vertex
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𝓔Fiber : ∀ x, CategoryTheory.IsIsomorphic (𝓔 ⁻¹' {x} : Type) (Fin 2)
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𝓥Fiber : ∀ x, CategoryTheory.IsIsomorphic (𝓥 ⁻¹' {x} : Type) (Fin 3)
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namespace VacuumFeynmanDiagram
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open CategoryTheory
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variable (F G : VacuumFeynmanDiagram)
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/-- The definition of a morphism between two `VacuumFeynmanDiagram`. -/
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structure Hom where
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halfEdgeMap : F.HalfEdge ⟶ G.HalfEdge
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edgeMap : F.Edge ⟶ G.Edge
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vertexMap : F.Vertex ⟶ G.Vertex
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namespace Hom
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/-- The identity morphism for an object in `VacuumFeynmanDiagram`. -/
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def id : Hom F F where
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halfEdgeMap := 𝟙 F.HalfEdge
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edgeMap := 𝟙 F.Edge
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vertexMap := 𝟙 F.Vertex
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/-- The composition of morphisms between two `VacuumFeynmanDiagram`. -/
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def comp {F G H : VacuumFeynmanDiagram} (f : Hom F G) (g : Hom G H) : Hom F H where
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halfEdgeMap := f.halfEdgeMap ≫ g.halfEdgeMap
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edgeMap := f.edgeMap ≫ g.edgeMap
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vertexMap := f.vertexMap ≫ g.vertexMap
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end Hom
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/-- `VacuumFeynmanDiagram` has the structure of a category. -/
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instance : Category VacuumFeynmanDiagram where
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Hom := Hom
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id := Hom.id
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comp := Hom.comp
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/-- The functor from the category `VacuumFeynmanDiagram` to `Type` defined by
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mapping to the `Type` of half edges. -/
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def toHalfEdge : VacuumFeynmanDiagram ⥤ Type where
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obj F := F.HalfEdge
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map f := f.halfEdgeMap
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@[simp]
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lemma 𝓔_preimage_nonempty (x : F.Edge) : (F.𝓔 ⁻¹' {x}).Nonempty := by
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obtain ⟨_, f, _, _⟩ := F.𝓔Fiber x
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exact ⟨f 0, Subtype.coe_prop (f 0)⟩
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@[simp]
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lemma 𝓥_preimage_nonempty (x : F.Vertex) : (F.𝓥 ⁻¹' {x}).Nonempty := by
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obtain ⟨_, f, _, _⟩ := F.𝓥Fiber x
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exact ⟨f 0, Subtype.coe_prop (f 0)⟩
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@[simp]
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lemma 𝓔_surjective : Function.Surjective F.𝓔 := by
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exact fun x => F.𝓔_preimage_nonempty x
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@[simp]
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lemma 𝓥_surjective : Function.Surjective F.𝓥 := by
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exact fun x => F.𝓥_preimage_nonempty x
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end VacuumFeynmanDiagram
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end PhiFour
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