73 lines
2.3 KiB
Text
73 lines
2.3 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 HepLean.FeynmanDiagrams.Basic
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import Mathlib.Analysis.InnerProductSpace.Adjoint
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import Mathlib.Algebra.Category.ModuleCat.Basic
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/-!
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# Momentum in Feynman diagrams
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The aim of this file is to associate with each half-edge of a Feynman diagram a momentum,
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and constrain the momentums based conservation at each vertex and edge.
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-/
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namespace FeynmanDiagram
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open CategoryTheory
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open PreFeynmanRule
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variable {P : PreFeynmanRule} (F : FeynmanDiagram P)
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variable (d : ℕ)
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/-- The momentum space for a `d`-dimensional field theory for a single particle.
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TODO: Move this definition, and define it as a four-vector. -/
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def SingleMomentumSpace : Type := Fin d → ℝ
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instance : AddCommGroup (SingleMomentumSpace d) := Pi.addCommGroup
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noncomputable instance : Module ℝ (SingleMomentumSpace d) := Pi.module _ _ _
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/-- The type which asociates to each half-edge a `d`-dimensional vector.
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This is to be interpreted as the momentum associated to that half-edge flowing from the
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corresponding `edge` to the corresponding `vertex`. So all momentums flow into vertices. -/
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def FullMomentumSpace : Type := F.𝓱𝓔 → Fin d → ℝ
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instance : AddCommGroup (F.FullMomentumSpace d) := Pi.addCommGroup
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noncomputable instance : Module ℝ (F.FullMomentumSpace d) := Pi.module _ _ _
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/-- The linear map taking a half-edge to its momentum.
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(defined as flowing from the `edge` to the vertex.) -/
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def toHalfEdgeMomentum (i : F.𝓱𝓔) : F.FullMomentumSpace d →ₗ[ℝ] SingleMomentumSpace d where
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toFun x := x i
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map_add' x y := by rfl
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map_smul' c x := by rfl
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namespace Hom
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variable {F G : FeynmanDiagram P}
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variable (f : F ⟶ G)
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/-- The linear map induced by a morphism of Feynman diagrams. -/
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def toLinearMap : G.FullMomentumSpace d →ₗ[ℝ] F.FullMomentumSpace d where
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toFun x := x ∘ f.𝓱𝓔
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map_add' x y := by rfl
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map_smul' c x := by rfl
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end Hom
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/-- The contravariant functor from Feynman diagrams to Modules over `ℝ`. -/
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noncomputable def funcFullMomentumSpace : FeynmanDiagram P ⥤ (ModuleCat ℝ)ᵒᵖ where
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obj F := Opposite.op $ ModuleCat.of ℝ (F.FullMomentumSpace d)
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map f := Opposite.op $ Hom.toLinearMap d f
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/-!
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## Edge constraints
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-/
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end FeynmanDiagram
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