96 lines
2.8 KiB
Text
96 lines
2.8 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.FeynmanDiagrams.Basic
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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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open CategoryTheory
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open FeynmanDiagram
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open PreFeynmanRule
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/-- The pre-Feynman rules for `Phi^4` theory. -/
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@[simps!]
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def phi4PreFeynmanRules : PreFeynmanRule where
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/- There is only 1 type of `half-edge`. -/
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HalfEdgeLabel := Fin 1
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/- There is only 1 type of `edge`. -/
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EdgeLabel := Fin 1
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/- There are two types of `vertex`, external `0` and internal `1`. -/
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VertexLabel := Fin 2
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edgeLabelMap x :=
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match x with
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| 0 => Over.mk ![0, 0]
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vertexLabelMap x :=
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match x with
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| 0 => Over.mk ![0]
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| 1 => Over.mk ![0, 0, 0, 0]
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/-- An instance allowing us to use integers for edge labels for Phi-4. -/
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instance (a : ℕ) : OfNat phi4PreFeynmanRules.EdgeLabel a where
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ofNat := (a : Fin _)
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/-- An instance allowing us to use integers for half edge labels for Phi-4. -/
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instance (a : ℕ) : OfNat phi4PreFeynmanRules.HalfEdgeLabel a where
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ofNat := (a : Fin _)
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/-- An instance allowing us to use integers for vertex labels for Phi-4. -/
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instance (a : ℕ) : OfNat phi4PreFeynmanRules.VertexLabel a where
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ofNat := (a : Fin _)
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/-- The pre feynman rules for Phi-4 are finite. -/
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instance : IsFinitePreFeynmanRule phi4PreFeynmanRules where
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edgeLabelDecidable := instDecidableEqFin _
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vertexLabelDecidable := instDecidableEqFin _
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halfEdgeLabelDecidable := instDecidableEqFin _
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vertexMapFintype := fun v =>
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match v with
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| 0 => Fin.fintype _
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| 1 => Fin.fintype _
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edgeMapFintype := fun v =>
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match v with
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| 0 => Fin.fintype _
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vertexMapDecidable := fun v =>
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match v with
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| 0 => instDecidableEqFin _
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| 1 => instDecidableEqFin _
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edgeMapDecidable := fun v =>
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match v with
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| 0 => instDecidableEqFin _
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/-!
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## The figure eight diagram
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This section provides an example of the use of Feynman diagrams in HepLean.
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-/
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section Example
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/-- The figure eight Feynman diagram. -/
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abbrev figureEight : FeynmanDiagram phi4PreFeynmanRules :=
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mk'
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![0, 0] -- edges
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![1] -- one internal vertex
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![⟨0, 0, 0⟩, ⟨0, 0, 0⟩, ⟨0, 1, 0⟩, ⟨0, 1, 0⟩] -- four half-edges
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(by decide) -- the condition to form a Feynman diagram.
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/-- `figureEight` is connected. We can get this from
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`#eval Connected figureEight`. -/
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lemma figureEight_connected : Connected figureEight := by decide
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/-- The symmetry factor of `figureEight` is 8. We can get this from
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`#eval symmetryFactor figureEight`. -/
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lemma figureEight_symmetryFactor : symmetryFactor figureEight = 8 := by rfl
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end Example
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end PhiFour
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