111 lines
4.3 KiB
Text
111 lines
4.3 KiB
Text
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/-
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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.FeynmanDiagrams.Basic
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/-!
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# Wick strings
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Currently this file is only for an example of Wick strings, correpsonding to a
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theory with two complex scalar fields. The concepts will however generalize.
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A wick string is defined to be a sequence of input fields,
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followed by a squence of vertices, followed by a sequence of output fields.
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A wick string can be combined with an appropriate map to spacetime to produce a specific
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term in the ring of operators. This has yet to be implemented.
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-/
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namespace TwoComplexScalar
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open CategoryTheory
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open FeynmanDiagram
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open PreFeynmanRule
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/-- The colors of edges which one can associate with a vertex for a theory
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with two complex scalar fields. -/
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inductive 𝓔 where
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/-- Corresponds to the first complex scalar field flowing out of a vertex. -/
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| complexScalarOut₁ : 𝓔
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/-- Corresponds to the first complex scalar field flowing into a vertex. -/
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| complexScalarIn₁ : 𝓔
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/-- Corresponds to the second complex scalar field flowing out of a vertex. -/
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| complexScalarOut₂ : 𝓔
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/-- Corresponds to the second complex scalar field flowing into a vertex. -/
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| complexScalarIn₂ : 𝓔
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/-- The map taking each color to it's dual, specifying how we can contract edges. -/
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def ξ : 𝓔 → 𝓔
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| 𝓔.complexScalarOut₁ => 𝓔.complexScalarIn₁
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| 𝓔.complexScalarIn₁ => 𝓔.complexScalarOut₁
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| 𝓔.complexScalarOut₂ => 𝓔.complexScalarIn₂
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| 𝓔.complexScalarIn₂ => 𝓔.complexScalarOut₂
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/-- The function `ξ` is an involution. -/
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lemma ξ_involutive : Function.Involutive ξ := by
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intro x
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match x with
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| 𝓔.complexScalarOut₁ => rfl
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| 𝓔.complexScalarIn₁ => rfl
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| 𝓔.complexScalarOut₂ => rfl
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| 𝓔.complexScalarIn₂ => rfl
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/-- The vertices associated with two complex scalars.
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We call this type, the type of vertex colors. -/
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inductive 𝓥 where
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| φ₁φ₁φ₂φ₂ : 𝓥
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| φ₁φ₁φ₁φ₁ : 𝓥
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| φ₂φ₂φ₂φ₂ : 𝓥
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/-- To each vertex, the association of the number of edges. -/
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@[nolint unusedArguments]
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def 𝓥NoEdges : 𝓥 → ℕ := fun _ => 4
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/-- To each vertex, associates the indexing map of half-edges associated with that edge. -/
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def 𝓥Edges (v : 𝓥) : Fin (𝓥NoEdges v) → 𝓔 :=
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match v with
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| 𝓥.φ₁φ₁φ₂φ₂ => fun i =>
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match i with
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| (0 : Fin 4)=> 𝓔.complexScalarOut₁
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| (1 : Fin 4) => 𝓔.complexScalarIn₁
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| (2 : Fin 4) => 𝓔.complexScalarOut₂
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| (3 : Fin 4) => 𝓔.complexScalarIn₂
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| 𝓥.φ₁φ₁φ₁φ₁ => fun i =>
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match i with
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| (0 : Fin 4)=> 𝓔.complexScalarOut₁
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| (1 : Fin 4) => 𝓔.complexScalarIn₁
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| (2 : Fin 4) => 𝓔.complexScalarOut₁
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| (3 : Fin 4) => 𝓔.complexScalarIn₁
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| 𝓥.φ₂φ₂φ₂φ₂ => fun i =>
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match i with
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| (0 : Fin 4)=> 𝓔.complexScalarOut₂
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| (1 : Fin 4) => 𝓔.complexScalarIn₂
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| (2 : Fin 4) => 𝓔.complexScalarOut₂
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| (3 : Fin 4) => 𝓔.complexScalarIn₂
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inductive WickStringLast where
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| incoming : WickStringLast
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| vertex : WickStringLast
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| outgoing : WickStringLast
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| final : WickStringLast
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open WickStringLast
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/-- A wick string is a representation of a string of fields from a theory.
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E.g. `φ(x1) φ(x2) φ(y) φ(y) φ(y) φ(x3)`. The use of vertices in the Wick string
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allows us to identify which fields have the same space-time coordinate. -/
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inductive WickString : {n : ℕ} → (c : Fin n → 𝓔) → WickStringLast → Type where
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| empty : WickString Fin.elim0 incoming
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| incoming {n : ℕ} {c : Fin n → 𝓔} (w : WickString c incoming) (e : 𝓔) :
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WickString (Fin.cons e c) incoming
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| endIncoming {n : ℕ} {c : Fin n → 𝓔} (w : WickString c incoming) : WickString c vertex
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| vertex {n : ℕ} {c : Fin n → 𝓔} (w : WickString c vertex) (v : 𝓥) :
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WickString (Fin.append (𝓥Edges v) c) vertex
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| endVertex {n : ℕ} {c : Fin n → 𝓔} (w : WickString c vertex) : WickString c outgoing
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| outgoing {n : ℕ} {c : Fin n → 𝓔} (w : WickString c outgoing) (e : 𝓔) :
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WickString (Fin.cons e c) outgoing
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| endOutgoing {n : ℕ} {c : Fin n → 𝓔} (w : WickString c outgoing) : WickString c final
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end TwoComplexScalar
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