423 lines
15 KiB
Text
423 lines
15 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.FeynmanDiagrams.Basic
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/-!
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# Feynman rules for a two complex scalar fields
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This file serves as a demonstration of a new approach to Feynman rules.
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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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def FieldString (n : ℕ) : Type := Fin n → 𝓔
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inductive WickString : {n : ℕ} → (c : FieldString n) → 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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inductive WickContract : {n : ℕ} → (f : FieldString n) →
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{k : ℕ} → (b1 : Fin k → Fin n) → (b2 : Fin k → Fin n) → Type where
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| string {n : ℕ} {c : Fin n → 𝓔} : WickContract c Fin.elim0 Fin.elim0
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| contr {n : ℕ} {c : Fin n → 𝓔} {k : ℕ}
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{b1 : Fin k → Fin n} {b2 : Fin k → Fin n}: (i : Fin n) →
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(j : Fin n) → (h : c j = ξ (c i)) →
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(hilej : i < j) → (hb1 : ∀ r, b1 r < i) → (hb2i : ∀ r, b2 r ≠ i) → (hb2j : ∀ r, b2 r ≠ j) →
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(w : WickContract c b1 b2) →
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WickContract c (Fin.snoc b1 i) (Fin.snoc b2 j)
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namespace WickContract
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/-- The number of nodes of a Wick contraction. -/
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def size {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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WickContract c b1 b2 → ℕ := fun
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| string => 1
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| contr _ _ _ _ _ _ _ w => w.size + 1
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def boundFst {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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WickContract c b1 b2 → Fin k → Fin n := fun _ => b1
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@[simp]
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lemma boundFst_contr_castSucc {n k : ℕ} {c : Fin n → 𝓔}
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{b1 b2 : Fin k → Fin n} (i j : Fin n)
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(h : c j = ξ (c i))
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(hilej : i < j)
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(hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i)
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(hb2j : ∀ r, b2 r ≠ j)
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(w : WickContract c b1 b2) (r : Fin k) :
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(contr i j h hilej hb1 hb2i hb2j w).boundFst r.castSucc = w.boundFst r := by
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simp only [boundFst, Fin.snoc_castSucc]
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@[simp]
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lemma boundFst_contr_last {n k : ℕ} {c : Fin n → 𝓔}
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{b1 b2 : Fin k → Fin n} (i j : Fin n)
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(h : c j = ξ (c i))
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(hilej : i < j)
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(hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i)
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(hb2j : ∀ r, b2 r ≠ j)
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(w : WickContract c b1 b2) :
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(contr i j h hilej hb1 hb2i hb2j w).boundFst (Fin.last k) = i := by
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simp only [boundFst, Fin.snoc_last]
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lemma boundFst_strictMono {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → StrictMono w.boundFst := fun
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| string => fun k => Fin.elim0 k
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| contr i j _ _ hb1 _ _ w => by
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intro r s hrs
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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· obtain ⟨r, hr⟩ := hr
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subst hr
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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simp
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apply w.boundFst_strictMono _
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simpa using hrs
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· subst hs
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simp
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exact hb1 r
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· subst hr
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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have hsp := s.prop
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rw [Fin.lt_def] at hrs
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simp at hrs
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omega
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· subst hs
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simp at hrs
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def boundSnd {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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WickContract c b1 b2 → Fin k → Fin n := fun _ => b2
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@[simp]
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lemma boundSnd_contr_castSucc {n k : ℕ} {c : Fin n → 𝓔}
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{b1 b2 : Fin k → Fin n} (i j : Fin n)
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(h : c j = ξ (c i))
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(hilej : i < j)
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(hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i)
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(hb2j : ∀ r, b2 r ≠ j)
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(w : WickContract c b1 b2) (r : Fin k) :
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(contr i j h hilej hb1 hb2i hb2j w).boundSnd r.castSucc = w.boundSnd r := by
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simp only [boundSnd, Fin.snoc_castSucc]
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@[simp]
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lemma boundSnd_contr_last {n k : ℕ} {c : Fin n → 𝓔}
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{b1 b2 : Fin k → Fin n} (i j : Fin n)
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(h : c j = ξ (c i))
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(hilej : i < j)
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(hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i)
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(hb2j : ∀ r, b2 r ≠ j)
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(w : WickContract c b1 b2) :
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(contr i j h hilej hb1 hb2i hb2j w).boundSnd (Fin.last k) = j := by
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simp only [boundSnd, Fin.snoc_last]
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lemma boundSnd_injective {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → Function.Injective w.boundSnd := fun
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| string => by
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intro i j _
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exact Fin.elim0 i
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| contr i j hij hilej hi h2i h2j w => by
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intro r s hrs
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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· obtain ⟨r, hr⟩ := hr
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subst hr
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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simp at hrs
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simpa using w.boundSnd_injective hrs
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· subst hs
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simp at hrs
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exact False.elim (h2j r hrs)
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· subst hr
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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simp at hrs
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exact False.elim (h2j s hrs.symm)
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· subst hs
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rfl
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lemma color_boundSnd_eq_dual_boundFst {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → (i : Fin k) → c (w.boundSnd i) = ξ (c (w.boundFst i)) := fun
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| string => fun i => Fin.elim0 i
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| contr i j hij hilej hi _ _ w => fun r => by
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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· obtain ⟨r, hr⟩ := hr
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subst hr
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simpa using w.color_boundSnd_eq_dual_boundFst r
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· subst hr
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simpa using hij
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lemma boundFst_lt_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → (i : Fin k) → w.boundFst i < w.boundSnd i := fun
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| string => fun i => Fin.elim0 i
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| contr i j hij hilej hi _ _ w => fun r => by
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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· obtain ⟨r, hr⟩ := hr
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subst hr
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simpa using w.boundFst_lt_boundSnd r
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· subst hr
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simp
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exact hilej
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lemma boundFst_neq_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → (r1 r2 : Fin k) → b1 r1 ≠ b2 r2 := fun
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| string => fun i => Fin.elim0 i
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| contr i j _ hilej h1 h2i h2j w => fun r s => by
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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<;> rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨r, hr⟩ := hr
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obtain ⟨s, hs⟩ := hs
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subst hr hs
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simpa using w.boundFst_neq_boundSnd r s
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· obtain ⟨r, hr⟩ := hr
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subst hr hs
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simp
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have hn := h1 r
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omega
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· obtain ⟨s, hs⟩ := hs
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subst hr hs
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simp
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exact (h2i s).symm
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· subst hr hs
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simp
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omega
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def castMaps {n k k' : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} {b1' b2' : Fin k' → Fin n}
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(hk : k = k')
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(hb1 : b1 = b1' ∘ Fin.cast hk) (hb2 : b2 = b2' ∘ Fin.cast hk) (w : WickContract c b1 b2) :
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WickContract c b1' b2' :=
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cast (by subst hk; rfl) (hb2 ▸ hb1 ▸ w)
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@[simp]
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lemma castMaps_rfl {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} (w : WickContract c b1 b2) :
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castMaps rfl rfl rfl w = w := rfl
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lemma mem_snoc' {n k : ℕ} {c : Fin n → 𝓔} {b1' b2' : Fin k → Fin n} :
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(w : WickContract c b1' b2') →
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{k' : ℕ} → (hk' : k'.succ = k ) →
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(b1 b2 : Fin k' → Fin n) → (i j : Fin n) → (h : c j = ξ (c i)) →
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(hilej : i < j) → (hb1 : ∀ r, b1 r < i) → (hb2i : ∀ r, b2 r ≠ i) → (hb2j : ∀ r, b2 r ≠ j) →
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(hb1' : Fin.snoc b1 i = b1' ∘ Fin.cast hk') →
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(hb2' : Fin.snoc b2 j = b2' ∘ Fin.cast hk') →
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∃ (w' : WickContract c b1 b2), w = castMaps hk' hb1' hb2' (contr i j h hilej hb1 hb2i hb2j w')
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:= fun
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| string => fun hk' => by
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simp at hk'
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| contr i' j' h' hilej' hb1' hb2i' hb2j' w' => by
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intro hk b1 b2 i j h hilej hb1 hb2i hb2j hb1' hb2'
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rename_i k' k b1' b2' f
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have hk2 : k' = k := Nat.succ_inj'.mp hk
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subst hk2
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simp_all
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have hb2'' : b2 = b2' := by
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funext k
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trans (@Fin.snoc k' (fun _ => Fin n) b2 j) (Fin.castSucc k)
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· simp
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· rw [hb2']
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simp
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have hb1'' : b1 = b1' := by
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funext k
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trans (@Fin.snoc k' (fun _ => Fin n) b1 i) (Fin.castSucc k)
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· simp
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· rw [hb1']
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simp
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have hi : i = i' := by
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trans (@Fin.snoc k' (fun _ => Fin n) b1 i) (Fin.last k')
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· simp
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· rw [hb1']
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simp
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have hj : j = j' := by
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trans (@Fin.snoc k' (fun _ => Fin n) b2 j) (Fin.last k')
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· simp
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· rw [hb2']
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simp
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subst hb1'' hb2'' hi hj
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simp
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lemma mem_snoc {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} (i j : Fin n)
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(h : c j = ξ (c i))
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(hilej : i < j)
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(hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i)
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(hb2j : ∀ r, b2 r ≠ j)
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(w : WickContract c (Fin.snoc b1 i) (Fin.snoc b2 j)) :
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∃ (w' : WickContract c b1 b2), w = contr i j h hilej hb1 hb2i hb2j w' := by
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exact mem_snoc' w rfl b1 b2 i j h hilej hb1 hb2i hb2j rfl rfl
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lemma is_subsingleton {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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Subsingleton (WickContract c b1 b2) := Subsingleton.intro fun w1 w2 => by
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induction k with
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| zero =>
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have hb1 : b1 = Fin.elim0 := Subsingleton.elim _ _
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have hb2 : b2 = Fin.elim0 := Subsingleton.elim _ _
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subst hb1 hb2
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match w1, w2 with
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| string, string => rfl
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| succ k hI =>
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match w1, w2 with
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| contr i j h hilej hb1 hb2i hb2j w, w2 =>
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let ⟨w', hw'⟩ := mem_snoc i j h hilej hb1 hb2i hb2j w2
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rw [hw']
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apply congrArg (contr i j _ _ _ _ _) (hI w w')
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lemma eq_snoc_castSucc {k n : ℕ} (b1 : Fin k.succ → Fin n) :
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b1 = Fin.snoc (b1 ∘ Fin.castSucc) (b1 (Fin.last k)) := by
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funext i
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rcases Fin.eq_castSucc_or_eq_last i with h1 | h1
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· obtain ⟨i, rfl⟩ := h1
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simp
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· subst h1
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simp
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def fromMaps {n k : ℕ} (c : Fin n → 𝓔) (b1 b2 : Fin k → Fin n)
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(hi : ∀ i, c (b2 i) = ξ (c (b1 i)))
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(hb1ltb2 : ∀ i, b1 i < b2 i)
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(hb1 : StrictMono b1)
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(hb1neb2 : ∀ r1 r2, b1 r1 ≠ b2 r2)
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(hb2 : Function.Injective b2) :
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WickContract c b1 b2 := by
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match k with
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| 0 =>
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refine castMaps ?_ ?_ ?_ string
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· rfl
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· exact funext (fun i => Fin.elim0 i)
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||
· exact funext (fun i => Fin.elim0 i)
|
||
| Nat.succ k =>
|
||
refine castMaps rfl (eq_snoc_castSucc b1).symm (eq_snoc_castSucc b2).symm
|
||
(contr (b1 (Fin.last k)) (b2 (Fin.last k)) (hi (Fin.last k)) (hb1ltb2 (Fin.last k)) (fun r => hb1 (Fin.castSucc_lt_last r)) ?_ ?_
|
||
(fromMaps c (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc) (fun i => hi (Fin.castSucc i))
|
||
(fun i => hb1ltb2 (Fin.castSucc i)) (StrictMono.comp hb1 Fin.strictMono_castSucc)
|
||
?_ ?_
|
||
))
|
||
· exact fun r a => hb1neb2 (Fin.last k) r.castSucc a.symm
|
||
· exact fun r => hb2.eq_iff.mp.mt (Fin.ne_last_of_lt (Fin.castSucc_lt_last r ))
|
||
· exact fun r1 r2 => hb1neb2 r1.castSucc r2.castSucc
|
||
· exact Function.Injective.comp hb2 (Fin.castSucc_injective k)
|
||
|
||
lemma eq_from_maps {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n}
|
||
(w : WickContract c b1 b2) :
|
||
w = fromMaps c w.boundFst w.boundSnd w.color_boundSnd_eq_dual_boundFst
|
||
w.boundFst_lt_boundSnd w.boundFst_strictMono w.boundFst_neq_boundSnd w.boundSnd_injective := by
|
||
exact is_subsingleton.allEq w _
|
||
|
||
structure struc {n : ℕ} (c : Fin n → 𝓔) where
|
||
k : ℕ
|
||
b1 : Fin k ↪o Fin n
|
||
b2 : Fin k ↪ Fin n
|
||
b2_color_eq_dual_b1 : ∀ i, c (b2 i) = ξ (c (b1 i))
|
||
b1_lt_b2 : ∀ i, b1 i < b2 i
|
||
b1_neq_b2 : ∀ r1 r2, b1 r1 ≠ b2 r2
|
||
|
||
def strucEquivSigma {n : ℕ} (c : Fin n → 𝓔) :
|
||
struc c ≃ Σ (k : ℕ) (b1 : Fin k → Fin n) (b2 : Fin k → Fin n), WickContract c b1 b2 where
|
||
toFun s := ⟨s.k, s.b1, s.b2, fromMaps c s.b1 s.b2 s.b2_color_eq_dual_b1
|
||
s.b1_lt_b2 s.b1.strictMono s.b1_neq_b2 s.b2.inj'⟩
|
||
invFun x :=
|
||
match x with
|
||
| ⟨k, b1, b2, w⟩ => ⟨k, OrderEmbedding.ofStrictMono b1 w.boundFst_strictMono,
|
||
⟨b2, w.boundSnd_injective⟩,
|
||
w.color_boundSnd_eq_dual_boundFst, w.boundFst_lt_boundSnd, w.boundFst_neq_boundSnd⟩
|
||
left_inv s := rfl
|
||
right_inv w := by
|
||
match w with
|
||
| ⟨k, b1, b2, w⟩ =>
|
||
simp only [OrderEmbedding.coe_ofStrictMono, Function.Embedding.coeFn_mk, Sigma.mk.inj_iff,
|
||
heq_eq_eq, true_and]
|
||
exact (eq_from_maps w).symm
|
||
|
||
|
||
end WickContract
|
||
|
||
end TwoComplexScalar
|