b5c987180a
* refactor: Fix field struct defn. * rename: FieldStruct to FieldSpecification * feat: Add examples of field specifications * docs: Slight improvement of module docs
67 lines
2.1 KiB
Text
67 lines
2.1 KiB
Text
/-
|
||
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.PerturbationTheory.WickContraction.Basic
|
||
/-!
|
||
|
||
# Uncontracted elements
|
||
|
||
-/
|
||
open FieldSpecification
|
||
variable {𝓕 : FieldSpecification}
|
||
|
||
namespace WickContraction
|
||
variable {n : ℕ} (c : WickContraction n)
|
||
open HepLean.List
|
||
|
||
/-- Given a Wick contraction, the finset of elements of `Fin n` which are not contracted. -/
|
||
def uncontracted : Finset (Fin n) := Finset.filter (fun i => c.getDual? i = none) (Finset.univ)
|
||
|
||
lemma congr_uncontracted {n m : ℕ} (c : WickContraction n) (h : n = m) :
|
||
(c.congr h).uncontracted = Finset.map (finCongr h).toEmbedding c.uncontracted := by
|
||
subst h
|
||
simp
|
||
|
||
/-- The equivalence of `Option c.uncontracted` for two propositionally equal Wick contractions. -/
|
||
def uncontractedCongr {c c': WickContraction n} (h : c = c') :
|
||
Option c.uncontracted ≃ Option c'.uncontracted :=
|
||
Equiv.optionCongr (Equiv.subtypeEquivRight (by rw [h]; simp))
|
||
|
||
@[simp]
|
||
lemma uncontractedCongr_none {c c': WickContraction n} (h : c = c') :
|
||
(uncontractedCongr h) none = none := by
|
||
simp [uncontractedCongr]
|
||
|
||
@[simp]
|
||
lemma uncontractedCongr_some {c c': WickContraction n} (h : c = c') (i : c.uncontracted) :
|
||
(uncontractedCongr h) (some i) = some (Equiv.subtypeEquivRight (by rw [h]; simp) i) := by
|
||
simp [uncontractedCongr]
|
||
|
||
lemma mem_uncontracted_iff_not_contracted (i : Fin n) :
|
||
i ∈ c.uncontracted ↔ ∀ p ∈ c.1, i ∉ p := by
|
||
simp only [uncontracted, getDual?, Finset.mem_filter, Finset.mem_univ, true_and]
|
||
apply Iff.intro
|
||
· intro h p hp
|
||
have hp := c.2.1 p hp
|
||
rw [Finset.card_eq_two] at hp
|
||
obtain ⟨a, b, ha, hb, hab⟩ := hp
|
||
rw [Fin.find_eq_none_iff] at h
|
||
by_contra hn
|
||
simp only [Finset.mem_insert, Finset.mem_singleton] at hn
|
||
rcases hn with hn | hn
|
||
· subst hn
|
||
exact h b hp
|
||
· subst hn
|
||
rw [Finset.pair_comm] at hp
|
||
exact h a hp
|
||
· intro h
|
||
rw [Fin.find_eq_none_iff]
|
||
by_contra hn
|
||
simp only [not_forall, Decidable.not_not] at hn
|
||
obtain ⟨j, hj⟩ := hn
|
||
apply h {i, j} hj
|
||
simp
|
||
|
||
end WickContraction
|