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Merge pull request #268 from HEPLean/WickContract
feat: Introduce field statistics
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| Original file line number | Diff line number | Diff line change |
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| /- | ||
| Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Joseph Tooby-Smith | ||
| -/ | ||
| import Mathlib.Algebra.FreeAlgebra | ||
| import Mathlib.Algebra.Lie.OfAssociative | ||
| import Mathlib.Analysis.Complex.Basic | ||
| /-! | ||
| # Field statistics | ||
| Basic properties related to whether a field, or list of fields, is bosonic or fermionic. | ||
| -/ | ||
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| /-- A field can either be bosonic or fermionic in nature. | ||
| That is to say, they can either have Bose-Einstein statistics or | ||
| Fermi-Dirac statistics. -/ | ||
| inductive FieldStatistic : Type where | ||
| | bosonic : FieldStatistic | ||
| | fermionic : FieldStatistic | ||
| deriving DecidableEq | ||
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| namespace FieldStatistic | ||
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| variable {𝓕 : Type} | ||
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| /-- Field statics form a finite type. -/ | ||
| instance : Fintype FieldStatistic where | ||
| elems := {bosonic, fermionic} | ||
| complete := by | ||
| intro c | ||
| cases c | ||
| · exact Finset.mem_insert_self bosonic {fermionic} | ||
| · refine Finset.insert_eq_self.mp ?_ | ||
| exact rfl | ||
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| @[simp] | ||
| lemma fermionic_not_eq_bonsic : ¬ fermionic = bosonic := by | ||
| intro h | ||
| exact FieldStatistic.noConfusion h | ||
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| lemma bonsic_eq_fermionic_false : bosonic = fermionic ↔ false := by | ||
| simp only [reduceCtorEq, Bool.false_eq_true] | ||
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| @[simp] | ||
| lemma neq_fermionic_iff_eq_bosonic (a : FieldStatistic) : ¬ a = fermionic ↔ a = bosonic := by | ||
| fin_cases a | ||
| · simp | ||
| · simp | ||
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| @[simp] | ||
| lemma bosonic_neq_iff_fermionic_eq (a : FieldStatistic) : ¬ bosonic = a ↔ fermionic = a := by | ||
| fin_cases a | ||
| · simp | ||
| · simp | ||
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| @[simp] | ||
| lemma fermionic_neq_iff_bosonic_eq (a : FieldStatistic) : ¬ fermionic = a ↔ bosonic = a := by | ||
| fin_cases a | ||
| · simp | ||
| · simp | ||
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| lemma eq_self_if_eq_bosonic {a : FieldStatistic} : | ||
| (if a = bosonic then bosonic else fermionic) = a := by | ||
| fin_cases a <;> rfl | ||
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| lemma eq_self_if_bosonic_eq {a : FieldStatistic} : | ||
| (if bosonic = a then bosonic else fermionic) = a := by | ||
| fin_cases a <;> rfl | ||
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| /-- The field statistics of a list of fields is fermionic if ther is an odd number of fermions, | ||
| otherwise it is bosonic. -/ | ||
| def ofList (s : 𝓕 → FieldStatistic) : (φs : List 𝓕) → FieldStatistic | ||
| | [] => bosonic | ||
| | φ :: φs => if s φ = ofList s φs then bosonic else fermionic | ||
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| @[simp] | ||
| lemma ofList_singleton (s : 𝓕 → FieldStatistic) (φ : 𝓕) : ofList s [φ] = s φ := by | ||
| simp only [ofList, Fin.isValue] | ||
| rw [eq_self_if_eq_bosonic] | ||
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| @[simp] | ||
| lemma ofList_freeMonoid (s : 𝓕 → FieldStatistic) (φ : 𝓕) : ofList s (FreeMonoid.of φ) = s φ := | ||
| ofList_singleton s φ | ||
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| @[simp] | ||
| lemma ofList_empty (s : 𝓕 → FieldStatistic) : ofList s [] = bosonic := rfl | ||
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| @[simp] | ||
| lemma ofList_append (s : 𝓕 → FieldStatistic) (l r : List 𝓕) : | ||
| ofList s (l ++ r) = if ofList s l = ofList s r then bosonic else fermionic := by | ||
| induction l with | ||
| | nil => | ||
| simp only [List.nil_append, ofList_empty, Fin.isValue, eq_self_if_bosonic_eq] | ||
| | cons a l ih => | ||
| have hab (a b c : FieldStatistic) : | ||
| (if a = (if b = c then bosonic else fermionic) then bosonic else fermionic) = | ||
| if (if a = b then bosonic else fermionic) = c then bosonic else fermionic := by | ||
| fin_cases a <;> fin_cases b <;> fin_cases c <;> rfl | ||
| simp only [ofList, List.append_eq, Fin.isValue, ih, hab] | ||
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| lemma ofList_eq_countP (s : 𝓕 → FieldStatistic) (φs : List 𝓕) : | ||
| ofList s φs = if Nat.mod (List.countP (fun i => decide (s i = fermionic)) φs) 2 = 0 then | ||
| bosonic else fermionic := by | ||
| induction φs with | ||
| | nil => simp | ||
| | cons r0 r ih => | ||
| simp only [ofList] | ||
| rw [List.countP_cons] | ||
| simp only [decide_eq_true_eq] | ||
| by_cases hr : s r0 = fermionic | ||
| · simp only [hr, ↓reduceIte] | ||
| simp_all only | ||
| split | ||
| next h => | ||
| simp_all only [↓reduceIte, fermionic_not_eq_bonsic] | ||
| split | ||
| next h_1 => | ||
| simp_all only [fermionic_not_eq_bonsic] | ||
| have ha (a : ℕ) (ha : a % 2 = 0) : (a + 1) % 2 ≠ 0 := by | ||
| omega | ||
| exact ha (List.countP (fun i => decide (s i = fermionic)) r) h h_1 | ||
| next h_1 => simp_all only | ||
| next h => | ||
| simp_all only [↓reduceIte] | ||
| split | ||
| next h_1 => rfl | ||
| next h_1 => | ||
| simp_all only [reduceCtorEq] | ||
| have ha (a : ℕ) (ha : ¬ a % 2 = 0) : (a + 1) % 2 = 0 := by | ||
| omega | ||
| exact h_1 (ha (List.countP (fun i => decide (s i = fermionic)) r) h) | ||
| · simp only [neq_fermionic_iff_eq_bosonic] at hr | ||
| by_cases hx : (List.countP (fun i => decide (s i = fermionic)) r).mod 2 = 0 | ||
| · simpa [hx, hr] using ih.symm | ||
| · simpa [hx, hr] using ih.symm | ||
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| lemma ofList_perm (s : 𝓕 → FieldStatistic) {l l' : List 𝓕} (h : l.Perm l') : | ||
| ofList s l = ofList s l' := by | ||
| rw [ofList_eq_countP, ofList_eq_countP, h.countP_eq] | ||
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| lemma ofList_orderedInsert (s : 𝓕 → FieldStatistic) (le1 : 𝓕 → 𝓕 → Prop) [DecidableRel le1] | ||
| (φs : List 𝓕) (φ : 𝓕) : ofList s (List.orderedInsert le1 φ φs) = ofList s (φ :: φs) := | ||
| ofList_perm s (List.perm_orderedInsert le1 φ φs) | ||
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| @[simp] | ||
| lemma ofList_insertionSort (s : 𝓕 → FieldStatistic) (le1 : 𝓕 → 𝓕 → Prop) [DecidableRel le1] | ||
| (φs : List 𝓕) : ofList s (List.insertionSort le1 φs) = ofList s φs := | ||
| ofList_perm s (List.perm_insertionSort le1 φs) | ||
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| end FieldStatistic |
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