feat: Property of time-order w.r.t. superCommute
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jstoobysmith
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8 changed files with 781 additions and 8 deletions
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@ -439,6 +439,40 @@ lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) :
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· simp
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· simp [Finset.mul_sum, smul_smul, ofStateList_cons, mul_assoc,
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FieldStatistic.ofList_cons_eq_mul, mul_comm]
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lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
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[ofCrAnList φs1, [ofCrAnList φs2, ofCrAnList φs3]ₛca]ₛca =
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𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
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(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3 ) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
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𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
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repeat rw [superCommute_ofCrAnList_ofCrAnList]
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simp
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repeat rw [superCommute_ofCrAnList_ofCrAnList]
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simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
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by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
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by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
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by_cases h3 : (𝓕 |>ₛ φs3) = bosonic
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· simp [h1, h2, exchangeSign_bosonic, h3, mul_one, one_smul]
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abel
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· simp [h1, h2, exchangeSign_bosonic, bosonic_exchangeSign, mul_one, one_smul]
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abel
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· simp [h1, bosonic_exchangeSign, h3, exchangeSign_bosonic, mul_one, one_smul]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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· simp at h1 h2 h3
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simp [h1, h2, h3]
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abel
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/-!
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## Interaction with grading.
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@ -39,6 +39,66 @@ lemma timeOrder_ofCrAnList (φs : List 𝓕.CrAnStates) :
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rw [← ofListBasis_eq_ofList]
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simp only [timeOrder, Basis.constr_basis]
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lemma timeOrder_timeOrder_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
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let pc (c : 𝓕.CrAnAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c)
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change pc c (Basis.mem_span _ c)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs, rfl⟩ := hx
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simp [pc]
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let pb (b : 𝓕.CrAnAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * b * ofCrAnList φs) = 𝓣ᶠ(a * 𝓣ᶠ(b) * ofCrAnList φs)
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change pb b (Basis.mem_span _ b)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs', rfl⟩ := hx
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simp [pb]
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let pa (a : 𝓕.CrAnAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * ofCrAnList φs' * ofCrAnList φs) = 𝓣ᶠ(a * 𝓣ᶠ(ofCrAnList φs') * ofCrAnList φs)
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change pa a (Basis.mem_span _ a)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs'', rfl⟩ := hx
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simp [pa]
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rw [timeOrder_ofCrAnList]
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simp only [← ofCrAnList_append, Algebra.mul_smul_comm,
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Algebra.smul_mul_assoc, map_smul]
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rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList, smul_smul]
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congr 1
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· simp only [crAnTimeOrderSign, crAnTimeOrderList]
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rw [Wick.koszulSign_of_append_eq_insertionSort, mul_comm]
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· congr 1
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simp only [crAnTimeOrderList]
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rw [insertionSort_append_insertionSort_append]
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· simp [pa]
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· intro x y hx hy h1 h2
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simp_all [pa, add_mul]
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· intro x hx h
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simp_all [pa]
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· simp [pb]
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· intro x y hx hy h1 h2
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simp_all [pb, mul_add, add_mul]
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· intro x hx h
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simp_all [pb]
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· simp [pc]
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· intro x y hx hy h1 h2
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simp_all [pc, mul_add]
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· intro x hx h hp
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simp_all [pc]
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lemma timeOrder_timeOrder_right (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
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trans 𝓣ᶠ(a * b * 1)
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· simp
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· rw [timeOrder_timeOrder_mid]
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simp
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lemma timeOrder_timeOrder_left (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
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trans 𝓣ᶠ(1 * a * b)
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· simp
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· rw [timeOrder_timeOrder_mid]
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simp
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lemma timeOrder_ofStateList (φs : List 𝓕.States) :
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𝓣ᶠ(ofStateList φs) = timeOrderSign φs • ofStateList (timeOrderList φs) := by
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conv_lhs =>
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@ -100,6 +160,119 @@ lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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· rw [crAnTimeOrderList_pair_ordered]
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simp_all
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [timeOrder_timeOrder_right,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
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rw [timeOrder_timeOrder_left,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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rw [timeOrder_timeOrder_mid,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel
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{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra):
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𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
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rw [← bosonicProj_add_fermionicProj a]
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simp
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rw [bosonic_superCommute (Submodule.coe_mem (bosonicProj a))]
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simp
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
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· rw [superCommute_bonsonic h']
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simp [ofCrAnList_singleton]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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· rw [superCommute_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
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simp [ofCrAnList_singleton]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
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(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [summerCommute_jacobi_ofCrAnList]
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simp [ofCrAnList_singleton]
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right
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12 ]
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rw [superCommute_ofCrAnState_ofCrAnState_symm φ3]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
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(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [summerCommute_jacobi_ofCrAnList]
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simp [ofCrAnList_singleton]
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right
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rw [superCommute_ofCrAnState_ofCrAnState_symm φ1]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12 ]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
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(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬ (
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crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
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crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2)) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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simp at h
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by_cases h23 : ¬ crAnTimeOrderRel φ2 φ3
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· simp_all
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h23]
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simp_all
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by_cases h32 : ¬ crAnTimeOrderRel φ3 φ2
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· simp_all
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rw [superCommute_ofCrAnState_ofCrAnState_symm]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h32]
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simp
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simp_all
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by_cases h12 : ¬ crAnTimeOrderRel φ1 φ2
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· have h13 : ¬ crAnTimeOrderRel φ1 φ3 := by
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intro h13
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apply h12
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exact IsTrans.trans φ1 φ3 φ2 h13 h32
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rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel h12 h13]
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simp_all
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have h13 : crAnTimeOrderRel φ1 φ3 := IsTrans.trans φ1 φ2 φ3 h12 h23
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simp_all
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by_cases h21 : ¬ crAnTimeOrderRel φ2 φ1
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· simp_all
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have h31 : ¬ crAnTimeOrderRel φ3 φ1 := by
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intro h31
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apply h21
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exact IsTrans.trans φ2 φ3 φ1 h23 h31
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rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel' h21 h31]
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simp_all
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refine False.elim (h ?_)
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exact IsTrans.trans φ3 φ2 φ1 h32 h21
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_eq_time
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{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = [ofCrAnState φ, ofCrAnState ψ]ₛca := by
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@ -92,7 +92,7 @@ lemma ι_superCommute_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
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left
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use φa, φa', hφa, hφa'
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lemma ι_superCommute_of_diff_statistic (φ ψ : 𝓕.CrAnStates)
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lemma ι_superCommute_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
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@ -107,7 +107,7 @@ lemma ι_superCommute_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton] at h ⊢
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rcases statistic_neq_of_superCommute_fermionic h with h | h
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· simp [ofCrAnList_singleton]
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apply ι_superCommute_of_diff_statistic _ _
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apply ι_superCommute_of_diff_statistic
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simpa using h
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· simp [h]
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@ -0,0 +1,175 @@
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/-
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Copyright (c) 2025 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.Algebras.CrAnAlgebra.TimeOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
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/-!
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# Time Ordering on Field operator algebra
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open HepLean.List
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open FieldStatistic
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_timeOrder_superCommute_time {φ ψ : 𝓕.CrAnStates}
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
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let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
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change pb b (Basis.mem_span _ b)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs, rfl⟩ := hx
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simp [pb]
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let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* ofCrAnList φs))
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change pa a (Basis.mem_span _ a)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs', rfl⟩ := hx
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simp [pa]
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conv_lhs =>
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [mul_sub, sub_mul, ← ofCrAnList_append]
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rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) = crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
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trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
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simp
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rw [crAnTimeOrderSign]
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have hp : List.Perm [φ,ψ] [ψ,φ] := by exact List.Perm.swap ψ φ []
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rw [Wick.koszulSign_perm_eq _ _ φ _ _ _ _ _ hp]
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simp
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rfl
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simp_all
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rw [h1]
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simp
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
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have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
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repeat rw [ofCrAnList_append]
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rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
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rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
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rw [← mul_smul_comm]
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||||
rw [mul_assoc, mul_assoc, mul_assoc ,mul_assoc ,mul_assoc ,mul_assoc]
|
||||
rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
|
||||
← smul_mul_assoc]
|
||||
rw [← sub_mul]
|
||||
have h1 : (ι (ofCrAnList [φ, ψ]) - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
|
||||
ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
|
||||
rw [superCommute_ofCrAnState_ofCrAnState]
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
|
||||
simp only [instCommGroup.eq_1, List.singleton_append, Algebra.smul_mul_assoc, map_sub,
|
||||
map_smul]
|
||||
rw [← ofCrAnList_append]
|
||||
simp
|
||||
rw [h1]
|
||||
have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
|
||||
apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
|
||||
rw [Subalgebra.mem_center_iff] at hc
|
||||
repeat rw [← mul_assoc]
|
||||
rw [hc]
|
||||
repeat rw [mul_assoc]
|
||||
rw [smul_mul_assoc]
|
||||
rw [← map_mul, ← map_mul, ← map_mul, ← map_mul]
|
||||
rw [← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append]
|
||||
have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
|
||||
simp at h1 ⊢
|
||||
rw [← h1]
|
||||
rw [← crAnTimeOrderList]
|
||||
by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
|
||||
· rw [ι_superCommute_of_diff_statistic hq]
|
||||
simp
|
||||
· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
|
||||
rw [timeOrder_ofCrAnList]
|
||||
simp
|
||||
exact hψφ
|
||||
exact hφψ
|
||||
simpa using hq
|
||||
· simp [pa]
|
||||
· intro x y hx hy hpx hpy
|
||||
simp_all [pa,mul_add, add_mul]
|
||||
· intro x hx hpx
|
||||
simp_all [pa, hpx]
|
||||
· simp [pb]
|
||||
· intro x y hx hy hpx hpy
|
||||
simp_all [pb,mul_add, add_mul]
|
||||
· intro x hx hpx
|
||||
simp_all [pb, hpx]
|
||||
|
||||
/-!
|
||||
|
||||
## Defining normal order for `FiedOpAlgebra`.
|
||||
|
||||
-/
|
||||
|
||||
lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
||||
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(a) = 0 := by
|
||||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
|
||||
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
|
||||
change p a h
|
||||
apply AddSubgroup.closure_induction
|
||||
· intro x hx
|
||||
obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
|
||||
obtain ⟨a, ha, c, hc, rfl⟩ := ha
|
||||
simp only [p]
|
||||
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq] at hc
|
||||
match hc with
|
||||
| Or.inl hc =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
| Or.inr (Or.inl hc) =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
| Or.inr (Or.inr (Or.inl hc)) =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
| Or.inr (Or.inr (Or.inr hc)) =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
sorry
|
||||
· simp [p]
|
||||
· intro x y hx hy
|
||||
simp only [map_add, p]
|
||||
intro h1 h2
|
||||
simp [h1, h2]
|
||||
· intro x hx
|
||||
simp [p]
|
||||
|
||||
lemma ι_timeOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
|
||||
ι 𝓣ᶠ(a) = ι 𝓣ᶠ(b) := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||||
simp only [LinearMap.mem_ker, ← map_sub]
|
||||
exact ι_timeOrder_zero_of_mem_ideal (a - b) h
|
||||
|
||||
/-- Normal ordering on `FieldOpAlgebra`. -/
|
||||
noncomputable def timeOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.timeOrder) ι_timeOrder_eq_of_equiv
|
||||
map_add' x y := by
|
||||
obtain ⟨x, hx⟩ := ι_surjective x
|
||||
obtain ⟨y, hy⟩ := ι_surjective y
|
||||
subst hx hy
|
||||
rw [← map_add, ι_apply, ι_apply, ι_apply]
|
||||
rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
|
||||
simp
|
||||
map_smul' c y := by
|
||||
obtain ⟨y, hy⟩ := ι_surjective y
|
||||
subst hy
|
||||
rw [← map_smul, ι_apply, ι_apply]
|
||||
simp
|
||||
|
||||
end FieldOpAlgebra
|
||||
end FieldSpecification
|
||||
|
|
@ -235,6 +235,10 @@ lemma ofList_map_eq_finset_prod (s : 𝓕 → FieldStatistic) :
|
|||
simp only [List.length_cons, List.nodup_cons] at hl
|
||||
exact hl.2
|
||||
|
||||
lemma ofList_pair (s : 𝓕 → FieldStatistic) (φ1 φ2 : 𝓕) :
|
||||
ofList s [φ1, φ2] = s φ1 * s φ2 := by
|
||||
rw [ofList_cons_eq_mul, ofList_singleton]
|
||||
|
||||
/-!
|
||||
|
||||
## ofList and take
|
||||
|
|
|
|||
|
|
@ -280,6 +280,40 @@ lemma koszulSign_swap_eq_rel {ψ φ : 𝓕} (h1 : le φ ψ) (h2 : le ψ φ) : (
|
|||
apply Wick.koszulSignInsert_eq_perm
|
||||
exact List.Perm.append_left φs (List.Perm.swap ψ φ φs')
|
||||
|
||||
lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
|
||||
koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs := by
|
||||
intro φs
|
||||
simp [koszulSign, ← mul_assoc]
|
||||
trans 1 * koszulSign q le φs
|
||||
swap
|
||||
simp
|
||||
congr
|
||||
simp [koszulSignInsert]
|
||||
simp_all
|
||||
rw [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hq]
|
||||
simp
|
||||
|
||||
lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs' φs : List 𝓕) →
|
||||
koszulSign q le (φs' ++ φ :: ψ :: φs) = koszulSign q le (φs' ++ φs)
|
||||
| [], φs => by
|
||||
simp
|
||||
exact koszulSign_eq_rel_eq_stat_append q le h1 h2 hq φs
|
||||
| φ'' :: φs', φs => by
|
||||
simp [koszulSign]
|
||||
rw [koszulSign_eq_rel_eq_stat h1 h2 hq φs' φs]
|
||||
simp
|
||||
left
|
||||
trans koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs) )
|
||||
apply koszulSignInsert_eq_perm
|
||||
refine List.Perm.symm (List.perm_cons_append_cons φ ?_)
|
||||
exact List.Perm.symm List.perm_middle
|
||||
rw [koszulSignInsert_eq_remove_same_stat_append q le ]
|
||||
simp_all
|
||||
simp_all
|
||||
simp_all
|
||||
|
||||
lemma koszulSign_of_sorted : (φs : List 𝓕)
|
||||
→ (hs : List.Sorted le φs) → koszulSign q le φs = 1
|
||||
| [], _ => by
|
||||
|
|
@ -351,4 +385,53 @@ lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le]
|
|||
refine List.Perm.append_left φs'' ?_
|
||||
exact List.Perm.symm (List.perm_insertionSort le φs)
|
||||
|
||||
/-!
|
||||
|
||||
# koszulSign with permutations
|
||||
|
||||
-/
|
||||
|
||||
lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) ( φs φs' φs2 : List 𝓕)
|
||||
(hp : φs.Perm φs') : (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
|
||||
koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2) := by
|
||||
let motive (φs φs' : List 𝓕) (hp : φs.Perm φs') : Prop :=
|
||||
(h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
|
||||
koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)
|
||||
change motive φs φs' hp
|
||||
apply List.Perm.recOn
|
||||
· simp [motive]
|
||||
· intro x l1 l2 h ih hxφ
|
||||
simp_all [motive]
|
||||
simp [koszulSign, ih]
|
||||
left
|
||||
apply koszulSignInsert_eq_perm
|
||||
exact (List.perm_append_right_iff φs2).mpr h
|
||||
· intro x y l h
|
||||
simp_all [motive]
|
||||
apply Wick.koszulSign_swap_eq_rel_cons
|
||||
exact IsTrans.trans y φ x h.1.2 h.2.1.1
|
||||
exact IsTrans.trans x φ y h.2.1.2 h.1.1
|
||||
· intro l1 l2 l3 h1 h2 ih1 ih2 h
|
||||
simp_all [motive]
|
||||
refine (ih2 ?_)
|
||||
intro φ' hφ
|
||||
refine h φ' ?_
|
||||
exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφ
|
||||
|
||||
lemma koszulSign_perm_eq [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
|
||||
(h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → (hp : φs.Perm φs') →
|
||||
koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)
|
||||
| [], φs, φs', φs2, h, hp => by
|
||||
simp
|
||||
exact koszulSign_perm_eq_append q le φ φs φs' φs2 hp h
|
||||
| φ1 :: φs1, φs, φs', φs2, h, hp => by
|
||||
simp only [koszulSign, List.append_eq]
|
||||
have ih := koszulSign_perm_eq φ φs1 φs φs' φs2 h hp
|
||||
rw [ih]
|
||||
congr 1
|
||||
apply koszulSignInsert_eq_perm
|
||||
refine (List.perm_append_right_iff φs2).mpr ?_
|
||||
exact List.Perm.append_left φs1 hp
|
||||
|
||||
|
||||
end Wick
|
||||
|
|
|
|||
|
|
@ -248,4 +248,42 @@ lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀
|
|||
· exact h φ1 (List.mem_cons_self _ _)
|
||||
|
||||
|
||||
lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
|
||||
koszulSignInsert q le φ φs = koszulSignInsert q le ψ φs
|
||||
| [] => by
|
||||
simp [koszulSignInsert]
|
||||
| φ' :: φs => by
|
||||
simp [koszulSignInsert]
|
||||
simp_all
|
||||
by_cases hr : le φ φ'
|
||||
· simp [hr]
|
||||
have h1' : le ψ φ' := by
|
||||
apply IsTrans.trans ψ φ φ' h2 hr
|
||||
simp [h1']
|
||||
exact koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs
|
||||
· have hψφ' : ¬ le ψ φ' := by
|
||||
intro hψφ'
|
||||
apply hr
|
||||
apply IsTrans.trans φ ψ φ' h1 hψφ'
|
||||
simp [hr, hψφ']
|
||||
rw [koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs]
|
||||
|
||||
lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : ( φs : List 𝓕) →
|
||||
koszulSignInsert q le φ' (φ :: ψ :: φs) = koszulSignInsert q le φ' φs := by
|
||||
intro φs
|
||||
simp_all [koszulSignInsert]
|
||||
by_cases hφ'φ : le φ' φ
|
||||
· have hφ'ψ : le φ' ψ := by
|
||||
apply IsTrans.trans φ' φ ψ hφ'φ h1
|
||||
simp [hφ'φ, hφ'ψ]
|
||||
· have hφ'ψ : ¬ le φ' ψ := by
|
||||
intro hφ'ψ
|
||||
apply hφ'φ
|
||||
apply IsTrans.trans φ' ψ φ hφ'ψ h2
|
||||
simp_all [hφ'φ, hφ'ψ]
|
||||
|
||||
|
||||
|
||||
end Wick
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue