feat: Add SuperCommute for FieldOpAlgebra
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jstoobysmith
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c2d89cc093
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9107115620
5 changed files with 708 additions and 9 deletions
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@ -19,7 +19,164 @@ 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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lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 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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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
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= 0 := by
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let l1 :=
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(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs1)
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let l2 := (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs2)
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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have h123 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)):= by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ2, φ3] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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simp only [List.append_assoc, List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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have h132 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)):= by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ3, φ2] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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congr 1
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have hp : List.Perm [φ1, φ3, φ2] [φ1, φ2, φ3] := by
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refine List.Perm.cons φ1 ?_
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exact List.Perm.swap φ2 φ3 []
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rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
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· simp
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· intro φ4 hφ4
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simp at hφ4
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rcases hφ4 with hφ4 | hφ4 | hφ4
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all_goals
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subst hφ4
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simp_all
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have hp231 : List.Perm [φ2, φ3, φ1] [φ1, φ2, φ3] := by
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refine List.Perm.trans (l₂ := [φ2, φ1, φ3]) ?_ ?_
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refine List.Perm.cons φ2 (List.Perm.swap φ1 φ3 [])
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exact List.Perm.swap φ1 φ2 [φ3]
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have h231 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)):= by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ2, φ3, φ1] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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congr 1
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rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp231, ← crAnTimeOrderSign]
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· simp
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· intro φ4 hφ4
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simp at hφ4
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rcases hφ4 with hφ4 | hφ4 | hφ4
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all_goals
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subst hφ4
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simp_all
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have h321 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)):= by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ3, φ2, φ1] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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congr 1
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have hp : List.Perm [φ3, φ2, φ1] [φ1, φ2, φ3] := by
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refine List.Perm.trans ?_ hp231
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exact List.Perm.swap φ2 φ3 [φ1]
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rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
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· simp
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· intro φ4 hφ4
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simp at hφ4
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rcases hφ4 with hφ4 | hφ4 | hφ4
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all_goals
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subst hφ4
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simp_all
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [superCommute_ofCrAnList_ofCrAnList]
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simp
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rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
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simp [mul_sub, sub_mul, ← ofCrAnList_append]
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rw [h123, h132, h231, h321]
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simp [smul_smul]
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rw [mul_comm, ← smul_smul, mul_comm, ← smul_smul]
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rw [← smul_sub, ← smul_sub, smul_smul, mul_comm, ← smul_smul, ← smul_sub]
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simp
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right
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rw [← smul_mul_assoc, ← mul_smul_comm, mul_assoc]
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rw [← smul_mul_assoc, ← mul_smul_comm]
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rw [smul_sub]
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rw [← smul_mul_assoc, ← mul_smul_comm]
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rw [← smul_mul_assoc, ← mul_smul_comm]
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repeat rw [mul_assoc]
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rw [← mul_sub, ← mul_sub, ← mul_sub]
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rw [← sub_mul, ← sub_mul, ← sub_mul]
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trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ι (ofCrAnList l2)
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rw [mul_assoc]
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congr
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [superCommute_ofCrAnList_ofCrAnList]
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simp
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rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
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simp [smul_sub]
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simp_all
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lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates):
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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
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= 0 := by
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by_cases 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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· exact ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList φs1 φs2 h
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· rw [timeOrder_timeOrder_mid]
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rw [timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel _ _ _ h]
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simp
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@[simp]
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lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra):
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ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := 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 φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
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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 φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs) = 0
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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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exact ι_timeOrder_superCommute_superCommute_ofCrAnList φs' φs
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· simp [pa]
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· intro x y hx hy hpx hpy
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simp_all [pa,mul_add, add_mul]
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· intro x hx hpx
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simp_all [pa, hpx]
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· simp [pb]
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· intro x y hx hy hpx hpy
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simp_all [pb,mul_add, add_mul]
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· intro x hx hpx
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simp_all [pb, hpx]
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lemma ι_timeOrder_superCommute_eq_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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@ -110,6 +267,25 @@ lemma ι_timeOrder_superCommute_time {φ ψ : 𝓕.CrAnStates}
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· intro x hx hpx
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simp_all [pb, hpx]
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lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
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(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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rw [timeOrder_timeOrder_mid]
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have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
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exact Decidable.not_and_iff_or_not.mp hφψ
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rcases hφψ with hφψ | hφψ
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· rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
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have ht := IsTotal.total (r := crAnTimeOrderRel) φ ψ
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simp_all
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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_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
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simp
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simp_all
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/-!
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## Defining normal order for `FiedOpAlgebra`.
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@ -130,16 +306,42 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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match hc with
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| Or.inl hc =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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simp
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| Or.inr (Or.inl hc) =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
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by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
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· rw [ι_timeOrder_superCommute_eq_time]
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simp
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rw [ι_superCommute_of_create_create]
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simp
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· exact hφa
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· exact hφb
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· exact heqt.1
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· exact heqt.2
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· rw [ι_timeOrder_superCommute_neq_time heqt]
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| Or.inr (Or.inr (Or.inl hc)) =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
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by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
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· rw [ι_timeOrder_superCommute_eq_time]
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simp
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rw [ι_superCommute_of_annihilate_annihilate]
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simp
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· exact hφa
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· exact hφb
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· exact heqt.1
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· exact heqt.2
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· rw [ι_timeOrder_superCommute_neq_time heqt]
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| Or.inr (Or.inr (Or.inr hc)) =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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obtain ⟨φa, φb, hdiff, rfl⟩ := hc
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by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
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· rw [ι_timeOrder_superCommute_eq_time]
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simp
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rw [ι_superCommute_of_diff_statistic]
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simp
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· exact hdiff
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· exact heqt.1
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· exact heqt.2
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· rw [ι_timeOrder_superCommute_neq_time heqt]
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· simp [p]
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· intro x y hx hy
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simp only [map_add, p]
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