feat: Property of time-order w.r.t. superCommute
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jstoobysmith
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a79d0f8fed
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c2d89cc093
8 changed files with 781 additions and 8 deletions
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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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