refactor: Normal order results
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jstoobysmith
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2 changed files with 209 additions and 149 deletions
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@ -10,17 +10,19 @@ import HepLean.PerturbationTheory.Koszul.KoszulSign
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# Normal Ordering in the CrAnAlgebra
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In the module
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`HepLean.PerturbationTheory.FieldSpecification.NormalOrder`
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we defined the normal ordering of a list of `CrAnStates`.
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In this module we extend the normal ordering to a linear map on `CrAnAlgebra`.
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We derive properties of this normal ordering.
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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open FieldStatistic
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/-!
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## Normal order on the CrAnAlgebra
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-/
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namespace CrAnAlgebra
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noncomputable section
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@ -51,6 +53,12 @@ lemma normalOrder_one : normalOrder (𝓕 := 𝓕) 1 = 1 := by
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rw [← ofCrAnList_nil, normalOrder_ofCrAnList]
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simp
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/-!
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## Normal ordering with a creation operator on the left or annihilation on the right
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-/
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lemma normalOrder_ofCrAnList_cons_create (φ : 𝓕.CrAnStates)
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(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (φs : List 𝓕.CrAnStates) :
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normalOrder (ofCrAnList (φ :: φs)) =
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@ -95,83 +103,6 @@ lemma normalOrder_mul_annihilate (φ : 𝓕.CrAnStates)
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rw [← ofCrAnList_singleton, ← ofCrAnList_append, ofCrAnList_singleton]
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rw [normalOrder_ofCrAnList_append_annihilate φ hφ]
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lemma normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(φs φs' : List 𝓕.CrAnStates) :
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normalOrder (ofCrAnList φs' * ofCrAnState φc * ofCrAnState φa * ofCrAnList φs) =
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𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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normalOrder (ofCrAnList φs' * ofCrAnState φa * ofCrAnState φc * ofCrAnList φs) := by
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rw [mul_assoc, mul_assoc, ← ofCrAnList_cons, ← ofCrAnList_cons, ← ofCrAnList_append]
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rw [normalOrder_ofCrAnList, normalOrderSign_swap_create_annihlate φc φa hφc hφa]
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rw [normalOrderList_swap_create_annihlate φc φa hφc hφa]
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rw [← smul_smul, ← normalOrder_ofCrAnList]
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congr
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rw [ofCrAnList_append, ofCrAnList_cons, ofCrAnList_cons]
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noncomm_ring
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lemma normalOrder_swap_create_annihlate_ofCrAnList (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(φs : List 𝓕.CrAnStates) (a : 𝓕.CrAnAlgebra) :
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normalOrder (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa * a) =
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𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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normalOrder (ofCrAnList φs * ofCrAnState φa * ofCrAnState φc * a) := by
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change (normalOrder ∘ₗ mulLinearMap (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa)) a =
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(smulLinearMap _ ∘ₗ normalOrder ∘ₗ
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mulLinearMap (ofCrAnList φs * ofCrAnState φa * ofCrAnState φc)) a
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refine LinearMap.congr_fun ?h a
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apply ofCrAnListBasis.ext
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intro l
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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normalOrder (a * ofCrAnState φc * ofCrAnState φa * b) =
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𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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normalOrder (a * ofCrAnState φa * ofCrAnState φc * b) := by
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rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc]
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change (normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φc * (ofCrAnState φa * b))) a =
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(smulLinearMap (𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa)) ∘ₗ
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normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φa * (ofCrAnState φc * b))) a
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apply LinearMap.congr_fun
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apply ofCrAnListBasis.ext
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intro l
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simp only [mulLinearMap, ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk, instCommGroup.eq_1]
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repeat rw [← mul_assoc]
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rw [normalOrder_swap_create_annihlate_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_superCommute_create_annihilate (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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normalOrder (a * superCommute (ofCrAnState φc) (ofCrAnState φa) * b) = 0 := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc]
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rw [← mul_assoc, ← mul_assoc]
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rw [normalOrder_swap_create_annihlate φc φa hφc hφa]
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simp only [FieldStatistic.instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc,
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map_smul, sub_self]
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lemma normalOrder_superCommute_annihilate_create (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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normalOrder (a * superCommute (ofCrAnState φa) (ofCrAnState φc) * b) = 0 := by
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rw [superCommute_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
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Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
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apply Or.inr
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exact normalOrder_superCommute_create_annihilate φc φa hφc hφa a b
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lemma normalOrder_crPart_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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normalOrder (crPart (StateAlgebra.ofState φ) * a) =
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crPart (StateAlgebra.ofState φ) * normalOrder a := by
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@ -205,6 +136,85 @@ lemma normalOrder_mul_anPart (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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refine normalOrder_mul_annihilate _ ?_ _
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simp [crAnStatesToCreateAnnihilate]
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/-!
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## Normal ordering for an adjacent creation and annihliation state
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The main result of this section is `normalOrder_superCommute_annihilate_create`.
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-/
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lemma normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(φs φs' : List 𝓕.CrAnStates) :
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normalOrder (ofCrAnList φs' * ofCrAnState φc * ofCrAnState φa * ofCrAnList φs) =
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𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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normalOrder (ofCrAnList φs' * ofCrAnState φa * ofCrAnState φc * ofCrAnList φs) := by
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rw [mul_assoc, mul_assoc, ← ofCrAnList_cons, ← ofCrAnList_cons, ← ofCrAnList_append]
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rw [normalOrder_ofCrAnList, normalOrderSign_swap_create_annihlate φc φa hφc hφa]
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rw [normalOrderList_swap_create_annihlate φc φa hφc hφa]
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rw [← smul_smul, ← normalOrder_ofCrAnList]
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congr
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rw [ofCrAnList_append, ofCrAnList_cons, ofCrAnList_cons]
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noncomm_ring
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lemma normalOrder_swap_create_annihlate_ofCrAnList (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(φs : List 𝓕.CrAnStates) (a : 𝓕.CrAnAlgebra) :
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normalOrder (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa * a) =
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𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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normalOrder (ofCrAnList φs * ofCrAnState φa * ofCrAnState φc * a) := by
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change (normalOrder ∘ₗ mulLinearMap (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa)) a =
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(smulLinearMap _ ∘ₗ normalOrder ∘ₗ
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mulLinearMap (ofCrAnList φs * ofCrAnState φa * ofCrAnState φc)) a
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refine LinearMap.congr_fun ?h a
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apply ofCrAnListBasis.ext
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intro l
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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normalOrder (a * ofCrAnState φc * ofCrAnState φa * b) =
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𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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normalOrder (a * ofCrAnState φa * ofCrAnState φc * b) := by
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rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc]
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change (normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φc * (ofCrAnState φa * b))) a =
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(smulLinearMap (𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa)) ∘ₗ
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normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φa * (ofCrAnState φc * b))) a
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apply LinearMap.congr_fun
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apply ofCrAnListBasis.ext
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intro l
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simp only [mulLinearMap, ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk, instCommGroup.eq_1]
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repeat rw [← mul_assoc]
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rw [normalOrder_swap_create_annihlate_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_superCommute_create_annihilate (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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normalOrder (a * superCommute (ofCrAnState φc) (ofCrAnState φa) * b) = 0 := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc]
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rw [← mul_assoc, ← mul_assoc]
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rw [normalOrder_swap_create_annihlate φc φa hφc hφa]
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simp only [FieldStatistic.instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc,
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map_smul, sub_self]
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lemma normalOrder_superCommute_annihilate_create (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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normalOrder (a * superCommute (ofCrAnState φa) (ofCrAnState φc) * b) = 0 := by
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rw [superCommute_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
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Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
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apply Or.inr
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exact normalOrder_superCommute_create_annihilate φc φa hφc hφa a b
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lemma normalOrder_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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normalOrder (a * (crPart (StateAlgebra.ofState φ)) * (anPart (StateAlgebra.ofState φ')) * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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@ -240,6 +250,15 @@ lemma normalOrder_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra
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rfl
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rfl
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/-!
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## Normal ordering for an anPart and crPart
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Using the results from above.
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-/
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lemma normalOrder_swap_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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normalOrder (a * (anPart (StateAlgebra.ofState φ)) * (crPart (StateAlgebra.ofState φ')) * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • normalOrder (a * (crPart (StateAlgebra.ofState φ')) *
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@ -290,11 +309,76 @@ lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnA
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simp only [anPart_posAsymp, crPart_position]
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refine normalOrder_superCommute_annihilate_create _ _ (by rfl) (by rfl) _ _
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/-!
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## The normal ordering of a product of two states
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-/
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@[simp]
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lemma normalOrder_crPart_mul_crPart (φ φ' : 𝓕.States) :
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normalOrder (crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
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crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') := by
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rw [normalOrder_crPart_mul]
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conv_lhs => rw [← mul_one (crPart (StateAlgebra.ofState φ'))]
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rw [normalOrder_crPart_mul, normalOrder_one]
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simp
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@[simp]
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lemma normalOrder_anPart_mul_anPart (φ φ' : 𝓕.States) :
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normalOrder (anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
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anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
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rw [normalOrder_mul_anPart]
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conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ))]
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rw [normalOrder_mul_anPart, normalOrder_one]
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simp
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@[simp]
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lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.States) :
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normalOrder (crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
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crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
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rw [normalOrder_crPart_mul]
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conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ'))]
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rw [normalOrder_mul_anPart, normalOrder_one]
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simp
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@[simp]
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lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.States) :
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normalOrder (anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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(crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) := by
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conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ'))]
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conv_lhs => rw [← mul_one (1 * (anPart (StateAlgebra.ofState φ) *
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crPart (StateAlgebra.ofState φ')))]
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rw [← mul_assoc, normalOrder_swap_anPart_crPart]
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simp
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lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
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normalOrder (ofState φ * ofState φ') =
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crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') +
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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(crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) +
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crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') +
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anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
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rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart]
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rw [mul_add, add_mul, add_mul]
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simp only [map_add, normalOrder_crPart_mul_crPart, normalOrder_anPart_mul_crPart,
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instCommGroup.eq_1, normalOrder_crPart_mul_anPart, normalOrder_anPart_mul_anPart]
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abel
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/-!
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## Normal order with super commutors
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-/
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/-! TODO: Split the following two lemmas up into smaller parts. -/
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lemma normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList
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(φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnStates) :
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(normalOrder (ofCrAnList φs *
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superCommute (ofCrAnState φc) (ofCrAnState φc') * ofCrAnList φs')) =
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superCommute (ofCrAnState φc) (ofCrAnState φc') * ofCrAnList φs')) =
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normalOrderSign (φs ++ φc' :: φc :: φs') •
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(ofCrAnList (createFilter φs) * superCommute (ofCrAnState φc) (ofCrAnState φc') *
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ofCrAnList (createFilter φs') * ofCrAnList (annihilateFilter (φs ++ φs'))) := by
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@ -440,56 +524,11 @@ lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
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rw [ofCrAnList_append, ofCrAnList_singleton, ofCrAnList_singleton]
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rw [ofCrAnList_append, ofCrAnList_singleton, ofCrAnList_singleton, smul_mul_assoc]
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||||
|
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@[simp]
|
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lemma normalOrder_crPart_mul_crPart (φ φ' : 𝓕.States) :
|
||||
normalOrder (crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
|
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crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') := by
|
||||
rw [normalOrder_crPart_mul]
|
||||
conv_lhs => rw [← mul_one (crPart (StateAlgebra.ofState φ'))]
|
||||
rw [normalOrder_crPart_mul, normalOrder_one]
|
||||
simp
|
||||
/-!
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_anPart_mul_anPart (φ φ' : 𝓕.States) :
|
||||
normalOrder (anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
|
||||
anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
|
||||
rw [normalOrder_mul_anPart]
|
||||
conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ))]
|
||||
rw [normalOrder_mul_anPart, normalOrder_one]
|
||||
simp
|
||||
## Super commututators involving a normal order.
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.States) :
|
||||
normalOrder (crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
|
||||
crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
|
||||
rw [normalOrder_crPart_mul]
|
||||
conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ'))]
|
||||
rw [normalOrder_mul_anPart, normalOrder_one]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.States) :
|
||||
normalOrder (anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||||
(crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) := by
|
||||
conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ'))]
|
||||
conv_lhs => rw [← mul_one (1 * (anPart (StateAlgebra.ofState φ) *
|
||||
crPart (StateAlgebra.ofState φ')))]
|
||||
rw [← mul_assoc, normalOrder_swap_anPart_crPart]
|
||||
simp
|
||||
|
||||
lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
|
||||
normalOrder (ofState φ * ofState φ') =
|
||||
crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') +
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||||
(crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) +
|
||||
crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') +
|
||||
anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
|
||||
rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart]
|
||||
rw [mul_add, add_mul, add_mul]
|
||||
simp only [map_add, normalOrder_crPart_mul_crPart, normalOrder_anPart_mul_crPart,
|
||||
instCommGroup.eq_1, normalOrder_crPart_mul_anPart, normalOrder_anPart_mul_anPart]
|
||||
abel
|
||||
-/
|
||||
|
||||
lemma ofCrAnList_superCommute_normalOrder_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
|
||||
⟨ofCrAnList φs, normalOrder (ofCrAnList φs')⟩ₛca =
|
||||
|
|
@ -509,6 +548,12 @@ lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnState
|
|||
rw [ofCrAnList_superCommute_normalOrder_ofCrAnList,
|
||||
CrAnSection.statistics_eq_state_statistics]
|
||||
|
||||
/-!
|
||||
|
||||
## Multiplications with normal order written in terms of super commute.
|
||||
|
||||
-/
|
||||
|
||||
lemma ofCrAnList_mul_normalOrder_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) :
|
||||
ofCrAnList φs * normalOrder (ofStateList φs') =
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue