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refactor: Change notation for normal order

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jstoobysmith 2025-01-27 12:19:21 +00:00
parent ec2e1e7df9
commit 835c47dbf8
7 changed files with 144 additions and 108 deletions
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30
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/NormalOrder.lean
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@ -33,7 +33,7 @@ is zero.
lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
ι 𝓝(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * ofCrAnList φs') = 0 := by
ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * ofCrAnList φs') = 0 := by
rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa) with hφa | hφa
<;> rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa') with hφa' | hφa'
· rw [normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList φa φa' hφa hφa' φs φs']
@ -52,7 +52,7 @@ lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero
lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
(a : 𝓕.CrAnAlgebra) : ι 𝓝(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
(a : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
have hf : ι.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap (ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca) = 0 := by
apply ofCrAnListBasis.ext
@ -67,7 +67,7 @@ lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero
lemma ι_normalOrder_superCommute_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
ι 𝓝(a * [ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
rw [mul_assoc]
change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ mulLinearMap.flip
([ofCrAnState φa, ofCrAnState φa']ₛca * b)) a = 0
@ -86,7 +86,7 @@ lemma ι_normalOrder_superCommute_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnS
lemma ι_normalOrder_superCommute_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
ι 𝓝(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_eq_sum]
rw [Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
@ -98,7 +98,7 @@ lemma ι_normalOrder_superCommute_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕
lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
ι 𝓝(a * [ofCrAnList φs, ofCrAnState φa]ₛca * b) = 0 := by
ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnState φa]ₛca * b) = 0 := by
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
Algebra.mul_smul_comm, neg_mul, Algebra.smul_mul_assoc, map_neg, map_smul]
@ -107,7 +107,7 @@ lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕
lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero_mul
(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
ι 𝓝(a * [ofCrAnList φs, ofCrAnList φs']ₛca * b) = 0 := by
ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnList φs']ₛca * b) = 0 := by
rw [superCommute_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
apply Fintype.sum_eq_zero
@ -119,7 +119,7 @@ lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero_mul
lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero_mul
(φs : List 𝓕.CrAnStates)
(a b c : 𝓕.CrAnAlgebra) :
ι 𝓝(a * [ofCrAnList φs, c]ₛca * b) = 0 := by
ι 𝓝ᶠ(a * [ofCrAnList φs, c]ₛca * b) = 0 := by
change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute (ofCrAnList φs)) c = 0
have hf : (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
@ -135,7 +135,7 @@ lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero_mul
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul
(a b c d : 𝓕.CrAnAlgebra) : ι 𝓝(a * [d, c]ₛca * b) = 0 := by
(a b c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute.flip c) d = 0
have hf : (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
@ -151,25 +151,25 @@ lemma ι_normalOrder_superCommute_eq_zero_mul
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul_right (b c d : 𝓕.CrAnAlgebra) :
ι 𝓝([d, c]ₛca * b) = 0 := by
ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul 1 b c d]
simp
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul_left (a c d : 𝓕.CrAnAlgebra) :
ι 𝓝(a * [d, c]ₛca) = 0 := by
ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul a 1 c d]
simp
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.CrAnAlgebra) :
ι 𝓝(a * [d, c]ₛca * b1 * b2) = 0 := by
ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul a (b1 * b2) c d]
congr 2
noncomm_ring
@[simp]
lemma ι_normalOrder_superCommute_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝([d, c]ₛca) = 0 := by
lemma ι_normalOrder_superCommute_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul 1 1 c d]
simp
@ -180,9 +180,9 @@ lemma ι_normalOrder_superCommute_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝([d,
-/
lemma ι_normalOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓝(a) = 0 := by
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓝ᶠ(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 𝓝(a) = 0
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
change p a h
apply AddSubgroup.closure_induction
· intro x hx
@ -212,7 +212,7 @@ lemma ι_normalOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
simp [p]
lemma ι_normalOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
ι 𝓝(a) = ι 𝓝(b) := by
ι 𝓝ᶠ(a) = ι 𝓝ᶠ(b) := by
rw [equiv_iff_sub_mem_ideal] at h
rw [LinearMap.sub_mem_ker_iff.mp]
simp only [LinearMap.mem_ker, ← map_sub]