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refactor: Rename ofCrAnState and ofCrAnList

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jstoobysmith 2025-02-03 11:21:11 +00:00
parent 93d06895c6
commit 171e80fc04
11 changed files with 601 additions and 601 deletions
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132
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/Basic.lean
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@ -24,13 +24,13 @@ variable (𝓕 : FieldSpecification)
def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
{ x |
(∃ (φ1 φ2 φ3 : 𝓕.CrAnStates),
x = [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca)
x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca)
∨ (∃ (φc φc' : 𝓕.CrAnStates) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
x = [ofCrAnState φc, ofCrAnState φc']ₛca)
x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛca)
∨ (∃ (φa φa' : 𝓕.CrAnStates) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
x = [ofCrAnState φa, ofCrAnState φa']ₛca)
x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛca)
∨ (∃ (φ φ' : 𝓕.CrAnStates) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
x = [ofCrAnState φ, ofCrAnState φ']ₛca)}
x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
/-- The algebra spanned by cr and an parts of fields, with appropriate super-commutors
set to zero. -/
@ -73,7 +73,7 @@ lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.f
simpa using hx
lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = .create)
(hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnState φc, ofCrAnState φc']ₛca = 0 := by
(hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnOpF φc, ofCrAnOpF φc']ₛca = 0 := by
apply ι_of_mem_fieldOpIdealSet
simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
simp only [exists_prop]
@ -83,7 +83,7 @@ lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc :
lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
(hφa : 𝓕 |>ᶜ φa = .annihilate) (hφa' : 𝓕 |>ᶜ φa' = .annihilate) :
ι [ofCrAnState φa, ofCrAnState φa']ₛca = 0 := by
ι [ofCrAnOpF φa, ofCrAnOpF φa']ₛca = 0 := by
apply ι_of_mem_fieldOpIdealSet
simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
simp only [exists_prop]
@ -93,7 +93,7 @@ lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
use φa, φa', hφa, hφa'
lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
apply ι_of_mem_fieldOpIdealSet
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
right
@ -102,21 +102,21 @@ lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
use φ, ψ
lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
(h : [ofCrAnState φ, ofCrAnState ψ]ₛca ∈ statisticSubmodule fermionic) :
ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton] at h ⊢
(h : [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule fermionic) :
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton] at h ⊢
rcases statistic_neq_of_superCommuteF_fermionic h with h | h
· simp only [ofCrAnList_singleton]
· simp only [ofCrAnListF_singleton]
apply ι_superCommuteF_of_diff_statistic
simpa using h
· simp [h]
lemma ι_superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_zero (φ ψ : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState ψ]ₛca ∈ statisticSubmodule bosonic ∨
ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [ψ] with h | h
· simp_all [ofCrAnList_singleton]
· simp_all only [ofCrAnList_singleton]
lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnStates) :
[ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule bosonic ∨
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [ψ] with h | h
· simp_all [ofCrAnListF_singleton]
· simp_all only [ofCrAnListF_singleton]
right
exact ι_superCommuteF_zero_of_fermionic _ _ h
@ -127,63 +127,63 @@ lemma ι_superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_zero (φ ψ : 𝓕.CrA
-/
@[simp]
lemma ι_superCommuteF_ofCrAnState_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca = 0 := by
lemma ι_superCommuteF_ofCrAnOpF_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnStates) :
ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca = 0 := by
apply ι_of_mem_fieldOpIdealSet
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
left
use φ1, φ2, φ3
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnState φ3]ₛca = 0 := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnStates) :
ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnOpF φ3]ₛca = 0 := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
· rw [bonsonic_superCommuteF_symm h]
simp [ofCrAnList_singleton]
· rcases ofCrAnList_bosonic_or_fermionic [φ3] with h' | h'
simp [ofCrAnListF_singleton]
· rcases ofCrAnListF_bosonic_or_fermionic [φ3] with h' | h'
· rw [superCommuteF_bonsonic_symm h']
simp [ofCrAnList_singleton]
simp [ofCrAnListF_singleton]
· rw [superCommuteF_fermionic_fermionic_symm h h']
simp [ofCrAnList_singleton]
simp [ofCrAnListF_singleton]
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2 : 𝓕.CrAnStates)
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates) :
ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnList φs]ₛca = 0 := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
· rw [superCommuteF_bosonic_ofCrAnList_eq_sum _ _ h]
simp [ofCrAnList_singleton, ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnState]
· rw [superCommuteF_fermionic_ofCrAnList_eq_sum _ _ h]
simp [ofCrAnList_singleton, ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnState]
ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnListF φs]ₛca = 0 := by
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
· rw [superCommuteF_bosonic_ofCrAnListF_eq_sum _ _ h]
simp [ofCrAnListF_singleton, ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF]
· rw [superCommuteF_fermionic_ofCrAnListF_eq_sum _ _ h]
simp [ofCrAnListF_singleton, ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF]
@[simp]
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnStates)
(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnState φ1, ofCrAnState φ2]ₛca) = 0 := by
apply (ofCrAnListBasis.ext fun l ↦ ?_)
simp [ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnList]
lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnStates)
(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca = 0 := by
change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) a = _
have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) = 0 := by
apply (ofCrAnListFBasis.ext fun l ↦ ?_)
simp [ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF]
rw [h1]
simp
lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
ι [ofCrAnState φ1, ofCrAnState φ2]ₛca * ι a = 0 := by
rcases ι_superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_zero φ1 φ2 with h | h
lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 : 𝓕.CrAnStates)
(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca -
ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca * ι a = 0 := by
rcases ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero φ1 φ2 with h | h
swap
· simp [h]
trans - ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca
trans - ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca
· rw [bosonic_superCommuteF h]
simp
· simp
lemma ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStates) :
ι [ofCrAnState φ, ofCrAnState ψ]ₛca ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnStates) :
ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
rw [Subalgebra.mem_center_iff]
intro a
obtain ⟨a, rfl⟩ := ι_surjective a
have h0 := ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnState_ofCrAnState φ ψ a
trans ι ((superCommuteF (ofCrAnState φ)) (ofCrAnState ψ)) * ι a + 0
have h0 := ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF φ ψ a
trans ι ((superCommuteF (ofCrAnOpF φ)) (ofCrAnOpF ψ)) * ι a + 0
swap
simp only [add_zero]
rw [← h0]
@ -209,25 +209,25 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
⟨φ, φ', hdiff, rfl⟩
· rcases superCommuteF_superCommuteF_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
· rcases superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic φ1 φ2 φ3 with h | h
· left
rw [bosonicProj_of_mem_bosonic _ h]
simpa using hx'
· right
rw [bosonicProj_of_mem_fermionic _ h]
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φc φc' with h | h
· left
rw [bosonicProj_of_mem_bosonic _ h]
simpa using hx'
· right
rw [bosonicProj_of_mem_fermionic _ h]
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φa φa' with h | h
· left
rw [bosonicProj_of_mem_bosonic _ h]
simpa using hx'
· right
rw [bosonicProj_of_mem_fermionic _ h]
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ φ' with h | h
· left
rw [bosonicProj_of_mem_bosonic _ h]
simpa using hx'
@ -240,25 +240,25 @@ lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (h
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
⟨φ, φ', hdiff, rfl⟩
· rcases superCommuteF_superCommuteF_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
· rcases superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic φ1 φ2 φ3 with h | h
· right
rw [fermionicProj_of_mem_bosonic _ h]
· left
rw [fermionicProj_of_mem_fermionic _ h]
simpa using hx'
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φc φc' with h | h
· right
rw [fermionicProj_of_mem_bosonic _ h]
· left
rw [fermionicProj_of_mem_fermionic _ h]
simpa using hx'
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φa φa' with h | h
· right
rw [fermionicProj_of_mem_bosonic _ h]
· left
rw [fermionicProj_of_mem_fermionic _ h]
simpa using hx'
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
· rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ φ' with h | h
· right
rw [fermionicProj_of_mem_bosonic _ h]
· left
@ -457,10 +457,10 @@ lemma ofFieldOpList_singleton (φ : 𝓕.States) :
simp only [ofFieldOpList, ofFieldOp, ofFieldOpListF_singleton]
/-- An element of `FieldOpAlgebra` from a `CrAnStates`. -/
def ofCrAnFieldOp (φ : 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (ofCrAnState φ)
def ofCrAnFieldOp (φ : 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (ofCrAnOpF φ)
lemma ofCrAnFieldOp_eq_ι_ofCrAnState (φ : 𝓕.CrAnStates) :
ofCrAnFieldOp φ = ι (ofCrAnState φ) := rfl
lemma ofCrAnFieldOp_eq_ι_ofCrAnOpF (φ : 𝓕.CrAnStates) :
ofCrAnFieldOp φ = ι (ofCrAnOpF φ) := rfl
lemma ofFieldOp_eq_sum (φ : 𝓕.States) :
ofFieldOp φ = (∑ i : 𝓕.statesToCrAnType φ, ofCrAnFieldOp ⟨φ, i⟩) := by
@ -469,20 +469,20 @@ lemma ofFieldOp_eq_sum (φ : 𝓕.States) :
rfl
/-- An element of `FieldOpAlgebra` from a list of `CrAnStates`. -/
def ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (ofCrAnList φs)
def ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (ofCrAnListF φs)
lemma ofCrAnFieldOpList_eq_ι_ofCrAnList (φs : List 𝓕.CrAnStates) :
ofCrAnFieldOpList φs = ι (ofCrAnList φs) := rfl
lemma ofCrAnFieldOpList_eq_ι_ofCrAnListF (φs : List 𝓕.CrAnStates) :
ofCrAnFieldOpList φs = ι (ofCrAnListF φs) := rfl
lemma ofCrAnFieldOpList_append (φs ψs : List 𝓕.CrAnStates) :
ofCrAnFieldOpList (φs ++ ψs) = ofCrAnFieldOpList φs * ofCrAnFieldOpList ψs := by
simp only [ofCrAnFieldOpList]
rw [ofCrAnList_append]
rw [ofCrAnListF_append]
simp
lemma ofCrAnFieldOpList_singleton (φ : 𝓕.CrAnStates) :
ofCrAnFieldOpList [φ] = ofCrAnFieldOp φ := by
simp only [ofCrAnFieldOpList, ofCrAnFieldOp, ofCrAnList_singleton]
simp only [ofCrAnFieldOpList, ofCrAnFieldOp, ofCrAnListF_singleton]
lemma ofFieldOpList_eq_sum (φs : List 𝓕.States) :
ofFieldOpList φs = ∑ s : CrAnSection φs, ofCrAnFieldOpList s.1 := by