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refactor: Rename superCommute for CrAnAlgebra

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jstoobysmith 2025-01-30 05:52:50 +00:00
parent 32aefb7eb7
commit 289050c829
13 changed files with 428 additions and 425 deletions
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2
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/Basic.lean
View file

@ -24,7 +24,7 @@ The main structures defined in this module are:
* `ofState` - Maps a state to a sum of creation and annihilation operators
* `crPart` - The creation part of a state in the algebra
* `anPart` - The annihilation part of a state in the algebra
* `superCommute` - The super commutator on the algebra
* `superCommuteF` - The super commutator on the algebra
The key lemmas show how these operators interact, particularly focusing on the
super commutation relations between creation and annihilation operators.

66
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/NormalOrder.lean
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@ -120,7 +120,7 @@ lemma normalOrder_mul_anPart (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
## Normal ordering for an adjacent creation and annihliation state
The main result of this section is `normalOrder_superCommute_annihilate_create`.
The main result of this section is `normalOrder_superCommuteF_annihilate_create`.
-/
lemma normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList (φc φa : 𝓕.CrAnStates)
@ -163,23 +163,23 @@ lemma normalOrder_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
normalOrder_swap_create_annihlate_ofCrAnList φc φa hφc hφa]
rfl
lemma normalOrder_superCommute_create_annihilate (φc φa : 𝓕.CrAnStates)
lemma normalOrder_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.CrAnAlgebra) :
𝓝ᶠ(a * [ofCrAnState φc, ofCrAnState φa]ₛca * b) = 0 := by
simp only [superCommute_ofCrAnState_ofCrAnState, instCommGroup.eq_1, Algebra.smul_mul_assoc]
simp only [superCommuteF_ofCrAnState_ofCrAnState, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc, ← mul_assoc, ← mul_assoc,
normalOrder_swap_create_annihlate φc φa hφc hφa]
simp
lemma normalOrder_superCommute_annihilate_create (φc φa : 𝓕.CrAnStates)
lemma normalOrder_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnStates)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.CrAnAlgebra) :
𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φc]ₛca * b) = 0 := by
rw [superCommute_ofCrAnState_ofCrAnState_symm]
rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
exact Or.inr (normalOrder_superCommute_create_annihilate φc φa hφc hφa ..)
exact Or.inr (normalOrder_superCommuteF_create_annihilate φc φa hφc hφa ..)
lemma normalOrder_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
𝓝ᶠ(a * (crPart φ) * (anPart φ') * b) =
@ -223,43 +223,43 @@ lemma normalOrder_swap_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra
(anPart φ) * b) := by
simp [normalOrder_swap_crPart_anPart, smul_smul]
lemma normalOrder_superCommute_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
𝓝ᶠ(a * superCommute
lemma normalOrder_superCommuteF_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
𝓝ᶠ(a * superCommuteF
(crPart φ) (anPart φ') * b) = 0 := by
match φ, φ' with
| _, .inAsymp φ' => simp
| .outAsymp φ', _ => simp
| .position φ, .position φ' =>
rw [crPart_position, anPart_position]
exact normalOrder_superCommute_create_annihilate _ _ rfl rfl ..
exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
| .inAsymp φ, .outAsymp φ' =>
rw [crPart_negAsymp, anPart_posAsymp]
exact normalOrder_superCommute_create_annihilate _ _ rfl rfl ..
exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
| .inAsymp φ, .position φ' =>
rw [crPart_negAsymp, anPart_position]
exact normalOrder_superCommute_create_annihilate _ _ rfl rfl ..
exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
| .position φ, .outAsymp φ' =>
rw [crPart_position, anPart_posAsymp]
exact normalOrder_superCommute_create_annihilate _ _ rfl rfl ..
exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
𝓝ᶠ(a * superCommute
lemma normalOrder_superCommuteF_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
𝓝ᶠ(a * superCommuteF
(anPart φ) (crPart φ') * b) = 0 := by
match φ, φ' with
| .inAsymp φ', _ => simp
| _, .outAsymp φ' => simp
| .position φ, .position φ' =>
rw [anPart_position, crPart_position]
exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
| .outAsymp φ', .inAsymp φ =>
simp only [anPart_posAsymp, crPart_negAsymp]
exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
| .position φ', .inAsymp φ =>
simp only [anPart_position, crPart_negAsymp]
exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
| .outAsymp φ, .position φ' =>
simp only [anPart_posAsymp, crPart_position]
exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
/-!
@ -325,14 +325,14 @@ lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
TODO "Split the following two lemmas up into smaller parts."
lemma normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList
lemma normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList
(φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnStates) :
(𝓝ᶠ(ofCrAnList φs * [ofCrAnState φc, ofCrAnState φc']ₛca * ofCrAnList φs')) =
normalOrderSign (φs ++ φc' :: φc :: φs') •
(ofCrAnList (createFilter φs) * [ofCrAnState φc, ofCrAnState φc']ₛca *
ofCrAnList (createFilter φs') * ofCrAnList (annihilateFilter (φs ++ φs'))) := by
rw [superCommute_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
rw [superCommuteF_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
conv_lhs =>
lhs; rhs
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append, ← ofCrAnList_append,
@ -384,7 +384,7 @@ lemma normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList
ofCrAnList_singleton]
rw [ofCrAnList_append, ofCrAnList_singleton, ofCrAnList_singleton, smul_mul_assoc]
lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
lemma normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
(φa φa' : 𝓕.CrAnStates)
(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
@ -394,7 +394,7 @@ lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
(ofCrAnList (createFilter (φs ++ φs'))
* ofCrAnList (annihilateFilter φs) * [ofCrAnState φa, ofCrAnState φa']ₛca
* ofCrAnList (annihilateFilter φs')) := by
rw [superCommute_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
rw [superCommuteF_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
conv_lhs =>
lhs; rhs
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append, ← ofCrAnList_append,
@ -458,14 +458,14 @@ lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
-/
lemma ofCrAnList_superCommute_normalOrder_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
lemma ofCrAnList_superCommuteF_normalOrder_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, 𝓝ᶠ(ofCrAnList φs')]ₛca =
ofCrAnList φs * 𝓝ᶠ(ofCrAnList φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnList φs') * ofCrAnList φs := by
simp [normalOrder_ofCrAnList, map_smul, superCommute_ofCrAnList_ofCrAnList, ofCrAnList_append,
simp [normalOrder_ofCrAnList, map_smul, superCommuteF_ofCrAnList_ofCrAnList, ofCrAnList_append,
smul_sub, smul_smul, mul_comm]
lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnStates)
lemma ofCrAnList_superCommuteF_normalOrder_ofStateList (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) : [ofCrAnList φs, 𝓝ᶠ(ofStateList φs')]ₛca =
ofCrAnList φs * 𝓝ᶠ(ofStateList φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnList φs := by
@ -473,7 +473,7 @@ lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnState
← Finset.sum_sub_distrib, map_sum]
congr
funext n
rw [ofCrAnList_superCommute_normalOrder_ofCrAnList,
rw [ofCrAnList_superCommuteF_normalOrder_ofCrAnList,
CrAnSection.statistics_eq_state_statistics]
/-!
@ -482,20 +482,20 @@ lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnState
-/
lemma ofCrAnList_mul_normalOrder_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates)
lemma ofCrAnList_mul_normalOrder_ofStateList_eq_superCommuteF (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) :
ofCrAnList φs * 𝓝ᶠ(ofStateList φs') =
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnList φs
+ [ofCrAnList φs, 𝓝ᶠ(ofStateList φs')]ₛca := by
simp [ofCrAnList_superCommute_normalOrder_ofStateList]
simp [ofCrAnList_superCommuteF_normalOrder_ofStateList]
lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.CrAnStates)
lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF (φ : 𝓕.CrAnStates)
(φs' : List 𝓕.States) : ofCrAnState φ * 𝓝ᶠ(ofStateList φs') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnState φ
+ [ofCrAnState φ, 𝓝ᶠ(ofStateList φs')]ₛca := by
simp [← ofCrAnList_singleton, ofCrAnList_mul_normalOrder_ofStateList_eq_superCommute]
simp [← ofCrAnList_singleton, ofCrAnList_mul_normalOrder_ofStateList_eq_superCommuteF]
lemma anPart_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.States)
lemma anPart_mul_normalOrder_ofStateList_eq_superCommuteF (φ : 𝓕.States)
(φs' : List 𝓕.States) :
anPart φ * 𝓝ᶠ(ofStateList φs') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs' * anPart φ)
@ -503,8 +503,8 @@ lemma anPart_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.States)
rw [normalOrder_mul_anPart]
match φ with
| .inAsymp φ => simp
| .position φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute, crAnStatistics]
| .outAsymp φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute, crAnStatistics]
| .position φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF, crAnStatistics]
| .outAsymp φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF, crAnStatistics]
end

262
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean
View file

@ -26,7 +26,7 @@ open FieldStatistic
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
noncomputable def superCommute : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
noncomputable def superCommuteF : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
Basis.constr ofCrAnListBasis ℂ fun φs =>
Basis.constr ofCrAnListBasis ℂ fun φs' =>
ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs)
@ -34,7 +34,7 @@ noncomputable def superCommute : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
scoped[FieldSpecification.CrAnAlgebra] notation "[" φs "," φs' "]ₛca" => superCommute φs φs'
scoped[FieldSpecification.CrAnAlgebra] notation "[" φs "," φs' "]ₛca" => superCommuteF φs φs'
/-!
@ -42,35 +42,35 @@ scoped[FieldSpecification.CrAnAlgebra] notation "[" φs "," φs' "]ₛca" => sup
-/
lemma superCommute_ofCrAnList_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnList_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList φs']ₛca =
ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs) := by
rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
simp only [superCommute, Basis.constr_basis]
simp only [superCommuteF, Basis.constr_basis]
lemma superCommute_ofCrAnState_ofCrAnState (φ φ' : 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnState_ofCrAnState (φ φ' : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState φ']ₛca =
ofCrAnState φ * ofCrAnState φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnState φ' * ofCrAnState φ := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
rw [superCommute_ofCrAnList_ofCrAnList, ofCrAnList_append]
rw [superCommuteF_ofCrAnList_ofCrAnList, ofCrAnList_append]
congr
rw [ofCrAnList_append]
rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
lemma superCommute_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
lemma superCommuteF_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
[ofCrAnList φcas, ofStateList φs]ₛca = ofCrAnList φcas * ofStateList φs -
𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofStateList φs * ofCrAnList φcas := by
conv_lhs => rw [ofStateList_sum]
rw [map_sum]
conv_lhs =>
enter [2, x]
rw [superCommute_ofCrAnList_ofCrAnList, CrAnSection.statistics_eq_state_statistics,
rw [superCommuteF_ofCrAnList_ofCrAnList, CrAnSection.statistics_eq_state_statistics,
ofCrAnList_append, ofCrAnList_append]
rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.smul_sum,
← Finset.sum_mul, ← ofStateList_sum]
simp
lemma superCommute_ofStateList_ofStatesList (φ : List 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_ofStateList_ofStatesList (φ : List 𝓕.States) (φs : List 𝓕.States) :
[ofStateList φ, ofStateList φs]ₛca = ofStateList φ * ofStateList φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs * ofStateList φ := by
conv_lhs => rw [ofStateList_sum]
@ -78,24 +78,24 @@ lemma superCommute_ofStateList_ofStatesList (φ : List 𝓕.States) (φs : List
Algebra.smul_mul_assoc]
conv_lhs =>
enter [2, x]
rw [superCommute_ofCrAnList_ofStatesList]
rw [superCommuteF_ofCrAnList_ofStatesList]
simp only [instCommGroup.eq_1, CrAnSection.statistics_eq_state_statistics,
Algebra.smul_mul_assoc, Finset.sum_sub_distrib]
rw [← Finset.sum_mul, ← Finset.smul_sum, ← Finset.mul_sum, ← ofStateList_sum]
lemma superCommute_ofState_ofStatesList (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_ofState_ofStatesList (φ : 𝓕.States) (φs : List 𝓕.States) :
[ofState φ, ofStateList φs]ₛca = ofState φ * ofStateList φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs * ofState φ := by
rw [← ofStateList_singleton, superCommute_ofStateList_ofStatesList, ofStateList_singleton]
rw [← ofStateList_singleton, superCommuteF_ofStateList_ofStatesList, ofStateList_singleton]
simp
lemma superCommute_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.States) :
lemma superCommuteF_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.States) :
[ofStateList φs, ofState φ]ₛca = ofStateList φs * ofState φ -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofStateList φs := by
rw [← ofStateList_singleton, superCommute_ofStateList_ofStatesList, ofStateList_singleton]
rw [← ofStateList_singleton, superCommuteF_ofStateList_ofStatesList, ofStateList_singleton]
simp
lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
lemma superCommuteF_anPart_crPart (φ φ' : 𝓕.States) :
[anPart φ, crPart φ']ₛca =
anPart φ * crPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
@ -107,24 +107,24 @@ lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
sub_self]
| States.position φ, States.position φ' =>
simp only [anPart_position, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.outAsymp φ, States.position φ' =>
simp only [anPart_posAsymp, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.position φ, States.inAsymp φ' =>
simp only [anPart_position, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp only [List.singleton_append, instCommGroup.eq_1, crAnStatistics,
FieldStatistic.ofList_singleton, Function.comp_apply, crAnStatesToStates_prod, ←
ofCrAnList_append]
| States.outAsymp φ, States.inAsymp φ' =>
simp only [anPart_posAsymp, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
lemma superCommuteF_crPart_anPart (φ φ' : 𝓕.States) :
[crPart φ, anPart φ']ₛca =
crPart φ * anPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
@ -138,22 +138,22 @@ lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
sub_self]
| States.position φ, States.position φ' =>
simp only [crPart_position, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.position φ, States.outAsymp φ' =>
simp only [crPart_position, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.inAsymp φ, States.position φ' =>
simp only [crPart_negAsymp, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.inAsymp φ, States.outAsymp φ' =>
simp only [crPart_negAsymp, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) :
lemma superCommuteF_crPart_crPart (φ φ' : 𝓕.States) :
[crPart φ, crPart φ']ₛca =
crPart φ * crPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
@ -166,22 +166,22 @@ lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) :
simp only [crPart_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul, sub_self]
| States.position φ, States.position φ' =>
simp only [crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.position φ, States.inAsymp φ' =>
simp only [crPart_position, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.inAsymp φ, States.position φ' =>
simp only [crPart_negAsymp, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.inAsymp φ, States.inAsymp φ' =>
simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) :
lemma superCommuteF_anPart_anPart (φ φ' : 𝓕.States) :
[anPart φ, anPart φ']ₛca =
anPart φ * anPart φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
@ -193,38 +193,38 @@ lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) :
simp
| States.position φ, States.position φ' =>
simp only [anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.position φ, States.outAsymp φ' =>
simp only [anPart_position, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.outAsymp φ, States.position φ' =>
simp only [anPart_posAsymp, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
| States.outAsymp φ, States.outAsymp φ' =>
simp only [anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [crAnStatistics, ← ofCrAnList_append]
lemma superCommute_crPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_crPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
[crPart φ, ofStateList φs]ₛca =
crPart φ * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs *
crPart φ := by
match φ with
| States.inAsymp φ =>
simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
simp [crAnStatistics]
| States.position φ =>
simp only [crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
simp [crAnStatistics]
| States.outAsymp φ =>
simp
lemma superCommute_anPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_anPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
[anPart φ, ofStateList φs]ₛca =
anPart φ * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
ofStateList φs * anPart φ := by
@ -233,97 +233,97 @@ lemma superCommute_anPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States
simp
| States.position φ =>
simp only [anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
simp [crAnStatistics]
| States.outAsymp φ =>
simp only [anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
simp [crAnStatistics]
lemma superCommute_crPart_ofState (φ φ' : 𝓕.States) :
lemma superCommuteF_crPart_ofState (φ φ' : 𝓕.States) :
[crPart φ, ofState φ']ₛca =
crPart φ * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * crPart φ := by
rw [← ofStateList_singleton, superCommute_crPart_ofStateList]
rw [← ofStateList_singleton, superCommuteF_crPart_ofStateList]
simp
lemma superCommute_anPart_ofState (φ φ' : 𝓕.States) :
lemma superCommuteF_anPart_ofState (φ φ' : 𝓕.States) :
[anPart φ, ofState φ']ₛca =
anPart φ * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * anPart φ := by
rw [← ofStateList_singleton, superCommute_anPart_ofStateList]
rw [← ofStateList_singleton, superCommuteF_anPart_ofStateList]
simp
/-!
## Mul equal superCommute
## Mul equal superCommuteF
Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
multiplication with a sign plus the super commutor.
-/
lemma ofCrAnList_mul_ofCrAnList_eq_superCommute (φs φs' : List 𝓕.CrAnStates) :
lemma ofCrAnList_mul_ofCrAnList_eq_superCommuteF (φs φs' : List 𝓕.CrAnStates) :
ofCrAnList φs * ofCrAnList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnList φs
+ [ofCrAnList φs, ofCrAnList φs']ₛca := by
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp [ofCrAnList_append]
lemma ofCrAnState_mul_ofCrAnList_eq_superCommute (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
lemma ofCrAnState_mul_ofCrAnList_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
ofCrAnState φ * ofCrAnList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnState φ
+ [ofCrAnState φ, ofCrAnList φs']ₛca := by
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommute]
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommuteF]
simp
lemma ofStateList_mul_ofStateList_eq_superCommute (φs φs' : List 𝓕.States) :
lemma ofStateList_mul_ofStateList_eq_superCommuteF (φs φs' : List 𝓕.States) :
ofStateList φs * ofStateList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofStateList φs' * ofStateList φs
+ [ofStateList φs, ofStateList φs']ₛca := by
rw [superCommute_ofStateList_ofStatesList]
rw [superCommuteF_ofStateList_ofStatesList]
simp
lemma ofState_mul_ofStateList_eq_superCommute (φ : 𝓕.States) (φs' : List 𝓕.States) :
lemma ofState_mul_ofStateList_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
ofState φ * ofStateList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofStateList φs' * ofState φ
+ [ofState φ, ofStateList φs']ₛca := by
rw [superCommute_ofState_ofStatesList]
rw [superCommuteF_ofState_ofStatesList]
simp
lemma ofStateList_mul_ofState_eq_superCommute (φs : List 𝓕.States) (φ : 𝓕.States) :
lemma ofStateList_mul_ofState_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
ofStateList φs * ofState φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofStateList φs
+ [ofStateList φs, ofState φ]ₛca := by
rw [superCommute_ofStateList_ofState]
rw [superCommuteF_ofStateList_ofState]
simp
lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
lemma crPart_mul_anPart_eq_superCommuteF (φ φ' : 𝓕.States) :
crPart φ * anPart φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ +
[crPart φ, anPart φ']ₛca := by
rw [superCommute_crPart_anPart]
rw [superCommuteF_crPart_anPart]
simp
lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
lemma anPart_mul_crPart_eq_superCommuteF (φ φ' : 𝓕.States) :
anPart φ * crPart φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
crPart φ' * anPart φ +
[anPart φ, crPart φ']ₛca := by
rw [superCommute_anPart_crPart]
rw [superCommuteF_anPart_crPart]
simp
lemma crPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
lemma crPart_mul_crPart_eq_superCommuteF (φ φ' : 𝓕.States) :
crPart φ * crPart φ' =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ +
[crPart φ, crPart φ']ₛca := by
rw [superCommute_crPart_crPart]
rw [superCommuteF_crPart_crPart]
simp
lemma anPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
lemma anPart_mul_anPart_eq_superCommuteF (φ φ' : 𝓕.States) :
anPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ +
[anPart φ, anPart φ']ₛca := by
rw [superCommute_anPart_anPart]
rw [superCommuteF_anPart_anPart]
simp
lemma ofCrAnList_mul_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
lemma ofCrAnList_mul_ofStateList_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
ofCrAnList φs * ofStateList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofStateList φs' * ofCrAnList φs
+ [ofCrAnList φs, ofStateList φs']ₛca := by
rw [superCommute_ofCrAnList_ofStatesList]
rw [superCommuteF_ofCrAnList_ofStatesList]
simp
/-!
@ -332,10 +332,10 @@ lemma ofCrAnList_mul_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates) (
-/
lemma superCommute_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList φs']ₛca =
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnList φs', ofCrAnList φs]ₛca := by
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList, smul_sub]
rw [superCommuteF_ofCrAnList_ofCrAnList, superCommuteF_ofCrAnList_ofCrAnList, smul_sub]
simp only [instCommGroup.eq_1, neg_smul, sub_neg_eq_add]
rw [smul_smul]
conv_rhs =>
@ -344,10 +344,10 @@ lemma superCommute_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnStates)
simp only [one_smul]
abel
lemma superCommute_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState φ']ₛca =
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnState φ', ofCrAnState φ]ₛca := by
rw [superCommute_ofCrAnState_ofCrAnState, superCommute_ofCrAnState_ofCrAnState]
rw [superCommuteF_ofCrAnState_ofCrAnState, superCommuteF_ofCrAnState_ofCrAnState]
rw [smul_sub]
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, neg_smul, sub_neg_eq_add]
rw [smul_smul]
@ -362,41 +362,41 @@ lemma superCommute_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
## Splitting the super commutor on lists into sums.
-/
lemma superCommute_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList (φ :: φs')]ₛca =
[ofCrAnList φs, ofCrAnState φ]ₛca * ofCrAnList φs' +
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
• ofCrAnState φ * [ofCrAnList φs, ofCrAnList φs']ₛca := by
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
conv_rhs =>
lhs
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList, sub_mul, ← ofCrAnList_append]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList, sub_mul, ← ofCrAnList_append]
rhs
rw [FieldStatistic.ofList_singleton, ofCrAnList_append, ofCrAnList_singleton, smul_mul_assoc,
mul_assoc, ← ofCrAnList_append]
conv_rhs =>
rhs
rw [superCommute_ofCrAnList_ofCrAnList, mul_sub, smul_mul_assoc]
rw [superCommuteF_ofCrAnList_ofCrAnList, mul_sub, smul_mul_assoc]
simp only [instCommGroup.eq_1, List.cons_append, List.append_assoc, List.nil_append,
Algebra.mul_smul_comm, Algebra.smul_mul_assoc, sub_add_sub_cancel, sub_right_inj]
rw [← ofCrAnList_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
simp only [instCommGroup, map_mul, mul_comm]
lemma superCommute_ofCrAnList_ofStateList_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_ofCrAnList_ofStateList_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) : [ofCrAnList φs, ofStateList (φ :: φs')]ₛca =
[ofCrAnList φs, ofState φ]ₛca * ofStateList φs' +
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * [ofCrAnList φs, ofStateList φs']ₛca := by
rw [superCommute_ofCrAnList_ofStatesList]
rw [superCommuteF_ofCrAnList_ofStatesList]
conv_rhs =>
lhs
rw [← ofStateList_singleton, superCommute_ofCrAnList_ofStatesList, sub_mul, mul_assoc,
rw [← ofStateList_singleton, superCommuteF_ofCrAnList_ofStatesList, sub_mul, mul_assoc,
← ofStateList_append]
rhs
rw [FieldStatistic.ofList_singleton, ofStateList_singleton, smul_mul_assoc,
smul_mul_assoc, mul_assoc]
conv_rhs =>
rhs
rw [superCommute_ofCrAnList_ofStatesList, mul_sub, smul_mul_assoc]
rw [superCommuteF_ofCrAnList_ofStatesList, mul_sub, smul_mul_assoc]
simp only [instCommGroup, Algebra.smul_mul_assoc, List.singleton_append, Algebra.mul_smul_comm,
sub_add_sub_cancel, sub_right_inj]
rw [ofStateList_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
@ -407,33 +407,33 @@ Within the creation and annihilation algebra, we have that
`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
-/
lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
(φs' : List 𝓕.CrAnStates) → [ofCrAnList φs, ofCrAnList φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofCrAnList (φs'.take n) * [ofCrAnList φs, ofCrAnState (φs'.get n)]ₛca *
ofCrAnList (φs'.drop (n + 1))
| [] => by
simp [← ofCrAnList_nil, superCommute_ofCrAnList_ofCrAnList]
simp [← ofCrAnList_nil, superCommuteF_ofCrAnList_ofCrAnList]
| φ :: φs' => by
rw [superCommute_ofCrAnList_ofCrAnList_cons, superCommute_ofCrAnList_ofCrAnList_eq_sum φs φs']
rw [superCommuteF_ofCrAnList_ofCrAnList_cons, superCommuteF_ofCrAnList_ofCrAnList_eq_sum φs φs']
conv_rhs => erw [Fin.sum_univ_succ]
congr 1
· simp
· simp [Finset.mul_sum, smul_smul, ofCrAnList_cons, mul_assoc,
FieldStatistic.ofList_cons_eq_mul, mul_comm]
lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
lemma superCommuteF_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
[ofCrAnList φs, ofStateList φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofStateList (φs'.take n) * [ofCrAnList φs, ofState (φs'.get n)]ₛca *
ofStateList (φs'.drop (n + 1))
| [] => by
simp only [superCommute_ofCrAnList_ofStatesList, instCommGroup, ofList_empty,
simp only [superCommuteF_ofCrAnList_ofStatesList, instCommGroup, ofList_empty,
exchangeSign_bosonic, one_smul, List.length_nil, Finset.univ_eq_empty, List.take_nil,
List.get_eq_getElem, List.drop_nil, Finset.sum_empty]
simp
| φ :: φs' => by
rw [superCommute_ofCrAnList_ofStateList_cons, superCommute_ofCrAnList_ofStateList_eq_sum φs φs']
rw [superCommuteF_ofCrAnList_ofStateList_cons, superCommuteF_ofCrAnList_ofStateList_eq_sum φs φs']
conv_rhs => erw [Fin.sum_univ_succ]
congr 1
· simp
@ -445,9 +445,9 @@ lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
repeat rw [superCommute_ofCrAnList_ofCrAnList]
repeat rw [superCommuteF_ofCrAnList_ofCrAnList]
simp only [instCommGroup, map_sub, map_smul, neg_smul]
repeat rw [superCommute_ofCrAnList_ofCrAnList]
repeat rw [superCommuteF_ofCrAnList_ofCrAnList]
simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
@ -489,7 +489,7 @@ lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
-/
lemma superCommute_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
lemma superCommuteF_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
(ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
[a, b]ₛca ∈ statisticSubmodule (f1 + f2) := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
@ -506,7 +506,7 @@ lemma superCommute_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
· intro x hx
obtain ⟨φs', rfl, hφs'⟩ := hx
simp only [add_eq_mul, instCommGroup, p]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
apply Submodule.sub_mem _
· apply ofCrAnList_mem_statisticSubmodule_of
rw [ofList_append_eq_mul, hφs, hφs']
@ -531,7 +531,7 @@ lemma superCommute_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
exact Submodule.smul_mem _ c hp1
· exact hb
lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
@ -547,7 +547,7 @@ lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
· intro x hx
obtain ⟨φs', rfl, hφs'⟩ := hx
simp only [p]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp [hφs, ofCrAnList_append]
· simp [p]
· intro x y hx hy hp1 hp2
@ -564,7 +564,7 @@ lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
@ -580,7 +580,7 @@ lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
· intro x hx
obtain ⟨φs', rfl, hφs'⟩ := hx
simp only [p]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp [hφs, hφs', ofCrAnList_append]
· simp [p]
· intro x y hx hy hp1 hp2
@ -597,7 +597,7 @@ lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
@ -613,7 +613,7 @@ lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
· intro x hx
obtain ⟨φs', rfl, hφs'⟩ := hx
simp only [p]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp [hφs, hφs', ofCrAnList_append]
· simp [p]
· intro x y hx hy hp1 hp2
@ -630,33 +630,33 @@ lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommuteF_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj a]
simp only [map_add, LinearMap.add_apply]
rw [superCommute_bosonic_bosonic (by simp) hb, superCommute_fermionic_bonsonic (by simp) hb]
rw [superCommuteF_bosonic_bosonic (by simp) hb, superCommuteF_fermionic_bonsonic (by simp) hb]
simp only [add_mul, mul_add]
abel
lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bosonic_superCommuteF {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj b]
simp only [map_add, LinearMap.add_apply]
rw [superCommute_bosonic_bosonic ha (by simp), superCommute_bosonic_fermionic ha (by simp)]
rw [superCommuteF_bosonic_bosonic ha (by simp), superCommuteF_bosonic_fermionic ha (by simp)]
simp only [add_mul, mul_add]
abel
lemma superCommute_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommuteF_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommute hb, superCommute_bonsonic hb]
rw [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
simp
lemma bonsonic_superCommute_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bonsonic_superCommuteF_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommute ha, superCommute_bonsonic ha]
rw [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
simp
lemma superCommute_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = a * b + b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
@ -672,7 +672,7 @@ lemma superCommute_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
· intro x hx
obtain ⟨φs', rfl, hφs'⟩ := hx
simp only [p]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp [hφs, hφs', ofCrAnList_append]
· simp [p]
· intro x y hx hy hp1 hp2
@ -689,27 +689,27 @@ lemma superCommute_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_fermionic_fermionic_symm {a b : 𝓕.CrAnAlgebra}
lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = [b, a]ₛca := by
rw [superCommute_fermionic_fermionic ha hb]
rw [superCommute_fermionic_fermionic hb ha]
rw [superCommuteF_fermionic_fermionic ha hb]
rw [superCommuteF_fermionic_fermionic hb ha]
abel
lemma superCommute_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
[a, b]ₛca = bosonicProj a * bosonicProj b - bosonicProj b * bosonicProj a +
bosonicProj a * fermionicProj b - fermionicProj b * bosonicProj a +
fermionicProj a * bosonicProj b - bosonicProj b * fermionicProj a +
fermionicProj a * fermionicProj b + fermionicProj b * fermionicProj a := by
conv_lhs => rw [← bosonicProj_add_fermionicProj a, ← bosonicProj_add_fermionicProj b]
simp only [map_add, LinearMap.add_apply]
rw [superCommute_bonsonic (by simp),
superCommute_fermionic_bonsonic (by simp) (by simp),
superCommute_bosonic_fermionic (by simp) (by simp),
superCommute_fermionic_fermionic (by simp) (by simp)]
rw [superCommuteF_bonsonic (by simp),
superCommuteF_fermionic_bonsonic (by simp) (by simp),
superCommuteF_bosonic_fermionic (by simp) (by simp),
superCommuteF_fermionic_fermionic (by simp) (by simp)]
abel
lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic := by
by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
@ -718,7 +718,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade
apply superCommuteF_grade
apply ofCrAnList_mem_statisticSubmodule_of _ _ h1
apply ofCrAnList_mem_statisticSubmodule_of _ _ h2
· right
@ -726,7 +726,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade
apply superCommuteF_grade
apply ofCrAnList_mem_statisticSubmodule_of _ _ h1
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
· right
@ -734,7 +734,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade
apply superCommuteF_grade
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
apply ofCrAnList_mem_statisticSubmodule_of _ _ h2
· left
@ -742,47 +742,47 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade
apply superCommuteF_grade
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
lemma superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule fermionic := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
exact superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
exact superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
lemma superCommute_superCommute_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
lemma superCommuteF_superCommuteF_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ2 φ3 with hs | hs
rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ2 φ3 with hs | hs
<;> rcases ofCrAnState_bosonic_or_fermionic φ1 with h1 | h1
· left
have h : bosonic = bosonic + bosonic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
apply superCommuteF_grade h1 hs
· right
have h : fermionic = fermionic + bosonic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
apply superCommuteF_grade h1 hs
· right
have h : fermionic = bosonic + fermionic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
apply superCommuteF_grade h1 hs
· left
have h : bosonic = fermionic + fermionic := by
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h]
apply superCommute_grade h1 hs
apply superCommuteF_grade h1 hs
lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
(ha : a ∈ statisticSubmodule bosonic) :
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
@ -796,7 +796,7 @@ lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
· intro a ha
obtain ⟨φs, rfl, hφs⟩ := ha
simp only [List.get_eq_getElem, p]
rw [superCommute_ofCrAnList_ofCrAnList_eq_sum]
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
congr
funext n
simp [hφs]
@ -811,7 +811,7 @@ lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
simp_all [p, Finset.smul_sum]
· exact ha
lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
lemma superCommuteF_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
(ha : a ∈ statisticSubmodule fermionic) :
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
@ -825,7 +825,7 @@ lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : Lis
· intro a ha
obtain ⟨φs, rfl, hφs⟩ := ha
simp only [instCommGroup, List.get_eq_getElem, Algebra.smul_mul_assoc, p]
rw [superCommute_ofCrAnList_ofCrAnList_eq_sum]
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
congr
funext n
simp [hφs]
@ -845,7 +845,7 @@ lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : Lis
simp [smul_smul, mul_comm]
· exact ha
lemma statistic_neq_of_superCommute_fermionic {φs φs' : List 𝓕.CrAnStates}
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnStates}
(h : [ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic) :
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnList φs, ofCrAnList φs']ₛca = 0 := by
by_cases h0 : [ofCrAnList φs, ofCrAnList φs']ₛca = 0
@ -858,7 +858,7 @@ lemma statistic_neq_of_superCommute_fermionic {φs φs' : List 𝓕.CrAnStates}
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h1]
apply superCommute_grade
apply superCommuteF_grade
apply ofCrAnList_mem_statisticSubmodule_of _ _ hc
apply ofCrAnList_mem_statisticSubmodule_of _ _
rw [← hn, hc]
@ -866,7 +866,7 @@ lemma statistic_neq_of_superCommute_fermionic {φs φs' : List 𝓕.CrAnStates}
simp only [add_eq_mul, instCommGroup, mul_self]
rfl
rw [h1]
apply superCommute_grade
apply superCommuteF_grade
apply ofCrAnList_mem_statisticSubmodule_of _ _
simpa using hc
apply ofCrAnList_mem_statisticSubmodule_of _ _

70
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/TimeOrder.lean
View file

@ -142,10 +142,10 @@ lemma timeOrder_ofState_ofState_not_ordered_eq_timeOrder {φ ψ : 𝓕.States} (
have hx := IsTotal.total (r := timeOrderRel) ψ φ
simp_all
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) :
𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
rw [superCommute_ofCrAnState_ofCrAnState]
rw [superCommuteF_ofCrAnState_ofCrAnState]
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
@ -160,51 +160,51 @@ lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
· rw [crAnTimeOrderList_pair_ordered]
simp_all
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
rw [timeOrder_timeOrder_right,
timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
rw [timeOrder_timeOrder_left,
timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.CrAnAlgebra) :
𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
rw [timeOrder_timeOrder_mid,
timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel
lemma timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel
{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra) :
𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
rw [← bosonicProj_add_fermionicProj a]
simp only [map_add, LinearMap.add_apply]
rw [bosonic_superCommute (Submodule.coe_mem (bosonicProj a))]
rw [bosonic_superCommuteF (Submodule.coe_mem (bosonicProj a))]
simp only [map_sub]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
simp only [sub_self, zero_add]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
· rw [superCommute_bonsonic h']
rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
· rw [superCommuteF_bonsonic h']
simp only [ofCrAnList_singleton, map_sub]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
simp
· rw [superCommute_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
· rw [superCommuteF_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
simp only [ofCrAnList_singleton, map_add]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
simp
lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel
{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
@ -213,14 +213,14 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnList_singleton, neg_smul, map_smul,
map_sub, map_neg, smul_eq_zero]
right
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
rw [superCommute_ofCrAnState_ofCrAnState_symm φ3]
rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
rw [superCommuteF_ofCrAnState_ofCrAnState_symm φ3]
simp only [smul_zero, neg_zero, instCommGroup.eq_1, neg_smul, map_neg, map_smul, smul_neg,
sub_neg_eq_add, zero_add, smul_eq_zero]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
simp
lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel'
{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
@ -229,14 +229,14 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnList_singleton, neg_smul, map_smul,
map_sub, map_neg, smul_eq_zero]
right
rw [superCommute_ofCrAnState_ofCrAnState_symm φ1]
rw [superCommuteF_ofCrAnState_ofCrAnState_symm φ1]
simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, smul_neg, neg_neg]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
simp only [smul_zero, zero_sub, neg_eq_zero, smul_eq_zero]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
simp
lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬
(crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
@ -245,13 +245,13 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
simp only [not_and] at h
by_cases h23 : ¬ crAnTimeOrderRel φ2 φ3
· simp_all only [IsEmpty.forall_iff, implies_true]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h23]
rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h23]
simp_all only [Decidable.not_not, forall_const]
by_cases h32 : ¬ crAnTimeOrderRel φ3 φ2
· simp_all only [not_false_eq_true, implies_true]
rw [superCommute_ofCrAnState_ofCrAnState_symm]
rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h32]
rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h32]
simp
simp_all only [imp_false, Decidable.not_not]
by_cases h12 : ¬ crAnTimeOrderRel φ1 φ2
@ -259,7 +259,7 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
intro h13
apply h12
exact IsTrans.trans φ1 φ3 φ2 h13 h32
rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel h12 h13]
rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel h12 h13]
simp_all only [Decidable.not_not, forall_const]
have h13 : crAnTimeOrderRel φ1 φ3 := IsTrans.trans φ1 φ2 φ3 h12 h23
simp_all only [forall_const]
@ -269,15 +269,15 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
intro h31
apply h21
exact IsTrans.trans φ2 φ3 φ1 h23 h31
rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel' h21 h31]
rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel' h21 h31]
simp_all only [Decidable.not_not, forall_const]
refine False.elim (h ?_)
exact IsTrans.trans φ3 φ2 φ1 h32 h21
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_eq_time
lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_eq_time
{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = [ofCrAnState φ, ofCrAnState ψ]ₛca := by
rw [superCommute_ofCrAnState_ofCrAnState]
rw [superCommuteF_ofCrAnState_ofCrAnState]
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]

84
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/Basic.lean
View file

@ -72,7 +72,7 @@ lemma ι_of_mem_fieldOpIdealSet (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpI
refine RingConGen.Rel.of x 0 ?_
simpa using hx
lemma ι_superCommute_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = .create)
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
apply ι_of_mem_fieldOpIdealSet
simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
@ -81,7 +81,7 @@ lemma ι_superCommute_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc : 𝓕
left
use φc, φc', hφc, hφc'
lemma ι_superCommute_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
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
apply ι_of_mem_fieldOpIdealSet
@ -92,7 +92,7 @@ lemma ι_superCommute_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
left
use φa, φa', hφa, hφa'
lemma ι_superCommute_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
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
apply ι_of_mem_fieldOpIdealSet
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
@ -101,24 +101,24 @@ lemma ι_superCommute_of_diff_statistic {φ ψ : 𝓕.CrAnStates}
right
use φ, ψ
lemma ι_superCommute_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
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 ⊢
rcases statistic_neq_of_superCommute_fermionic h with h | h
rcases statistic_neq_of_superCommuteF_fermionic h with h | h
· simp only [ofCrAnList_singleton]
apply ι_superCommute_of_diff_statistic
apply ι_superCommuteF_of_diff_statistic
simpa using h
· simp [h]
lemma ι_superCommute_ofCrAnState_ofCrAnState_bosonic_or_zero (φ ψ : 𝓕.CrAnStates) :
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 superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [ψ] with h | h
rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [ψ] with h | h
· simp_all [ofCrAnList_singleton]
· simp_all only [ofCrAnList_singleton]
right
exact ι_superCommute_zero_of_fermionic _ _ h
exact ι_superCommuteF_zero_of_fermionic _ _ h
/-!
@ -127,63 +127,63 @@ lemma ι_superCommute_ofCrAnState_ofCrAnState_bosonic_or_zero (φ ψ : 𝓕.CrAn
-/
@[simp]
lemma ι_superCommute_ofCrAnState_superCommute_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
lemma ι_superCommuteF_ofCrAnState_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState 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_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
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 superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
· rw [bonsonic_superCommute_symm h]
rcases superCommuteF_ofCrAnList_ofCrAnList_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'
· rw [superCommute_bonsonic_symm h']
· rw [superCommuteF_bonsonic_symm h']
simp [ofCrAnList_singleton]
· rw [superCommute_fermionic_fermionic_symm h h']
· rw [superCommuteF_fermionic_fermionic_symm h h']
simp [ofCrAnList_singleton]
lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2 : 𝓕.CrAnStates)
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnList (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 superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
· rw [superCommute_bosonic_ofCrAnList_eq_sum _ _ h]
simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
· rw [superCommute_fermionic_ofCrAnList_eq_sum _ _ h]
simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
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]
@[simp]
lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
(a : 𝓕.CrAnAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
change (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
have h1 : (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]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_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList]
simp [ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnList]
rw [h1]
simp
lemma ι_commute_crAnAlgebra_superCommute_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
lemma ι_commute_crAnAlgebra_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
(a : 𝓕.CrAnAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
ι [ofCrAnState φ1, ofCrAnState φ2]ₛca * ι a = 0 := by
rcases ι_superCommute_ofCrAnState_ofCrAnState_bosonic_or_zero φ1 φ2 with h | h
rcases ι_superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_zero φ1 φ2 with h | h
swap
· simp [h]
trans - ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca
· rw [bosonic_superCommute h]
· rw [bosonic_superCommuteF h]
simp
· simp
lemma ι_superCommute_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStates) :
lemma ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStates) :
ι [ofCrAnState φ, ofCrAnState 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_crAnAlgebra_superCommute_ofCrAnState_ofCrAnState φ ψ a
trans ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) * ι a + 0
have h0 := ι_commute_crAnAlgebra_superCommuteF_ofCrAnState_ofCrAnState φ ψ a
trans ι ((superCommuteF (ofCrAnState φ)) (ofCrAnState ψ)) * ι a + 0
swap
simp only [add_zero]
rw [← h0]
@ -209,25 +209,25 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈
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 superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
· rcases superCommuteF_superCommuteF_ofCrAnState_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 superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
· rcases superCommuteF_ofCrAnState_ofCrAnState_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 superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
· rcases superCommuteF_ofCrAnState_ofCrAnState_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 superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
· rcases superCommuteF_ofCrAnState_ofCrAnState_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 : CrAnAlgebra 𝓕) (hx : x
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 superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
· rcases superCommuteF_superCommuteF_ofCrAnState_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 superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
· rcases superCommuteF_ofCrAnState_ofCrAnState_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 superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
· rcases superCommuteF_ofCrAnState_ofCrAnState_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 superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
· rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
· right
rw [fermionicProj_of_mem_bosonic _ h]
· left
@ -396,8 +396,8 @@ lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.spa
· intro x y hx hy hpx hpy
simp_all only [map_add, Submodule.coe_add, p]
apply TwoSidedIdeal.add_mem
exact hpx
exact hpy
· exact hpx
· exact hpy
· intro x hx
simp [p]
@ -413,7 +413,7 @@ lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : CrAnAlgebra 𝓕) :
ι x = 0 ↔ ι x.bosonicProj.1 = 0 ∧ ι x.fermionicProj.1 = 0 := by
apply Iff.intro
· intro h
rw [@ι_eq_zero_iff_mem_ideal] at h ⊢
rw [ι_eq_zero_iff_mem_ideal] at h ⊢
rw [ι_eq_zero_iff_mem_ideal]
apply And.intro
· exact bosonicProj_mem_ideal x h

74
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/NormalOrder.lean
View file

@ -24,32 +24,32 @@ variable {𝓕 : FieldSpecification}
## Normal order on super-commutators.
The main result of this is
`ι_normalOrder_superCommute_eq_zero_mul`
`ι_normalOrder_superCommuteF_eq_zero_mul`
which states that applying `ι` to the normal order of something containing a super-commutator
is zero.
-/
lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero
lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
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']
rw [map_smul, map_mul, map_mul, map_mul, ι_superCommute_of_create_create φa φa' hφa hφa']
· rw [normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, ι_superCommuteF_of_create_create φa φa' hφa hφa']
simp
· rw [normalOrder_superCommute_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
· rw [normalOrder_superCommuteF_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
(ofCrAnList φs')]
simp
· rw [normalOrder_superCommute_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
· rw [normalOrder_superCommuteF_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
(ofCrAnList φs')]
simp
· rw [normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
· rw [normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul,
ι_superCommute_of_annihilate_annihilate φa φa' hφa hφa']
ι_superCommuteF_of_annihilate_annihilate φa φa' hφa hφa']
simp
lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero
lemma ι_normalOrder_superCommuteF_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
(a : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
have hf : ι.toLinearMap ∘ₗ normalOrder ∘ₗ
@ -58,13 +58,13 @@ lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero
intro l
simp only [CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
AlgHom.toLinearMap_apply, LinearMap.zero_apply]
exact ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
exact ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
change (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap ((ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca))) a = 0
rw [hf]
simp
lemma ι_normalOrder_superCommute_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
lemma ι_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
rw [mul_assoc]
@ -78,98 +78,98 @@ lemma ι_normalOrder_superCommute_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnS
Function.comp_apply, LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk,
AlgHom.toLinearMap_apply, LinearMap.zero_apply]
rw [← mul_assoc]
exact ι_normalOrder_superCommute_ofCrAnList_eq_zero φa φa' _ _
exact ι_normalOrder_superCommuteF_ofCrAnList_eq_zero φa φa' _ _
rw [hf]
simp
lemma ι_normalOrder_superCommute_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
lemma ι_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_eq_sum]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
rw [Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
apply Fintype.sum_eq_zero
intro n
rw [← mul_assoc, ← mul_assoc]
rw [mul_assoc _ _ b, ofCrAnList_singleton]
rw [ι_normalOrder_superCommute_ofCrAnState_eq_zero_mul]
rw [ι_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul]
lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnState φa]ₛca * b) = 0 := by
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
rw [← ofCrAnList_singleton, superCommuteF_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]
rw [ι_normalOrder_superCommute_ofCrAnState_ofCrAnList_eq_zero_mul]
rw [ι_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul]
simp
lemma ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero_mul
lemma ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnList φs']ₛca * b) = 0 := by
rw [superCommute_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
apply Fintype.sum_eq_zero
intro n
rw [← mul_assoc, ← mul_assoc]
rw [mul_assoc _ _ b]
rw [ι_normalOrder_superCommute_ofCrAnList_ofCrAnState_eq_zero_mul]
rw [ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul]
lemma ι_normalOrder_superCommute_ofCrAnList_eq_zero_mul
lemma ι_normalOrder_superCommuteF_ofCrAnList_eq_zero_mul
(φs : List 𝓕.CrAnStates)
(a b c : 𝓕.CrAnAlgebra) :
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
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) c = 0
have hf : (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute (ofCrAnList φs)) = 0 := by
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) = 0 := by
apply ofCrAnListBasis.ext
intro φs'
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [ι_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero_mul]
rw [ι_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
rw [hf]
simp
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul
lemma ι_normalOrder_superCommuteF_eq_zero_mul
(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
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
have hf : (ι.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute.flip c) = 0 := by
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
apply ofCrAnListBasis.ext
intro φs
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [ι_normalOrder_superCommute_ofCrAnList_eq_zero_mul]
rw [ι_normalOrder_superCommuteF_ofCrAnList_eq_zero_mul]
rw [hf]
simp
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul_right (b c d : 𝓕.CrAnAlgebra) :
lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul 1 b c d]
rw [← ι_normalOrder_superCommuteF_eq_zero_mul 1 b c d]
simp
@[simp]
lemma ι_normalOrder_superCommute_eq_zero_mul_left (a c d : 𝓕.CrAnAlgebra) :
lemma ι_normalOrder_superCommuteF_eq_zero_mul_left (a c d : 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul a 1 c d]
rw [← ι_normalOrder_superCommuteF_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) :
lemma ι_normalOrder_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.CrAnAlgebra) :
ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
rw [← ι_normalOrder_superCommute_eq_zero_mul a (b1 * b2) c d]
rw [← ι_normalOrder_superCommuteF_eq_zero_mul a (b1 * b2) c d]
congr 2
noncomm_ring
@[simp]
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]
lemma ι_normalOrder_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
rw [← ι_normalOrder_superCommuteF_eq_zero_mul 1 1 c d]
simp
/-!

31
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/SuperCommute.lean
View file

@ -19,15 +19,15 @@ open FieldStatistic
namespace FieldOpAlgebra
variable {𝓕 : FieldSpecification}
lemma ι_superCommute_eq_zero_of_ι_right_zero (a b : 𝓕.CrAnAlgebra) (h : ι b = 0) :
lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.CrAnAlgebra) (h : ι b = 0) :
ι [a, b]ₛca = 0 := by
rw [superCommute_expand_bosonicProj_fermionicProj]
rw [superCommuteF_expand_bosonicProj_fermionicProj]
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
simp_all
lemma ι_superCommute_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a = 0) :
lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a = 0) :
ι [a, b]ₛca = 0 := by
rw [superCommute_expand_bosonicProj_fermionicProj]
rw [superCommuteF_expand_bosonicProj_fermionicProj]
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
simp_all
@ -37,23 +37,23 @@ lemma ι_superCommute_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a
-/
lemma ι_superCommute_right_zero_of_mem_ideal (a b : 𝓕.CrAnAlgebra)
lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.CrAnAlgebra)
(h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛca = 0 := by
apply ι_superCommute_eq_zero_of_ι_right_zero
apply ι_superCommuteF_eq_zero_of_ι_right_zero
exact (ι_eq_zero_iff_mem_ideal b).mpr h
lemma ι_superCommute_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈ b2) :
lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈ b2) :
ι [a, b1]ₛca = ι [a, b2]ₛca := by
rw [equiv_iff_sub_mem_ideal] at h
rw [LinearMap.sub_mem_ker_iff.mp]
simp only [LinearMap.mem_ker, ← map_sub]
exact ι_superCommute_right_zero_of_mem_ideal a (b1 - b2) h
exact ι_superCommuteF_right_zero_of_mem_ideal a (b1 - b2) h
/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.superCommute a)
(ι_superCommute_eq_of_equiv_right a)
toFun := Quotient.lift (ι.toLinearMap ∘ₗ superCommuteF a)
(ι_superCommuteF_eq_of_equiv_right a)
map_add' x y := by
obtain ⟨x, hx⟩ := ι_surjective x
obtain ⟨y, hy⟩ := ι_surjective y
@ -80,10 +80,10 @@ lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
obtain ⟨b, rfl⟩ := ι_surjective b
have ha1b1 : (superCommuteRight (a1 - a2)) (ι b) = 0 := by
rw [superCommuteRight_apply_ι]
apply ι_superCommute_eq_zero_of_ι_left_zero
apply ι_superCommuteF_eq_zero_of_ι_left_zero
exact (ι_eq_zero_iff_mem_ideal (a1 - a2)).mpr h
simp_all only [superCommuteRight_apply_ι, map_sub, LinearMap.sub_apply]
trans ι ((superCommute a2) b) + 0
trans ι ((superCommuteF a2) b) + 0
rw [← ha1b1]
simp only [add_sub_cancel]
simp
@ -111,8 +111,11 @@ noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
rw [superCommuteRight_apply_quot, superCommuteRight_apply_quot]
simp
lemma ι_superCommute (a b : 𝓕.CrAnAlgebra) : ι [a, b]ₛca = superCommute (ι a) (ι b) := by
rfl
@[inherit_doc superCommute]
scoped[FieldSpecification.FieldOpAlgebra] notation "[" a "," b "]ₛ" => superCommute a b
lemma ι_superCommuteF_eq_superCommute (a b : 𝓕.CrAnAlgebra) :
ι [a, b]ₛca = [ι a, ι b]ₛ := rfl
end FieldOpAlgebra
end FieldSpecification

62
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean
View file

@ -19,7 +19,7 @@ open FieldStatistic
namespace FieldOpAlgebra
variable {𝓕 : FieldSpecification}
lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
lemma ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
(φs1 φs2 : List 𝓕.CrAnStates) (h :
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
@ -107,9 +107,9 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
subst hφ4
simp_all
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp only [List.singleton_append, instCommGroup.eq_1, ofList_singleton, map_sub, map_smul]
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList, superCommuteF_ofCrAnList_ofCrAnList]
simp only [List.cons_append, List.nil_append, instCommGroup.eq_1, ofList_singleton, mul_sub, ←
ofCrAnList_append, Algebra.mul_smul_comm, sub_mul, List.append_assoc, Algebra.smul_mul_assoc,
map_sub, map_smul]
@ -132,14 +132,14 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
rw [mul_assoc]
congr
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
rw [superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList]
simp only [List.singleton_append, instCommGroup.eq_1, ofList_singleton, map_sub, map_smul]
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
rw [superCommuteF_ofCrAnList_ofCrAnList, superCommuteF_ofCrAnList_ofCrAnList]
simp only [List.cons_append, List.nil_append, instCommGroup.eq_1, ofList_singleton, map_sub,
map_smul, smul_sub]
simp_all
lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
lemma ι_timeOrder_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
(φs1 φs2 : List 𝓕.CrAnStates) :
ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
= 0 := by
@ -147,13 +147,13 @@ lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAn
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2
· exact ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList φs1 φs2 h
· exact ι_timeOrder_superCommuteF_superCommuteF_eq_time_ofCrAnList φs1 φs2 h
· rw [timeOrder_timeOrder_mid]
rw [timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel _ _ _ h]
rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel _ _ _ h]
simp
@[simp]
lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
lemma ι_timeOrder_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
@ -169,7 +169,7 @@ lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a
· intro x hx
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pa]
exact ι_timeOrder_superCommute_superCommute_ofCrAnList φs' φs
exact ι_timeOrder_superCommuteF_superCommuteF_ofCrAnList φs' φs
· simp [pa]
· intro x y hx hy hpx hpy
simp_all [pa,mul_add, add_mul]
@ -181,7 +181,7 @@ lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a
· intro x hx hpx
simp_all [pb, hpx]
lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
lemma ι_timeOrder_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
@ -202,7 +202,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, map_mul, pa]
conv_lhs =>
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
simp [mul_sub, sub_mul, ← ofCrAnList_append]
rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) =
@ -234,16 +234,16 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
have h1 : (ι (ofCrAnList [φ, ψ]) -
(exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
rw [superCommute_ofCrAnState_ofCrAnState]
rw [superCommuteF_ofCrAnState_ofCrAnState]
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
simp only [instCommGroup.eq_1, List.singleton_append, Algebra.smul_mul_assoc, map_sub,
map_smul]
rw [← ofCrAnList_append]
simp
rw [h1]
have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈
have hc : ι ((superCommuteF (ofCrAnState φ)) (ofCrAnState ψ)) ∈
Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
apply ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center
rw [Subalgebra.mem_center_iff] at hc
repeat rw [← mul_assoc]
rw [hc]
@ -257,7 +257,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
rw [← h1]
rw [← crAnTimeOrderList]
by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
· rw [ι_superCommute_of_diff_statistic hq]
· rw [ι_superCommuteF_of_diff_statistic hq]
simp
· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
rw [timeOrder_ofCrAnList]
@ -277,20 +277,20 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
· intro x hx hpx
simp_all [pb, hpx]
lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
lemma ι_timeOrder_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
rw [timeOrder_timeOrder_mid]
have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
exact Decidable.not_and_iff_or_not.mp hφψ
rcases hφψ with hφψ | hφψ
· rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
· rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
simp_all only [false_and, not_false_eq_true, false_or, mul_zero, zero_mul, map_zero]
simp_all
· rw [superCommute_ofCrAnState_ofCrAnState_symm]
· rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, mul_neg, Algebra.mul_smul_comm,
neg_mul, Algebra.smul_mul_assoc, neg_eq_zero, smul_eq_zero]
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
simp only [mul_zero, zero_mul, map_zero, or_true]
simp_all
@ -314,42 +314,42 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
match hc with
| Or.inl hc =>
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
simp only [ι_timeOrder_superCommute_superCommute]
simp only [ι_timeOrder_superCommuteF_superCommuteF]
| Or.inr (Or.inl hc) =>
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
· rw [ι_timeOrder_superCommute_eq_time]
· rw [ι_timeOrder_superCommuteF_eq_time]
simp only [map_mul]
rw [ι_superCommute_of_create_create]
rw [ι_superCommuteF_of_create_create]
simp only [zero_mul]
· exact hφa
· exact hφb
· exact heqt.1
· exact heqt.2
· rw [ι_timeOrder_superCommute_neq_time heqt]
· rw [ι_timeOrder_superCommuteF_neq_time heqt]
| Or.inr (Or.inr (Or.inl hc)) =>
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
· rw [ι_timeOrder_superCommute_eq_time]
· rw [ι_timeOrder_superCommuteF_eq_time]
simp only [map_mul]
rw [ι_superCommute_of_annihilate_annihilate]
rw [ι_superCommuteF_of_annihilate_annihilate]
simp only [zero_mul]
· exact hφa
· exact hφb
· exact heqt.1
· exact heqt.2
· rw [ι_timeOrder_superCommute_neq_time heqt]
· rw [ι_timeOrder_superCommuteF_neq_time heqt]
| Or.inr (Or.inr (Or.inr hc)) =>
obtain ⟨φa, φb, hdiff, rfl⟩ := hc
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
· rw [ι_timeOrder_superCommute_eq_time]
· rw [ι_timeOrder_superCommuteF_eq_time]
simp only [map_mul]
rw [ι_superCommute_of_diff_statistic]
rw [ι_superCommuteF_of_diff_statistic]
simp only [zero_mul]
· exact hdiff
· exact heqt.1
· exact heqt.2
· rw [ι_timeOrder_superCommute_neq_time heqt]
· rw [ι_timeOrder_superCommuteF_neq_time heqt]
· simp [p]
· intro x y hx hy
simp only [map_add, p]
@ -365,7 +365,7 @@ lemma ι_timeOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
simp only [LinearMap.mem_ker, ← map_sub]
exact ι_timeOrder_zero_of_mem_ideal (a - b) h
/-- Normal ordering on `FieldOpAlgebra`. -/
/-- Time ordering on `FieldOpAlgebra`. -/
noncomputable def timeOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.timeOrder) ι_timeOrder_eq_of_equiv
map_add' x y := by

64
HepLean/PerturbationTheory/Algebras/ProtoOperatorAlgebra/Basic.lean
View file

@ -31,15 +31,15 @@ structure ProtoOperatorAlgebra where
/-- An algebra map from the creation and annihilation free algebra to the
algebra A. -/
crAnF : 𝓕.CrAnAlgebra →ₐ[ℂ] A
superCommute_crAn_center : ∀ (φ ψ : 𝓕.CrAnStates),
superCommuteF_crAn_center : ∀ (φ ψ : 𝓕.CrAnStates),
crAnF [ofCrAnState φ, ofCrAnState ψ]ₛca ∈ Subalgebra.center ℂ A
superCommute_create_create : ∀ (φc φc' : 𝓕.CrAnStates)
superCommuteF_create_create : ∀ (φc φc' : 𝓕.CrAnStates)
(_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
crAnF [ofCrAnState φc, ofCrAnState φc']ₛca = 0
superCommute_annihilate_annihilate : ∀ (φa φa' : 𝓕.CrAnStates)
superCommuteF_annihilate_annihilate : ∀ (φa φa' : 𝓕.CrAnStates)
(_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
crAnF [ofCrAnState φa, ofCrAnState φa']ₛca = 0
superCommute_different_statistics : ∀ (φ φ' : 𝓕.CrAnStates) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
superCommuteF_different_statistics : ∀ (φ φ' : 𝓕.CrAnStates) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
crAnF [ofCrAnState φ, ofCrAnState φ']ₛca = 0
namespace ProtoOperatorAlgebra
@ -52,14 +52,14 @@ instance : Semiring 𝓞.A := 𝓞.A_semiRing
/-- The algebra `𝓞.A` carries the instance of aan algebra over `ℂ` induced via `A_algebra`. -/
instance : Algebra ℂ 𝓞.A := 𝓞.A_algebra
lemma crAnF_superCommute_ofCrAnState_ofState_mem_center (φ : 𝓕.CrAnStates) (ψ : 𝓕.States) :
lemma crAnF_superCommuteF_ofCrAnState_ofState_mem_center (φ : 𝓕.CrAnStates) (ψ : 𝓕.States) :
𝓞.crAnF [ofCrAnState φ, ofState ψ]ₛca ∈ Subalgebra.center ℂ 𝓞.A := by
rw [ofState, map_sum, map_sum]
refine Subalgebra.sum_mem (Subalgebra.center ℂ 𝓞.A) ?h
intro x _
exact 𝓞.superCommute_crAn_center φ ⟨ψ, x⟩
exact 𝓞.superCommuteF_crAn_center φ ⟨ψ, x⟩
lemma crAnF_superCommute_anPart_ofState_mem_center (φ ψ : 𝓕.States) :
lemma crAnF_superCommuteF_anPart_ofState_mem_center (φ ψ : 𝓕.States) :
𝓞.crAnF [anPart φ, ofState ψ]ₛca ∈ Subalgebra.center ℂ 𝓞.A := by
match φ with
| States.inAsymp _ =>
@ -67,21 +67,21 @@ lemma crAnF_superCommute_anPart_ofState_mem_center (φ ψ : 𝓕.States) :
exact Subalgebra.zero_mem (Subalgebra.center ℂ 𝓞.A)
| States.position φ =>
simp only [anPart_position]
exact 𝓞.crAnF_superCommute_ofCrAnState_ofState_mem_center _ _
exact 𝓞.crAnF_superCommuteF_ofCrAnState_ofState_mem_center _ _
| States.outAsymp _ =>
simp only [anPart_posAsymp]
exact 𝓞.crAnF_superCommute_ofCrAnState_ofState_mem_center _ _
exact 𝓞.crAnF_superCommuteF_ofCrAnState_ofState_mem_center _ _
lemma crAnF_superCommute_ofCrAnState_ofState_diff_grade_zero (φ : 𝓕.CrAnStates) (ψ : 𝓕.States)
lemma crAnF_superCommuteF_ofCrAnState_ofState_diff_grade_zero (φ : 𝓕.CrAnStates) (ψ : 𝓕.States)
(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) :
𝓞.crAnF [ofCrAnState φ, ofState ψ]ₛca = 0 := by
rw [ofState, map_sum, map_sum]
rw [Finset.sum_eq_zero]
intro x hx
apply 𝓞.superCommute_different_statistics
apply 𝓞.superCommuteF_different_statistics
simpa [crAnStatistics] using h
lemma crAnF_superCommute_anPart_ofState_diff_grade_zero (φ ψ : 𝓕.States)
lemma crAnF_superCommuteF_anPart_ofState_diff_grade_zero (φ ψ : 𝓕.States)
(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) :
𝓞.crAnF [anPart φ, ofState ψ]ₛca = 0 := by
match φ with
@ -89,22 +89,22 @@ lemma crAnF_superCommute_anPart_ofState_diff_grade_zero (φ ψ : 𝓕.States)
simp
| States.position φ =>
simp only [anPart_position]
apply 𝓞.crAnF_superCommute_ofCrAnState_ofState_diff_grade_zero _ _ _
apply 𝓞.crAnF_superCommuteF_ofCrAnState_ofState_diff_grade_zero _ _ _
simpa [crAnStatistics] using h
| States.outAsymp _ =>
simp only [anPart_posAsymp]
apply 𝓞.crAnF_superCommute_ofCrAnState_ofState_diff_grade_zero _ _
apply 𝓞.crAnF_superCommuteF_ofCrAnState_ofState_diff_grade_zero _ _
simpa [crAnStatistics] using h
lemma crAnF_superCommute_ofState_ofState_mem_center (φ ψ : 𝓕.States) :
lemma crAnF_superCommuteF_ofState_ofState_mem_center (φ ψ : 𝓕.States) :
𝓞.crAnF [ofState φ, ofState ψ]ₛca ∈ Subalgebra.center ℂ 𝓞.A := by
rw [ofState, map_sum]
simp only [LinearMap.coeFn_sum, Finset.sum_apply, map_sum]
refine Subalgebra.sum_mem (Subalgebra.center ℂ 𝓞.A) ?h
intro x _
exact crAnF_superCommute_ofCrAnState_ofState_mem_center 𝓞 ⟨φ, x⟩ ψ
exact crAnF_superCommuteF_ofCrAnState_ofState_mem_center 𝓞 ⟨φ, x⟩ ψ
lemma crAnF_superCommute_anPart_anPart (φ ψ : 𝓕.States) :
lemma crAnF_superCommuteF_anPart_anPart (φ ψ : 𝓕.States) :
𝓞.crAnF [anPart φ, anPart ψ]ₛca = 0 := by
match φ, ψ with
| _, States.inAsymp _ =>
@ -113,26 +113,26 @@ lemma crAnF_superCommute_anPart_anPart (φ ψ : 𝓕.States) :
simp
| States.position φ, States.position ψ =>
simp only [anPart_position]
rw [𝓞.superCommute_annihilate_annihilate]
rw [𝓞.superCommuteF_annihilate_annihilate]
rfl
rfl
| States.position φ, States.outAsymp _ =>
simp only [anPart_position, anPart_posAsymp]
rw [𝓞.superCommute_annihilate_annihilate]
rw [𝓞.superCommuteF_annihilate_annihilate]
rfl
rfl
| States.outAsymp _, States.outAsymp _ =>
simp only [anPart_posAsymp]
rw [𝓞.superCommute_annihilate_annihilate]
rw [𝓞.superCommuteF_annihilate_annihilate]
rfl
rfl
| States.outAsymp _, States.position _ =>
simp only [anPart_posAsymp, anPart_position]
rw [𝓞.superCommute_annihilate_annihilate]
rw [𝓞.superCommuteF_annihilate_annihilate]
rfl
rfl
lemma crAnF_superCommute_crPart_crPart (φ ψ : 𝓕.States) :
lemma crAnF_superCommuteF_crPart_crPart (φ ψ : 𝓕.States) :
𝓞.crAnF [crPart φ, crPart ψ]ₛca = 0 := by
match φ, ψ with
| _, States.outAsymp _ =>
@ -141,56 +141,56 @@ lemma crAnF_superCommute_crPart_crPart (φ ψ : 𝓕.States) :
simp
| States.position φ, States.position ψ =>
simp only [crPart_position]
rw [𝓞.superCommute_create_create]
rw [𝓞.superCommuteF_create_create]
rfl
rfl
| States.position φ, States.inAsymp _ =>
simp only [crPart_position, crPart_negAsymp]
rw [𝓞.superCommute_create_create]
rw [𝓞.superCommuteF_create_create]
rfl
rfl
| States.inAsymp _, States.inAsymp _ =>
simp only [crPart_negAsymp]
rw [𝓞.superCommute_create_create]
rw [𝓞.superCommuteF_create_create]
rfl
rfl
| States.inAsymp _, States.position _ =>
simp only [crPart_negAsymp, crPart_position]
rw [𝓞.superCommute_create_create]
rw [𝓞.superCommuteF_create_create]
rfl
rfl
lemma crAnF_superCommute_ofCrAnState_ofCrAnList_eq_sum (φ : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates) :
lemma crAnF_superCommuteF_ofCrAnState_ofCrAnList_eq_sum (φ : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates) :
𝓞.crAnF [ofCrAnState φ, ofCrAnList φs]ₛca
= 𝓞.crAnF (∑ (n : Fin φs.length), 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (List.take n φs)) •
[ofCrAnState φ, ofCrAnState (φs.get n)]ₛca * ofCrAnList (φs.eraseIdx n)) := by
conv_lhs =>
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_eq_sum]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
rw [map_sum, map_sum]
congr
funext x
repeat rw [map_mul]
rw [map_smul, map_smul, ofCrAnList_singleton]
have h := Subalgebra.mem_center_iff.mp (𝓞.superCommute_crAn_center φ (φs.get x))
have h := Subalgebra.mem_center_iff.mp (𝓞.superCommuteF_crAn_center φ (φs.get x))
rw [h, mul_smul_comm, smul_mul_assoc, smul_mul_assoc, mul_assoc]
congr 1
· simp
· congr
rw [← map_mul, ← ofCrAnList_append, ← List.eraseIdx_eq_take_drop_succ]
lemma crAnF_superCommute_ofCrAnState_ofStateList_eq_sum (φ : 𝓕.CrAnStates) (φs : List 𝓕.States) :
lemma crAnF_superCommuteF_ofCrAnState_ofStateList_eq_sum (φ : 𝓕.CrAnStates) (φs : List 𝓕.States) :
𝓞.crAnF [ofCrAnState φ, ofStateList φs]ₛca
= 𝓞.crAnF (∑ (n : Fin φs.length), 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (List.take n φs)) •
[ofCrAnState φ, ofState (φs.get n)]ₛca * ofStateList (φs.eraseIdx n)) := by
conv_lhs =>
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStateList_eq_sum]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStateList_eq_sum]
rw [map_sum, map_sum]
congr
funext x
repeat rw [map_mul]
rw [map_smul, map_smul, ofCrAnList_singleton]
have h := Subalgebra.mem_center_iff.mp
(crAnF_superCommute_ofCrAnState_ofState_mem_center 𝓞 φ (φs.get x))
(crAnF_superCommuteF_ofCrAnState_ofState_mem_center 𝓞 φ (φs.get x))
rw [h, mul_smul_comm, smul_mul_assoc, smul_mul_assoc, mul_assoc]
congr 1
· simp

122
HepLean/PerturbationTheory/Algebras/ProtoOperatorAlgebra/NormalOrder.lean
View file

@ -24,46 +24,46 @@ open FieldStatistic
## Normal order of super-commutators.
The main result of this section is
`crAnF_normalOrder_superCommute_eq_zero_mul`.
`crAnF_normalOrder_superCommuteF_eq_zero_mul`.
-/
lemma crAnF_normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList
(φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnStates) :
𝓞.crAnF (𝓝ᶠ(ofCrAnList φs * [ofCrAnState φc, ofCrAnState φc']ₛca * ofCrAnList φs')) = 0 := by
rw [normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList φc φc' hφc hφc' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommute_create_create φc φc' hφc hφc']
rw [normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList φc φc' hφc hφc' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_create_create φc φc' hφc hφc']
simp
lemma crAnF_normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
(φa φa' : 𝓕.CrAnStates) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate) (φs φs' : List 𝓕.CrAnStates) :
𝓞.crAnF (𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * ofCrAnList φs')) = 0 := by
rw [normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommute_annihilate_annihilate φa φa' hφa hφa']
rw [normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_annihilate_annihilate φa φa' hφa hφa']
simp
lemma crAnF_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
𝓞.crAnF (normalOrder
(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']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommute_create_create φa φa' hφa hφa']
· rw [normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_create_create φa φa' hφa hφa']
simp
· rw [normalOrder_superCommute_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
· rw [normalOrder_superCommuteF_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
(ofCrAnList φs')]
simp
· rw [normalOrder_superCommute_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
· rw [normalOrder_superCommuteF_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
(ofCrAnList φs')]
simp
· rw [normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommute_annihilate_annihilate φa φa' hφa hφa']
· rw [normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_annihilate_annihilate φa φa' hφa hφa']
simp
lemma crAnF_normalOrder_superCommute_ofCrAnList_eq_zero
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_eq_zero
(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
(a : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrder (ofCrAnList φs *
[ofCrAnState φa, ofCrAnState φa']ₛca * a)) = 0 := by
@ -75,11 +75,11 @@ lemma crAnF_normalOrder_superCommute_ofCrAnList_eq_zero
intro l
simp only [ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
AlgHom.toLinearMap_apply, LinearMap.zero_apply]
exact crAnF_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
exact crAnF_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
rw [hf]
simp
lemma crAnF_normalOrder_superCommute_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
lemma crAnF_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
𝓞.crAnF (normalOrder (a * [ofCrAnState φa, ofCrAnState φa']ₛca * b)) = 0 := by
rw [mul_assoc]
@ -93,94 +93,94 @@ lemma crAnF_normalOrder_superCommute_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.Cr
LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [← mul_assoc]
exact crAnF_normalOrder_superCommute_ofCrAnList_eq_zero φa φa' _ _
exact crAnF_normalOrder_superCommuteF_ofCrAnList_eq_zero φa φa' _ _
rw [hf]
simp
lemma crAnF_normalOrder_superCommute_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
lemma crAnF_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
𝓞.crAnF (normalOrder (a * [ofCrAnState φa, ofCrAnList φs]ₛca * b)) = 0 := by
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_eq_sum]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
rw [Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
apply Fintype.sum_eq_zero
intro n
rw [← mul_assoc, ← mul_assoc]
rw [mul_assoc _ _ b, ofCrAnList_singleton]
rw [crAnF_normalOrder_superCommute_ofCrAnState_eq_zero_mul]
rw [crAnF_normalOrder_superCommuteF_ofCrAnState_eq_zero_mul]
lemma crAnF_normalOrder_superCommute_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
𝓞.crAnF (normalOrder (a * [ofCrAnList φs, ofCrAnState φa]ₛca * b)) = 0 := by
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
rw [← ofCrAnList_singleton, superCommuteF_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]
rw [crAnF_normalOrder_superCommute_ofCrAnState_ofCrAnList_eq_zero_mul]
rw [crAnF_normalOrder_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul]
simp
lemma crAnF_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero_mul
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
(φs φs' : List 𝓕.CrAnStates)
(a b : 𝓕.CrAnAlgebra) :
𝓞.crAnF (normalOrder (a * [ofCrAnList φs, ofCrAnList φs']ₛca * b)) = 0 := by
rw [superCommute_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
rw [map_sum, map_sum]
apply Fintype.sum_eq_zero
intro n
rw [← mul_assoc, ← mul_assoc]
rw [mul_assoc _ _ b]
rw [crAnF_normalOrder_superCommute_ofCrAnList_ofCrAnState_eq_zero_mul]
rw [crAnF_normalOrder_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul]
lemma crAnF_normalOrder_superCommute_ofCrAnList_eq_zero_mul
lemma crAnF_normalOrder_superCommuteF_ofCrAnList_eq_zero_mul
(φs : List 𝓕.CrAnStates)
(a b c : 𝓕.CrAnAlgebra) :
𝓞.crAnF (normalOrder (a * [ofCrAnList φs, c]ₛca * b)) = 0 := by
change (𝓞.crAnF.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute (ofCrAnList φs)) c = 0
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) c = 0
have hf : (𝓞.crAnF.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute (ofCrAnList φs)) = 0 := by
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) = 0 := by
apply ofCrAnListBasis.ext
intro φs'
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [crAnF_normalOrder_superCommute_ofCrAnList_ofCrAnList_eq_zero_mul]
rw [crAnF_normalOrder_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
rw [hf]
simp
@[simp]
lemma crAnF_normalOrder_superCommute_eq_zero_mul
lemma crAnF_normalOrder_superCommuteF_eq_zero_mul
(a b c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrder (a * [d, c]ₛca * b)) = 0 := by
change (𝓞.crAnF.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute.flip c) d = 0
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
have hf : (𝓞.crAnF.toLinearMap ∘ₗ normalOrder ∘ₗ
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommute.flip c) = 0 := by
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
apply ofCrAnListBasis.ext
intro φs
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [crAnF_normalOrder_superCommute_ofCrAnList_eq_zero_mul]
rw [crAnF_normalOrder_superCommuteF_ofCrAnList_eq_zero_mul]
rw [hf]
simp
@[simp]
lemma crAnF_normalOrder_superCommute_eq_zero_mul_right
lemma crAnF_normalOrder_superCommuteF_eq_zero_mul_right
(b c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrder ([d, c]ₛca * b)) = 0 := by
rw [← crAnF_normalOrder_superCommute_eq_zero_mul 1 b c d]
rw [← crAnF_normalOrder_superCommuteF_eq_zero_mul 1 b c d]
simp
@[simp]
lemma crAnF_normalOrder_superCommute_eq_zero_mul_left
lemma crAnF_normalOrder_superCommuteF_eq_zero_mul_left
(a c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrder (a * [d, c]ₛca)) = 0 := by
rw [← crAnF_normalOrder_superCommute_eq_zero_mul a 1 c d]
rw [← crAnF_normalOrder_superCommuteF_eq_zero_mul a 1 c d]
simp
@[simp]
lemma crAnF_normalOrder_superCommute_eq_zero
lemma crAnF_normalOrder_superCommuteF_eq_zero
(c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrder [d, c]ₛca) = 0 := by
rw [← crAnF_normalOrder_superCommute_eq_zero_mul 1 1 c d]
rw [← crAnF_normalOrder_superCommuteF_eq_zero_mul 1 1 c d]
simp
/-!
@ -193,14 +193,14 @@ lemma crAnF_normalOrder_ofState_ofState_swap (φ φ' : 𝓕.States) :
𝓞.crAnF (normalOrder (ofState φ * ofState φ')) =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓞.crAnF (normalOrder (ofState φ' * ofState φ)) := by
rw [← ofStateList_singleton, ← ofStateList_singleton,
ofStateList_mul_ofStateList_eq_superCommute]
ofStateList_mul_ofStateList_eq_superCommuteF]
simp
lemma crAnF_normalOrder_ofCrAnState_ofCrAnList_swap (φ : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates) :
𝓞.crAnF (normalOrder (ofCrAnState φ * ofCrAnList φs)) =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • 𝓞.crAnF (normalOrder (ofCrAnList φs * ofCrAnState φ)) := by
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommute]
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommuteF]
simp
lemma crAnF_normalOrder_ofCrAnState_ofStatesList_swap (φ : 𝓕.CrAnStates)
@ -208,7 +208,7 @@ lemma crAnF_normalOrder_ofCrAnState_ofStatesList_swap (φ : 𝓕.CrAnStates)
𝓞.crAnF (normalOrder (ofCrAnState φ * ofStateList φ')) =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
𝓞.crAnF (normalOrder (ofStateList φ' * ofCrAnState φ)) := by
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofStateList_eq_superCommute]
rw [← ofCrAnList_singleton, ofCrAnList_mul_ofStateList_eq_superCommuteF]
simp
lemma crAnF_normalOrder_anPart_ofStatesList_swap (φ : 𝓕.States)
@ -248,13 +248,13 @@ lemma crAnF_normalOrder_ofStatesList_mul_anPart_swap (φ : 𝓕.States)
## Super commutators with a normal ordered term as sums
-/
lemma crAnF_ofCrAnState_superCommute_normalOrder_ofCrAnList_eq_sum (φ : 𝓕.CrAnStates)
lemma crAnF_ofCrAnState_superCommuteF_normalOrder_ofCrAnList_eq_sum (φ : 𝓕.CrAnStates)
(φs : List 𝓕.CrAnStates) : 𝓞.crAnF ([ofCrAnState φ, normalOrder (ofCrAnList φs)]ₛca) =
∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
𝓞.crAnF ([ofCrAnState φ, ofCrAnState φs[n]]ₛca)
* 𝓞.crAnF (normalOrder (ofCrAnList (φs.eraseIdx n))) := by
rw [normalOrder_ofCrAnList, map_smul, map_smul]
rw [crAnF_superCommute_ofCrAnState_ofCrAnList_eq_sum, sum_normalOrderList_length]
rw [crAnF_superCommuteF_ofCrAnState_ofCrAnList_eq_sum, sum_normalOrderList_length]
simp only [instCommGroup.eq_1, List.get_eq_getElem, normalOrderList_get_normalOrderEquiv,
normalOrderList_eraseIdx_normalOrderEquiv, Algebra.smul_mul_assoc, map_sum, map_smul, map_mul,
Finset.smul_sum, Fin.getElem_fin]
@ -272,10 +272,10 @@ lemma crAnF_ofCrAnState_superCommute_normalOrder_ofCrAnList_eq_sum (φ : 𝓕.Cr
rw [hs]
rfl
· simp [hs]
· erw [𝓞.superCommute_different_statistics _ _ hs]
· erw [𝓞.superCommuteF_different_statistics _ _ hs]
simp
lemma crAnF_ofCrAnState_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.CrAnStates)
lemma crAnF_ofCrAnState_superCommuteF_normalOrder_ofStateList_eq_sum (φ : 𝓕.CrAnStates)
(φs : List 𝓕.States) : 𝓞.crAnF ([ofCrAnState φ, normalOrder (ofStateList φs)]ₛca) =
∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
𝓞.crAnF ([ofCrAnState φ, ofState φs[n]]ₛca)
@ -283,7 +283,7 @@ lemma crAnF_ofCrAnState_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.C
conv_lhs =>
rw [ofStateList_sum, map_sum, map_sum, map_sum]
enter [2, s]
rw [crAnF_ofCrAnState_superCommute_normalOrder_ofCrAnList_eq_sum,
rw [crAnF_ofCrAnState_superCommuteF_normalOrder_ofCrAnList_eq_sum,
CrAnSection.sum_over_length]
enter [2, n]
rw [CrAnSection.take_statistics_eq_take_state_statistics, smul_mul_assoc]
@ -304,9 +304,9 @@ Within a proto-operator algebra we have that
where `sᵢ` is the exchange sign for `φ` and `φ₀…φᵢ₋₁`.
The origin of this result is
- `superCommute_ofCrAnList_ofCrAnList_eq_sum`
- `superCommuteF_ofCrAnList_ofCrAnList_eq_sum`
-/
lemma crAnF_anPart_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.States) (φs : List 𝓕.States) :
lemma crAnF_anPart_superCommuteF_normalOrder_ofStateList_eq_sum (φ : 𝓕.States) (φs : List 𝓕.States) :
𝓞.crAnF ([anPart φ, 𝓝ᶠ(φs)]ₛca) =
∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
𝓞.crAnF ([anPart φ, ofState φs[n]]ₛca) * 𝓞.crAnF 𝓝ᶠ(φs.eraseIdx n) := by
@ -315,11 +315,11 @@ lemma crAnF_anPart_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.States
simp
| .position φ =>
simp only [anPart_position, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
rw [crAnF_ofCrAnState_superCommute_normalOrder_ofStateList_eq_sum]
rw [crAnF_ofCrAnState_superCommuteF_normalOrder_ofStateList_eq_sum]
simp [crAnStatistics]
| .outAsymp φ =>
simp only [anPart_posAsymp, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
rw [crAnF_ofCrAnState_superCommute_normalOrder_ofStateList_eq_sum]
rw [crAnF_ofCrAnState_superCommuteF_normalOrder_ofStateList_eq_sum]
simp [crAnStatistics]
/-!
@ -331,12 +331,12 @@ lemma crAnF_anPart_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.States
Within a proto-operator algebra we have that
`anPart φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ((anPart φ)φ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛca`.
-/
lemma crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommute (φ : 𝓕.States)
lemma crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommuteF (φ : 𝓕.States)
(φ' : List 𝓕.States) :
𝓞.crAnF (anPart φ * normalOrder (ofStateList φ')) =
𝓞.crAnF (normalOrder (anPart φ * ofStateList φ')) +
𝓞.crAnF ([anPart φ, normalOrder (ofStateList φ')]ₛca) := by
rw [anPart_mul_normalOrder_ofStateList_eq_superCommute]
rw [anPart_mul_normalOrder_ofStateList_eq_superCommuteF]
simp only [instCommGroup.eq_1, map_add, map_smul]
congr
rw [crAnF_normalOrder_anPart_ofStatesList_swap]
@ -345,12 +345,12 @@ lemma crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommute (φ : 𝓕.State
Within a proto-operator algebra we have that
`φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ(φφ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛca`.
-/
lemma crAnF_ofState_mul_normalOrder_ofStatesList_eq_superCommute (φ : 𝓕.States)
lemma crAnF_ofState_mul_normalOrder_ofStatesList_eq_superCommuteF (φ : 𝓕.States)
(φs : List 𝓕.States) : 𝓞.crAnF (ofState φ * 𝓝ᶠ(φs)) =
𝓞.crAnF (normalOrder (ofState φ * ofStateList φs)) +
𝓞.crAnF ([anPart φ, 𝓝ᶠ(φs)]ₛca) := by
conv_lhs => rw [ofState_eq_crPart_add_anPart]
rw [add_mul, map_add, crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommute, ← add_assoc,
rw [add_mul, map_add, crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommuteF, ← add_assoc,
← normalOrder_crPart_mul, ← map_add]
conv_lhs =>
lhs
@ -376,8 +376,8 @@ lemma crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum (φ : 𝓕.States)
∑ n : Option (Fin φs.length),
contractStateAtIndex φ φs n *
𝓞.crAnF (normalOrder (ofStateList (HepLean.List.optionEraseZ φs φ n))) := by
rw [crAnF_ofState_mul_normalOrder_ofStatesList_eq_superCommute]
rw [crAnF_anPart_superCommute_normalOrder_ofStateList_eq_sum]
rw [crAnF_ofState_mul_normalOrder_ofStatesList_eq_superCommuteF]
rw [crAnF_anPart_superCommuteF_normalOrder_ofStateList_eq_sum]
simp only [instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc, contractStateAtIndex,
Fintype.sum_option, one_mul]
rfl
@ -401,9 +401,9 @@ lemma crAnF_ofState_normalOrder_insert (φ : 𝓕.States) (φs : List 𝓕.State
simp
rw [hl]
rw [ofStateList_append, ofStateList_append]
rw [ofStateList_mul_ofStateList_eq_superCommute, add_mul]
rw [ofStateList_mul_ofStateList_eq_superCommuteF, add_mul]
simp only [instCommGroup.eq_1, Nat.succ_eq_add_one, ofList_singleton, Algebra.smul_mul_assoc,
map_add, map_smul, crAnF_normalOrder_superCommute_eq_zero_mul_right, add_zero, smul_smul,
map_add, map_smul, crAnF_normalOrder_superCommuteF_eq_zero_mul_right, add_zero, smul_smul,
exchangeSign_mul_self_swap, one_smul]
rw [← ofStateList_append, ← ofStateList_append]
simp

10
HepLean/PerturbationTheory/Algebras/ProtoOperatorAlgebra/TimeContraction.lean
View file

@ -38,7 +38,7 @@ lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.States) (h : timeOrderRel φ ψ
𝓞.timeContract φ ψ = 𝓞.crAnF ([anPart φ, ofState ψ]ₛca) := by
conv_rhs =>
rw [ofState_eq_crPart_add_anPart]
rw [map_add, map_add, crAnF_superCommute_anPart_anPart, superCommute_anPart_crPart]
rw [map_add, map_add, crAnF_superCommuteF_anPart_anPart, superCommuteF_anPart_crPart]
simp only [timeContract, instCommGroup.eq_1, Algebra.smul_mul_assoc, add_zero]
rw [timeOrder_ofState_ofState_ordered h]
rw [normalOrder_ofState_mul_ofState]
@ -61,11 +61,11 @@ lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.States) (h : ¬ timeOrderRe
lemma timeContract_mem_center (φ ψ : 𝓕.States) : 𝓞.timeContract φ ψ ∈ Subalgebra.center ℂ 𝓞.A := by
by_cases h : timeOrderRel φ ψ
· rw [timeContract_of_timeOrderRel _ _ _ h]
exact 𝓞.crAnF_superCommute_anPart_ofState_mem_center _ _
exact 𝓞.crAnF_superCommuteF_anPart_ofState_mem_center _ _
· rw [timeContract_of_not_timeOrderRel _ _ _ h]
refine Subalgebra.smul_mem (Subalgebra.center ℂ 𝓞.A) ?_ 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ)
rw [timeContract_of_timeOrderRel]
exact 𝓞.crAnF_superCommute_anPart_ofState_mem_center _ _
exact 𝓞.crAnF_superCommuteF_anPart_ofState_mem_center _ _
have h1 := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
simp_all
@ -73,11 +73,11 @@ lemma timeContract_zero_of_diff_grade (φ ψ : 𝓕.States) (h : (𝓕 |>ₛ φ)
𝓞.timeContract φ ψ = 0 := by
by_cases h1 : timeOrderRel φ ψ
· rw [timeContract_of_timeOrderRel _ _ _ h1]
rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
rw [crAnF_superCommuteF_anPart_ofState_diff_grade_zero]
exact h
· rw [timeContract_of_not_timeOrderRel _ _ _ h1]
rw [timeContract_of_timeOrderRel _ _ _]
rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
rw [crAnF_superCommuteF_anPart_ofState_diff_grade_zero]
simp only [instCommGroup.eq_1, smul_zero]
exact h.symm
have ht := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ

2
HepLean/PerturbationTheory/WickContraction/TimeContract.lean
View file

@ -87,7 +87,7 @@ lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt
Algebra.smul_mul_assoc, uncontractedListGet]
· simp only [hik, ↓reduceIte, MulMemClass.coe_mul]
rw [𝓞.timeContract_of_timeOrderRel]
trans (1 : ℂ) • (𝓞.crAnF ((CrAnAlgebra.superCommute
trans (1 : ℂ) • (𝓞.crAnF ((CrAnAlgebra.superCommuteF
(CrAnAlgebra.anPart φ)) (CrAnAlgebra.ofState φs[k.1])) *
↑(timeContract 𝓞 φsΛ))
· simp

4
HepLean/PerturbationTheory/WicksTheorem.lean
View file

@ -245,14 +245,14 @@ lemma wick_term_some_eq_wick_term_optionEraseZ (φ : 𝓕.States) (φs : List
· simp only [h1, ↓reduceIte, MulMemClass.coe_mul]
rw [timeContract_zero_of_diff_grade]
simp only [zero_mul, smul_zero]
rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
rw [crAnF_superCommuteF_anPart_ofState_diff_grade_zero]
simp only [zero_mul, smul_zero]
exact hg
exact hg
· simp only [h1, ↓reduceIte, MulMemClass.coe_mul]
rw [timeContract_zero_of_diff_grade]
simp only [zero_mul, smul_zero]
rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
rw [crAnF_superCommuteF_anPart_ofState_diff_grade_zero]
simp only [zero_mul, smul_zero]
exact hg
exact fun a => hg (id (Eq.symm a))