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refactor: ofState rename to ofFieldOpF

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jstoobysmith 2025-02-03 11:13:23 +00:00
parent 08260e709c
commit 93d06895c6
12 changed files with 85 additions and 85 deletions
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8
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/Basic.lean
View file

@ -429,9 +429,9 @@ lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : FieldOpFreeAlgebra
-/
/-- An element of `FieldOpAlgebra` from a `States`. -/
def ofFieldOp (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (ofState φ)
def ofFieldOp (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (ofFieldOpF φ)
lemma ofFieldOp_eq_ι_ofState (φ : 𝓕.States) : ofFieldOp φ = ι (ofState φ) := rfl
lemma ofFieldOp_eq_ι_ofFieldOpF (φ : 𝓕.States) : ofFieldOp φ = ι (ofFieldOpF φ) := rfl
/-- An element of `FieldOpAlgebra` from a list of `States`. -/
def ofFieldOpList (φs : List 𝓕.States) : 𝓕.FieldOpAlgebra := ι (ofFieldOpListF φs)
@ -464,7 +464,7 @@ lemma ofCrAnFieldOp_eq_ι_ofCrAnState (φ : 𝓕.CrAnStates) :
lemma ofFieldOp_eq_sum (φ : 𝓕.States) :
ofFieldOp φ = (∑ i : 𝓕.statesToCrAnType φ, ofCrAnFieldOp ⟨φ, i⟩) := by
rw [ofFieldOp, ofState]
rw [ofFieldOp, ofFieldOpF]
simp only [map_sum]
rfl
@ -534,7 +534,7 @@ lemma crPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
lemma ofFieldOp_eq_crPart_add_anPart (φ : 𝓕.States) :
ofFieldOp φ = crPart φ + anPart φ := by
rw [ofFieldOp, crPart, anPart, ofState_eq_crPartF_add_anPartF]
rw [ofFieldOp, crPart, anPart, ofFieldOpF_eq_crPartF_add_anPartF]
simp only [map_add]
end FieldOpAlgebra

6
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/NormalOrder.lean
View file

@ -473,7 +473,7 @@ where `sᵢ` is the exchange sign for `φ` and `φ₀…φᵢ₋₁`.
-/
lemma anPart_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.States) (φs : List 𝓕.States) :
[anPart φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
[anPart φ, ofState φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := by
[anPart φ, ofFieldOpF φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := by
match φ with
| .inAsymp φ =>
simp
@ -520,7 +520,7 @@ lemma ofFieldOp_mul_normalOrder_ofFieldOpList_eq_superCommute (φ : 𝓕.States)
lhs
rw [← add_mul, ← ofFieldOp_eq_crPart_add_anPart]
/-- In the expansion of `ofState φ * normalOrderF (ofFieldOpListF φs)` the element
/-- In the expansion of `ofFieldOpF φ * normalOrderF (ofFieldOpListF φs)` the element
of `𝓞.A` associated with contracting `φ` with the (optional) `n`th element of `φs`. -/
noncomputable def contractStateAtIndex (φ : 𝓕.States) (φs : List 𝓕.States)
(n : Option (Fin φs.length)) : 𝓕.FieldOpAlgebra :=
@ -599,7 +599,7 @@ lemma normalOrder_ofFieldOp_mul_ofFieldOp (φ φ' : 𝓕.States) : 𝓝(ofFieldO
crPart φ * crPart φ' + 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • (crPart φ' * anPart φ) +
crPart φ * anPart φ' + anPart φ * anPart φ' := by
rw [ofFieldOp, ofFieldOp, ← map_mul, normalOrder_eq_ι_normalOrderF,
normalOrderF_ofState_mul_ofState]
normalOrderF_ofFieldOpF_mul_ofFieldOpF]
rfl
end FieldOpAlgebra

2
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/StaticWickTheorem.lean
View file

@ -103,7 +103,7 @@ theorem static_wick_theorem : (φs : List 𝓕.States) →
right
simp only [uncontractedListGet, List.getElem_map,
uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem]
rw [superCommute_anPart_ofState_diff_grade_zero]
rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
exact hn
rw [h1]
simp

14
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/SuperCommute.lean
View file

@ -154,7 +154,7 @@ lemma superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero (φ : 𝓕.CrAnStates)
apply superCommute_diff_statistic
simpa [crAnStatistics] using h
lemma superCommute_anPart_ofState_diff_grade_zero (φ ψ : 𝓕.States)
lemma superCommute_anPart_ofFieldOpF_diff_grade_zero (φ ψ : 𝓕.States)
(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : [anPart φ, ofFieldOp ψ]ₛ = 0 := by
match φ with
| States.inAsymp _ =>
@ -231,28 +231,28 @@ lemma superCommute_ofCrAnFieldOpList_ofFieldOpList (φcas : List 𝓕.CrAnStates
[ofCrAnFieldOpList φcas, ofFieldOpList φs]ₛ = ofCrAnFieldOpList φcas * ofFieldOpList φs -
𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofCrAnFieldOpList φcas := by
rw [ofCrAnFieldOpList, ofFieldOpList]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnList_ofStatesList]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnList_ofFieldOpFsList]
rfl
lemma superCommute_ofFieldOpList_ofFieldOpList (φs φs' : List 𝓕.States) :
[ofFieldOpList φs, ofFieldOpList φs']ₛ = ofFieldOpList φs * ofFieldOpList φs' -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs := by
rw [ofFieldOpList, ofFieldOpList]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofStatesList]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofFieldOpFsList]
rfl
lemma superCommute_ofFieldOp_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
[ofFieldOp φ, ofFieldOpList φs]ₛ = ofFieldOp φ * ofFieldOpList φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofFieldOp φ := by
rw [ofFieldOp, ofFieldOpList]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofState_ofStatesList]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpF_ofFieldOpFsList]
rfl
lemma superCommute_ofFieldOpList_ofFieldOp (φs : List 𝓕.States) (φ : 𝓕.States) :
[ofFieldOpList φs, ofFieldOp φ]ₛ = ofFieldOpList φs * ofFieldOp φ -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs := by
rw [ofFieldOpList, ofFieldOp]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofState]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofFieldOpF]
rfl
lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
@ -343,14 +343,14 @@ lemma superCommute_crPart_ofFieldOp (φ φ' : 𝓕.States) :
[crPart φ, ofFieldOp φ']ₛ = crPart φ * ofFieldOp φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * crPart φ := by
rw [crPart, ofFieldOp]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofState]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofFieldOpF]
rfl
lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.States) :
[anPart φ, ofFieldOp φ']ₛ = anPart φ * ofFieldOp φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * anPart φ := by
rw [anPart, ofFieldOp]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofState]
rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofFieldOpF]
rfl
/-!

4
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeContraction.lean
View file

@ -78,11 +78,11 @@ lemma timeContract_zero_of_diff_grade (φ ψ : 𝓕.States) (h : (𝓕 |>ₛ φ)
timeContract φ ψ = 0 := by
by_cases h1 : timeOrderRel φ ψ
· rw [timeContract_of_timeOrderRel _ _ h1]
rw [superCommute_anPart_ofState_diff_grade_zero]
rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
exact h
· rw [timeContract_of_not_timeOrderRel _ _ h1]
rw [timeContract_of_timeOrderRel _ _ _]
rw [superCommute_anPart_ofState_diff_grade_zero]
rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
simp only [instCommGroup.eq_1, smul_zero]
exact h.symm
have ht := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ

6
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean
View file

@ -396,19 +396,19 @@ lemma timeOrder_eq_ι_timeOrderF (a : 𝓕.FieldOpFreeAlgebra) :
lemma timeOrder_ofFieldOp_ofFieldOp_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
𝓣(ofFieldOp φ * ofFieldOp ψ) = ofFieldOp φ * ofFieldOp ψ := by
rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
timeOrderF_ofState_ofState_ordered h]
timeOrderF_ofFieldOpF_ofFieldOpF_ordered h]
lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
𝓣(ofFieldOp φ * ofFieldOp ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOp ψ * ofFieldOp φ := by
rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
timeOrderF_ofState_ofState_not_ordered h]
timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
simp
lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder {φ ψ : 𝓕.States}
(h : ¬ timeOrderRel φ ψ) :
𝓣(ofFieldOp φ * ofFieldOp ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣(ofFieldOp ψ * ofFieldOp φ) := by
rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
timeOrderF_ofState_ofState_not_ordered_eq_timeOrderF h]
timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF h]
simp only [instCommGroup.eq_1, map_smul]
rfl

16
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Basic.lean
View file

@ -21,7 +21,7 @@ The main structures defined in this module are:
* `FieldOpFreeAlgebra` - The creation and annihilation algebra
* `ofCrAnState` - Maps a creation/annihilation state to the algebra
* `ofCrAnList` - Maps a list of creation/annihilation states to the algebra
* `ofState` - Maps a state to a sum of creation and annihilation operators
* `ofFieldOpF` - Maps a state to a sum of creation and annihilation operators
* `crPartF` - The creation part of a state in the algebra
* `anPartF` - The annihilation part of a state in the algebra
* `superCommuteF` - The super commutator on the algebra
@ -66,13 +66,13 @@ lemma ofCrAnList_singleton (φ : 𝓕.CrAnStates) :
/-- Maps a state to the sum of creation and annihilation operators in
creation and annihilation free-algebra. -/
def ofState (φ : 𝓕.States) : FieldOpFreeAlgebra 𝓕 :=
def ofFieldOpF (φ : 𝓕.States) : FieldOpFreeAlgebra 𝓕 :=
∑ (i : 𝓕.statesToCrAnType φ), ofCrAnState ⟨φ, i⟩
/-- Maps a list of states to the creation and annihilation free-algebra by taking
the product of their sums of creation and annihlation operators.
Roughly `[φ1, φ2]` gets sent to `(φ1ᶜ+ φ1ᵃ) * (φ2ᶜ+ φ2ᵃ)` etc. -/
def ofFieldOpListF (φs : List 𝓕.States) : FieldOpFreeAlgebra 𝓕 := (List.map ofState φs).prod
def ofFieldOpListF (φs : List 𝓕.States) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
/-- Coercion from `List 𝓕.States` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
instance : Coe (List 𝓕.States) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
@ -81,10 +81,10 @@ instance : Coe (List 𝓕.States) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF
lemma ofFieldOpListF_nil : ofFieldOpListF ([] : List 𝓕.States) = 1 := rfl
lemma ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.States) :
ofFieldOpListF (φ :: φs) = ofState φ * ofFieldOpListF φs := rfl
ofFieldOpListF (φ :: φs) = ofFieldOpF φ * ofFieldOpListF φs := rfl
lemma ofFieldOpListF_singleton (φ : 𝓕.States) :
ofFieldOpListF [φ] = ofState φ := by simp [ofFieldOpListF]
ofFieldOpListF [φ] = ofFieldOpF φ := by simp [ofFieldOpListF]
lemma ofFieldOpListF_append (φs φs' : List 𝓕.States) :
ofFieldOpListF (φs ++ φs') = ofFieldOpListF φs * ofFieldOpListF φs' := by
@ -160,9 +160,9 @@ lemma anPartF_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
anPartF (States.outAsymp φ) = ofCrAnState ⟨States.outAsymp φ, ()⟩ := by
simp [anPartF]
lemma ofState_eq_crPartF_add_anPartF (φ : 𝓕.States) :
ofState φ = crPartF φ + anPartF φ := by
rw [ofState]
lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.States) :
ofFieldOpF φ = crPartF φ + anPartF φ := by
rw [ofFieldOpF]
cases φ with
| inAsymp φ => simp [statesToCrAnType]
| position φ => simp [statesToCrAnType, CreateAnnihilate.sum_eq]

6
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/NormalOrder.lean
View file

@ -367,14 +367,14 @@ lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.States) :
rw [← mul_assoc, normalOrderF_swap_anPartF_crPartF]
simp
lemma normalOrderF_ofState_mul_ofState (φ φ' : 𝓕.States) :
𝓝ᶠ(ofState φ * ofState φ') =
lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.States) :
𝓝ᶠ(ofFieldOpF φ * ofFieldOpF φ') =
crPartF φ * crPartF φ' +
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
(crPartF φ' * anPartF φ) +
crPartF φ * anPartF φ' +
anPartF φ * anPartF φ' := by
rw [ofState_eq_crPartF_add_anPartF, ofState_eq_crPartF_add_anPartF, mul_add, add_mul, add_mul]
rw [ofFieldOpF_eq_crPartF_add_anPartF, ofFieldOpF_eq_crPartF_add_anPartF, mul_add, add_mul, add_mul]
simp only [map_add, normalOrderF_crPartF_mul_crPartF, normalOrderF_anPartF_mul_crPartF,
instCommGroup.eq_1, normalOrderF_crPartF_mul_anPartF, normalOrderF_anPartF_mul_anPartF]
abel

80
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/SuperCommute.lean
View file

@ -57,7 +57,7 @@ lemma superCommuteF_ofCrAnState_ofCrAnState (φ φ' : 𝓕.CrAnStates) :
rw [ofCrAnList_append]
rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
lemma superCommuteF_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
lemma superCommuteF_ofCrAnList_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
[ofCrAnList φcas, ofFieldOpListF φs]ₛca = ofCrAnList φcas * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofCrAnList φcas := by
conv_lhs => rw [ofFieldOpListF_sum]
@ -70,7 +70,7 @@ lemma superCommuteF_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs
← Finset.sum_mul, ← ofFieldOpListF_sum]
simp
lemma superCommuteF_ofFieldOpListF_ofStatesList (φ : List 𝓕.States) (φs : List 𝓕.States) :
lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs : List 𝓕.States) :
[ofFieldOpListF φ, ofFieldOpListF φs]ₛca = ofFieldOpListF φ * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpListF φ := by
conv_lhs => rw [ofFieldOpListF_sum]
@ -78,21 +78,21 @@ lemma superCommuteF_ofFieldOpListF_ofStatesList (φ : List 𝓕.States) (φs : L
Algebra.smul_mul_assoc]
conv_lhs =>
enter [2, x]
rw [superCommuteF_ofCrAnList_ofStatesList]
rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
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, ← ofFieldOpListF_sum]
lemma superCommuteF_ofState_ofStatesList (φ : 𝓕.States) (φs : List 𝓕.States) :
[ofState φ, ofFieldOpListF φs]ₛca = ofState φ * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofState φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofStatesList, ofFieldOpListF_singleton]
lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.States) (φs : List 𝓕.States) :
[ofFieldOpF φ, ofFieldOpListF φs]ₛca = ofFieldOpF φ * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
simp
lemma superCommuteF_ofFieldOpListF_ofState (φs : List 𝓕.States) (φ : 𝓕.States) :
[ofFieldOpListF φs, ofState φ]ₛca = ofFieldOpListF φs * ofState φ -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofFieldOpListF φs := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofStatesList, ofFieldOpListF_singleton]
lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.States) (φ : 𝓕.States) :
[ofFieldOpListF φs, ofFieldOpF φ]ₛca = ofFieldOpListF φs * ofFieldOpF φ -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
simp
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
@ -209,11 +209,11 @@ lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.S
match φ with
| States.inAsymp φ =>
simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
simp [crAnStatistics]
| States.position φ =>
simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
simp [crAnStatistics]
| States.outAsymp φ =>
simp
@ -227,24 +227,24 @@ lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.S
simp
| States.position φ =>
simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
simp [crAnStatistics]
| States.outAsymp φ =>
simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
simp [crAnStatistics]
lemma superCommuteF_crPartF_ofState (φ φ' : 𝓕.States) :
[crPartF φ, ofState φ']ₛca =
crPartF φ * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * crPartF φ := by
lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.States) :
[crPartF φ, ofFieldOpF φ']ₛca =
crPartF φ * ofFieldOpF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * crPartF φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_crPartF_ofFieldOpListF]
simp
lemma superCommuteF_anPartF_ofState (φ φ' : 𝓕.States) :
[anPartF φ, ofState φ']ₛca =
anPartF φ * ofState φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * anPartF φ := by
lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.States) :
[anPartF φ, ofFieldOpF φ']ₛca =
anPartF φ * ofFieldOpF φ' -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * anPartF φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_anPartF_ofFieldOpListF]
simp
@ -271,19 +271,19 @@ lemma ofCrAnState_mul_ofCrAnList_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' :
lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.States) :
ofFieldOpListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
+ [ofFieldOpListF φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofFieldOpListF_ofStatesList]
rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
simp
lemma ofState_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
ofState φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofState φ
+ [ofState φ, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofState_ofStatesList]
lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ
+ [ofFieldOpF φ, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofFieldOpF_ofFieldOpFsList]
simp
lemma ofFieldOpListF_mul_ofState_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
ofFieldOpListF φs * ofState φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofFieldOpListF φs
+ [ofFieldOpListF φs, ofState φ]ₛca := by
rw [superCommuteF_ofFieldOpListF_ofState]
lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
ofFieldOpListF φs * ofFieldOpF φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs
+ [ofFieldOpListF φs, ofFieldOpF φ]ₛca := by
rw [superCommuteF_ofFieldOpListF_ofFieldOpF]
simp
lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
@ -317,7 +317,7 @@ lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
lemma ofCrAnList_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
ofCrAnList φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnList φs
+ [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofCrAnList_ofStatesList]
rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
simp
/-!
@ -379,19 +379,19 @@ lemma superCommuteF_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs'
lemma superCommuteF_ofCrAnList_ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
(φs' : List 𝓕.States) : [ofCrAnList φs, ofFieldOpListF (φ :: φs')]ₛca =
[ofCrAnList φs, ofState φ]ₛca * ofFieldOpListF φs' +
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofCrAnList_ofStatesList]
[ofCrAnList φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
conv_rhs =>
lhs
rw [← ofFieldOpListF_singleton, superCommuteF_ofCrAnList_ofStatesList, sub_mul, mul_assoc,
rw [← ofFieldOpListF_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList, sub_mul, mul_assoc,
← ofFieldOpListF_append]
rhs
rw [FieldStatistic.ofList_singleton, ofFieldOpListF_singleton, smul_mul_assoc,
smul_mul_assoc, mul_assoc]
conv_rhs =>
rhs
rw [superCommuteF_ofCrAnList_ofStatesList, mul_sub, smul_mul_assoc]
rw [superCommuteF_ofCrAnList_ofFieldOpFsList, 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 [ofFieldOpListF_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
@ -420,10 +420,10 @@ lemma superCommuteF_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
lemma superCommuteF_ofCrAnList_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
[ofCrAnList φs, ofFieldOpListF φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofFieldOpListF (φs'.take n) * [ofCrAnList φs, ofState (φs'.get n)]ₛca *
ofFieldOpListF (φs'.take n) * [ofCrAnList φs, ofFieldOpF (φs'.get n)]ₛca *
ofFieldOpListF (φs'.drop (n + 1))
| [] => by
simp only [superCommuteF_ofCrAnList_ofStatesList, instCommGroup, ofList_empty,
simp only [superCommuteF_ofCrAnList_ofFieldOpFsList, 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

20
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/TimeOrder.lean
View file

@ -119,27 +119,27 @@ lemma timeOrderF_ofFieldOpListF_nil : timeOrderF (𝓕 := 𝓕) (ofFieldOpListF
lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
simp [timeOrderF_ofFieldOpListF, timeOrderSign, timeOrderList]
lemma timeOrderF_ofState_ofState_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
𝓣ᶠ(ofState φ * ofState ψ) = ofState φ * ofState ψ := by
lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = ofFieldOpF φ * ofFieldOpF ψ := by
rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton, ← ofFieldOpListF_append,
timeOrderF_ofFieldOpListF]
simp only [List.singleton_append]
rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
simp
lemma timeOrderF_ofState_ofState_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofState ψ * ofState φ := by
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOpF ψ * ofFieldOpF φ := by
rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton,
← ofFieldOpListF_append, timeOrderF_ofFieldOpListF]
simp only [List.singleton_append, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
simp [← ofFieldOpListF_append]
lemma timeOrderF_ofState_ofState_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States}
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States}
(h : ¬ timeOrderRel φ ψ) :
𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofState ψ * ofState φ) := by
rw [timeOrderF_ofState_ofState_not_ordered h]
rw [timeOrderF_ofState_ofState_ordered]
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofFieldOpF ψ * ofFieldOpF φ) := by
rw [timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
rw [timeOrderF_ofFieldOpF_ofFieldOpF_ordered]
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
have hx := IsTotal.total (r := timeOrderRel) ψ φ
simp_all
@ -300,7 +300,7 @@ lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_eq_time
lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
𝓣ᶠ(ofFieldOpListF (φ :: φs)) =
𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
ofState (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
ofFieldOpF (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
rw [timeOrderF_ofFieldOpListF, timeOrderList_eq_maxTimeField_timeOrderList]
rw [ofFieldOpListF_cons, timeOrderF_ofFieldOpListF]
simp only [instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul]
@ -316,7 +316,7 @@ lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.
𝓣ᶠ(ofFieldOpListF (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
(Finset.filter (fun x =>
(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
ofState (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
ofFieldOpF (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
rw [timeOrderF_eq_maxTimeField_mul]
congr 3
apply FieldStatistic.ofList_perm

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

@ -108,7 +108,7 @@ lemma staticContract_of_not_gradingCompliant (φs : List 𝓕.States)
simp only [Finset.univ_eq_attach, Finset.mem_attach]
apply Subtype.eq
simp only [List.get_eq_getElem, ZeroMemClass.coe_zero]
rw [superCommute_anPart_ofState_diff_grade_zero]
rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
simp [ha2]
end WickContraction

6
HepLean/PerturbationTheory/WicksTheorem.lean
View file

@ -247,14 +247,14 @@ 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 [superCommute_anPart_ofState_diff_grade_zero]
rw [superCommute_anPart_ofFieldOpF_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 [superCommute_anPart_ofState_diff_grade_zero]
rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
simp only [zero_mul, smul_zero]
exact hg
exact fun a => hg (id (Eq.symm a))
@ -272,7 +272,7 @@ is equal to the product of
over all `k` in `Option φsΛ.uncontracted`.
The proof of this result primarily depends on
- `crAnF_ofState_mul_normalOrderF_ofStatesList_eq_sum` to rewrite `𝓞.crAnF (φ * 𝓝ᶠ([φsΛ]ᵘᶜ))`
- `crAnF_ofFieldOpF_mul_normalOrderF_ofFieldOpFsList_eq_sum` to rewrite `𝓞.crAnF (φ * 𝓝ᶠ([φsΛ]ᵘᶜ))`
- `wick_term_none_eq_wick_term_cons`
- `wick_term_some_eq_wick_term_optionEraseZ`
-/