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refactor: Lint

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jstoobysmith 2025-02-03 05:39:48 +00:00
parent 006e29fd08
commit fca3f02eca
16 changed files with 416 additions and 352 deletions
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14
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/NormalOrder.lean
View file

@ -262,12 +262,12 @@ lemma ofCrAnFieldOpList_eq_normalOrder (φs : List 𝓕.CrAnStates) :
one_smul]
lemma normalOrder_normalOrder_mid (a b c : 𝓕.FieldOpAlgebra) :
𝓝(a * b * c) = 𝓝(a * 𝓝(b) * c) := by
𝓝(a * b * c) = 𝓝(a * 𝓝(b) * c) := by
obtain ⟨a, rfl⟩ := ι_surjective a
obtain ⟨b, rfl⟩ := ι_surjective b
obtain ⟨c, rfl⟩ := ι_surjective c
rw [normalOrder_eq_ι_normalOrderF]
simp [← map_mul]
simp only [← map_mul]
rw [normalOrder_eq_ι_normalOrderF]
rw [normalOrderF_normalOrderF_mid]
rfl
@ -277,7 +277,7 @@ lemma normalOrder_normalOrder_left (a b : 𝓕.FieldOpAlgebra) :
obtain ⟨a, rfl⟩ := ι_surjective a
obtain ⟨b, rfl⟩ := ι_surjective b
rw [normalOrder_eq_ι_normalOrderF]
simp [← map_mul]
simp only [← map_mul]
rw [normalOrder_eq_ι_normalOrderF]
rw [normalOrderF_normalOrderF_left]
rfl
@ -287,7 +287,7 @@ lemma normalOrder_normalOrder_right (a b : 𝓕.FieldOpAlgebra) :
obtain ⟨a, rfl⟩ := ι_surjective a
obtain ⟨b, rfl⟩ := ι_surjective b
rw [normalOrder_eq_ι_normalOrderF]
simp [← map_mul]
simp only [← map_mul]
rw [normalOrder_eq_ι_normalOrderF]
rw [normalOrderF_normalOrderF_right]
rfl
@ -503,7 +503,7 @@ lemma anPart_mul_normalOrder_ofFieldOpList_eq_superCommute_reorder (φ : 𝓕.St
(φs : List 𝓕.States) : anPart φ * 𝓝(ofFieldOpList φs) =
𝓝(anPart φ * ofFieldOpList φs) + [anPart φ, 𝓝(ofFieldOpList φs)]ₛ := by
rw [anPart_mul_normalOrder_ofFieldOpList_eq_superCommute]
simp [instCommGroup.eq_1, map_add, map_smul]
simp only [instCommGroup.eq_1, add_left_inj]
rw [normalOrder_anPart_ofFieldOpList_swap]
/--
@ -562,8 +562,8 @@ lemma ofFieldOpList_normalOrder_insert (φ : 𝓕.States) (φs : List 𝓕.State
rw [hl]
rw [ofFieldOpList_append, ofFieldOpList_append]
rw [ofFieldOpList_mul_ofFieldOpList_eq_superCommute, add_mul]
simp [instCommGroup.eq_1, Nat.succ_eq_add_one, ofList_singleton, Algebra.smul_mul_assoc,
map_add, map_smul, add_zero, smul_smul,
simp only [instCommGroup.eq_1, Nat.succ_eq_add_one, ofList_singleton, Algebra.smul_mul_assoc,
map_add, map_smul, normalOrder_superCommute_left_eq_zero, add_zero, smul_smul,
exchangeSign_mul_self_swap, one_smul]
rw [← ofFieldOpList_append, ← ofFieldOpList_append]
simp

21
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/StaticWickTheorem.lean
View file

@ -34,15 +34,15 @@ theorem static_wick_theorem : (φs : List 𝓕.States) →
| φ :: φs => by
rw [ofFieldOpList_cons]
rw [static_wick_theorem φs]
rw [show (φ :: φs) = φs.insertIdx (⟨0, Nat.zero_lt_succ φs.length⟩ : Fin φs.length.succ) φ
rw [show (φ :: φs) = φs.insertIdx (⟨0, Nat.zero_lt_succ φs.length⟩ : Fin φs.length.succ) φ
from rfl]
conv_rhs => rw [insertLift_sum ]
conv_rhs => rw [insertLift_sum]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro c _
trans (sign φs c • ↑c.staticContract * (ofFieldOp φ * normalOrder (ofFieldOpList [c]ᵘᶜ)))
trans (sign φs c • ↑c.staticContract * (ofFieldOp φ * normalOrder (ofFieldOpList [c]ᵘᶜ)))
· have ht := Subalgebra.mem_center_iff.mp (Subalgebra.smul_mem (Subalgebra.center ℂ _)
(c.staticContract).2 c.sign )
(c.staticContract).2 c.sign)
conv_rhs => rw [← mul_assoc, ← ht]
simp [mul_assoc]
rw [ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum]
@ -61,7 +61,7 @@ theorem static_wick_theorem : (φs : List 𝓕.States) →
simp only [Algebra.smul_mul_assoc, Nat.succ_eq_add_one, Fin.zero_eta, Fin.val_zero,
List.insertIdx_zero]
rw [normalOrder_uncontracted_some]
simp [← mul_assoc]
simp only [← mul_assoc]
rw [← smul_mul_assoc]
conv_rhs => rw [← smul_mul_assoc]
congr 1
@ -69,7 +69,7 @@ theorem static_wick_theorem : (φs : List 𝓕.States) →
swap
· simp
rw [smul_smul]
by_cases hn : GradingCompliant φs c ∧ (𝓕|>ₛφ) = (𝓕|>ₛ φs[n.1])
by_cases hn : GradingCompliant φs c ∧ (𝓕|>ₛφ) = (𝓕|>ₛ φs[n.1])
· congr 1
swap
· have h1 := c.staticContract.2
@ -82,19 +82,20 @@ theorem static_wick_theorem : (φs : List 𝓕.States) →
ofFinset_empty, map_one, one_mul]
simp only [Fin.zero_succAbove, Fin.not_lt_zero, not_false_eq_true]
exact hn
· simp at hn
· simp only [Fin.getElem_fin, not_and] at hn
by_cases h0 : ¬ GradingCompliant φs c
· rw [staticContract_of_not_gradingCompliant]
simp only [ZeroMemClass.coe_zero, zero_mul, smul_zero, instCommGroup.eq_1, mul_zero]
exact h0
· simp_all
have h1 : contractStateAtIndex φ [c]ᵘᶜ
· simp_all only [Finset.mem_univ, not_not, instCommGroup.eq_1, forall_const]
have h1 : contractStateAtIndex φ [c]ᵘᶜ
((uncontractedStatesEquiv φs c) (some n)) = 0 := by
simp only [contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply,
instCommGroup.eq_1, Fin.coe_cast, Fin.getElem_fin, smul_eq_zero]
right
simp [uncontractedListGet]
simp only [uncontractedListGet, List.getElem_map,
uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem]
rw [superCommute_anPart_ofState_diff_grade_zero]
exact hn
rw [h1]

24
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeContraction.lean
View file

@ -94,7 +94,7 @@ lemma normalOrder_timeContract (φ ψ : 𝓕.States) :
· rw [timeContract_of_timeOrderRel _ _ h]
simp
· rw [timeContract_of_not_timeOrderRel _ _ h]
simp
simp only [instCommGroup.eq_1, map_smul, smul_eq_zero]
have h1 : timeOrderRel ψ φ := by
have ht : timeOrderRel φ ψ ∨ timeOrderRel ψ φ := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
simp_all
@ -103,10 +103,10 @@ lemma normalOrder_timeContract (φ ψ : 𝓕.States) :
lemma timeOrder_timeContract_eq_time_mid {φ ψ : 𝓕.States}
(h1 : timeOrderRel φ ψ) (h2 : timeOrderRel ψ φ) (a b : 𝓕.FieldOpAlgebra) :
𝓣(a * timeContract φ ψ * b) = timeContract φ ψ * 𝓣(a * b):= by
𝓣(a * timeContract φ ψ * b) = timeContract φ ψ * 𝓣(a * b) := by
rw [timeContract_of_timeOrderRel _ _ h1]
rw [ofFieldOp_eq_sum]
simp [Finset.mul_sum, Finset.sum_mul]
simp only [map_sum, Finset.mul_sum, Finset.sum_mul]
congr
funext x
match φ with
@ -115,19 +115,19 @@ lemma timeOrder_timeContract_eq_time_mid {φ ψ : 𝓕.States}
| .position φ =>
simp only [anPart_position, instCommGroup.eq_1]
apply timeOrder_superCommute_eq_time_mid _ _
simp [crAnTimeOrderRel, h1]
simp only [crAnTimeOrderRel, h1]
simp [crAnTimeOrderRel, h2]
| .outAsymp φ =>
simp only [anPart_posAsymp, instCommGroup.eq_1]
apply timeOrder_superCommute_eq_time_mid _ _
simp [crAnTimeOrderRel, h1]
simp only [crAnTimeOrderRel, h1]
simp [crAnTimeOrderRel, h2]
lemma timeOrder_timeContract_eq_time_left {φ ψ : 𝓕.States}
(h1 : timeOrderRel φ ψ) (h2 : timeOrderRel ψ φ) (b : 𝓕.FieldOpAlgebra) :
𝓣(timeContract φ ψ * b) = timeContract φ ψ * 𝓣(b):= by
𝓣(timeContract φ ψ * b) = timeContract φ ψ * 𝓣(b) := by
trans 𝓣(1 * timeContract φ ψ * b)
simp
simp only [one_mul]
rw [timeOrder_timeContract_eq_time_mid h1 h2]
simp
@ -135,11 +135,11 @@ lemma timeOrder_timeContract_neq_time {φ ψ : 𝓕.States}
(h1 : ¬ (timeOrderRel φ ψ ∧ timeOrderRel ψ φ)) :
𝓣(timeContract φ ψ) = 0 := by
by_cases h2 : timeOrderRel φ ψ
· simp_all
· simp_all only [true_and]
rw [timeContract_of_timeOrderRel _ _ h2]
simp
simp only
rw [ofFieldOp_eq_sum]
simp [Finset.mul_sum, Finset.sum_mul]
simp only [map_sum]
apply Finset.sum_eq_zero
intro x hx
match φ with
@ -154,10 +154,10 @@ lemma timeOrder_timeContract_neq_time {φ ψ : 𝓕.States}
apply timeOrder_superCommute_neq_time
simp_all [crAnTimeOrderRel]
· rw [timeContract_of_not_timeOrderRel_expand _ _ h2]
simp
simp only [instCommGroup.eq_1, map_smul, smul_eq_zero]
right
rw [ofFieldOp_eq_sum]
simp [Finset.mul_sum, Finset.sum_mul]
simp only [map_sum]
apply Finset.sum_eq_zero
intro x hx
match ψ with

24
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean
View file

@ -440,15 +440,15 @@ lemma timeOrder_superCommute_eq_time_mid {φ ψ : 𝓕.CrAnStates}
rw [← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF]
rw [ι_timeOrderF_superCommuteF_eq_time]
rfl
simp_all
simp_all
· simp_all
· simp_all
lemma timeOrder_superCommute_eq_time_left {φ ψ : 𝓕.CrAnStates}
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (b : 𝓕.FieldOpAlgebra) :
𝓣([ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * b) =
[ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * 𝓣(b) := by
trans 𝓣(1 * [ofCrAnFieldOp φ, ofCrAnFieldOp ψ]ₛ * b)
simp
simp only [one_mul]
rw [timeOrder_superCommute_eq_time_mid hφψ hψφ]
simp
@ -458,16 +458,16 @@ lemma timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
rw [ofCrAnFieldOp, ofCrAnFieldOp]
rw [superCommute_eq_ι_superCommuteF]
rw [timeOrder_eq_ι_timeOrderF]
trans ι (timeOrderF (1 * (superCommuteF (ofCrAnState φ)) (ofCrAnState ψ) * 1))
simp
rw [ι_timeOrderF_superCommuteF_neq_time ]
trans ι (timeOrderF (1 * (superCommuteF (ofCrAnState φ)) (ofCrAnState ψ) * 1))
simp only [one_mul, mul_one]
rw [ι_timeOrderF_superCommuteF_neq_time]
exact hφψ
lemma timeOrder_superCommute_anPart_ofFieldOp_neq_time {φ ψ : 𝓕.States}
(hφψ : ¬ (timeOrderRel φ ψ ∧ timeOrderRel ψ φ)) :
𝓣([anPart φ,ofFieldOp ψ]ₛ) = 0 := by
rw [ofFieldOp_eq_sum]
simp
simp only [map_sum]
apply Finset.sum_eq_zero
intro a ha
match φ with
@ -483,24 +483,24 @@ lemma timeOrder_superCommute_anPart_ofFieldOp_neq_time {φ ψ : 𝓕.States}
simp_all [crAnTimeOrderRel]
lemma timeOrder_timeOrder_mid (a b c : 𝓕.FieldOpAlgebra) :
𝓣(a * b * c) = 𝓣(a * 𝓣(b) * c):= by
𝓣(a * b * c) = 𝓣(a * 𝓣(b) * c) := by
obtain ⟨a, rfl⟩ := ι_surjective a
obtain ⟨b, rfl⟩ := ι_surjective b
obtain ⟨c, rfl⟩ := ι_surjective c
rw [← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF, timeOrder_eq_ι_timeOrderF,
← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF, timeOrderF_timeOrderF_mid]
← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF, timeOrderF_timeOrderF_mid]
lemma timeOrder_timeOrder_left (b c : 𝓕.FieldOpAlgebra) :
𝓣(b * c) = 𝓣(𝓣(b) * c):= by
𝓣(b * c) = 𝓣(𝓣(b) * c) := by
trans 𝓣(1 * b * c)
simp
simp only [one_mul]
rw [timeOrder_timeOrder_mid]
simp
lemma timeOrder_timeOrder_right (a b : 𝓕.FieldOpAlgebra) :
𝓣(a * b) = 𝓣(a * 𝓣(b)) := by
trans 𝓣(a * b * 1)
simp
simp only [mul_one]
rw [timeOrder_timeOrder_mid]
simp

65
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/WicksTheoremNormal.lean
View file

@ -20,11 +20,13 @@ open WickContraction
open EqTimeOnly
lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.States) :
timeOrder (ofFieldOpList φs) = ∑ (φsΛ : EqTimeOnly φs),
timeOrder (ofFieldOpList φs) = ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs)}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)):= by
rw [static_wick_theorem φs]
let e2 : WickContraction φs.length ≃ {φsΛ // φsΛ ∈ EqTimeOnly φs} ⊕ {φsΛ // ¬ φsΛ ∈ EqTimeOnly φs} :=
(Equiv.sumCompl (Membership.mem (EqTimeOnly φs))).symm
let e2 : WickContraction φs.length ≃
{φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} ⊕
{φsΛ : WickContraction φs.length // ¬ φsΛ.EqTimeOnly} :=
(Equiv.sumCompl _).symm
rw [← e2.symm.sum_comp]
simp only [Equiv.symm_symm, Algebra.smul_mul_assoc, Fintype.sum_sum_type,
Equiv.sumCompl_apply_inl, Equiv.sumCompl_apply_inr, map_add, map_sum, map_smul, e2]
@ -32,31 +34,35 @@ lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.States) :
enter [2, 2, x]
rw [timeOrder_timeOrder_left]
rw [timeOrder_staticContract_of_not_mem _ x.2]
simp
simp only [Finset.univ_eq_attach, zero_mul, map_zero, smul_zero, Finset.sum_const_zero, add_zero]
congr
funext x
rw [staticContract_eq_timeContract]
rw [timeOrder_timeContract_mul_of_mem_left]
rw [staticContract_eq_timeContract_of_eqTimeOnly]
rw [timeOrder_timeContract_mul_of_eqTimeOnly_left]
exact x.2
exact x.2
lemma timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.States) :
timeOrder (ofFieldOpList φs) = 𝓣(𝓝(ofFieldOpList φs)) +
∑ (φsΛ : {φsΛ // φsΛ ∈ EqTimeOnly φs ∧ φsΛ ≠ empty}),
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
let e1 : EqTimeOnly φs ≃ {φsΛ : EqTimeOnly φs // φsΛ.1 = empty} ⊕ {φsΛ : EqTimeOnly φs // ¬ φsΛ.1 = empty} :=
(Equiv.sumCompl fun (a : EqTimeOnly φs) => a.1 = empty).symm
let e1 : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} ≃
{φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} // φsΛ.1 = empty} ⊕
{φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} // ¬ φsΛ.1 = empty} :=
(Equiv.sumCompl _).symm
rw [timeOrder_ofFieldOpList_eqTimeOnly, ← e1.symm.sum_comp]
simp [e1]
simp only [Equiv.symm_symm, Algebra.smul_mul_assoc, Fintype.sum_sum_type,
Equiv.sumCompl_apply_inl, Equiv.sumCompl_apply_inr, ne_eq, e1]
congr 1
· let e2 : { φsΛ : EqTimeOnly φs // φsΛ.1 = empty } ≃ Unit := {
· let e2 : { φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} // φsΛ.1 = empty } ≃ Unit := {
toFun := fun x => (), invFun := fun x => ⟨⟨empty, by simp⟩, rfl⟩,
left_inv a := by
ext
simp [a.2], right_inv a := by simp }
rw [← e2.symm.sum_comp]
simp [e2, sign_empty]
· let e2 : { φsΛ : EqTimeOnly φs // ¬ φsΛ.1 = empty } ≃
{φsΛ // φsΛ ∈ EqTimeOnly φs ∧ φsΛ ≠ empty} := {
· let e2 : { φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} // ¬ φsΛ.1 = empty } ≃
{φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty} := {
toFun := fun x => ⟨x, ⟨x.1.2, x.2⟩⟩, invFun := fun x => ⟨⟨x.1, x.2.1⟩, x.2.2⟩,
left_inv a := by rfl, right_inv a := by rfl }
rw [← e2.symm.sum_comp]
@ -64,7 +70,7 @@ lemma timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.States) :
lemma normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.States) :
𝓣(𝓝(ofFieldOpList φs)) = 𝓣(ofFieldOpList φs) -
∑ (φsΛ : {φsΛ // φsΛ ∈ EqTimeOnly φs ∧ φsΛ ≠ empty}),
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
rw [timeOrder_ofFieldOpList_eq_eqTimeOnly_empty]
simp
@ -74,20 +80,21 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime (φs
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
+ (∑ (φsΛ : {φsΛ : WickContraction φs.length // HaveEqTime φsΛ}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
- ∑ (φsΛ : {φsΛ // φsΛ ∈ EqTimeOnly φs ∧ φsΛ ≠ empty}),
- ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
rw [normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty]
rw [wicks_theorem]
let e1 : WickContraction φs.length ≃ {φsΛ // HaveEqTime φsΛ} ⊕ {φsΛ // ¬ HaveEqTime φsΛ} := by
exact (Equiv.sumCompl HaveEqTime).symm
rw [← e1.symm.sum_comp]
simp [e1]
simp only [Equiv.symm_symm, Algebra.smul_mul_assoc, Fintype.sum_sum_type,
Equiv.sumCompl_apply_inl, Equiv.sumCompl_apply_inr, ne_eq, sub_left_inj, e1]
rw [add_comm]
lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.States) :
(∑ (φsΛ : {φsΛ : WickContraction φs.length // HaveEqTime φsΛ}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) =
∑ (φsΛ : {φsΛ // φsΛ ∈ EqTimeOnly φs ∧ φsΛ ≠ empty}), (sign φs ↑φsΛ • (φsΛ.1).timeContract *
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}), (sign φs ↑φsΛ • (φsΛ.1).timeContract *
∑ φssucΛ : { φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬φssucΛ.HaveEqTime },
sign [φsΛ.1]ᵘᶜ φssucΛ •
(φssucΛ.1).timeContract * normalOrder (ofFieldOpList [φssucΛ.1]ᵘᶜ)) := by
@ -108,7 +115,7 @@ lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.States) :
rw [← Finset.mul_sum]
congr
funext φsΛ'
simp
simp only [ne_eq, Algebra.smul_mul_assoc]
congr 1
rw [@join_uncontractedListGet]
@ -116,7 +123,7 @@ lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.States) :
lemma normalOrder_timeOrder_ofFieldOpList_eq_not_haveEqTime_sub_inductive (φs : List 𝓕.States) :
𝓣(𝓝(ofFieldOpList φs)) = (∑ (φsΛ : {φsΛ : WickContraction φs.length // ¬ HaveEqTime φsΛ}),
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
+ ∑ (φsΛ : {φsΛ // φsΛ ∈ EqTimeOnly φs ∧ φsΛ ≠ empty}),
+ ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
sign φs ↑φsΛ • (φsΛ.1).timeContract *
(∑ φssucΛ : { φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬ φssucΛ.HaveEqTime },
sign [φsΛ.1]ᵘᶜ φssucΛ • (φssucΛ.1).timeContract * normalOrder (ofFieldOpList [φssucΛ.1]ᵘᶜ) -
@ -125,11 +132,11 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_not_haveEqTime_sub_inductive (φs :
rw [add_sub_assoc]
congr 1
rw [haveEqTime_wick_sum_eq_split]
simp
simp only [ne_eq, Algebra.smul_mul_assoc]
rw [← Finset.sum_sub_distrib]
congr 1
funext x
simp
simp only
rw [← smul_sub, ← mul_sub]
lemma wicks_theorem_normal_order_empty : 𝓣(𝓝(ofFieldOpList [])) = ∑ (φsΛ : {φsΛ : WickContraction ([] : List 𝓕.States).length // ¬ HaveEqTime φsΛ}),
@ -140,12 +147,12 @@ lemma wicks_theorem_normal_order_empty : 𝓣(𝓝(ofFieldOpList [])) = ∑ (φs
invFun := fun x => ⟨empty, by simp⟩,
left_inv := by
intro a
simp
simp only [List.length_nil]
apply Subtype.eq
apply Subtype.eq
simp [empty]
simp only [empty]
ext i
simp
simp only [Finset.not_mem_empty, false_iff]
by_contra hn
have h2 := a.1.2.1 i hn
rw [@Finset.card_eq_two] at h2
@ -153,11 +160,13 @@ lemma wicks_theorem_normal_order_empty : 𝓣(𝓝(ofFieldOpList [])) = ∑ (φs
exact Fin.elim0 a,
right_inv := by intro a; simp}
rw [← e2.symm.sum_comp]
simp [e2, sign_empty]
simp only [Finset.univ_unique, PUnit.default_eq_unit, List.length_nil, Equiv.coe_fn_symm_mk,
sign_empty, timeContract_empty, OneMemClass.coe_one, one_smul, uncontractedListGet_empty,
one_mul, Finset.sum_const, Finset.card_singleton, e2]
have h1' : ofFieldOpList (𝓕 := 𝓕) [] = ofCrAnFieldOpList [] := by rfl
rw [h1']
rw [normalOrder_ofCrAnFieldOpList]
simp
simp only [normalOrderSign_nil, normalOrderList_nil, one_smul]
rw [ofCrAnFieldOpList, timeOrder_eq_ι_timeOrderF]
rw [timeOrderF_ofCrAnList]
simp
@ -181,9 +190,9 @@ decreasing_by
simp only [uncontractedListGet, List.length_cons, List.length_map, gt_iff_lt]
rw [uncontractedList_length_eq_card]
have hc := uncontracted_card_eq_iff φsΛ.1
simp [φsΛ.2.2] at hc
simp only [List.length_cons, φsΛ.2.2, iff_false] at hc
have hc' := uncontracted_card_le φsΛ.1
simp_all
simp_all only [Algebra.smul_mul_assoc, List.length_cons, Finset.mem_univ, gt_iff_lt]
omega