refactor: Lint
This commit is contained in:
jstoobysmith
parent
fca3f02eca
commit
c8e9c285a3
8 changed files with 229 additions and 194 deletions
|
|
@ -21,11 +21,11 @@ open EqTimeOnly
|
|||
|
||||
lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.States) :
|
||||
timeOrder (ofFieldOpList φs) = ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs)}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)):= by
|
||||
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 : WickContraction φs.length // φsΛ.EqTimeOnly} ⊕
|
||||
{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,
|
||||
|
|
@ -43,22 +43,23 @@ lemma timeOrder_ofFieldOpList_eqTimeOnly (φs : List 𝓕.States) :
|
|||
exact x.2
|
||||
|
||||
lemma timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.States) :
|
||||
timeOrder (ofFieldOpList φs) = 𝓣(𝓝(ofFieldOpList φs)) +
|
||||
timeOrder (ofFieldOpList φs) = 𝓣(𝓝(ofFieldOpList φs)) +
|
||||
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
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
|
||||
(Equiv.sumCompl _).symm
|
||||
rw [timeOrder_ofFieldOpList_eqTimeOnly, ← e1.symm.sum_comp]
|
||||
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 : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} // 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 }
|
||||
simp [a.2], right_inv a := by simp}
|
||||
rw [← e2.symm.sum_comp]
|
||||
simp [e2, sign_empty]
|
||||
· let e2 : { φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly} // ¬ φsΛ.1 = empty } ≃
|
||||
|
|
@ -69,22 +70,22 @@ lemma timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.States) :
|
|||
rfl
|
||||
|
||||
lemma normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty (φs : List 𝓕.States) :
|
||||
𝓣(𝓝(ofFieldOpList φs)) = 𝓣(ofFieldOpList φs) -
|
||||
𝓣(𝓝(ofFieldOpList φs)) = 𝓣(ofFieldOpList φs) -
|
||||
∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
rw [timeOrder_ofFieldOpList_eq_eqTimeOnly_empty]
|
||||
simp
|
||||
|
||||
lemma normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime (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 : WickContraction φs.length // HaveEqTime φsΛ}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
|
||||
- ∑ (φsΛ : {φsΛ // φsΛ.EqTimeOnly (φs := φs) ∧ φsΛ ≠ empty}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) := by
|
||||
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 : WickContraction φs.length // HaveEqTime φsΛ}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
|
||||
- ∑ (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
|
||||
let e1 : WickContraction φs.length ≃ {φsΛ // HaveEqTime φsΛ} ⊕ {φsΛ // ¬ HaveEqTime φsΛ} := by
|
||||
exact (Equiv.sumCompl HaveEqTime).symm
|
||||
rw [← e1.symm.sum_comp]
|
||||
simp only [Equiv.symm_symm, Algebra.smul_mul_assoc, Fintype.sum_sum_type,
|
||||
|
|
@ -93,8 +94,9 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime (φs
|
|||
|
||||
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 := φs) ∧ φsΛ ≠ empty}), (sign φs ↑φsΛ • (φsΛ.1).timeContract *
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)) =
|
||||
∑ (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
|
||||
|
|
@ -104,7 +106,7 @@ lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.States) :
|
|||
rw [sum_haveEqTime]
|
||||
congr
|
||||
funext φsΛ
|
||||
simp only [ f]
|
||||
simp only [f]
|
||||
conv_lhs =>
|
||||
enter [2, φsucΛ]
|
||||
enter [1]
|
||||
|
|
@ -119,15 +121,14 @@ lemma haveEqTime_wick_sum_eq_split (φs : List 𝓕.States) :
|
|||
congr 1
|
||||
rw [@join_uncontractedListGet]
|
||||
|
||||
|
||||
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.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))
|
||||
+ ∑ (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) -
|
||||
𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))) := by
|
||||
𝓣(𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ))) := by
|
||||
rw [normalOrder_timeOrder_ofFieldOpList_eq_haveEqTime_sum_not_haveEqTime]
|
||||
rw [add_sub_assoc]
|
||||
congr 1
|
||||
|
|
@ -139,8 +140,9 @@ lemma normalOrder_timeOrder_ofFieldOpList_eq_not_haveEqTime_sub_inductive (φs :
|
|||
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}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ) := by
|
||||
lemma wicks_theorem_normal_order_empty : 𝓣(𝓝(ofFieldOpList [])) =
|
||||
∑ (φsΛ : {φsΛ : WickContraction ([] : List 𝓕.States).length // ¬ HaveEqTime φsΛ}),
|
||||
φsΛ.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ) := by
|
||||
let e2 : {φsΛ : WickContraction ([] : List 𝓕.States).length // ¬ HaveEqTime φsΛ} ≃ Unit :=
|
||||
{
|
||||
toFun := fun x => (),
|
||||
|
|
@ -173,11 +175,11 @@ lemma wicks_theorem_normal_order_empty : 𝓣(𝓝(ofFieldOpList [])) = ∑ (φs
|
|||
|
||||
theorem wicks_theorem_normal_order : (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.1.sign • φsΛ.1.timeContract.1 * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)
|
||||
| [] => wicks_theorem_normal_order_empty
|
||||
| φ :: φs => by
|
||||
rw [normalOrder_timeOrder_ofFieldOpList_eq_not_haveEqTime_sub_inductive]
|
||||
simp only [ Algebra.smul_mul_assoc, ne_eq, add_right_eq_self]
|
||||
simp only [Algebra.smul_mul_assoc, ne_eq, add_right_eq_self]
|
||||
apply Finset.sum_eq_zero
|
||||
intro φsΛ hφsΛ
|
||||
simp only [smul_eq_zero]
|
||||
|
|
@ -195,6 +197,5 @@ decreasing_by
|
|||
simp_all only [Algebra.smul_mul_assoc, List.length_cons, Finset.mem_univ, gt_iff_lt]
|
||||
omega
|
||||
|
||||
|
||||
end FieldOpAlgebra
|
||||
end FieldSpecification
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue