refactor: More notes on Wick's thoerem
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jstoobysmith
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6 changed files with 115 additions and 77 deletions
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@ -32,8 +32,19 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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∨ (∃ (φ φ' : 𝓕.CrAnFieldOp) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
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x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
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/-- The algebra spanned by cr and an parts of fields, with appropriate super-commutors
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set to zero. -/
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/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
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of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by elements of the form
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- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`, which (with the other conditions)
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gives us that super-commutors are in the center.
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- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` creation operators. I.e two creation
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operators always super-commute.
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- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` annihilation operators. I.e two
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annihilation operators always super-commute.
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- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` field operators with different statistics.
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I.e. Fermions super-commute with bosons.
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The algebra `𝓕.FieldOpAlgebra` is the most general (in the correct sense) algebra
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satisfying these properties.
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-/
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abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
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namespace FieldOpAlgebra
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@ -469,9 +469,10 @@ lemma ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnFi
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rw [← Finset.sum_mul, ← map_sum, ← map_sum, ← ofFieldOp_eq_sum, ← ofFieldOpList_eq_sum]
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/--
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Within a proto-operator algebra we have that
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`[anPartF φ, 𝓝(φs)] = ∑ i, sᵢ • [anPartF φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ)`
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where `sᵢ` is the exchange sign for `φ` and `φ₀…φᵢ₋₁`.
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The commutor of the annihilation part of a field operator with a normal ordered list of field
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operators can be decomponsed into the sum of the commutators of the annihilation part with each
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element of the list of field operators, i.e.
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`[anPart φ, 𝓝(φ₀…φₙ)]ₛ= ∑ i, 𝓢(φ, φ₀…φᵢ₋₁) • [anPart φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ)`.
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-/
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lemma anPart_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[anPart φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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@ -67,13 +67,13 @@ def statesIsPosition : 𝓕.FieldOp → Bool
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| FieldOp.position _ => true
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| _ => false
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/-- The statistics associated to a state. -/
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/-- The statistics associated to a field operator. -/
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def statesStatistic : 𝓕.FieldOp → FieldStatistic := 𝓕.statistics ∘ 𝓕.fieldOpToField
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/-- The field statistics associated with a state. -/
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/-- The field statistics associated with a field operator. -/
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scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => statesStatistic 𝓕 φ
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/-- The field statistics associated with a list states. -/
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/-- The field statistics associated with a list field operators. -/
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scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => FieldStatistic.ofList
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(statesStatistic 𝓕) φ
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@ -48,6 +48,11 @@ def isFullInvolutionEquiv : {c : WickContraction n // IsFull c} ≃
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left_inv c := by simp
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right_inv f := by simp
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remark card_in_general := "The cardinality of `WickContraction n` in general
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follows OEIS:A000085. I.e.
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1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480,
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10349536, 46206736, 211799312, 997313824,...."
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/-- If `n` is even then the number of full contractions is `(n-1)!!`. -/
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theorem card_of_isfull_even (he : Even n) :
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Fintype.card {c : WickContraction n // IsFull c} = (n - 1)‼ := by
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@ -20,10 +20,9 @@ open HepLean.List
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open FieldStatistic
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lemma signFinset_insertAndContract_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (i1 i2 : Fin φs.length) :
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(φsΛ ↩Λ φ i none).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
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(i.succAbove i1)) (finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove i2)) =
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (i1 i2 : Fin φs.length) :
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(φsΛ ↩Λ φ i none).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
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(i.succAbove i1)) (finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove i2)) =
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if i.succAbove i1 < i ∧ i < i.succAbove i2 then
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Insert.insert (finCongr (insertIdx_length_fin φ φs i).symm i)
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(insertAndContractLiftFinset φ i (φsΛ.signFinset i1 i2))
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@ -214,8 +213,18 @@ lemma signInsertNone_eq_filter_map (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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exact List.nodup_finRange φs.length
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· exact hG
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/-- The change in sign when inserting a field `φ` at `i` into `φsΛ` is equal
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to the sign got by moving `φ` through each field `φ₀…φᵢ₋₁` which has a dual. -/
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/-- The following signs for a grading compliant Wick contraction are equal:
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- The sign `φsΛ.signInsertNone φ φs i` which is given by the following: For each
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contracted pair `{a1, a2}` in `φsΛ` if `a1 < a2`
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such that `i` is within the range `a1 < i < a2` we pick up a sign equal to `𝓢(φ, φs[a2])`.
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- The sign got by moving `φ` through `φ₀…φᵢ₋₁` and only picking up a sign when `φᵢ` has a dual.
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These are equal since: Both ignore uncontracted fields, and for a contracted pair `{a1, a2}`
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with `a1 < a2`
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- if `i < a1 < a2` then we don't pick up a sign from either `φₐ₁` or `φₐ₂`.
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- if `a1 < i < a2` then we pick up a sign from `φₐ₁` cases which is equal to `𝓢(φ, φs[a2])`
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(since `φsΛ` is grading compliant).
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- if `a1 < a2 < i` then we pick up a sign from both `φₐ₁` and `φₐ₂` which cancel each other out.
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-/
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lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
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φsΛ.signInsertNone φ φs i = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter
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@ -198,11 +198,13 @@ lemma signFinset_insertAndContract_some (φ : 𝓕.FieldOp) (φs : List 𝓕.Fie
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simp only [Fin.coe_cast, Option.get_map, Function.comp_apply, Fin.val_fin_lt]
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rw [Fin.succAbove_lt_succAbove_iff]
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/-- Given a Wick contraction `c` associated with a list of states `φs`
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and an `i : Fin φs.length.succ`, the change in sign of the contraction associated with
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inserting `φ` into `φs` at position `i` and contracting it with `j : c.uncontracted`
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coming from contractions other then the `i` and `j` contraction but which
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are effected by this new contraction. -/
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/--
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Given a Wick contraction `φsΛ` the sign defined in the followin way,
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related to inserting a field `φ` at position `i` and contracting it with `j : φsΛ.uncontracted`.
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- For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
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`𝓢(φ, φₐ₂)` if `a₁ < i ≤ a₂` and `a₁ < j`.
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- For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
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`𝓢(φⱼ, φₐ₂)` if `a₁ < j < a₂` and `i < a₁`. -/
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def signInsertSomeProd (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : φsΛ.uncontracted) : ℂ :=
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∏ (a : φsΛ.1),
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@ -216,10 +218,12 @@ def signInsertSomeProd (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : Wi
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else
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1
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/-- Given a Wick contraction `c` associated with a list of states `φs`
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and an `i : Fin φs.length.succ`, the change in sign of the contraction associated with
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inserting `φ` into `φs` at position `i` and contracting it with `j : c.uncontracted`
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coming from putting `i` next to `j`. -/
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/-- Given a Wick contraction `φsΛ` the sign defined in the followin way,
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related to inserting a field `φ` at position `i` and contracting it with `j : φsΛ.uncontracted`.
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- If `j < i`, for each field `φₐ` in `φⱼ₊₁…φᵢ₋₁` without a dual at position `< j`
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the sign `𝓢(φₐ, φᵢ)`.
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- Else, for each field `φₐ` in `φᵢ…φⱼ₋₁` of `φs` without dual at position `< i` the sign
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`𝓢(φₐ, φⱼ)`. -/
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def signInsertSomeCoef (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : φsΛ.uncontracted) : ℂ :=
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let a : (φsΛ ↩Λ φ i (some j)).1 := congrLift (insertIdx_length_fin φ φs i).symm
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@ -239,8 +243,7 @@ def signInsertSome (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickCo
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lemma sign_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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(φsΛ ↩Λ φ i (some j)).sign = (φsΛ.signInsertSome φ φs i j) * φsΛ.sign := by
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rw [sign]
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rw [signInsertSome, signInsertSomeProd, sign, mul_assoc, ← Finset.prod_mul_distrib]
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rw [sign, signInsertSome, signInsertSomeProd, sign, mul_assoc, ← Finset.prod_mul_distrib]
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rw [insertAndContract_some_prod_contractions]
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congr
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funext a
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@ -377,40 +380,6 @@ lemma signInsertSomeProd_eq_prod_fin (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldO
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simp only [hφj, Fin.getElem_fin]
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exact hg
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lemma signInsertSomeProd_eq_list (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1]))
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(hg : GradingCompliant φs φsΛ) :
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φsΛ.signInsertSomeProd φ φs i j =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (List.filter (fun x => (φsΛ.getDual? x).isSome ∧
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∀ (h : (φsΛ.getDual? x).isSome), x < j ∧ (i.succAbove x < i ∧
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i < i.succAbove ((φsΛ.getDual? x).get h)
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∨ j < ((φsΛ.getDual? x).get h) ∧ ¬ i.succAbove x < i))
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(List.finRange φs.length)).map φs.get) := by
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rw [signInsertSomeProd_eq_prod_fin]
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rw [FieldStatistic.ofList_map_eq_finset_prod]
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rw [map_prod]
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congr
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funext x
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split
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· rename_i h
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simp only [Nat.succ_eq_add_one, not_lt, instCommGroup.eq_1, Bool.decide_and,
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Bool.decide_eq_true, List.mem_filter, List.mem_finRange, h, forall_true_left, Bool.decide_or,
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Bool.true_and, Bool.and_eq_true, decide_eq_true_eq, Bool.or_eq_true, true_and,
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Fin.getElem_fin]
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split
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· rename_i h1
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simp [h1]
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· rename_i h1
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simp [h1]
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· rename_i h
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simp [h]
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refine
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List.Nodup.filter _ ?_
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exact List.nodup_finRange φs.length
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simp only [hφj, Fin.getElem_fin]
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exact hg
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lemma signInsertSomeProd_eq_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1]))
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@ -630,10 +599,33 @@ lemma signInsertSomeCoef_eq_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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stat_signFinset_insert_some_self_fst]
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simp [hφj]
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/-- The change in sign when inserting a field `φ` at `i` into `φsΛ` and
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contracting it with `k` (`k < i`) is equal
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to the sign got by moving `φ` through each field `φ₀…φᵢ₋₁`
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multiplied by the sign got moving `φ` through each uncontracted field `φ₀…φₖ`. -/
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/--
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The following two signs are equal for `i.succAbove k < i`. The sign `signInsertSome φ φs φsΛ i k` which is constructed
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as follows:
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1a. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
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`𝓢(φ, φₐ₂)` if `a₁ < i ≤ a₂` and `a₁ < k`.
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1b. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
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`𝓢(φⱼ, φₐ₂)` if `a₁ < k < a₂` and `i < a₁`.
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1c. For each field `φₐ` in `φₖ₊₁…φᵢ₋₁` without a dual at position `< k`
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the sign `𝓢(φₐ, φᵢ)`.
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and the sign constructed as follows:
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2a. For each uncontracted field `φₐ` in `φ₀…φₖ` in `φsΛ` the sign `𝓢(φ, φₐ)`
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2b. For each field in `φₐ` in `φ₀…φᵢ₋₁` the sign `𝓢(φ, φₐ)`.
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The outline of why this is true can be got by considering contributions of fields.
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- `φₐ`, `i ≤ a`. No contributions.
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- `φₖ`, `k -> 2a`, `k -> 2b`
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- `φₐ`, `a ≤ k` uncontracted `a -> 2a`, `a -> 2b`.
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- `φₐ`, `k < a < i` uncontracted `a -> 1c`, `a -> 2b`.
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For contracted fields `{a₁, a₂}` in `φsΛ` with `a₁ < a₂` we have the following cases:
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- `φₐ₁` `φₐ₂` `a₁ < a₂ < k < i`, `a₁ -> 2b`, `a₂ -> 2b`,
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- `φₐ₁` `φₐ₂` `a₁ < k < a₂ < i`, `a₁ -> 2b`, `a₂ -> 2b`,
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- `φₐ₁` `φₐ₂` `a₁ < k < i ≤ a₂`, `a₁ -> 2b`, `a₂ -> 1a`
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- `φₐ₁` `φₐ₂` `k < a₁ < a₂ < i`, `a₁ -> 2b`, `a₂ -> 2b`, `a₁ -> 1c`, `a₂ -> 1c`
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- `φₐ₁` `φₐ₂` `k < a₁ < i ≤ a₂ `,`a₁ -> 2b`, `a₁ -> 1c`
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- `φₐ₁` `φₐ₂` `k < i ≤ a₁ < a₂ `, No contributions.
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-/
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lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
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(hk : i.succAbove k < i) (hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
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@ -669,14 +661,12 @@ lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.FieldOp) (φs : List
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simp only [not_lt] at hkj
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have h2 := h.2 hkj
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apply Fin.ne_succAbove i j
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have hij : i.succAbove j ≤ i.succAbove k.1 :=
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Fin.succAbove_le_succAbove_iff.mpr hkj
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have hij : i.succAbove j ≤ i.succAbove k.1 := Fin.succAbove_le_succAbove_iff.mpr hkj
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omega
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· have h1' := h.1
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rcases h1' with h1' | h1'
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· have hl := h.2 h1'
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have hij : i.succAbove j ≤ i.succAbove k.1 :=
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Fin.succAbove_le_succAbove_iff.mpr h1'
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have hij : i.succAbove j ≤ i.succAbove k.1 := Fin.succAbove_le_succAbove_iff.mpr h1'
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by_contra hn
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apply Fin.ne_succAbove i j
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omega
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@ -737,10 +727,35 @@ lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.FieldOp) (φs : List
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or_true, imp_self]
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omega
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/-- The change in sign when inserting a field `φ` at `i` into `φsΛ` and
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contracting it with `k` (`i < k`) is equal
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to the sign got by moving `φ` through each field `φ₀…φᵢ₋₁`
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multiplied by the sign got moving `φ` through each uncontracted field `φ₀…φₖ₋₁`. -/
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/--
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The following two signs are equal for `i < i.succAbove k`.
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The sign `signInsertSome φ φs φsΛ i k` which is constructed
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as follows:
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1a. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
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`𝓢(φ, φₐ₂)` if `a₁ < i ≤ a₂` and `a₁ < k`.
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1b. For each contracted pair `{a1, a2}` in `φsΛ` with `a1 < a2` the sign
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`𝓢(φⱼ, φₐ₂)` if `a₁ < k < a₂` and `i < a₁`.
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1c. For each field `φₐ` in `φᵢ…φₖ₋₁` of `φs` without dual at position `< i` the sign
|
||||
`𝓢(φₐ, φⱼ)`.
|
||||
and the sign constructed as follows:
|
||||
2a. For each uncontracted field `φₐ` in `φ₀…φₖ₋₁` in `φsΛ` the sign `𝓢(φ, φₐ)`
|
||||
2b. For each field in `φₐ` in `φ₀…φᵢ₋₁` the sign `𝓢(φ, φₐ)`.
|
||||
|
||||
The outline of why this is true can be got by considering contributions of fields.
|
||||
- `φₐ`, `k < a`. No contributions.
|
||||
- `φₖ`, No Contributes
|
||||
- `φₐ`, `a < i` uncontracted `a -> 2a`, `a -> 2b`.
|
||||
- `φₐ`, `i ≤ a < k` uncontracted `a -> 1c`, `a -> 2a`.
|
||||
|
||||
For contracted fields `{a₁, a₂}` in `φsΛ` with `a₁ < a₂` we have the following cases:
|
||||
- `φₐ₁` `φₐ₂` `a₁ < a₂ < i ≤ k`, `a₁ -> 2b`, `a₂ -> 2b`
|
||||
- `φₐ₁` `φₐ₂` `a₁ < i ≤ a₂ < k`, `a₁ -> 2b`, `a₂ -> 1a`
|
||||
- `φₐ₁` `φₐ₂` `a₁ < i ≤ k < a₂`, `a₁ -> 2b`, `a₂ -> 1a`
|
||||
- `φₐ₁` `φₐ₂` `i ≤ a₁ < a₂ < k`, `a₂ -> 1c`, `a₁ -> 1c`
|
||||
- `φₐ₁` `φₐ₂` `i ≤ a₁ < k < a₂ `, `a₁ -> 1c`, `a₁ -> 1b`
|
||||
- `φₐ₁` `φₐ₂` `i ≤ k ≤ a₁ < a₂ `, No contributions
|
||||
-/
|
||||
lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
|
||||
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
|
||||
(hk : ¬ i.succAbove k < i) (hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
|
||||
|
|
@ -804,9 +819,7 @@ lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.FieldOp) (φs :
|
|||
imp_false, not_lt, true_and, implies_true, imp_self, and_true, forall_const, hik,
|
||||
imp_forall_iff_forall]
|
||||
· have hikn : j < k.1 := by omega
|
||||
have hksucc : i.succAbove j < i.succAbove k.1 := by
|
||||
rw [Fin.succAbove_lt_succAbove_iff]
|
||||
omega
|
||||
have hksucc : i.succAbove j < i.succAbove k.1 := Fin.succAbove_lt_succAbove_iff.mpr hikn
|
||||
simp only [hikn, true_and, forall_const, hik, false_and, or_false, IsEmpty.forall_iff,
|
||||
and_true]
|
||||
by_cases hij: i < i.succAbove j
|
||||
|
|
@ -822,9 +835,8 @@ lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.FieldOp) (φs :
|
|||
· apply Or.inl
|
||||
omega
|
||||
· apply Or.inl
|
||||
have hi : i.succAbove k.1 < i.succAbove ((φsΛ.getDual? j).get hj) := by
|
||||
rw [Fin.succAbove_lt_succAbove_iff]
|
||||
omega
|
||||
have hi : i.succAbove k.1 < i.succAbove ((φsΛ.getDual? j).get hj) :=
|
||||
Fin.succAbove_lt_succAbove_iff.mpr h1
|
||||
apply And.intro
|
||||
· apply Or.inr
|
||||
apply And.intro
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue