refactor: Change notation for normal order
This commit is contained in:
jstoobysmith
parent
ec2e1e7df9
commit
835c47dbf8
7 changed files with 144 additions and 108 deletions
|
|
@ -32,18 +32,18 @@ open FieldStatistic
|
|||
|
||||
/--
|
||||
Let `c` be a Wick Contraction for `φs := φ₀φ₁…φₙ`.
|
||||
We have (roughly) `𝓝([φsΛ ↩Λ φ i none]ᵘᶜ) = s • 𝓝(φ :: [φsΛ]ᵘᶜ)`
|
||||
We have (roughly) `𝓝ᶠ([φsΛ ↩Λ φ i none]ᵘᶜ) = s • 𝓝ᶠ(φ :: [φsΛ]ᵘᶜ)`
|
||||
Where `s` is the exchange sign for `φ` and the uncontracted fields in `φ₀φ₁…φᵢ₋₁`.
|
||||
-/
|
||||
lemma normalOrder_uncontracted_none (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) :
|
||||
𝓞.crAnF (𝓝([φsΛ ↩Λ φ i none]ᵘᶜ))
|
||||
𝓞.crAnF (𝓝ᶠ([φsΛ ↩Λ φ i none]ᵘᶜ))
|
||||
= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, φsΛ.uncontracted.filter (fun x => i.succAbove x < i)⟩) •
|
||||
𝓞.crAnF 𝓝(φ :: [φsΛ]ᵘᶜ) := by
|
||||
𝓞.crAnF 𝓝ᶠ(φ :: [φsΛ]ᵘᶜ) := by
|
||||
simp only [Nat.succ_eq_add_one, instCommGroup.eq_1]
|
||||
rw [crAnF_ofState_normalOrder_insert φ [φsΛ]ᵘᶜ
|
||||
⟨(φsΛ.uncontractedListOrderPos i), by simp [uncontractedListGet]⟩, smul_smul]
|
||||
trans (1 : ℂ) • 𝓞.crAnF (𝓝(ofStateList [φsΛ ↩Λ φ i none]ᵘᶜ))
|
||||
trans (1 : ℂ) • 𝓞.crAnF (𝓝ᶠ(ofStateList [φsΛ ↩Λ φ i none]ᵘᶜ))
|
||||
· simp
|
||||
congr 1
|
||||
simp only [instCommGroup.eq_1, uncontractedListGet]
|
||||
|
|
@ -107,8 +107,8 @@ where `k'` is the position in `c.uncontractedList` corresponding to `k`.
|
|||
-/
|
||||
lemma normalOrder_uncontracted_some (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted) :
|
||||
𝓞.crAnF 𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)
|
||||
= 𝓞.crAnF 𝓝((optionEraseZ [φsΛ]ᵘᶜ φ ((uncontractedStatesEquiv φs φsΛ) k))) := by
|
||||
𝓞.crAnF 𝓝ᶠ([φsΛ ↩Λ φ i (some k)]ᵘᶜ)
|
||||
= 𝓞.crAnF 𝓝ᶠ((optionEraseZ [φsΛ]ᵘᶜ φ ((uncontractedStatesEquiv φs φsΛ) k))) := by
|
||||
simp only [Nat.succ_eq_add_one, insertAndContract, optionEraseZ, uncontractedStatesEquiv,
|
||||
Equiv.optionCongr_apply, Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply,
|
||||
Fin.coe_cast, uncontractedListGet]
|
||||
|
|
@ -127,24 +127,24 @@ lemma normalOrder_uncontracted_some (φ : 𝓕.States) (φs : List 𝓕.States)
|
|||
|
||||
remark wick_term_terminology := "
|
||||
Let `φsΛ` be a Wick contraction. We informally call the term
|
||||
`(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝([φsΛ]ᵘᶜ)` the Wick term
|
||||
`(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝ᶠ([φsΛ]ᵘᶜ)` the Wick term
|
||||
associated with `φsΛ`. We do not make this a fully-fledge definition, as
|
||||
in most cases we want to consider slight modifications of this term."
|
||||
|
||||
/--
|
||||
Let `φsΛ` be a Wick Contraction for `φs = φ₀φ₁…φₙ`. Then the wick-term of ` (φsΛ ↩Λ φ i none)`
|
||||
|
||||
```(φsΛ ↩Λ φ i none).sign • (φsΛ ↩Λ φ i none).timeContract 𝓞 * 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i none]ᵘᶜ)```
|
||||
```(φsΛ ↩Λ φ i none).sign • (φsΛ ↩Λ φ i none).timeContract 𝓞 * 𝓞.crAnF 𝓝ᶠ([φsΛ ↩Λ φ i none]ᵘᶜ)```
|
||||
|
||||
is equal to
|
||||
|
||||
`s • (φsΛ.sign • φsΛ.timeContract 𝓞 * 𝓞.crAnF 𝓝(φ :: [φsΛ]ᵘᶜ))`
|
||||
`s • (φsΛ.sign • φsΛ.timeContract 𝓞 * 𝓞.crAnF 𝓝ᶠ(φ :: [φsΛ]ᵘᶜ))`
|
||||
|
||||
where `s` is the exchange sign of `φ` and the uncontracted fields in `φ₀φ₁…φᵢ₋₁`.
|
||||
|
||||
The proof of this result relies primarily on:
|
||||
- `normalOrder_uncontracted_none` which replaces `𝓝([φsΛ ↩Λ φ i none]ᵘᶜ)` with
|
||||
`𝓝(φ :: [φsΛ]ᵘᶜ)` up to a sign.
|
||||
- `normalOrder_uncontracted_none` which replaces `𝓝ᶠ([φsΛ ↩Λ φ i none]ᵘᶜ)` with
|
||||
`𝓝ᶠ(φ :: [φsΛ]ᵘᶜ)` up to a sign.
|
||||
- `timeContract_insertAndContract_none` which replaces `(φsΛ ↩Λ φ i none).timeContract 𝓞` with
|
||||
`φsΛ.timeContract 𝓞`.
|
||||
- `sign_insert_none` and `signInsertNone_eq_filterset` which are used to take account of
|
||||
|
|
@ -153,9 +153,9 @@ The proof of this result relies primarily on:
|
|||
lemma wick_term_none_eq_wick_term_cons (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) :
|
||||
(φsΛ ↩Λ φ i none).sign • (φsΛ ↩Λ φ i none).timeContract 𝓞
|
||||
* 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i none]ᵘᶜ) =
|
||||
* 𝓞.crAnF 𝓝ᶠ([φsΛ ↩Λ φ i none]ᵘᶜ) =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (Finset.univ.filter (fun k => i.succAbove k < i))⟩)
|
||||
• (φsΛ.sign • φsΛ.timeContract 𝓞 * 𝓞.crAnF 𝓝(φ :: [φsΛ]ᵘᶜ)) := by
|
||||
• (φsΛ.sign • φsΛ.timeContract 𝓞 * 𝓞.crAnF 𝓝ᶠ(φ :: [φsΛ]ᵘᶜ)) := by
|
||||
by_cases hg : GradingCompliant φs φsΛ
|
||||
· rw [normalOrder_uncontracted_none, sign_insert_none]
|
||||
simp only [Nat.succ_eq_add_one, timeContract_insertAndContract_none, instCommGroup.eq_1,
|
||||
|
|
@ -203,11 +203,11 @@ lemma wick_term_some_eq_wick_term_optionEraseZ (φ : 𝓕.States) (φs : List
|
|||
(hlt : ∀ (k : Fin φs.length), timeOrderRel φ φs[k])
|
||||
(hn : ∀ (k : Fin φs.length), i.succAbove k < i → ¬ timeOrderRel φs[k] φ) :
|
||||
(φsΛ ↩Λ φ i (some k)).sign • (φsΛ ↩Λ φ i (some k)).timeContract 𝓞
|
||||
* 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ) =
|
||||
* 𝓞.crAnF 𝓝ᶠ([φsΛ ↩Λ φ i (some k)]ᵘᶜ) =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (Finset.univ.filter (fun x => i.succAbove x < i))⟩)
|
||||
• (φsΛ.sign • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ
|
||||
((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract 𝓞)
|
||||
* 𝓞.crAnF 𝓝((optionEraseZ [φsΛ]ᵘᶜ φ (uncontractedStatesEquiv φs φsΛ k)))) := by
|
||||
* 𝓞.crAnF 𝓝ᶠ((optionEraseZ [φsΛ]ᵘᶜ φ (uncontractedStatesEquiv φs φsΛ k)))) := by
|
||||
by_cases hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[k.1])
|
||||
· by_cases hk : i.succAbove k < i
|
||||
· rw [WickContraction.timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_not_lt]
|
||||
|
|
@ -263,24 +263,24 @@ lemma wick_term_some_eq_wick_term_optionEraseZ (φ : 𝓕.States) (φs : List
|
|||
|
||||
/--
|
||||
Given a Wick contraction `φsΛ` of `φs = φ₀φ₁…φₙ` and an `i`, we have that
|
||||
`(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF (φ * 𝓝([φsΛ]ᵘᶜ))`
|
||||
`(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF (φ * 𝓝ᶠ([φsΛ]ᵘᶜ))`
|
||||
is equal to the product of
|
||||
- the exchange sign of `φ` and `φ₀φ₁…φᵢ₋₁`,
|
||||
- the sum of `((φsΛ ↩Λ φ i k).sign • (φsΛ ↩Λ φ i k).timeContract 𝓞) * 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i k]ᵘᶜ)`
|
||||
- the sum of `((φsΛ ↩Λ φ i k).sign • (φsΛ ↩Λ φ i k).timeContract 𝓞) * 𝓞.crAnF 𝓝ᶠ([φsΛ ↩Λ φ i k]ᵘᶜ)`
|
||||
over all `k` in `Option φsΛ.uncontracted`.
|
||||
|
||||
The proof of this result primarily depends on
|
||||
- `crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum` to rewrite `𝓞.crAnF (φ * 𝓝([φsΛ]ᵘᶜ))`
|
||||
- `crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum` to rewrite `𝓞.crAnF (φ * 𝓝ᶠ([φsΛ]ᵘᶜ))`
|
||||
- `wick_term_none_eq_wick_term_cons`
|
||||
- `wick_term_some_eq_wick_term_optionEraseZ`
|
||||
-/
|
||||
lemma wick_term_cons_eq_sum_wick_term (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin φs.length.succ)
|
||||
(φsΛ : WickContraction φs.length) (hlt : ∀ (k : Fin φs.length), timeOrderRel φ φs[k])
|
||||
(hn : ∀ (k : Fin φs.length), i.succAbove k < i → ¬timeOrderRel φs[k] φ) :
|
||||
(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF ((CrAnAlgebra.ofState φ) * 𝓝([φsΛ]ᵘᶜ)) =
|
||||
(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF ((CrAnAlgebra.ofState φ) * 𝓝ᶠ([φsΛ]ᵘᶜ)) =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (Finset.univ.filter (fun x => i.succAbove x < i))⟩) •
|
||||
∑ (k : Option φsΛ.uncontracted), ((φsΛ ↩Λ φ i k).sign •
|
||||
(φsΛ ↩Λ φ i k).timeContract 𝓞 * 𝓞.crAnF (𝓝([φsΛ ↩Λ φ i k]ᵘᶜ))) := by
|
||||
(φsΛ ↩Λ φ i k).timeContract 𝓞 * 𝓞.crAnF (𝓝ᶠ([φsΛ ↩Λ φ i k]ᵘᶜ))) := by
|
||||
rw [crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum, Finset.mul_sum,
|
||||
uncontractedStatesEquiv_list_sum, Finset.smul_sum]
|
||||
simp only [instCommGroup.eq_1, Nat.succ_eq_add_one]
|
||||
|
|
@ -318,7 +318,7 @@ lemma wick_term_cons_eq_sum_wick_term (φ : 𝓕.States) (φs : List 𝓕.States
|
|||
/-- Wick's theorem for the empty list. -/
|
||||
lemma wicks_theorem_nil :
|
||||
𝓞.crAnF (𝓣ᶠ(ofStateList [])) = ∑ (nilΛ : WickContraction [].length),
|
||||
(nilΛ.sign • nilΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝([nilΛ]ᵘᶜ) := by
|
||||
(nilΛ.sign • nilΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝ᶠ([nilΛ]ᵘᶜ) := by
|
||||
rw [timeOrder_ofStateList_nil]
|
||||
simp only [map_one, List.length_nil, Algebra.smul_mul_assoc]
|
||||
rw [sum_WickContraction_nil, uncontractedListGet, nil_zero_uncontractedList]
|
||||
|
|
@ -329,9 +329,9 @@ lemma wicks_theorem_nil :
|
|||
|
||||
lemma wicks_theorem_congr {φs φs' : List 𝓕.States} (h : φs = φs') :
|
||||
∑ (φsΛ : WickContraction φs.length), (φsΛ.sign • φsΛ.timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝([φsΛ]ᵘᶜ)
|
||||
𝓞.crAnF 𝓝ᶠ([φsΛ]ᵘᶜ)
|
||||
= ∑ (φs'Λ : WickContraction φs'.length), (φs'Λ.sign • φs'Λ.timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝([φs'Λ]ᵘᶜ) := by
|
||||
𝓞.crAnF 𝓝ᶠ([φs'Λ]ᵘᶜ) := by
|
||||
subst h
|
||||
simp
|
||||
|
||||
|
|
@ -352,7 +352,7 @@ remark wicks_theorem_context := "
|
|||
- The normal-ordering of the uncontracted fields in `c`.
|
||||
-/
|
||||
theorem wicks_theorem : (φs : List 𝓕.States) → 𝓞.crAnF (𝓣ᶠ(ofStateList φs)) =
|
||||
∑ (φsΛ : WickContraction φs.length), (φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝([φsΛ]ᵘᶜ)
|
||||
∑ (φsΛ : WickContraction φs.length), (φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝ᶠ([φsΛ]ᵘᶜ)
|
||||
| [] => wicks_theorem_nil
|
||||
| φ :: φs => by
|
||||
have ih := wicks_theorem (eraseMaxTimeField φ φs)
|
||||
|
|
@ -375,7 +375,7 @@ theorem wicks_theorem : (φs : List 𝓕.States) → 𝓞.crAnF (𝓣ᶠ(ofState
|
|||
(List.insertIdx (↑(maxTimeFieldPosFin φ φs)) (maxTimeField φ φs) (eraseMaxTimeField φ φs))
|
||||
(c ↩Λ (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k) •
|
||||
↑((c ↩Λ (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k).timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝(ofStateList (List.map (List.insertIdx (↑(maxTimeFieldPosFin φ φs))
|
||||
𝓞.crAnF 𝓝ᶠ(ofStateList (List.map (List.insertIdx (↑(maxTimeFieldPosFin φ φs))
|
||||
(maxTimeField φ φs) (eraseMaxTimeField φ φs)).get
|
||||
(c ↩Λ (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k).uncontractedList))
|
||||
swap
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue