118 lines
4.5 KiB
Text
118 lines
4.5 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.FieldOpAlgebra.NormalOrder.Lemmas
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import HepLean.PerturbationTheory.WickContraction.InsertAndContract
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/-!
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# Normal ordering with relation to Wick contractions
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-/
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namespace FieldSpecification
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open FieldOpFreeAlgebra
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open HepLean.List
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open FieldStatistic
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open WickContraction
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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/-!
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## Normal order of uncontracted terms within proto-algebra.
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-/
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/--
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Let `c` be a Wick Contraction for `φs := φ₀φ₁…φₙ`.
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We have (roughly) `𝓝ᶠ([φsΛ ↩Λ φ i none]ᵘᶜ) = s • 𝓝ᶠ(φ :: [φsΛ]ᵘᶜ)`
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Where `s` is the exchange sign for `φ` and the uncontracted fields in `φ₀φ₁…φᵢ₋₁`.
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-/
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lemma normalOrder_uncontracted_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) :
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𝓝(ofFieldOpList [φsΛ ↩Λ φ i none]ᵘᶜ)
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= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, φsΛ.uncontracted.filter (fun x => i.succAbove x < i)⟩) •
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𝓝(ofFieldOpList (φ :: [φsΛ]ᵘᶜ)) := by
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simp only [Nat.succ_eq_add_one, instCommGroup.eq_1]
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rw [ofFieldOpList_normalOrder_insert φ [φsΛ]ᵘᶜ
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⟨(φsΛ.uncontractedListOrderPos i), by simp [uncontractedListGet]⟩, smul_smul]
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trans (1 : ℂ) • (𝓝(ofFieldOpList [φsΛ ↩Λ φ i none]ᵘᶜ))
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· simp
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congr 1
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simp only [instCommGroup.eq_1, uncontractedListGet]
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rw [← List.map_take, take_uncontractedListOrderPos_eq_filter]
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have h1 : (𝓕 |>ₛ List.map φs.get (List.filter (fun x => decide (↑x < i.1)) φsΛ.uncontractedList))
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= 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x.val < i.1))⟩ := by
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simp only [Nat.succ_eq_add_one, ofFinset]
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congr
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rw [uncontractedList_eq_sort]
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have hdup : (List.filter (fun x => decide (x.1 < i.1))
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(Finset.sort (fun x1 x2 => x1 ≤ x2) φsΛ.uncontracted)).Nodup := by
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exact List.Nodup.filter _ (Finset.sort_nodup (fun x1 x2 => x1 ≤ x2) φsΛ.uncontracted)
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have hsort : (List.filter (fun x => decide (x.1 < i.1))
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(Finset.sort (fun x1 x2 => x1 ≤ x2) φsΛ.uncontracted)).Sorted (· ≤ ·) := by
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exact List.Sorted.filter _ (Finset.sort_sorted (fun x1 x2 => x1 ≤ x2) φsΛ.uncontracted)
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rw [← (List.toFinset_sort (· ≤ ·) hdup).mpr hsort]
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congr
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ext a
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simp
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rw [h1]
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simp only [Nat.succ_eq_add_one]
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have h2 : (Finset.filter (fun x => x.1 < i.1) φsΛ.uncontracted) =
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(Finset.filter (fun x => i.succAbove x < i) φsΛ.uncontracted) := by
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ext a
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simp only [Nat.succ_eq_add_one, Finset.mem_filter, and_congr_right_iff]
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intro ha
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simp only [Fin.succAbove]
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split
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· apply Iff.intro
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· intro h
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omega
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· intro h
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rename_i h
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rw [Fin.lt_def] at h
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simp only [Fin.coe_castSucc] at h
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omega
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· apply Iff.intro
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· intro h
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rename_i h'
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rw [Fin.lt_def]
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simp only [Fin.val_succ]
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rw [Fin.lt_def] at h'
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simp only [Fin.coe_castSucc, not_lt] at h'
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omega
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· intro h
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rename_i h
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rw [Fin.lt_def] at h
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simp only [Fin.val_succ] at h
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omega
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rw [h2]
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simp only [exchangeSign_mul_self]
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congr
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simp only [Nat.succ_eq_add_one]
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rw [insertAndContract_uncontractedList_none_map]
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/--
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Let `c` be a Wick Contraction for `φ₀φ₁…φₙ`.
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We have (roughly) `N(c ↩Λ φ i k).uncontractedList`
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is equal to `N((c.uncontractedList).eraseIdx k')`
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where `k'` is the position in `c.uncontractedList` corresponding to `k`.
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-/
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lemma normalOrder_uncontracted_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted) :
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𝓝(ofFieldOpList [φsΛ ↩Λ φ i (some k)]ᵘᶜ)
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= 𝓝(ofFieldOpList (optionEraseZ [φsΛ]ᵘᶜ φ ((uncontractedFieldOpEquiv φs φsΛ) k))) := by
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simp only [Nat.succ_eq_add_one, insertAndContract, optionEraseZ, uncontractedFieldOpEquiv,
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Equiv.optionCongr_apply, Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply,
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Fin.coe_cast, uncontractedListGet]
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congr
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rw [congr_uncontractedList]
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erw [uncontractedList_extractEquiv_symm_some]
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simp only [Fin.coe_succAboveEmb, List.map_eraseIdx, List.map_map]
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congr
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conv_rhs => rw [get_eq_insertIdx_succAbove φ φs i]
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end FieldOpAlgebra
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end FieldSpecification
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