refactor: Rename OperatorAlgebra
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jstoobysmith
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commit
c86974a617
13 changed files with 76 additions and 99 deletions
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@ -121,9 +121,9 @@ import HepLean.Meta.TransverseTactics
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
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import HepLean.PerturbationTheory.Algebras.OperatorAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.OperatorAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.OperatorAlgebra.TimeContraction
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.TimeContraction
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import HepLean.PerturbationTheory.Algebras.StateAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.StateAlgebra.TimeOrder
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import HepLean.PerturbationTheory.CreateAnnihilate
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@ -18,7 +18,7 @@ open CrAnAlgebra
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to be isomorphic to the actual operator algebra of a field theory.
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These properties are sufficent to prove certain theorems about the Operator algebra
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in particular Wick's theorem. -/
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structure OperatorAlgebra where
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structure ProtoOperatorAlgebra where
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/-- The algebra representing the operator algebra. -/
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A : Type
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/-- The instance of the type `A` as a semi-ring. -/
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@ -43,9 +43,9 @@ structure OperatorAlgebra where
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(_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
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crAnF (superCommute (ofCrAnState φ) (ofCrAnState φ')) = 0
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namespace OperatorAlgebra
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namespace ProtoOperatorAlgebra
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open FieldStatistic
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variable {𝓕 : FieldSpecification} (𝓞 : 𝓕.OperatorAlgebra)
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variable {𝓕 : FieldSpecification} (𝓞 : 𝓕.ProtoOperatorAlgebra)
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/-- The algebra `𝓞.A` carries the instance of a semi-ring induced via `A_seimRing`. -/
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instance : Semiring 𝓞.A := 𝓞.A_semiRing
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@ -198,5 +198,5 @@ lemma crAnF_superCommute_ofCrAnState_ofStateList_eq_sum (φ : 𝓕.CrAnStates) (
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· congr
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rw [← map_mul, ← ofStateList_append, ← List.eraseIdx_eq_take_drop_succ]
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end OperatorAlgebra
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end ProtoOperatorAlgebra
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end FieldSpecification
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@ -14,8 +14,8 @@ import HepLean.PerturbationTheory.Koszul.KoszulSign
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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namespace OperatorAlgebra
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variable {𝓞 : OperatorAlgebra 𝓕}
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namespace ProtoOperatorAlgebra
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variable {𝓞 : ProtoOperatorAlgebra 𝓕}
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open CrAnAlgebra
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open FieldStatistic
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@ -390,6 +390,6 @@ lemma crAnF_ofState_normalOrder_insert (φ : 𝓕.States) (φs : List 𝓕.State
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rw [← ofStateList_append, ← ofStateList_append]
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simp
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end OperatorAlgebra
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end ProtoOperatorAlgebra
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end FieldSpecification
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@ -3,7 +3,7 @@ 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.Algebras.OperatorAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.StateAlgebra.TimeOrder
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/-!
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@ -19,9 +19,9 @@ variable {𝓕 : FieldSpecification}
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open CrAnAlgebra
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noncomputable section
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namespace OperatorAlgebra
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namespace ProtoOperatorAlgebra
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variable (𝓞 : 𝓕.OperatorAlgebra)
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variable (𝓞 : 𝓕.ProtoOperatorAlgebra)
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open FieldStatistic
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/-- The time contraction of two States as an element of `𝓞.A` defined
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@ -86,7 +86,7 @@ lemma timeContract_zero_of_diff_grade (φ ψ : 𝓕.States) (h : (𝓕 |>ₛ φ)
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have ht := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
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simp_all
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end OperatorAlgebra
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end ProtoOperatorAlgebra
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end
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end FieldSpecification
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@ -3,7 +3,7 @@ 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.Algebras.OperatorAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.Basic
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import HepLean.PerturbationTheory.Koszul.KoszulSign
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/-!
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@ -3,7 +3,7 @@ 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.Algebras.OperatorAlgebra.Basic
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.Basic
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import HepLean.PerturbationTheory.Koszul.KoszulSign
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import HepLean.PerturbationTheory.FieldSpecification.Filters
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/-!
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@ -3,7 +3,7 @@ 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.Algebras.OperatorAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.NormalOrder
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import HepLean.Mathematics.List.InsertIdx
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/-!
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@ -5,7 +5,7 @@ Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.WickContraction.Uncontracted
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import HepLean.PerturbationTheory.Algebras.StateAlgebra.TimeOrder
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import HepLean.PerturbationTheory.Algebras.OperatorAlgebra.TimeContraction
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import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.TimeContraction
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import HepLean.PerturbationTheory.WickContraction.InsertList
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/-!
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@ -19,6 +19,7 @@ namespace WickContraction
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variable {n : ℕ} (c : WickContraction n)
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open HepLean.List
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open FieldStatistic
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open Nat
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/-- A contraction is full if there are no uncontracted fields, i.e. the finite set
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of uncontracted fields is empty. -/
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@ -46,8 +47,6 @@ 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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open Nat
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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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@ -85,8 +85,7 @@ lemma signFinset_insertList_none (φ : 𝓕.States) (φs : List 𝓕.States)
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lemma stat_ofFinset_of_insertListLiftFinset (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (a : Finset (Fin φs.length)) :
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(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get, insertListLiftFinset φ i a⟩) =
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𝓕 |>ₛ ⟨φs.get, a⟩ := by
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(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get, insertListLiftFinset φ i a⟩) = 𝓕 |>ₛ ⟨φs.get, a⟩ := by
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simp only [ofFinset, Nat.succ_eq_add_one]
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congr 1
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rw [get_eq_insertIdx_succAbove φ _ i]
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@ -823,8 +822,7 @@ lemma stat_signFinset_insert_some_self_fst
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simpa [Option.get_map] using hy2
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lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : c.uncontracted) :
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
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(signFinset (c.insertList φ φs i (some j))
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(finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
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@ -832,8 +830,7 @@ lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.States) (φs : List 𝓕.S
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𝓕 |>ₛ ⟨φs.get,
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(Finset.univ.filter (fun x => j < x ∧ i.succAbove x < i ∧ ((c.getDual? x = none) ∨
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∀ (h : (c.getDual? x).isSome), j < ((c.getDual? x).get h))))⟩ := by
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rw [get_eq_insertIdx_succAbove φ _ i]
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rw [ofFinset_finset_map]
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rw [get_eq_insertIdx_succAbove φ _ i, ofFinset_finset_map]
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swap
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refine
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(Equiv.comp_injective i.succAbove (finCongr (Eq.symm (insertIdx_length_fin φ φs i)))).mpr ?hi.a
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@ -905,55 +902,43 @@ lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.States) (φs : List 𝓕.S
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exact Fin.succAbove_lt_succAbove_iff.mpr hy2
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lemma signInsertSomeCoef_eq_finset (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : c.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1])) :
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c.signInsertSomeCoef φ φs i j =
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted)
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(hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1])) : c.signInsertSomeCoef φ φs i j =
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if i < i.succAbove j then
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get,
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(Finset.univ.filter (fun x => i < i.succAbove x ∧ x < j ∧ ((c.getDual? x = none) ∨
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∀ (h : (c.getDual? x).isSome), i < i.succAbove ((c.getDual? x).get h))))⟩) else
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get,
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(Finset.univ.filter (fun x => j < x ∧ i.succAbove x < i ∧ ((c.getDual? x = none) ∨
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∀ (h : (c.getDual? x).isSome), j < ((c.getDual? x).get h))))⟩) := by
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rw [signInsertSomeCoef_if]
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rw [stat_signFinset_insert_some_self_snd]
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rw [stat_signFinset_insert_some_self_fst]
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get,
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(Finset.univ.filter (fun x => i < i.succAbove x ∧ x < j ∧ ((c.getDual? x = none) ∨
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∀ (h : (c.getDual? x).isSome), i < i.succAbove ((c.getDual? x).get h))))⟩)
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else
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get,
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(Finset.univ.filter (fun x => j < x ∧ i.succAbove x < i ∧ ((c.getDual? x = none) ∨
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∀ (h : (c.getDual? x).isSome), j < ((c.getDual? x).get h))))⟩) := by
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rw [signInsertSomeCoef_if, stat_signFinset_insert_some_self_snd,
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stat_signFinset_insert_some_self_fst]
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simp [hφj]
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lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
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(hk : i.succAbove k < i)
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(hg : GradingCompliant φs c ∧ 𝓕.statesStatistic φ = 𝓕.statesStatistic φs[k.1]) :
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(hk : i.succAbove k < i) (hg : GradingCompliant φs c ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
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signInsertSome φ φs c i k *
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, c.uncontracted.filter (fun x => x ≤ ↑k)⟩)
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= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter (fun x => i.succAbove x < i)⟩) := by
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rw [signInsertSome, signInsertSomeProd_eq_finset, signInsertSomeCoef_eq_finset]
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rw [if_neg (by omega), ← map_mul, ← map_mul]
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rw [signInsertSome, signInsertSomeProd_eq_finset (hφj := hg.2) (hg := hg.1),
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signInsertSomeCoef_eq_finset (hφj := hg.2), if_neg (by omega), ← map_mul, ← map_mul]
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congr 1
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rw [mul_eq_iff_eq_mul, ofFinset_union_disjoint]
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swap
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· rw [Finset.disjoint_filter]
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· /- Disjointness needed for `ofFinset_union_disjoint`. -/
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rw [Finset.disjoint_filter]
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intro j _ h
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simp only [Nat.succ_eq_add_one, not_lt, not_and, not_forall, not_or, not_le]
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intro h1
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use h1
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omega
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trans 𝓕 |>ₛ ⟨φs.get, (Finset.filter (fun x =>
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(↑k < x ∧ i.succAbove x < i ∧ (c.getDual? x = none ∨
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∀ (h : (c.getDual? x).isSome = true), ↑k < (c.getDual? x).get h))
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∨ ((c.getDual? x).isSome = true ∧
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∀ (h : (c.getDual? x).isSome = true), x < ↑k ∧
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(i.succAbove x < i ∧ i < i.succAbove ((c.getDual? x).get h) ∨
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↑k < (c.getDual? x).get h ∧ ¬i.succAbove x < i))) Finset.univ)⟩
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· congr
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ext j
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simp
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rw [ofFinset_union, ← mul_eq_one_iff, ofFinset_union]
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simp only [Nat.succ_eq_add_one, not_lt]
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apply stat_ofFinset_eq_one_of_gradingCompliant
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· exact hg.1
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/- the getDual? is none case-/
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· intro j hn
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apply stat_ofFinset_eq_one_of_gradingCompliant _ _ _ hg.1
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· /- The `c.getDual? i = none` case for `stat_ofFinset_eq_one_of_gradingCompliant`. -/
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intro j hn
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simp only [uncontracted, Finset.mem_sdiff, Finset.mem_union, Finset.mem_filter, Finset.mem_univ,
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hn, Option.isSome_none, Bool.false_eq_true, IsEmpty.forall_iff, or_self, and_true, or_false,
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true_and, and_self, Finset.mem_inter, not_and, not_lt, Classical.not_imp, not_le, and_imp]
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@ -978,8 +963,8 @@ lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.States) (φs : List
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apply Fin.ne_succAbove i j
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omega
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· exact h1'
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/- the getDual? is some case-/
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· intro j hj
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· /- The `(c.getDual? i).isSome` case for `stat_ofFinset_eq_one_of_gradingCompliant`. -/
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intro j hj
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have hn : ¬ c.getDual? j = none := by exact Option.isSome_iff_ne_none.mp hj
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simp only [uncontracted, Finset.mem_sdiff, Finset.mem_union, Finset.mem_filter, Finset.mem_univ,
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hn, hj, forall_true_left, false_or, true_and, and_false, false_and, Finset.mem_inter,
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@ -1033,28 +1018,24 @@ lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.States) (φs : List
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· simp_all only [Fin.getElem_fin, ne_eq, not_lt, false_and, false_or, or_false, and_self,
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or_true, imp_self]
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omega
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· exact hg.2
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· exact hg.2
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· exact hg.1
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lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
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(hk : ¬ i.succAbove k < i)
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(hg : GradingCompliant φs c ∧ 𝓕.statesStatistic φ = 𝓕.statesStatistic φs[k.1]) :
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(hk : ¬ i.succAbove k < i) (hg : GradingCompliant φs c ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
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signInsertSome φ φs c i k *
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, c.uncontracted.filter (fun x => x < ↑k)⟩)
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= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter (fun x => i.succAbove x < i)⟩) := by
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have hik : i.succAbove ↑k ≠ i := by exact Fin.succAbove_ne i ↑k
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rw [signInsertSome, signInsertSomeProd_eq_finset, signInsertSomeCoef_eq_finset]
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rw [if_pos (by omega), ← map_mul, ← map_mul]
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rw [signInsertSome, signInsertSomeProd_eq_finset (hφj := hg.2) (hg := hg.1),
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signInsertSomeCoef_eq_finset (hφj := hg.2), if_pos (by omega), ← map_mul, ← map_mul]
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congr 1
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rw [mul_eq_iff_eq_mul, ofFinset_union, ofFinset_union]
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apply (mul_eq_one_iff _ _).mp
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rw [ofFinset_union]
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simp only [Nat.succ_eq_add_one, not_lt]
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apply stat_ofFinset_eq_one_of_gradingCompliant
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· exact hg.1
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· intro j hj
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apply stat_ofFinset_eq_one_of_gradingCompliant _ _ _ hg.1
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· /- The `c.getDual? i = none` case for `stat_ofFinset_eq_one_of_gradingCompliant`. -/
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intro j hj
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have hijsucc : i.succAbove j ≠ i := by exact Fin.succAbove_ne i j
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simp only [uncontracted, Finset.mem_sdiff, Finset.mem_union, Finset.mem_filter, Finset.mem_univ,
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hj, Option.isSome_none, Bool.false_eq_true, IsEmpty.forall_iff, or_self, and_true, true_and,
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@ -1072,7 +1053,8 @@ lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.States) (φs : L
|
|||
omega
|
||||
simp only [hij, true_and] at h ⊢
|
||||
omega
|
||||
· intro j hj
|
||||
· /- The `(c.getDual? i).isSome` case for `stat_ofFinset_eq_one_of_gradingCompliant`. -/
|
||||
intro j hj
|
||||
have hn : ¬ c.getDual? j = none := by exact Option.isSome_iff_ne_none.mp hj
|
||||
have hijSuc : i.succAbove j ≠ i := by exact Fin.succAbove_ne i j
|
||||
have hkneqj : ↑k ≠ j := by
|
||||
|
|
@ -1136,8 +1118,5 @@ lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.States) (φs : L
|
|||
simp only [hijn, true_and, hijn', and_false, or_false, or_true, imp_false, not_lt,
|
||||
forall_const]
|
||||
exact fun h => lt_of_le_of_ne h (Fin.succAbove_ne i ((c.getDual? j).get hj))
|
||||
· exact hg.2
|
||||
· exact hg.2
|
||||
· exact hg.1
|
||||
|
||||
end WickContraction
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.PerturbationTheory.WickContraction.Sign
|
||||
import HepLean.PerturbationTheory.Algebras.OperatorAlgebra.TimeContraction
|
||||
import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.TimeContraction
|
||||
/-!
|
||||
|
||||
# Time contractions
|
||||
|
|
@ -27,14 +27,14 @@ open HepLean.List
|
|||
/-- Given a Wick contraction `c` associated with a list `φs`, the
|
||||
product of all time-contractions of pairs of contracted elements in `φs`,
|
||||
as a member of the center of `𝓞.A`. -/
|
||||
noncomputable def timeContract (𝓞 : 𝓕.OperatorAlgebra) {φs : List 𝓕.States}
|
||||
noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List 𝓕.States}
|
||||
(c : WickContraction φs.length) :
|
||||
Subalgebra.center ℂ 𝓞.A :=
|
||||
∏ (a : c.1), ⟨𝓞.timeContract (φs.get (c.fstFieldOfContract a)) (φs.get (c.sndFieldOfContract a)),
|
||||
𝓞.timeContract_mem_center _ _⟩
|
||||
|
||||
@[simp]
|
||||
lemma timeContract_insertList_none (𝓞 : 𝓕.OperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
lemma timeContract_insertList_none (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(c : WickContraction φs.length) (i : Fin φs.length.succ) :
|
||||
(c.insertList φ φs i none).timeContract 𝓞 = c.timeContract 𝓞 := by
|
||||
rw [timeContract, insertList_none_prod_contractions]
|
||||
|
|
@ -42,7 +42,7 @@ lemma timeContract_insertList_none (𝓞 : 𝓕.OperatorAlgebra) (φ : 𝓕.Stat
|
|||
ext a
|
||||
simp
|
||||
|
||||
lemma timeConract_insertList_some (𝓞 : 𝓕.OperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
lemma timeConract_insertList_some (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
|
||||
(c.insertList φ φs i (some j)).timeContract 𝓞 =
|
||||
(if i < i.succAbove j then
|
||||
|
|
@ -62,7 +62,7 @@ lemma timeConract_insertList_some (𝓞 : 𝓕.OperatorAlgebra) (φ : 𝓕.State
|
|||
open FieldStatistic
|
||||
|
||||
lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_lt
|
||||
(𝓞 : 𝓕.OperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
|
||||
(ht : 𝓕.timeOrderRel φ φs[k.1]) (hik : i < i.succAbove k) :
|
||||
(c.insertList φ φs i (some k)).timeContract 𝓞 =
|
||||
|
|
@ -71,7 +71,7 @@ lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_lt
|
|||
((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞) := by
|
||||
rw [timeConract_insertList_some]
|
||||
simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
|
||||
OperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
|
||||
ProtoOperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
|
||||
Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply, Fin.coe_cast,
|
||||
List.getElem_map, uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem,
|
||||
Algebra.smul_mul_assoc]
|
||||
|
|
@ -97,7 +97,7 @@ lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_lt
|
|||
· exact ht
|
||||
|
||||
lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_not_lt
|
||||
(𝓞 : 𝓕.OperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
|
||||
(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
|
||||
(ht : ¬ 𝓕.timeOrderRel φs[k.1] φ) (hik : ¬ i < i.succAbove k) :
|
||||
(c.insertList φ φs i (some k)).timeContract 𝓞 =
|
||||
|
|
@ -106,7 +106,7 @@ lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_not_lt
|
|||
((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞) := by
|
||||
rw [timeConract_insertList_some]
|
||||
simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
|
||||
OperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
|
||||
ProtoOperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
|
||||
Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply, Fin.coe_cast,
|
||||
List.getElem_map, uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem,
|
||||
Algebra.smul_mul_assoc]
|
||||
|
|
@ -148,7 +148,7 @@ lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_not_lt
|
|||
simp_all only [Fin.getElem_fin, Nat.succ_eq_add_one, not_lt, false_or]
|
||||
exact ht
|
||||
|
||||
lemma timeContract_of_not_gradingCompliant (𝓞 : 𝓕.OperatorAlgebra) (φs : List 𝓕.States)
|
||||
lemma timeContract_of_not_gradingCompliant (𝓞 : 𝓕.ProtoOperatorAlgebra) (φs : List 𝓕.States)
|
||||
(c : WickContraction φs.length) (h : ¬ GradingCompliant φs c) :
|
||||
c.timeContract 𝓞 = 0 := by
|
||||
rw [timeContract]
|
||||
|
|
@ -159,7 +159,7 @@ lemma timeContract_of_not_gradingCompliant (𝓞 : 𝓕.OperatorAlgebra) (φs :
|
|||
simp only [Finset.univ_eq_attach, Finset.mem_attach]
|
||||
apply Subtype.eq
|
||||
simp only [List.get_eq_getElem, ZeroMemClass.coe_zero]
|
||||
rw [OperatorAlgebra.timeContract_zero_of_diff_grade]
|
||||
rw [ProtoOperatorAlgebra.timeContract_zero_of_diff_grade]
|
||||
simp [ha2]
|
||||
|
||||
end WickContraction
|
||||
|
|
|
|||
|
|
@ -497,19 +497,18 @@ lemma orderedInsert_succAboveEmb_uncontractedList_eq_insertIdx (c : WickContract
|
|||
(List.orderedInsert (· ≤ ·) i (List.map i.succAboveEmb c.uncontractedList)) =
|
||||
(List.map i.succAboveEmb c.uncontractedList).insertIdx (uncontractedListOrderPos c i) i := by
|
||||
rw [orderedInsert_eq_insertIdx_of_fin_list_sorted]
|
||||
swap
|
||||
exact uncontractedList_succAboveEmb_sorted c i
|
||||
congr 1
|
||||
simp only [Nat.succ_eq_add_one, Fin.val_fin_lt, Fin.coe_succAboveEmb, uncontractedListOrderPos]
|
||||
rw [List.filter_map]
|
||||
simp only [List.length_map]
|
||||
congr
|
||||
funext x
|
||||
simp only [Function.comp_apply, Fin.succAbove, decide_eq_decide]
|
||||
split
|
||||
simp only [Fin.lt_def, Fin.coe_castSucc]
|
||||
rename_i h
|
||||
simp_all only [Fin.lt_def, Fin.coe_castSucc, not_lt, Fin.val_succ]
|
||||
omega
|
||||
· congr 1
|
||||
simp only [Nat.succ_eq_add_one, Fin.val_fin_lt, Fin.coe_succAboveEmb, uncontractedListOrderPos]
|
||||
rw [List.filter_map]
|
||||
simp only [List.length_map]
|
||||
congr
|
||||
funext x
|
||||
simp only [Function.comp_apply, Fin.succAbove, decide_eq_decide]
|
||||
split
|
||||
· simp only [Fin.lt_def, Fin.coe_castSucc]
|
||||
· rename_i h
|
||||
simp_all only [Fin.lt_def, Fin.coe_castSucc, not_lt, Fin.val_succ]
|
||||
omega
|
||||
· exact uncontractedList_succAboveEmb_sorted c i
|
||||
|
||||
end WickContraction
|
||||
|
|
|
|||
|
|
@ -16,10 +16,10 @@ Wick's theorem is related to Isserlis' theorem in mathematics.
|
|||
-/
|
||||
|
||||
namespace FieldSpecification
|
||||
variable {𝓕 : FieldSpecification} {𝓞 : 𝓕.OperatorAlgebra}
|
||||
variable {𝓕 : FieldSpecification} {𝓞 : 𝓕.ProtoOperatorAlgebra}
|
||||
open CrAnAlgebra
|
||||
open StateAlgebra
|
||||
open OperatorAlgebra
|
||||
open ProtoOperatorAlgebra
|
||||
open HepLean.List
|
||||
open WickContraction
|
||||
open FieldStatistic
|
||||
|
|
|
|||
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Reference in a new issue