refactor: Split files
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9 changed files with 1721 additions and 1481 deletions
194
HepLean/PerturbationTheory/Wick/Koszul/OperatorMap.lean
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194
HepLean/PerturbationTheory/Wick/Koszul/OperatorMap.lean
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/-
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Copyright (c) 2024 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.Wick.Species
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import HepLean.Lorentz.RealVector.Basic
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import HepLean.Mathematics.Fin
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import HepLean.SpaceTime.Basic
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import HepLean.Mathematics.SuperAlgebra.Basic
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import HepLean.Mathematics.List
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import HepLean.Meta.Notes.Basic
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import Init.Data.List.Sort.Basic
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import Mathlib.Data.Fin.Tuple.Take
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import HepLean.PerturbationTheory.Wick.Koszul.SuperCommuteM
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/-!
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# Koszul signs and ordering for lists and algebras
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-/
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namespace Wick
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noncomputable section
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class SuperCommuteCenterMap {A : Type} [Semiring A] [Algebra ℂ A]
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(f : FreeAlgebra ℂ I →ₐ[ℂ] A) : Prop where
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prop : ∀ i j, f (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A
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namespace SuperCommuteCenterMap
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variable {I : Type} {A : Type} [Semiring A] [Algebra ℂ A]
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(f : FreeAlgebra ℂ I →ₐ[ℂ] A) [SuperCommuteCenterMap f]
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lemma ofList_fst (q : I → Fin 2) (i j : I) :
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f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A := by
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have h1 : f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) =
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xa • f (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) := by
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rw [← map_smul]
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congr
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rw [ofList_eq_smul_one, ofList_singleton]
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rw [map_smul]
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rfl
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rw [h1]
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refine Subalgebra.smul_mem (Subalgebra.center ℂ A) ?_ xa
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exact prop i j
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lemma ofList_freeAlgebraMap {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (c : (Σ i, f i)) (x : ℂ)
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{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
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[SuperCommuteCenterMap F] (b : I) :
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F ((superCommute fun i => q i.fst) (ofList [c] x) ((freeAlgebraMap f) (FreeAlgebra.ι ℂ b)))
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∈ Subalgebra.center ℂ A := by
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rw [freeAlgebraMap_ι]
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rw [map_sum, map_sum]
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refine Subalgebra.sum_mem (Subalgebra.center ℂ A) ?h
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intro n hn
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exact ofList_fst F (fun i => q i.fst) c ⟨b, n⟩
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end SuperCommuteCenterMap
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lemma superCommuteTake_superCommuteCenterMap {I : Type} (q : I → Fin 2) (lb : List I) (xa xb : ℂ) (n : ℕ)
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(hn : n < lb.length) {A : Type} [Semiring A] [Algebra ℂ A] (f : FreeAlgebra ℂ I →ₐ[ℂ] A)
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[SuperCommuteCenterMap f] (i : I) :
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f (superCommuteTake q [i] lb xa xb n hn) =
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f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ (lb.get ⟨n, hn⟩)))
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* (superCommuteCoef q [i] (List.take n lb) •
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f (ofList (List.eraseIdx lb n) xb)) := by
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have hn : f ((superCommute q) (ofList [i] xa) (FreeAlgebra.ι ℂ (lb.get ⟨n, hn⟩))) ∈
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Subalgebra.center ℂ A := SuperCommuteCenterMap.ofList_fst f q i (lb.get ⟨n, hn⟩)
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rw [Subalgebra.mem_center_iff] at hn
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rw [superCommuteTake, map_mul, map_mul, map_smul, hn, mul_assoc, smul_mul_assoc,
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← map_mul, ← ofList_pair]
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congr
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· exact Eq.symm (List.eraseIdx_eq_take_drop_succ lb n)
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· exact one_mul xb
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lemma superCommuteTakeM_F {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (c : (Σ i, f i)) (r : List I) (x y : ℂ) (n : ℕ)
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(hn : n < r.length)
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{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
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[SuperCommuteCenterMap F] :
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F (superCommuteTakeM q [c] r x y n hn) = superCommuteCoefM q [c] (List.take n r) •
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(F (superCommute (fun i => q i.1) (ofList [c] x) (freeAlgebraMap f (FreeAlgebra.ι ℂ (r.get ⟨n, hn⟩))))
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* F (ofListM f (List.eraseIdx r n) y)) := by
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rw [superCommuteTakeM]
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rw [map_smul]
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congr
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rw [map_mul, map_mul]
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have h1 : F ((superCommute fun i => q i.fst) (ofList [c] x) ((freeAlgebraMap f) (FreeAlgebra.ι ℂ (r.get ⟨n, hn⟩))))
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∈ Subalgebra.center ℂ A :=
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SuperCommuteCenterMap.ofList_freeAlgebraMap q c x F (r.get ⟨n, hn⟩)
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rw [Subalgebra.mem_center_iff] at h1
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rw [h1, mul_assoc, ← map_mul]
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congr
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rw [ofListM, ofListM, ofListM, ← map_mul]
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congr
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rw [← ofList_pair, one_mul]
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congr
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exact Eq.symm (List.eraseIdx_eq_take_drop_succ r n)
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lemma koszulOrder_superCommuteM_le {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (r : List I) (x : ℂ)
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(le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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(i : (Σ i, f i)) (hi : ∀ j, le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap F] :
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F (koszulOrder le1 (fun i => q i.fst)
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(superCommute (fun i => q i.1) (FreeAlgebra.ι ℂ i) (ofListM f r x))) = 0 := by
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sorry
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lemma koszulOrder_of_le_all {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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(i : (Σ i, f i)) (hi : ∀ j, le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap F] :
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F (koszulOrder le1 (fun i => q i.fst)
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(ofListM f r x * FreeAlgebra.ι ℂ i))
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= superCommuteCoefM q [i] r • F (koszulOrder le1 (fun i => q i.fst)
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(FreeAlgebra.ι ℂ i * ofListM f r x)) := by
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conv_lhs =>
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rhs
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rhs
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rw [← ofList_singleton]
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rw [ofListM_ofList_superCommute q]
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rw [map_sub]
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rw [sub_eq_add_neg]
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rw [map_add]
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conv_lhs =>
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rhs
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rhs
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rw [map_smul]
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rw [← neg_smul]
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rw [map_smul, map_smul, map_smul]
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rw [ofList_singleton, koszulOrder_superCommuteM_le]
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· simp
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· exact fun j => hi j
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lemma le_all_mul_koszulOrder {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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(i : (Σ i, f i)) (hi : ∀ j, le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder le1 (fun i => q i.fst)
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(ofListM f r x)) = F ((koszulOrder le1 fun i => q i.fst) (FreeAlgebra.ι ℂ i * ofListM f r x)) +
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F (((superCommute fun i => q i.fst) (ofList [i] 1))
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((koszulOrder le1 fun i => q i.fst) (ofListM f r x))) := by
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sorry
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/-
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rw [map_smul]
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rw [Algebra.mul_smul_comm, map_smul]
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change koszulSign le1 q r • F (FreeAlgebra.ι ℂ i * (ofListM f (List.insertionSort le1 r) x)) = _
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rw [← ofList_singleton]
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rw [ofList_ofListM_superCommute q]
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rw [map_add]
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rw [smul_add]
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rw [← map_smul]
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conv_lhs =>
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lhs
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rhs
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rw [← Algebra.smul_mul_assoc]
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rw [smul_smul, mul_comm, ← smul_smul]
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rw [ ofListM, ← map_smul, ← koszulOrder_ofList, ← koszulOrder_ofListM, ofList_singleton]
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rw [Algebra.smul_mul_assoc]
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rw [koszulOrder_mul_ge]
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rw [map_smul]
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rw [koszulOrder_of_le_all]
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rw [smul_smul]
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have h1 : (superCommuteCoefM q [i] (List.insertionSort le1 r) * superCommuteCoefM q [i] r) = 1 := by
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simp [superCommuteCoefM]
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have ha (a b : Fin 2): (if a = 1 ∧ b = 1 then
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-if a = 1 ∧ b = 1 then -1 else 1
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else if a = 1 ∧ b = 1 then -1 else (1 : ℂ)) = 1 := by
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fin_cases a <;> fin_cases b
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· rfl
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· rfl
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· rfl
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· simp only [Fin.mk_one, Fin.isValue, and_self, ↓reduceIte, neg_neg]
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exact ha _ _
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rw [h1]
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simp only [one_smul]
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conv_lhs =>
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rhs
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rw [← map_smul, ← map_smul]
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rw [ ofListM, ← map_smul, ← koszulOrder_ofList, ← koszulOrder_ofListM]
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congr
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rw [ofList_singleton]
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· exact fun j => hi j
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· exact fun j => hi j.fst
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-/
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end
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end Wick
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