refactor: Rename CrAnAlgebra
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jstoobysmith
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16 changed files with 214 additions and 214 deletions
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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.CrAnAlgebra.SuperCommute
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import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.SuperCommute
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import Mathlib.Algebra.RingQuot
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import Mathlib.RingTheory.TwoSidedIdeal.Operations
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/-!
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@ -13,7 +13,7 @@ import Mathlib.RingTheory.TwoSidedIdeal.Operations
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open FieldOpFreeAlgebra
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open HepLean.List
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open FieldStatistic
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@ -21,7 +21,7 @@ variable (𝓕 : FieldSpecification)
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/-- The set contains the super-commutors equal to zero in the operator algebra.
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This contains e.g. the super-commutor of two creation operators. -/
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def fieldOpIdealSet : Set (CrAnAlgebra 𝓕) :=
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def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
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{ x |
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(∃ (φ1 φ2 φ3 : 𝓕.CrAnStates),
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x = [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca)
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@ -39,16 +39,16 @@ abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCo
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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/-- The instance of a setoid on `CrAnAlgebra` from the ideal `TwoSidedIdeal`. -/
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instance : Setoid (CrAnAlgebra 𝓕) := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.toSetoid
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/-- The instance of a setoid on `FieldOpFreeAlgebra` from the ideal `TwoSidedIdeal`. -/
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instance : Setoid (FieldOpFreeAlgebra 𝓕) := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.toSetoid
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lemma equiv_iff_sub_mem_ideal (x y : CrAnAlgebra 𝓕) :
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lemma equiv_iff_sub_mem_ideal (x y : FieldOpFreeAlgebra 𝓕) :
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x ≈ y ↔ x - y ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [← TwoSidedIdeal.rel_iff]
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rfl
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/-- The projection of `CrAnAlgebra` down to `FieldOpAlgebra` as an algebra map. -/
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def ι : CrAnAlgebra 𝓕 →ₐ[ℂ] FieldOpAlgebra 𝓕 where
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/-- The projection of `FieldOpFreeAlgebra` down to `FieldOpAlgebra` as an algebra map. -/
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def ι : FieldOpFreeAlgebra 𝓕 →ₐ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.mk'
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map_one' := by rfl
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map_mul' x y := by rfl
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@ -62,9 +62,9 @@ lemma ι_surjective : Function.Surjective (@ι 𝓕) := by
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use x
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rfl
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lemma ι_apply (x : CrAnAlgebra 𝓕) : ι x = Quotient.mk _ x := rfl
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lemma ι_apply (x : FieldOpFreeAlgebra 𝓕) : ι x = Quotient.mk _ x := rfl
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lemma ι_of_mem_fieldOpIdealSet (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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ι x = 0 := by
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rw [ι_apply]
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change ⟦x⟧ = ⟦0⟧
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@ -157,8 +157,8 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2
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simp [ofCrAnList_singleton, ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_ofCrAnState]
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@[simp]
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lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
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lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnStates)
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(a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
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change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
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have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnState φ1, ofCrAnState φ2]ₛca) = 0 := by
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apply (ofCrAnListBasis.ext fun l ↦ ?_)
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@ -166,8 +166,8 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ
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rw [h1]
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simp
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lemma ι_commute_crAnAlgebra_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
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lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
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(a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
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ι [ofCrAnState φ1, ofCrAnState φ2]ₛca * ι a = 0 := by
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rcases ι_superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_zero φ1 φ2 with h | h
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swap
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@ -182,7 +182,7 @@ lemma ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStat
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rw [Subalgebra.mem_center_iff]
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intro a
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obtain ⟨a, rfl⟩ := ι_surjective a
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have h0 := ι_commute_crAnAlgebra_superCommuteF_ofCrAnState_ofCrAnState φ ψ a
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have h0 := ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnState_ofCrAnState φ ψ a
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trans ι ((superCommuteF (ofCrAnState φ)) (ofCrAnState ψ)) * ι a + 0
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swap
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simp only [add_zero]
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@ -194,7 +194,7 @@ lemma ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStat
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## The kernal of ι
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-/
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lemma ι_eq_zero_iff_mem_ideal (x : CrAnAlgebra 𝓕) :
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lemma ι_eq_zero_iff_mem_ideal (x : FieldOpFreeAlgebra 𝓕) :
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ι x = 0 ↔ x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [ι_apply]
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change ⟦x⟧ = ⟦0⟧ ↔ _
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@ -203,7 +203,7 @@ lemma ι_eq_zero_iff_mem_ideal (x : CrAnAlgebra 𝓕) :
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simp only
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rfl
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lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.bosonicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProj = 0 := by
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have hx' := hx
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simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
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@ -234,7 +234,7 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
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have hx' := hx
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simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
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@ -265,10 +265,10 @@ lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
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lemma bosonicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
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x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
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let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
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let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
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a.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
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change p x hx
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apply AddSubgroup.closure_induction
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@ -401,7 +401,7 @@ lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.spa
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· intro x hx
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simp [p]
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lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
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lemma fermionicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
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x.fermionicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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have hb := bosonicProj_mem_ideal x hx
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rw [← ι_eq_zero_iff_mem_ideal] at hx hb ⊢
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@ -409,7 +409,7 @@ lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.s
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simp only [map_add] at hx
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simp_all
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lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : CrAnAlgebra 𝓕) :
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lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : FieldOpFreeAlgebra 𝓕) :
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ι x = 0 ↔ ι x.bosonicProj.1 = 0 ∧ ι x.fermionicProj.1 = 0 := by
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apply Iff.intro
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· intro h
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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.CrAnAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
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/-!
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@ -12,7 +12,7 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open FieldOpFreeAlgebra
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open HepLean.List
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open FieldStatistic
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@ -52,12 +52,12 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero
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(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
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(a : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * a) = 0 := by
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have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap (ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca) = 0 := by
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apply ofCrAnListBasis.ext
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intro l
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simp only [CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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simp only [FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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AlgHom.toLinearMap_apply, LinearMap.zero_apply]
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exact ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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@ -66,7 +66,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero
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simp
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lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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(a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
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rw [mul_assoc]
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
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@ -75,7 +75,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrA
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([ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
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apply ofCrAnListBasis.ext
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intro l
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simp only [mulLinearMap, CrAnAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
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simp only [mulLinearMap, FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
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Function.comp_apply, LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk,
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AlgHom.toLinearMap_apply, LinearMap.zero_apply]
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rw [← mul_assoc]
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@ -85,7 +85,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrA
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lemma ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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(a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnState φa, ofCrAnList φs]ₛca * b) = 0 := by
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
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rw [Finset.mul_sum, Finset.sum_mul]
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@ -97,7 +97,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa :
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rw [ι_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul]
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
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(φs : List 𝓕.CrAnStates) (a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnState φa]ₛca * b) = 0 := by
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
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simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
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@ -106,7 +106,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa :
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simp
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
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(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.CrAnAlgebra) :
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(φs φs' : List 𝓕.CrAnStates) (a b : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, ofCrAnList φs']ₛca * b) = 0 := by
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rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
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rw [map_sum, map_sum]
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@ -118,7 +118,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
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lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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(φs : List 𝓕.CrAnStates)
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(a b c : 𝓕.CrAnAlgebra) :
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(a b c : 𝓕.FieldOpFreeAlgebra) :
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ι 𝓝ᶠ(a * [ofCrAnList φs, c]ₛca * b) = 0 := by
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) c = 0
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@ -126,7 +126,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) = 0 := by
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apply ofCrAnListBasis.ext
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intro φs'
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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rw [ι_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
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@ -135,14 +135,14 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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@[simp]
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lemma ι_normalOrderF_superCommuteF_eq_zero_mul
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(a b c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
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(a b c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
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change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
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have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
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apply ofCrAnListBasis.ext
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intro φs
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, CrAnAlgebra.ofListBasis_eq_ofList,
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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||||
rw [ι_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul]
|
||||
|
|
@ -150,26 +150,26 @@ lemma ι_normalOrderF_superCommuteF_eq_zero_mul
|
|||
simp
|
||||
|
||||
@[simp]
|
||||
lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.CrAnAlgebra) :
|
||||
lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.FieldOpFreeAlgebra) :
|
||||
ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
|
||||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 b c d]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul_left (a c d : 𝓕.CrAnAlgebra) :
|
||||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul_left (a c d : 𝓕.FieldOpFreeAlgebra) :
|
||||
ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
|
||||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a 1 c d]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.CrAnAlgebra) :
|
||||
lemma ι_normalOrderF_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.FieldOpFreeAlgebra) :
|
||||
ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
|
||||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a (b1 * b2) c d]
|
||||
congr 2
|
||||
noncomm_ring
|
||||
|
||||
@[simp]
|
||||
lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
|
||||
lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
|
||||
rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 1 c d]
|
||||
simp
|
||||
|
||||
|
|
@ -179,10 +179,10 @@ lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.CrAnAlgebra) : ι 𝓝
|
|||
|
||||
-/
|
||||
|
||||
lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
||||
lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
|
||||
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓝ᶠ(a) = 0 := by
|
||||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
|
||||
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
|
||||
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
|
||||
change p a h
|
||||
apply AddSubgroup.closure_induction
|
||||
· intro x hx
|
||||
|
|
@ -211,7 +211,7 @@ lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
|||
· intro x hx
|
||||
simp [p]
|
||||
|
||||
lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
|
||||
lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
|
||||
ι 𝓝ᶠ(a) = ι 𝓝ᶠ(b) := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||||
|
|
@ -241,7 +241,7 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓝(" a ")" => normalOrder
|
|||
|
||||
-/
|
||||
|
||||
lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.CrAnAlgebra) :
|
||||
lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.FieldOpFreeAlgebra) :
|
||||
𝓝(ι a) = ι 𝓝ᶠ(a) := rfl
|
||||
|
||||
lemma normalOrder_ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) :
|
||||
|
|
|
|||
|
|
@ -14,7 +14,7 @@ import HepLean.Meta.Remark.Basic
|
|||
|
||||
namespace FieldSpecification
|
||||
variable {𝓕 : FieldSpecification}
|
||||
open CrAnAlgebra
|
||||
open FieldOpFreeAlgebra
|
||||
|
||||
open HepLean.List
|
||||
open WickContraction
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@ Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.TimeOrder
|
||||
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.TimeOrder
|
||||
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
|
||||
/-!
|
||||
|
||||
|
|
@ -12,20 +12,20 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
|
|||
-/
|
||||
|
||||
namespace FieldSpecification
|
||||
open CrAnAlgebra
|
||||
open FieldOpFreeAlgebra
|
||||
open HepLean.List
|
||||
open FieldStatistic
|
||||
|
||||
namespace FieldOpAlgebra
|
||||
variable {𝓕 : FieldSpecification}
|
||||
|
||||
lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.CrAnAlgebra) (h : ι b = 0) :
|
||||
lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.FieldOpFreeAlgebra) (h : ι b = 0) :
|
||||
ι [a, b]ₛca = 0 := by
|
||||
rw [superCommuteF_expand_bosonicProj_fermionicProj]
|
||||
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
|
||||
simp_all
|
||||
|
||||
lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a = 0) :
|
||||
lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.FieldOpFreeAlgebra) (h : ι a = 0) :
|
||||
ι [a, b]ₛca = 0 := by
|
||||
rw [superCommuteF_expand_bosonicProj_fermionicProj]
|
||||
rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
|
||||
|
|
@ -37,12 +37,12 @@ lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι
|
|||
|
||||
-/
|
||||
|
||||
lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.CrAnAlgebra)
|
||||
lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.FieldOpFreeAlgebra)
|
||||
(h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛca = 0 := by
|
||||
apply ι_superCommuteF_eq_zero_of_ι_right_zero
|
||||
exact (ι_eq_zero_iff_mem_ideal b).mpr h
|
||||
|
||||
lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈ b2) :
|
||||
lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.FieldOpFreeAlgebra) (h : b1 ≈ b2) :
|
||||
ι [a, b1]ₛca = ι [a, b2]ₛca := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||||
|
|
@ -50,7 +50,7 @@ lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1
|
|||
exact ι_superCommuteF_right_zero_of_mem_ideal a (b1 - b2) h
|
||||
|
||||
/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
|
||||
noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
|
||||
noncomputable def superCommuteRight (a : 𝓕.FieldOpFreeAlgebra) :
|
||||
FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ superCommuteF a)
|
||||
(ι_superCommuteF_eq_of_equiv_right a)
|
||||
|
|
@ -67,13 +67,13 @@ noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
|
|||
rw [← map_smul, ι_apply, ι_apply]
|
||||
simp
|
||||
|
||||
lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) :
|
||||
lemma superCommuteRight_apply_ι (a b : 𝓕.FieldOpFreeAlgebra) :
|
||||
superCommuteRight a (ι b) = ι [a, b]ₛca := by rfl
|
||||
|
||||
lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) :
|
||||
lemma superCommuteRight_apply_quot (a b : 𝓕.FieldOpFreeAlgebra) :
|
||||
superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by rfl
|
||||
|
||||
lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
|
||||
lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.FieldOpFreeAlgebra) (h : a1 ≈ a2) :
|
||||
superCommuteRight a1 = superCommuteRight a2 := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
ext b
|
||||
|
|
@ -114,7 +114,7 @@ noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
|
|||
@[inherit_doc superCommute]
|
||||
scoped[FieldSpecification.FieldOpAlgebra] notation "[" a "," b "]ₛ" => superCommute a b
|
||||
|
||||
lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.CrAnAlgebra) :
|
||||
lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.FieldOpFreeAlgebra) :
|
||||
[ι a, ι b]ₛ = ι [a, b]ₛca := rfl
|
||||
|
||||
/-!
|
||||
|
|
|
|||
|
|
@ -16,7 +16,7 @@ generated by the states.
|
|||
|
||||
namespace FieldSpecification
|
||||
variable {𝓕 : FieldSpecification}
|
||||
open CrAnAlgebra
|
||||
open FieldOpFreeAlgebra
|
||||
noncomputable section
|
||||
|
||||
namespace FieldOpAlgebra
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@ Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.TimeOrder
|
||||
import HepLean.PerturbationTheory.Algebras.FieldOpFreeAlgebra.TimeOrder
|
||||
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
|
||||
/-!
|
||||
|
||||
|
|
@ -12,7 +12,7 @@ import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
|
|||
-/
|
||||
|
||||
namespace FieldSpecification
|
||||
open CrAnAlgebra
|
||||
open FieldOpFreeAlgebra
|
||||
open HepLean.List
|
||||
open FieldStatistic
|
||||
|
||||
|
|
@ -153,16 +153,16 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnList {φ1 φ2 φ3 : 𝓕.C
|
|||
simp
|
||||
|
||||
@[simp]
|
||||
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
|
||||
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.FieldOpFreeAlgebra) :
|
||||
ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
|
||||
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
|
||||
change pb b (Basis.mem_span _ b)
|
||||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs, rfl⟩ := hx
|
||||
simp only [ofListBasis_eq_ofList, pb]
|
||||
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs) = 0
|
||||
change pa a (Basis.mem_span _ a)
|
||||
apply Submodule.span_induction
|
||||
|
|
@ -182,10 +182,10 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnStates}
|
|||
simp_all [pb, hpx]
|
||||
|
||||
lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
|
||||
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
|
||||
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :
|
||||
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
|
||||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
|
||||
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
|
||||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
|
||||
change pb b (Basis.mem_span _ b)
|
||||
|
|
@ -193,7 +193,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
|
|||
· intro x hx
|
||||
obtain ⟨φs, rfl⟩ := hx
|
||||
simp only [ofListBasis_eq_ofList, map_mul, pb]
|
||||
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
|
||||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* ofCrAnList φs))
|
||||
change pa a (Basis.mem_span _ a)
|
||||
|
|
@ -278,7 +278,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnStates}
|
|||
simp_all [pb, hpx]
|
||||
|
||||
lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
|
||||
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
|
||||
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.FieldOpFreeAlgebra) :
|
||||
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
|
||||
rw [timeOrderF_timeOrderF_mid]
|
||||
have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
|
||||
|
|
@ -300,10 +300,10 @@ lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnStates}
|
|||
|
||||
-/
|
||||
|
||||
lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
||||
lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
|
||||
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(a) = 0 := by
|
||||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
|
||||
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
|
||||
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
|
||||
change p a h
|
||||
apply AddSubgroup.closure_induction
|
||||
· intro x hx
|
||||
|
|
@ -358,7 +358,7 @@ lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
|||
· intro x hx
|
||||
simp [p]
|
||||
|
||||
lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
|
||||
lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
|
||||
ι 𝓣ᶠ(a) = ι 𝓣ᶠ(b) := by
|
||||
rw [equiv_iff_sub_mem_ideal] at h
|
||||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||||
|
|
@ -390,7 +390,7 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓣(" a ")" => timeOrder a
|
|||
|
||||
-/
|
||||
|
||||
lemma timeOrder_eq_ι_timeOrderF (a : 𝓕.CrAnAlgebra) :
|
||||
lemma timeOrder_eq_ι_timeOrderF (a : 𝓕.FieldOpFreeAlgebra) :
|
||||
𝓣(ι a) = ι 𝓣ᶠ(a) := rfl
|
||||
|
||||
lemma timeOrder_ofFieldOp_ofFieldOp_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
|
||||
|
|
|
|||
|
|
@ -15,7 +15,7 @@ import HepLean.Meta.Remark.Basic
|
|||
|
||||
namespace FieldSpecification
|
||||
variable {𝓕 : FieldSpecification}
|
||||
open CrAnAlgebra
|
||||
open FieldOpFreeAlgebra
|
||||
namespace FieldOpAlgebra
|
||||
open WickContraction
|
||||
open EqTimeOnly
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue