520 lines
21 KiB
Text
520 lines
21 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.Algebras.CrAnAlgebra.TimeOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
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/-!
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# SuperCommute on Field operator algebra
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open HepLean.List
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open FieldStatistic
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.CrAnAlgebra) (h : ι b = 0) :
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ι [a, b]ₛca = 0 := by
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rw [superCommuteF_expand_bosonicProj_fermionicProj]
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rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
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simp_all
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lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.CrAnAlgebra) (h : ι a = 0) :
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ι [a, b]ₛca = 0 := by
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rw [superCommuteF_expand_bosonicProj_fermionicProj]
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rw [ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero] at h
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simp_all
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/-!
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## Defining normal order for `FiedOpAlgebra`.
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-/
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lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.CrAnAlgebra)
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(h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛca = 0 := by
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apply ι_superCommuteF_eq_zero_of_ι_right_zero
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exact (ι_eq_zero_iff_mem_ideal b).mpr h
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lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈ b2) :
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ι [a, b1]ₛca = ι [a, b2]ₛca := by
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rw [equiv_iff_sub_mem_ideal] at h
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rw [LinearMap.sub_mem_ker_iff.mp]
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simp only [LinearMap.mem_ker, ← map_sub]
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exact ι_superCommuteF_right_zero_of_mem_ideal a (b1 - b2) h
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/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
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noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
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FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ superCommuteF a)
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(ι_superCommuteF_eq_of_equiv_right a)
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map_add' x y := by
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obtain ⟨x, hx⟩ := ι_surjective x
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obtain ⟨y, hy⟩ := ι_surjective y
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subst hx hy
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rw [← map_add, ι_apply, ι_apply, ι_apply]
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rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
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simp
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map_smul' c y := by
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obtain ⟨y, hy⟩ := ι_surjective y
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subst hy
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rw [← map_smul, ι_apply, ι_apply]
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simp
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lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) :
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superCommuteRight a (ι b) = ι [a, b]ₛca := by rfl
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lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) :
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superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by rfl
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lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
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superCommuteRight a1 = superCommuteRight a2 := by
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rw [equiv_iff_sub_mem_ideal] at h
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ext b
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obtain ⟨b, rfl⟩ := ι_surjective b
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have ha1b1 : (superCommuteRight (a1 - a2)) (ι b) = 0 := by
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rw [superCommuteRight_apply_ι]
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apply ι_superCommuteF_eq_zero_of_ι_left_zero
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exact (ι_eq_zero_iff_mem_ideal (a1 - a2)).mpr h
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simp_all only [superCommuteRight_apply_ι, map_sub, LinearMap.sub_apply]
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trans ι ((superCommuteF a2) b) + 0
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rw [← ha1b1]
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simp only [add_sub_cancel]
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simp
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/-- The super commutor on the `FieldOpAlgebra`. -/
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noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
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FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift superCommuteRight superCommuteRight_eq_of_equiv
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map_add' x y := by
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obtain ⟨x, rfl⟩ := ι_surjective x
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obtain ⟨y, rfl⟩ := ι_surjective y
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ext b
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obtain ⟨b, rfl⟩ := ι_surjective b
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rw [← map_add, ι_apply, ι_apply, ι_apply, ι_apply]
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rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
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simp only [LinearMap.add_apply]
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rw [superCommuteRight_apply_quot, superCommuteRight_apply_quot, superCommuteRight_apply_quot]
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simp
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map_smul' c y := by
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obtain ⟨y, rfl⟩ := ι_surjective y
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ext b
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obtain ⟨b, rfl⟩ := ι_surjective b
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rw [← map_smul, ι_apply, ι_apply, ι_apply]
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simp only [Quotient.lift_mk, RingHom.id_apply, LinearMap.smul_apply]
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rw [superCommuteRight_apply_quot, superCommuteRight_apply_quot]
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simp
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@[inherit_doc superCommute]
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scoped[FieldSpecification.FieldOpAlgebra] notation "[" a "," b "]ₛ" => superCommute a b
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lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.CrAnAlgebra) :
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[ι a, ι b]ₛ = ι [a, b]ₛca := rfl
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/-!
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## Properties of `superCommute`.
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-/
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/-!
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## Properties from the definition of FieldOpAlgebra
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-/
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lemma superCommute_create_create {φ φ' : 𝓕.CrAnStates}
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(h : 𝓕 |>ᶜ φ = .create) (h' : 𝓕 |>ᶜ φ' = .create) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_create_create _ _ h h']
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lemma superCommute_annihilate_annihilate {φ φ' : 𝓕.CrAnStates}
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(h : 𝓕 |>ᶜ φ = .annihilate) (h' : 𝓕 |>ᶜ φ' = .annihilate) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_annihilate_annihilate _ _ h h']
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lemma superCommute_diff_statistic {φ φ' : 𝓕.CrAnStates} (h : (𝓕 |>ₛ φ) ≠ 𝓕 |>ₛ φ') :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_diff_statistic h]
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero (φ : 𝓕.CrAnStates) (ψ : 𝓕.States)
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : [ofCrAnFieldOp φ, ofFieldOp ψ]ₛ = 0 := by
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rw [ofFieldOp_eq_sum, map_sum]
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rw [Finset.sum_eq_zero]
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intro x hx
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apply superCommute_diff_statistic
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simpa [crAnStatistics] using h
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lemma superCommute_anPart_ofState_diff_grade_zero (φ ψ : 𝓕.States)
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : [anPart φ, ofFieldOp ψ]ₛ = 0 := by
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match φ with
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| States.inAsymp _ =>
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simp
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| States.position φ =>
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simp only [anPartF_position]
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apply superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero _ _ _
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simpa [crAnStatistics] using h
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| States.outAsymp _ =>
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simp only [anPartF_posAsymp]
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apply superCommute_ofCrAnFieldOp_ofFieldOp_diff_stat_zero _ _
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simpa [crAnStatistics] using h
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp, superCommute_eq_ι_superCommuteF]
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exact ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center φ φ'
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_commute (φ φ' : 𝓕.CrAnStates) (a : FieldOpAlgebra 𝓕) :
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a * [ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = [ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ * a := by
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have h1 := superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center φ φ'
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rw [@Subalgebra.mem_center_iff] at h1
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exact h1 a
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_mem_center (φ : 𝓕.CrAnStates) (φ' : 𝓕.States) :
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[ofCrAnFieldOp φ, ofFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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rw [ofFieldOp_eq_sum]
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simp
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refine Subalgebra.sum_mem (Subalgebra.center ℂ 𝓕.FieldOpAlgebra) ?_
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intro x hx
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exact superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center φ ⟨φ', x⟩
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lemma superCommute_ofCrAnFieldOp_ofFieldOp_commute (φ : 𝓕.CrAnStates) (φ' : 𝓕.States)
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(a : FieldOpAlgebra 𝓕) : a * [ofCrAnFieldOp φ, ofFieldOp φ']ₛ =
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[ofCrAnFieldOp φ, ofFieldOp φ']ₛ * a := by
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have h1 := superCommute_ofCrAnFieldOp_ofFieldOp_mem_center φ φ'
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rw [@Subalgebra.mem_center_iff] at h1
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exact h1 a
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lemma superCommute_anPart_ofFieldOp_mem_center (φ φ' : 𝓕.States) :
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[anPart φ, ofFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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match φ with
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| States.inAsymp _ =>
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simp only [anPart_negAsymp, map_zero, LinearMap.zero_apply]
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exact Subalgebra.zero_mem (Subalgebra.center ℂ _)
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| States.position φ =>
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exact superCommute_ofCrAnFieldOp_ofFieldOp_mem_center _ _
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| States.outAsymp _ =>
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exact superCommute_ofCrAnFieldOp_ofFieldOp_mem_center _ _
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/-!
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### `superCommute` on different constructors.
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-/
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lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
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ofCrAnFieldOpList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnFieldOpList (φs' ++ φs) := by
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rw [ofCrAnFieldOpList_eq_ι_ofCrAnList, ofCrAnFieldOpList_eq_ι_ofCrAnList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnList_ofCrAnList]
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rfl
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = ofCrAnFieldOp φ * ofCrAnFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnFieldOp φ' * ofCrAnFieldOp φ := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnState_ofCrAnState]
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rfl
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lemma superCommute_ofCrAnFieldOpList_ofFieldOpList (φcas : List 𝓕.CrAnStates)
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(φs : List 𝓕.States) :
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[ofCrAnFieldOpList φcas, ofFieldOpList φs]ₛ = ofCrAnFieldOpList φcas * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofCrAnFieldOpList φcas := by
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rw [ofCrAnFieldOpList, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnList_ofStatesList]
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rfl
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lemma superCommute_ofFieldOpList_ofFieldOpList (φs φs' : List 𝓕.States) :
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[ofFieldOpList φs, ofFieldOpList φs']ₛ = ofFieldOpList φs * ofFieldOpList φs' -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs := by
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rw [ofFieldOpList, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofStateList_ofStatesList]
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rfl
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lemma superCommute_ofFieldOp_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[ofFieldOp φ, ofFieldOpList φs]ₛ = ofFieldOp φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofFieldOp φ := by
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rw [ofFieldOp, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofState_ofStatesList]
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rfl
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lemma superCommute_ofFieldOpList_ofFieldOp (φs : List 𝓕.States) (φ : 𝓕.States) :
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[ofFieldOpList φs, ofFieldOp φ]ₛ = ofFieldOpList φs * ofFieldOp φ -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs := by
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rw [ofFieldOpList, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofStateList_ofState]
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rfl
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lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
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[anPart φ, crPart φ']ₛ = anPart φ * crPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
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rw [anPart, crPart]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_crPartF]
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rfl
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lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
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[crPart φ, anPart φ']ₛ = crPart φ * anPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ := by
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rw [anPart, crPart]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_anPartF]
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rfl
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@[simp]
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lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) : [crPart φ, crPart φ']ₛ = 0 := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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simp
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| _, States.outAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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simp only [crPart_position]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.position φ, States.inAsymp φ' =>
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simp only [crPart_position, crPart_negAsymp]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.inAsymp φ, States.inAsymp φ' =>
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simp only [crPart_negAsymp]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.inAsymp φ, States.position φ' =>
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simp only [crPart_negAsymp, crPart_position]
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apply superCommute_create_create
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· rfl
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· rfl
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@[simp]
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lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) : [anPart φ, anPart φ']ₛ = 0 := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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simp
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| _, States.inAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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simp only [anPart_position]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.position φ, States.outAsymp φ' =>
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simp only [anPart_position, anPart_posAsymp]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.outAsymp φ, States.outAsymp φ' =>
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simp only [anPart_posAsymp]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.outAsymp φ, States.position φ' =>
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simp only [anPart_posAsymp, anPart_position]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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lemma superCommute_crPart_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[crPart φ, ofFieldOpList φs]ₛ = crPart φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * crPart φ := by
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rw [crPart, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofStateList]
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rfl
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lemma superCommute_anPart_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[anPart φ, ofFieldOpList φs]ₛ = anPart φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * anPart φ := by
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rw [anPart, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofStateList]
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rfl
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lemma superCommute_crPart_ofFieldOp (φ φ' : 𝓕.States) :
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[crPart φ, ofFieldOp φ']ₛ = crPart φ * ofFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * crPart φ := by
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rw [crPart, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofState]
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rfl
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lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.States) :
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[anPart φ, ofFieldOp φ']ₛ = anPart φ * ofFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * anPart φ := by
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rw [anPart, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofState]
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rfl
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/-!
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## Mul equal superCommute
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Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
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multiplication with a sign plus the super commutor.
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-/
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lemma ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute (φs φs' : List 𝓕.CrAnStates) :
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ofCrAnFieldOpList φs * ofCrAnFieldOpList φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnFieldOpList φs' * ofCrAnFieldOpList φs
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+ [ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ := by
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rw [superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList]
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simp [ofCrAnFieldOpList_append]
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lemma ofCrAnFieldOp_mul_ofCrAnFieldOpList_eq_superCommute (φ : 𝓕.CrAnStates)
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(φs' : List 𝓕.CrAnStates) : ofCrAnFieldOp φ * ofCrAnFieldOpList φs' =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnFieldOpList φs' * ofCrAnFieldOp φ
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+ [ofCrAnFieldOp φ, ofCrAnFieldOpList φs']ₛ := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute]
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simp
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lemma ofFieldOpList_mul_ofFieldOpList_eq_superCommute (φs φs' : List 𝓕.States) :
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ofFieldOpList φs * ofFieldOpList φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs
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+ [ofFieldOpList φs, ofFieldOpList φs']ₛ := by
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rw [superCommute_ofFieldOpList_ofFieldOpList]
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simp
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lemma ofFieldOp_mul_ofFieldOpList_eq_superCommute (φ : 𝓕.States) (φs' : List 𝓕.States) :
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ofFieldOp φ * ofFieldOpList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOp φ
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+ [ofFieldOp φ, ofFieldOpList φs']ₛ := by
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rw [superCommute_ofFieldOp_ofFieldOpList]
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simp
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lemma ofFieldOp_mul_ofFieldOp_eq_superCommute (φ φ' : 𝓕.States) :
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ofFieldOp φ * ofFieldOp φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * ofFieldOp φ
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+ [ofFieldOp φ, ofFieldOp φ']ₛ := by
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rw [← ofFieldOpList_singleton, ← ofFieldOpList_singleton]
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rw [ofFieldOpList_mul_ofFieldOpList_eq_superCommute, ofFieldOpList_singleton]
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simp
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lemma ofFieldOpList_mul_ofFieldOp_eq_superCommute (φs : List 𝓕.States) (φ : 𝓕.States) :
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ofFieldOpList φs * ofFieldOp φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs
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+ [ofFieldOpList φs, ofFieldOp φ]ₛ := by
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rw [superCommute_ofFieldOpList_ofFieldOp]
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simp
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lemma ofCrAnFieldOpList_mul_ofFieldOpList_eq_superCommute (φs : List 𝓕.CrAnStates)
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(φs' : List 𝓕.States) : ofCrAnFieldOpList φs * ofFieldOpList φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofCrAnFieldOpList φs
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+ [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ := by
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rw [superCommute_ofCrAnFieldOpList_ofFieldOpList]
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simp
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lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
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crPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ
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+ [crPart φ, anPart φ']ₛ := by
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rw [superCommute_crPart_anPart]
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simp
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lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
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anPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ
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+ [anPart φ, crPart φ']ₛ := by
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rw [superCommute_anPart_crPart]
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simp
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lemma crPart_mul_crPart_swap (φ φ' : 𝓕.States) :
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crPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ := by
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trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ + [crPart φ, crPart φ']ₛ
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· rw [crPart, crPart, superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_crPartF]
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simp
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· simp
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lemma anPart_mul_anPart_swap (φ φ' : 𝓕.States) :
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anPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ := by
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trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ + [anPart φ, anPart φ']ₛ
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· rw [anPart, anPart, superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_anPartF]
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simp
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· simp
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/-!
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## Symmetry of the super commutor.
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-/
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lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_symm (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
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(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnFieldOpList φs', ofCrAnFieldOpList φs]ₛ := by
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rw [ofCrAnFieldOpList, ofCrAnFieldOpList, superCommute_eq_ι_superCommuteF,
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superCommuteF_ofCrAnList_ofCrAnList_symm]
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rfl
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||
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_symm (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ =
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(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnFieldOp φ', ofCrAnFieldOp φ]ₛ := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp, superCommute_eq_ι_superCommuteF,
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superCommuteF_ofCrAnState_ofCrAnState_symm]
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rfl
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/-!
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||
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## splitting the super commute into sums
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||
|
||
-/
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||
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lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_eq_sum (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
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ofCrAnFieldOpList (φs'.take n) * [ofCrAnFieldOpList φs, ofCrAnFieldOp (φs'.get n)]ₛ *
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ofCrAnFieldOpList (φs'.drop (n + 1)) := by
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conv_lhs =>
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rw [ofCrAnFieldOpList, ofCrAnFieldOpList, superCommute_eq_ι_superCommuteF,
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superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
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rw [map_sum]
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rfl
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||
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOpList_eq_sum (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
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[ofCrAnFieldOp φ, ofCrAnFieldOpList φs']ₛ =
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs'.take n) •
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[ofCrAnFieldOp φ, ofCrAnFieldOp (φs'.get n)]ₛ * ofCrAnFieldOpList (φs'.eraseIdx n) := by
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conv_lhs =>
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rw [← ofCrAnFieldOpList_singleton, superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_eq_sum]
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congr
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||
funext n
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simp
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||
congr 1
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||
rw [ofCrAnFieldOpList_singleton, superCommute_ofCrAnFieldOp_ofCrAnFieldOp_commute]
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rw [mul_assoc, ← ofCrAnFieldOpList_append]
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||
congr
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exact Eq.symm (List.eraseIdx_eq_take_drop_succ φs' ↑n)
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||
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||
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lemma superCommute_ofCrAnFieldOpList_ofFieldOpList_eq_sum (φs : List 𝓕.CrAnStates)
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||
(φs' : List 𝓕.States) : [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ =
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||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||
ofFieldOpList (φs'.take n) * [ofCrAnFieldOpList φs, ofFieldOp (φs'.get n)]ₛ *
|
||
ofFieldOpList (φs'.drop (n + 1)) := by
|
||
conv_lhs =>
|
||
rw [ofCrAnFieldOpList, ofFieldOpList, superCommute_eq_ι_superCommuteF,
|
||
superCommuteF_ofCrAnList_ofStateList_eq_sum]
|
||
rw [map_sum]
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||
rfl
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||
|
||
lemma superCommute_ofCrAnFieldOp_ofFieldOpList_eq_sum (φ : 𝓕.CrAnStates) (φs' : List 𝓕.States) :
|
||
[ofCrAnFieldOp φ, ofFieldOpList φs']ₛ =
|
||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs'.take n) •
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||
[ofCrAnFieldOp φ, ofFieldOp (φs'.get n)]ₛ * ofFieldOpList (φs'.eraseIdx n) := by
|
||
conv_lhs =>
|
||
rw [← ofCrAnFieldOpList_singleton, superCommute_ofCrAnFieldOpList_ofFieldOpList_eq_sum]
|
||
congr
|
||
funext n
|
||
simp
|
||
congr 1
|
||
rw [ofCrAnFieldOpList_singleton, superCommute_ofCrAnFieldOp_ofFieldOp_commute]
|
||
rw [mul_assoc, ← ofFieldOpList_append]
|
||
congr
|
||
exact Eq.symm (List.eraseIdx_eq_take_drop_succ φs' ↑n)
|
||
|
||
end FieldOpAlgebra
|
||
end FieldSpecification
|