118 lines
4.2 KiB
Text
118 lines
4.2 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_superCommute_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 [superCommute_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_superCommute_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 [superCommute_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_superCommute_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_superCommute_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_superCommute_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_superCommute_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 CrAnAlgebra.superCommute a)
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(ι_superCommute_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_superCommute_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 ((superCommute 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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lemma ι_superCommute (a b : 𝓕.CrAnAlgebra) : ι [a, b]ₛca = superCommute (ι a) (ι b) := by
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rfl
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end FieldOpAlgebra
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end FieldSpecification
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