feat: Add SuperCommute for FieldOpAlgebra
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jstoobysmith
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c2d89cc093
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9107115620
5 changed files with 708 additions and 9 deletions
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@ -191,5 +191,241 @@ lemma ι_superCommute_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnState
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rw [← h0]
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abel
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/-!
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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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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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simp only [ringConGen, Quotient.eq]
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rw [TwoSidedIdeal.mem_iff]
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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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x.bosonicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProj = 0 := by
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have hx' := hx
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simp [fieldOpIdealSet] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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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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x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
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have hx' := hx
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simp [fieldOpIdealSet] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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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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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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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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· intro x hx
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simp [p]
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obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
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obtain ⟨d, hd, y, hy, rfl⟩ := Set.mem_mul.mp ha
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rw [bosonicProj_mul, bosonicProj_mul, fermionicProj_mul]
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simp [mul_add, add_mul]
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rcases fermionicProj_mem_fieldOpIdealSet_or_zero y hy with hfy | hfy
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<;> rcases bosonicProj_mem_fieldOpIdealSet_or_zero y hy with hby | hby
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· apply TwoSidedIdeal.add_mem
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apply TwoSidedIdeal.add_mem
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· /- boson, boson, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (bosonicProj y).1
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simp [hby]
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· use ↑(bosonicProj b)
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simp
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· /- fermion, fermion, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(bosonicProj b)
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simp
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apply TwoSidedIdeal.add_mem
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· /- boson, fermion, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· /- fermion, boson, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (bosonicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· simp [hby]
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apply TwoSidedIdeal.add_mem
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· /- fermion, fermion, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(bosonicProj b)
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simp
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· /- boson, fermion, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· simp [hfy]
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apply TwoSidedIdeal.add_mem
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· /- boson, boson, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp
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use (bosonicProj y).1
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simp [hby]
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· use ↑(bosonicProj b)
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simp
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· /- fermion, boson, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp
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use (bosonicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· simp [hfy, hby]
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· simp [p]
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· intro x y hx hy hpx hpy
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simp_all [p]
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apply TwoSidedIdeal.add_mem
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exact hpx
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exact hpy
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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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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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rw [← bosonicProj_add_fermionicProj x] at hx
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simp 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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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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rw [@ι_eq_zero_iff_mem_ideal] at h ⊢
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rw [ι_eq_zero_iff_mem_ideal]
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apply And.intro
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· exact bosonicProj_mem_ideal x h
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· exact fermionicProj_mem_ideal x h
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· intro h
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rw [← bosonicProj_add_fermionicProj x]
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simp_all
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end FieldOpAlgebra
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end FieldSpecification
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