| import OAI.Analysis.InterpolatedFactors.Gaussian |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| universe u_1 u_2 u_3 |
|
|
| namespace FreeGroupFactorMain.WeightedFock |
|
|
| section |
| open FreeFock |
| variable {ι : Type u_1} [DecidableEq ι] |
|
|
| theorem P_mem_old (u v : ℝ) (J : Set ι) : (P : Operator ι)∈oldAlgebra u v J := |
| Bicommutant.subset _ (Set.mem_insert _ _) |
|
|
| theorem S_mem_old (u v : ℝ) (J : Set ι) (i : ι) (hi : i∈J) : S u v i∈oldAlgebra u v J := |
| Bicommutant.subset _ (Set.mem_insert_of_mem _ ⟨i,hi,rfl⟩) |
|
|
| theorem offDiagonal_mem_old (u v : ℝ) (J : Set ι) (i : ι) (hi : i∈J) : |
| offDiagonal u v i∈oldAlgebra u v J := by |
| unfold offDiagonal |
| exact mul_mem (mul_mem (P_mem_old u v J) (S_mem_old u v J i hi)) |
| (sub_mem (one_mem _) (P_mem_old u v J)) |
|
|
| theorem edgeAlgebra_le_old (u v : ℝ) (J : Set ι) (i : ι) (hi : i∈J) : |
| edgeAlgebra u v i≤oldAlgebra u v J := by |
| apply Bicommutant.generated_le |
| rintro T (rfl|hT) |
| · exact P_mem_old u v J |
| · have he : T=offDiagonal u v i+star (offDiagonal u v i) := hT |
| rw [he] |
| exact add_mem (offDiagonal_mem_old u v J i hi) (star_mem (offDiagonal_mem_old u v J i hi)) |
|
|
| def weightedDefectCreation (u v : ℝ) (i j : ι) : Operator ι := |
| edgeDefect u v i*C u v j*edgeDefect u v i |
|
|
| def defectCovariance (u v : ℝ) (i : ι) (a : edgeAlgebra u v i) : edgeAlgebra u v i := |
| ⟨vacuumTrace u v (edgeDefect u v i*(a : Operator ι)*edgeDefect u v i) • edgeDefect u v i, |
| (edgeAlgebra u v i).smul_mem (edgeDefect_mem_edge u v i) _⟩ |
|
|
| theorem weightedDefect_contraction (u v : ℝ) (i j : ι) (hji : j≠i) (a : edgeAlgebra u v i) : |
| star (weightedDefectCreation u v i j)*(a : Operator ι)*weightedDefectCreation u v i j= |
| (defectCovariance u v i a : Operator ι) := by |
| let d : edgeAlgebra u v i := ⟨edgeDefect u v i,edgeDefect_mem_edge u v i⟩ |
| let b : oldAlgebra u v {i} := ⟨(d*a*d : edgeAlgebra u v i), |
| edgeAlgebra_le_old u v {i} i (Set.mem_singleton i) (d*a*d).property⟩ |
| have hc := fresh_C_contraction u v {i} j j hji hji b |
| simp only [ite_true] at hc |
| change star (C u v j)*(edgeDefect u v i*(a : Operator ι)*edgeDefect u v i)*C u v j=_ at hc |
| calc |
| _ = edgeDefect u v i*(star (C u v j)*(edgeDefect u v i*(a : Operator ι)*edgeDefect u v i)* |
| C u v j)*edgeDefect u v i := by |
| simp only [weightedDefectCreation,star_mul,(edgeDefect_projection u v i).isSelfAdjoint.star_eq] |
| noncomm_ring |
| _ = _ := by |
| rw [hc,mul_smul_comm,smul_mul_assoc,mul_one,(edgeDefect_projection u v i).isIdempotentElem.eq] |
| rfl |
|
|
| theorem weightedDefect_annihilates_edge (u v : ℝ) (i j : ι) (hji : j≠i) (a : edgeAlgebra u v i) : |
| star (weightedDefectCreation u v i j) ((a : Operator ι) (vacuum u v))=0 := by |
| let b : oldAlgebra u v {i} := ⟨edgeDefect u v i*(a : Operator ι), |
| edgeAlgebra_le_old u v {i} i (Set.mem_singleton i) |
| (mul_mem (edgeDefect_mem_edge u v i) a.property)⟩ |
| have hc := fresh_C_annihilates_old_vacuum u v {i} j hji b |
| rw [weightedDefectCreation,star_mul,star_mul,(edgeDefect_projection u v i).isSelfAdjoint.star_eq] |
| change edgeDefect u v i (star (C u v j) (edgeDefect u v i ((a : Operator ι) (vacuum u v))))=0 |
| rw [show star (C u v j) (edgeDefect u v i ((a : Operator ι) (vacuum u v)))=0 from hc,map_zero] |
|
|
| theorem fresh_C_weightedDefect (u v : ℝ) (i j n : ι) (hji : j≠i) (hni : n≠i) (hnj : n≠j) |
| (a : edgeAlgebra u v i) : star (C u v n)*(a : Operator ι)*weightedDefectCreation u v i j=0 := by |
| let b : oldAlgebra u v {i} := ⟨(a : Operator ι)*edgeDefect u v i, |
| edgeAlgebra_le_old u v {i} i (Set.mem_singleton i) |
| (mul_mem a.property (edgeDefect_mem_edge u v i))⟩ |
| have hc := fresh_C_contraction u v {i} n j hni hji b |
| rw [ite_eq_right hnj] at hc |
| change star (C u v n)*((a : Operator ι)*edgeDefect u v i)*C u v j=0 at hc |
| rw [weightedDefectCreation] |
| calc |
| _ = (star (C u v n)*((a : Operator ι)*edgeDefect u v i)*C u v j)*edgeDefect u v i := by |
| simp only [mul_assoc] |
| _ = 0 := by rw [hc,zero_mul] |
|
|
| def filledEdgeAlgebra (u v : ℝ) (i j : ι) : StarSubalgebra ℂ (Operator ι) := |
| Bicommutant.algebra (insert (weightedDefectCreation u v i j+star (weightedDefectCreation u v i j)) |
| (edgeAlgebra u v i : Set (Operator ι))) |
|
|
| theorem fresh_C_filled_contraction (u v : ℝ) (i j n m : ι) (hji : j≠i) |
| (hni : n≠i) (hnj : n≠j) (hmi : m≠i) (hmj : m≠j) (T : filledEdgeAlgebra u v i j) : |
| star (C u v n)*(T : Operator ι)*C u v m= |
| if n=m then vacuumTrace u v (T : Operator ι) • 1 else 0 := by |
| have hD := weightedDefect_annihilates_edge u v i j hji |
| have hR (a : edgeAlgebra u v i) : star (weightedDefectCreation u v i j)*(a : Operator ι)*C u v m=0 := by |
| have hh := congrArg star (fresh_C_weightedDefect u v i j m hji hmi hmj (star a)) |
| simpa only [star_mul,star_star,StarMemClass.coe_star,star_zero (Operator ι),mul_assoc] using hh |
| have hb (a : edgeAlgebra u v i) : star (C u v n)*(a : Operator ι)*C u v m= |
| (inner ℂ (vacuum u v) ((a : Operator ι) (vacuum u v))) • (if n=m then (1 : Operator ι) else 0) := by |
| have hh := fresh_C_contraction u v {i} n m hni hmi |
| ⟨a,edgeAlgebra_le_old u v {i} i (Set.mem_singleton i) a.property⟩ |
| split_ifs with h <;> simpa only [vacuumTrace,h,ite_true,ite_false, |
| smul_zero (M := ℂ) (A := Operator ι)] using hh |
| have hh := GaussianExtension.covariance_extension (edgeAlgebra u v i) (weightedDefectCreation u v i j) |
| (defectCovariance u v i) (weightedDefect_contraction u v i j hji) |
| (C u v n) (C u v m) (if n=m then (1 : Operator ι) else 0) (vacuum u v) (vacuum u v) |
| hD hD (fresh_C_weightedDefect u v i j n hji hni hnj) hR hb (T : Operator ι) T.property |
| split_ifs with h <;> simpa only [vacuumTrace,h,ite_true,ite_false, |
| smul_zero (M := ℂ) (A := Operator ι)] using hh |
|
|
| theorem fresh_C_filled_annihilation (u v : ℝ) (i j n : ι) (hji : j≠i) (hni : n≠i) (hnj : n≠j) |
| (T : filledEdgeAlgebra u v i j) : star (C u v n) ((T : Operator ι) (vacuum u v))=0 := by |
| apply GaussianExtension.annihilation_extension (edgeAlgebra u v i) (weightedDefectCreation u v i j) |
| (defectCovariance u v i) (weightedDefect_contraction u v i j hji) |
| (C u v n) (vacuum u v) (weightedDefect_annihilates_edge u v i j hji) |
| _ (fresh_C_weightedDefect u v i j n hji hni hnj) (T : Operator ι) T.property |
| intro a |
| exact fresh_C_annihilates_old_vacuum u v {i} n hni |
| ⟨a,edgeAlgebra_le_old u v {i} i (Set.mem_singleton i) a.property⟩ |
| end |
| section |
| open FreeFock |
| variable {ι : Type u_2} [DecidableEq ι] |
|
|
| theorem edgeAlgebra_le_ambient (u v : ℝ) (i : ι) : edgeAlgebra u v i≤ambientAlgebra u v := by |
| apply Bicommutant.generated_le |
| rintro T (rfl|hT) |
| · exact P_mem_ambient u v |
| · have he : T=offDiagonal u v i+star (offDiagonal u v i) := hT |
| rw [he] |
| exact add_mem (offDiagonal_mem_ambient u v i) (star_mem (offDiagonal_mem_ambient u v i)) |
|
|
| theorem edgeAlgebra_le_filled (u v : ℝ) (i j : ι) : edgeAlgebra u v i≤filledEdgeAlgebra u v i j := |
| fun _ h => Bicommutant.subset _ (Set.mem_insert_of_mem _ h) |
|
|
| theorem weightedDefect_semicircle_mem_filled (u v : ℝ) (i j : ι) : |
| weightedDefectCreation u v i j+star (weightedDefectCreation u v i j)∈filledEdgeAlgebra u v i j := |
| Bicommutant.subset _ (Set.mem_insert _ _) |
|
|
| theorem filledEdgeAlgebra_le_ambient (u v : ℝ) (i j : ι) : filledEdgeAlgebra u v i j≤ambientAlgebra u v := by |
| apply Bicommutant.generated_le |
| rintro T (rfl|hT) |
| · have he : weightedDefectCreation u v i j+star (weightedDefectCreation u v i j)= |
| edgeDefect u v i*S u v j*edgeDefect u v i := by |
| simp only [weightedDefectCreation,S_eq,star_mul,(edgeDefect_projection u v i).isSelfAdjoint.star_eq, |
| mul_add,add_mul,mul_assoc] |
| rw [he] |
| have hd := edgeAlgebra_le_ambient u v i (edgeDefect_mem_edge u v i) |
| exact mul_mem (mul_mem hd (S_mem_ambient u v j)) hd |
| · exact edgeAlgebra_le_ambient u v i hT |
|
|
| theorem weightedDefectCreation_eq (u v : ℝ) (i j : ι) : |
| weightedDefectCreation u v i j=(u:ℂ) • defectCreation u v i j := by |
| have he : P*C u v j*P=(u:ℂ) • edgeCreation j true true := by |
| simp only [C,mul_add,mul_smul_comm,smul_mul_assoc,P_creation_true,P_creation_false, |
| smul_zero (M := ℂ) (A := Operator ι),add_zero,edgeCreation,vertex_true] |
| calc |
| _ = edgeDefect u v i*(P*C u v j*P)*edgeDefect u v i := by |
| simp only [weightedDefectCreation,mul_assoc,P_mul_edgeDefect] |
| rw [← mul_assoc (edgeDefect u v i) P,edgeDefect_mul_P] |
| _ = _ := by rw [he,mul_smul_comm,smul_mul_assoc]; rfl |
|
|
| theorem defectSemicircle_mem_filled (u v : ℝ) (hu : u≠0) (i j : ι) : |
| defectSemicircle u v i j∈filledEdgeAlgebra u v i j := by |
| have he : weightedDefectCreation u v i j+star (weightedDefectCreation u v i j)= |
| (u:ℂ) • defectSemicircle u v i j := by |
| simp only [weightedDefectCreation_eq,star_smul,Complex.star_def,Complex.conj_ofReal, |
| defectSemicircle,smul_add] |
| have hh := (filledEdgeAlgebra u v i j).smul_mem (weightedDefect_semicircle_mem_filled u v i j) (u:ℂ)⁻¹ |
| rw [he,smul_smul,inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr hu),one_smul] at hh |
| exact hh |
|
|
| theorem diffuseBase_le_filled (u v : ℝ) (hu : u≠0) (i j : ι) : |
| diffuseBase u v i j≤filledEdgeAlgebra u v i j := by |
| apply Bicommutant.generated_le |
| rintro T (rfl|hT) |
| · exact edgeAlgebra_le_filled u v i j (P_mem_edge u v i) |
| · have he : T=oldBaseGenerator u v i j := hT |
| rw [he,oldBaseGenerator] |
| apply add_mem |
| · apply add_mem |
| · exact edgeAlgebra_le_filled u v i j |
| (mul_mem (offDiagonal_mem_edge u v i) (star_mem (edgePolar_mem_edge u v i))) |
| · exact defectSemicircle_mem_filled u v hu i j |
| · exact (filledEdgeAlgebra u v i j).smul_mem |
| (edgeAlgebra_le_filled u v i j (edgeRange_mem_edge u v i)) _ |
| end |
|
|
| open FreeFock |
| variable {ι : Type u_3} [DecidableEq ι] |
|
|
| theorem edgePolar_mul_P (u v : ℝ) (i : ι) : edgePolar u v i*P=0 := |
| CornerRerouting.right_zero (edgePolar_right u v i) |
|
|
| theorem P_mul_star_edgePolar (u v : ℝ) (i : ι) : P*star (edgePolar u v i)=0 := |
| CornerRerouting.star_left_zero projection_P.isSelfAdjoint (edgePolar_mul_P u v i) |
|
|
| def supportedLeg (u v : ℝ) (i₀ : ι) (r : Bool) : Operator ι := |
| if r then edgePolar u v i₀ else P |
|
|
| def legSupport (u v : ℝ) (i₀ : ι) (r : Bool) : Operator ι := |
| if r then edgeRange u v i₀ else P |
|
|
| theorem supportedLeg_mem_edge (u v : ℝ) (i : ι) (r : Bool) : |
| supportedLeg u v i r∈edgeAlgebra u v i := by |
| cases r |
| · exact P_mem_edge u v i |
| · exact edgePolar_mem_edge u v i |
|
|
| theorem supportedLeg_mem_ambient (u v : ℝ) (i : ι) (r : Bool) : |
| supportedLeg u v i r∈ambientAlgebra u v := |
| edgeAlgebra_le_ambient u v i (supportedLeg_mem_edge u v i r) |
|
|
| theorem supportedLeg_mul_star (u v : ℝ) (i : ι) (r s : Bool) : |
| supportedLeg u v i r*star (supportedLeg u v i s)= |
| if r=s then legSupport u v i r else 0 := by |
| cases r <;> cases s <;> |
| simp only [supportedLeg,legSupport,Bool.false_eq_true,ite_false,ite_true, |
| projection_P.isSelfAdjoint.star_eq,projection_P.isIdempotentElem.eq, |
| P_mul_star_edgePolar,edgePolar_mul_P,Bool.true_eq_false,edgeRange] |
|
|
| theorem supportedLeg_trace_covariance (u v : ℝ) (hu : u≠0) (hv : v≠0) |
| (i : ι) (r s : Bool) (T : ambientAlgebra (ι := ι) u v) : |
| vacuumTrace u v (star (supportedLeg u v i r)*(T : Operator ι)*supportedLeg u v i s)= |
| if r=s then vacuumTrace u v ((T : Operator ι)*legSupport u v i r) else 0 := by |
| have ht := ambient_trace_mul u v hu hv |
| ⟨star (supportedLeg u v i r),star_mem (supportedLeg_mem_ambient u v i r)⟩ |
| (T*⟨supportedLeg u v i s,supportedLeg_mem_ambient u v i s⟩) |
| change vacuumTrace u v (star (supportedLeg u v i r)*((T : Operator ι)*supportedLeg u v i s))= |
| vacuumTrace u v (((T : Operator ι)*supportedLeg u v i s)*star (supportedLeg u v i r)) at ht |
| rw [mul_assoc,ht,mul_assoc,supportedLeg_mul_star] |
| by_cases h : r=s |
| · subst s |
| simp only [ite_true] |
| · simp only [h,Ne.symm h,ite_false,mul_zero,vacuumTrace,zero_apply,inner_zero_right] |
|
|
| def supportedCreation (u v : ℝ) (i₀ i : ι) (r s : Bool) : Operator ι := |
| supportedLeg u v i₀ r*C u v i*star (supportedLeg u v i₀ s) |
|
|
| theorem supportedCreation_filled_covariance (u v : ℝ) (hu : u≠0) (hv : v≠0) |
| (i j n m : ι) (hji : j≠i) (hni : n≠i) (hnj : n≠j) (hmi : m≠i) (hmj : m≠j) |
| (r s r' s' : Bool) (T : filledEdgeAlgebra u v i j) : |
| star (supportedCreation u v i n r s)*(T : Operator ι)*supportedCreation u v i m r' s'= |
| if n=m ∧ r=r' ∧ s=s' then |
| vacuumTrace u v ((T : Operator ι)*legSupport u v i r) • legSupport u v i s else 0 := by |
| let a (b : Bool) : filledEdgeAlgebra u v i j := |
| ⟨supportedLeg u v i b,edgeAlgebra_le_filled u v i j (supportedLeg_mem_edge u v i b)⟩ |
| have hc := fresh_C_filled_contraction u v i j n m hji hni hnj hmi hmj (star (a r)*T*a r') |
| change star (C u v n)*(star (supportedLeg u v i r)*(T : Operator ι)*supportedLeg u v i r')*C u v m=_ at hc |
| calc |
| _ = supportedLeg u v i s*(star (C u v n)* |
| (star (supportedLeg u v i r)*(T : Operator ι)*supportedLeg u v i r')*C u v m)* |
| star (supportedLeg u v i s') := by simp only [supportedCreation,star_mul,star_star,mul_assoc] |
| _ = _ := by |
| rw [hc] |
| change (supportedLeg u v i s*(if n=m then |
| vacuumTrace u v (star (supportedLeg u v i r)*(T : Operator ι)*supportedLeg u v i r') • 1 else 0)* |
| star (supportedLeg u v i s'))=_ |
| rw [supportedLeg_trace_covariance u v hu hv i r r' |
| ⟨T,filledEdgeAlgebra_le_ambient u v i j T.property⟩] |
| by_cases hnm : n=m |
| · rw [ite_eq_left hnm,mul_smul_comm,smul_mul_assoc,mul_one,supportedLeg_mul_star] |
| by_cases hr : r=r' <;> by_cases hs : s=s' <;> |
| simp only [hnm,hr,hs,ite_true,ite_false,zero_smul ℂ (A := Operator ι), |
| smul_zero (M := ℂ) (A := Operator ι),true_and,false_and,and_false] |
| · simp only [hnm,ite_false,mul_zero,zero_mul,false_and] |
|
|
| theorem supportedCreation_filled_annihilation (u v : ℝ) |
| (i j n : ι) (hji : j≠i) (hni : n≠i) (hnj : n≠j) |
| (r s : Bool) (T : filledEdgeAlgebra u v i j) : |
| star (supportedCreation u v i n r s) ((T : Operator ι) (vacuum u v))=0 := by |
| let a : filledEdgeAlgebra u v i j := |
| ⟨star (supportedLeg u v i r),star_mem |
| (edgeAlgebra_le_filled u v i j (supportedLeg_mem_edge u v i r))⟩*T |
| have hh := fresh_C_filled_annihilation u v i j n hji hni hnj a |
| simp only [supportedCreation,star_mul,star_star,mul_apply_eq_comp] |
| change supportedLeg u v i s (star (C u v n) ((a : Operator ι) (vacuum u v)))=0 |
| rw [hh,map_zero] |
|
|
| def pVacuumTrace (u v : ℝ) (T : Operator ι) : ℂ := (u:ℂ)⁻¹*(u:ℂ)⁻¹*vacuumTrace u v T |
|
|
| def normalizedCreation (u v : ℝ) (i n : ι) (r s : Bool) : Operator ι := |
| (u:ℂ)⁻¹ • supportedCreation u v i n r s |
|
|
| theorem normalizedCreation_filled_covariance (u v : ℝ) (hu : u≠0) (hv : v≠0) |
| (i j n m : ι) (hji : j≠i) (hni : n≠i) (hnj : n≠j) (hmi : m≠i) (hmj : m≠j) |
| (r s r' s' : Bool) (T : filledEdgeAlgebra u v i j) : |
| star (normalizedCreation u v i n r s)*(T : Operator ι)*normalizedCreation u v i m r' s'= |
| if n=m ∧ r=r' ∧ s=s' then |
| pVacuumTrace u v ((T : Operator ι)*legSupport u v i r) • legSupport u v i s else 0 := by |
| simp only [normalizedCreation,star_smul,Complex.star_def,map_inv₀,Complex.conj_ofReal, |
| smul_mul_assoc,mul_smul_comm] |
| rw [supportedCreation_filled_covariance u v hu hv i j n m hji hni hnj hmi hmj] |
| split_ifs <;> simp only [smul_zero (M := ℂ) (A := Operator ι),smul_smul,pVacuumTrace,mul_assoc] |
|
|
| theorem supportedCreation_paired (u v : ℝ) (i n : ι) (r s : Bool) : |
| supportedCreation u v i n r s+star (supportedCreation u v i n s r)= |
| supportedLeg u v i r*S u v n*star (supportedLeg u v i s) := by |
| simp only [supportedCreation,S_eq,star_mul,star_star,mul_add,add_mul,mul_assoc] |
|
|
| theorem normalizedCreation_paired_mem (u v : ℝ) (i n : ι) (r s : Bool) : |
| normalizedCreation u v i n r s+star (normalizedCreation u v i n s r)∈ambientAlgebra u v := by |
| simp only [normalizedCreation,star_smul,Complex.star_def,map_inv₀,Complex.conj_ofReal, |
| ← smul_add (M := ℂ) (A := Operator ι)] |
| rw [supportedCreation_paired] |
| exact (ambientAlgebra u v).smul_mem |
| (mul_mem (mul_mem (supportedLeg_mem_ambient u v i r) (S_mem_ambient u v n)) |
| (star_mem (supportedLeg_mem_ambient u v i s))) _ |
| end FreeGroupFactorMain.WeightedFock |
|
|
| end |
|
|
| end OAI |
|
|
