| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.OscillatorLimits |
|
|
| namespace OAI |
|
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| /-! Strong core and closed energy extraction with lower limits. -/ |
|
|
| namespace BFSSQuantum.AlgebraData.GaugeData |
| noncomputable section |
| open MeasureTheory Set Filter Topology |
| open scoped NNReal |
| variable {N : ℕ} {M : AlgebraData N} (G : M.GaugeData) |
| attribute [local irreducible] AlgebraData.normalSlice AlgebraData.gaugeAction coreToL2 |
|
|
| structure NormalCoreEnergyData (h : ℕ → ℝ) (f : ℕ → G.invariantCore) (L : ℝ) where |
| S : Finset (Boson N) |
| zero : ∀ q ∈ S, q ∈ M.massiveZeros |
| cover : M.massiveZeros = ⋃ q ∈ S, M.gaugeOrbit q |
| orbitDisjoint : Pairwise (fun i j : S => Disjoint (M.gaugeOrbit i.val) (M.gaugeOrbit j.val)) |
| P : (q : S) → G.MeasuredNormalPatch q.val |
| patchDisjoint : Pairwise (fun i j : S => Disjoint (Function.support (P i).cutoff) (Function.support (P j).cutoff)) |
| u : (q : S) → Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q.val)) |
| v : (q : S) → M.NormalGradient q.val |
| σ : ℕ → ℕ |
| mono : StrictMono σ |
| equivariant : ∀ q : S, u q ∈ G.normalEquivariantSpace q.val |
| domain : ∀ q : S, M.NormalFormDomain q.val (u q) (v q) |
| profiles : ∀ q : S, Tendsto (fun n => (P q).profile (h (σ n)) (f (σ n))) atTop (𝓝 (u q)) |
| norm : Tendsto (fun n => ‖coreToL2 (f (σ n)).val‖^2) atTop (𝓝 (∑ q : S, ‖u q‖^2)) |
| energy : (∑ q : S, M.normalOscillatorForm q.val (u q) (v q)) ≤ L |
|
|
| def NormalCoreEnergyConclusion (h : ℕ → ℝ) (f : ℕ → G.invariantCore) (L : ℝ) : Prop := |
| Nonempty (G.NormalCoreEnergyData h f L) |
|
|
| theorem normalCore_energy_extraction (hN : 2 ≤ N) |
| (h : ℕ → ℝ) (hh : ∀ n, 0 < h n ∧ h n ≤ 1) (hzero : Tendsto h atTop (𝓝 0)) |
| (f : ℕ → G.invariantCore) (E : ℝ≥0) |
| (hnorm : ∀ n, ‖coreToL2 (f n).val‖^2 ≤ E) |
| (henergy : ∀ n, M.deformedCoreEnergy (h n) 1 (f n).val ≤ E) |
| (L : ℝ) (hL : Tendsto (fun n => M.deformedCoreEnergy (h n) 1 (f n).val) atTop (𝓝 L)) : |
| G.NormalCoreEnergyConclusion h f L := by |
| classical |
| unfold NormalCoreEnergyConclusion |
| obtain ⟨S,hS,hcov,hdis,P,hPd,δ,c,K,hδ,hc,hK,hbound⟩ := G.normalPatch_norm_exhaustion hN |
| let C : ℝ := ((E : ℝ)+K*E)/(c/2*δ^2) |
| have hden : 0 < c/2*δ^2 := mul_pos (half_pos hc) (sq_pos_of_pos hδ) |
| let r : ℕ → ℝ := fun n => ‖coreToL2 (f n).val‖^2 - ∑ q : S, ‖(P q).profile (h n) (f n)‖^2 |
| have hr (n : ℕ) : 0 ≤ r n ∧ r n ≤ (h n)^2*C := by |
| have hb := hbound (h n) (hh n).1 (hh n).2 (f n) |
| simp_rw [←MeasuredNormalPatch.profile_norm_square] at hb |
| refine ⟨hb.1,?_⟩ |
| change r n ≤ (h n)^2 * (((E : ℝ)+K*E)/(c/2*δ^2)) |
| rw [←mul_div_assoc] |
| apply (le_div_iff₀ hden).2 |
| have he : M.deformedCoreEnergy (h n) 1 (f n).val + K*‖coreToL2 (f n).val‖^2 ≤ (E : ℝ)+K*E := |
| add_le_add (henergy n) (mul_le_mul_of_nonneg_left (hnorm n) hK) |
| have hs := hb.2.trans (mul_le_mul_of_nonneg_left he (sq_nonneg _)) |
| change r n * (c/2*δ^2) ≤ _ |
| rw [mul_comm (r n)] |
| exact hs |
| have hrzero : Tendsto r atTop (nhds 0) := |
| squeeze_zero (fun n => (hr n).1) (fun n => (hr n).2) |
| (by simpa using (hzero.pow 2).mul_const C) |
| let X (q : S) := Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q.val)) |
| let a : ℕ → ((q : S) → X q) := fun n q => (P q).profile (h n) (f n) |
| have hcompact (q : S) : IsCompact (closure (range (fun n => a n q))) := |
| ((P q).profile_totallyBounded hN h hh hzero f E hnorm henergy).closure.isCompact_of_isClosed isClosed_closure |
| have hprod := isCompact_pi_infinite hcompact |
| obtain ⟨v,hv,σ,hσ,ht⟩ := hprod.tendsto_subseq (fun n q => subset_closure (mem_range_self n)) |
| have hcoord (q : S) : Tendsto (fun n => (P q).profile (h (σ n)) (f (σ n))) atTop (nhds (v q)) := |
| (tendsto_pi_nhds.mp ht) q |
| have hsum : Tendsto (fun n => ∑ q : S, ‖(P q).profile (h (σ n)) (f (σ n))‖^2) atTop |
| (nhds (∑ q : S, ‖v q‖^2)) := tendsto_finsetSum _ (fun q _ => (hcoord q).norm.pow 2) |
| have hgrad := G.normalPatch_summed_energy_limit (fun q : S => q.val) P hN |
| (fun q => hS q.val q.prop) hPd (fun n => h (σ n)) (fun n => hh (σ n)) |
| (hzero.comp hσ.tendsto_atTop) (fun n => f (σ n)) E |
| (fun n => hnorm (σ n)) (fun n => henergy (σ n)) v hcoord |
| (hrzero.comp hσ.tendsto_atTop) L (hL.comp hσ.tendsto_atTop) |
| obtain ⟨w,hw,hwe⟩ := hgrad |
| refine ⟨⟨S,hS,hcov,hdis,P,hPd,v,w,σ,hσ,?_,hw,hcoord,?_,hwe⟩⟩ |
| · intro q |
| exact (G.normalEquivariantSpace_closed q.val).mem_of_tendsto (hcoord q) |
| (Filter.Eventually.of_forall fun n => (P q).profile_mem_equivariantSpace _ _) |
| · have ht' := (hrzero.comp hσ.tendsto_atTop).add hsum |
| simpa only [Function.comp_apply,r,zero_add,sub_add_cancel] using ht' |
| end |
| end BFSSQuantum.AlgebraData.GaugeData |
| namespace BFSSQuantum.AlgebraData.GaugeData |
| noncomputable section |
| open MeasureTheory Set Filter Topology |
| open scoped NNReal |
| variable {N : ℕ} {M : AlgebraData N} (G : M.GaugeData) |
| attribute [local irreducible] AlgebraData.normalSlice AlgebraData.gaugeAction coreToL2 |
| MeasuredNormalPatch.closedProfile |
|
|
| structure NormalClosedEnergyData (h : ℕ → ℝ) (x : ℕ → G.coreNormClosure) (L : ℝ) where |
| S : Finset (Boson N) |
| zero : ∀ q ∈ S, q ∈ M.massiveZeros |
| cover : M.massiveZeros = ⋃ q ∈ S, M.gaugeOrbit q |
| orbitDisjoint : Pairwise (fun i j : S => Disjoint (M.gaugeOrbit i.val) (M.gaugeOrbit j.val)) |
| P : (q : S) → G.MeasuredNormalPatch q.val |
| patchDisjoint : Pairwise (fun i j : S => Disjoint (Function.support (P i).cutoff) (Function.support (P j).cutoff)) |
| u : (q : S) → Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q.val)) |
| v : (q : S) → M.NormalGradient q.val |
| σ : ℕ → ℕ |
| mono : StrictMono σ |
| equivariant : ∀ q : S, u q ∈ G.normalEquivariantSpace q.val |
| domain : ∀ q : S, M.NormalFormDomain q.val (u q) (v q) |
| profiles : ∀ q : S, Tendsto (fun n => (P q).closedProfile (h (σ n)) (x (σ n))) atTop (𝓝 (u q)) |
| norm : Tendsto (fun n => ‖x (σ n)‖^2) atTop (𝓝 (∑ q : S, ‖u q‖^2)) |
| energy : (∑ q : S, M.normalOscillatorForm q.val (u q) (v q)) ≤ L |
|
|
| def NormalClosedEnergyConclusion (h : ℕ → ℝ) (x : ℕ → G.coreNormClosure) (L : ℝ) : Prop := |
| Nonempty (G.NormalClosedEnergyData h x L) |
|
|
| theorem normalClosed_energy_extraction (hN : 2 ≤ N) |
| (h : ℕ → ℝ) (hh : ∀ n, 0 < h n ∧ h n ≤ 1) (hzero : Tendsto h atTop (𝓝 0)) |
| (x : ℕ → G.coreNormClosure) (y : ℕ → SpinIndex → FullL2 N) (B : ℝ≥0) |
| (hg : ∀ n, ((x n : FullL2 N),y n) ∈ G.deformedModelGraph (h n) 1) |
| (hx : ∀ n, ‖x n‖ ≤ B) (hy : ∀ n α, ‖y n α‖ ≤ B) |
| (L : ℝ) (hL : Tendsto (fun n => jointEnergy (y n)) atTop (𝓝 L)) : |
| G.NormalClosedEnergyConclusion h x L := by |
| classical |
| unfold NormalClosedEnergyConclusion |
| let ε : ℕ → ℝ := fun n => 1/((n : ℝ)+1) |
| have hε (n : ℕ) : 0 < ε n := one_div_pos.mpr (by positivity) |
| have hε1 (n : ℕ) : ε n ≤ 1 := by |
| apply (div_le_iff₀ (show 0 < (n : ℝ)+1 by positivity)).2 |
| simp only [one_mul] |
| linarith [Nat.cast_nonneg (α := ℝ) n] |
| have hεzero : Tendsto ε atTop (nhds 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| choose f hf hQ using (fun n => G.deformedModelGraph_approx (h n) 1 (hg n) (ε n) (hε n)) |
| let C : ℝ≥0 := B+1 |
| have hcf (n : ℕ) : ‖coreToL2 (f n).val‖ ≤ C := by |
| have hd : dist (coreToL2 (f n).val) (x n : FullL2 N) ≤ 1 := |
| by rw [dist_comm]; exact (hf n).le.trans (hε1 n) |
| have ha := norm_le_norm_add_const_of_dist_le hd |
| change _ ≤ (B : ℝ)+1 |
| exact ha.trans (add_le_add (show ‖(x n : FullL2 N)‖ ≤ B from hx n) le_rfl) |
| have hqf (n : ℕ) (α : SpinIndex) : ‖coreToL2 (M.deformedCoreCharge (h n) 1 α (f n).val)‖ ≤ C := by |
| have hd : dist (coreToL2 (M.deformedCoreCharge (h n) 1 α (f n).val)) (y n α) ≤ 1 := |
| by rw [dist_comm]; exact (hQ n α).le.trans (hε1 n) |
| have ha := norm_le_norm_add_const_of_dist_le hd |
| change _ ≤ (B : ℝ)+1 |
| linarith [hy n α] |
| have hfs (n : ℕ) : ‖coreToL2 (f n).val‖^2 ≤ (C^2 : ℝ≥0) := |
| pow_le_pow_left₀ (norm_nonneg _) (hcf n) 2 |
| have he (n : ℕ) : M.deformedCoreEnergy (h n) 1 (f n).val ≤ (C^2 : ℝ≥0) := by |
| have hs := Finset.sum_le_sum (s := Finset.univ) (fun α _ => |
| pow_le_pow_left₀ (norm_nonneg _) (hqf n α) 2) |
| unfold AlgebraData.deformedCoreEnergy |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at hs |
| norm_num only [Nat.cast_ofNat] at hs |
| change _ ≤ (C : ℝ)^2 |
| linarith |
| have henergyDist : Tendsto (fun n => dist (jointEnergy (y n)) |
| (M.deformedCoreEnergy (h n) 1 (f n).val)) atTop (𝓝 0) := by |
| apply squeeze_zero (fun n => dist_nonneg) (g := fun n => ε n*(2*(C : ℝ))) _ |
| (by simpa only [zero_mul] using hεzero.mul_const (2*(C : ℝ))) |
| intro n |
| exact jointEnergy_distance_le (y n) (M.deformedChargeVector (h n) 1 (f n).val) |
| (hε n).le (fun α => (hy n α).trans (by change (B : ℝ) ≤ (B : ℝ)+1; linarith)) |
| (hqf n) (fun α => (hQ n α).le) |
| have hdata := G.normalCore_energy_extraction hN h hh hzero f (C^2) hfs he L |
| (hL.congr_dist henergyDist) |
| unfold NormalCoreEnergyConclusion at hdata |
| obtain ⟨⟨S,hS,hcov,hdis,P,hPd,v,w,σ,hσ,hve,hw,hcoord,hnorm,hlim⟩⟩ := hdata |
| have hdist (q : S) : Tendsto (fun n => dist ((P q).profile (h n) (f n)) |
| ((P q).closedProfile (h n) (x n))) atTop (nhds 0) := by |
| apply squeeze_zero (fun n => dist_nonneg) _ hεzero |
| intro n |
| exact ((P q).closedProfile_core_dist_le (hh n).1 (hh n).2 (f n) (x n)).trans |
| (by rw [dist_comm]; exact (hf n).le) |
| refine ⟨⟨S,hS,hcov,hdis,P,hPd,v,w,σ,hσ,hve,hw, |
| fun q => (hcoord q).congr_dist ((hdist q).comp hσ.tendsto_atTop),?_,hlim⟩⟩ |
| have hnormdist : Tendsto (fun n => dist (‖coreToL2 (f n).val‖^2) (‖x n‖^2)) atTop (nhds 0) := by |
| apply squeeze_zero (fun n => dist_nonneg) |
| (g := fun n => ε n*((C : ℝ)+B)) _ (by simpa only [zero_mul] using hεzero.mul_const ((C : ℝ)+B)) |
| intro n |
| have hd := BFSSRellich.normSquare_distance_le (coreToL2 (f n).val) (x n : FullL2 N) |
| apply hd.trans |
| apply mul_le_mul |
| · rw [dist_comm]; exact (hf n).le |
| · exact add_le_add (hcf n) (hx n) |
| · exact add_nonneg (norm_nonneg _) (norm_nonneg _) |
| · exact (hε n).le |
| exact hnorm.congr_dist (hnormdist.comp hσ.tendsto_atTop) |
| end |
| end BFSSQuantum.AlgebraData.GaugeData |
| namespace BFSSNormalAnalysis |
| noncomputable section |
| open Set Filter Topology |
|
|
| lemma nonnegative_bounded_liminf_subsequence (a : ℕ → ℝ) (C : ℝ) |
| (ha : ∀ n, 0 ≤ a n) (hb : ∀ n, a n ≤ C) : |
| ∃ τ : ℕ → ℕ, StrictMono τ ∧ Tendsto (a ∘ τ) atTop (𝓝 (liminf a atTop)) := by |
| have hlow : IsBoundedUnder (· ≥ ·) atTop a := by |
| refine ⟨0, ?_⟩ |
| change ∀ᶠ n in atTop, 0 ≤ a n |
| exact Eventually.of_forall ha |
| have hupp : IsBoundedUnder (· ≤ ·) atTop a := by |
| refine ⟨C, ?_⟩ |
| change ∀ᶠ n in atTop, a n ≤ C |
| exact Eventually.of_forall hb |
| obtain ⟨t, ht, htTop⟩ := exists_seq_tendsto_liminf hupp.isCoboundedUnder_flip hlow |
| obtain ⟨s, hs, hts⟩ := strictMono_subseq_of_tendsto_atTop htTop |
| exact ⟨t ∘ s, hts, ht.comp hs.tendsto_atTop⟩ |
| end |
| end BFSSNormalAnalysis |
|
|
| namespace BFSSQuantum |
| noncomputable section |
| open Set Filter Topology Finset |
| open scoped NNReal |
| variable {N : ℕ} |
|
|
| lemma charge_norm_bound_of_energy (y : SpinIndex → FullL2 N) (E : ℝ≥0) |
| (he : jointEnergy y ≤ E) (α : SpinIndex) : ‖y α‖ ≤ 16*(E : ℝ)+1 := by |
| have hs : ‖y α‖^2 ≤ ∑ β : SpinIndex, ‖y β‖^2 := |
| Finset.single_le_sum (fun β _ => sq_nonneg ‖y β‖) (Finset.mem_univ α) |
| unfold jointEnergy at he |
| have hy := norm_nonneg (y α) |
| have hE : (0 : ℝ) ≤ E := E.coe_nonneg |
| nlinarith [sq_nonneg (‖y α‖-1)] |
| end |
| end BFSSQuantum |
|
|
| namespace BFSSQuantum.AlgebraData.GaugeData |
| noncomputable section |
| open Set Filter Topology MeasureTheory |
| open scoped NNReal |
| variable {N : ℕ} {M : AlgebraData N} (G : M.GaugeData) |
| attribute [local irreducible] AlgebraData.normalSlice AlgebraData.gaugeAction coreToL2 |
| MeasuredNormalPatch.closedProfile |
|
|
| def NormalClosedEnergyData.reindex {h : ℕ → ℝ} {x : ℕ → G.coreNormClosure} {L : ℝ} |
| {τ : ℕ → ℕ} (D : G.NormalClosedEnergyData (h ∘ τ) (x ∘ τ) L) (hτ : StrictMono τ) : |
| G.NormalClosedEnergyData h x L where |
| S := D.S |
| zero := D.zero |
| cover := D.cover |
| orbitDisjoint := D.orbitDisjoint |
| P := D.P |
| patchDisjoint := D.patchDisjoint |
| u := D.u |
| v := D.v |
| σ := τ ∘ D.σ |
| mono := hτ.comp D.mono |
| equivariant := D.equivariant |
| domain := D.domain |
| profiles := D.profiles |
| norm := D.norm |
| energy := D.energy |
|
|
| theorem normalClosed_liminf_extraction (hN : 2 ≤ N) |
| (h : ℕ → ℝ) (hh : ∀ n, 0 < h n ∧ h n ≤ 1) (hzero : Tendsto h atTop (𝓝 0)) |
| (x : ℕ → G.coreNormClosure) (y : ℕ → SpinIndex → FullL2 N) (A E : ℝ≥0) |
| (hg : ∀ n, ((x n : FullL2 N),y n) ∈ G.deformedModelGraph (h n) 1) |
| (hx : ∀ n, ‖x n‖ ≤ A) (he : ∀ n, jointEnergy (y n) ≤ E) : |
| G.NormalClosedEnergyConclusion h x (liminf (fun n => jointEnergy (y n)) atTop) := by |
| obtain ⟨τ,hτ,ht⟩ := BFSSNormalAnalysis.nonnegative_bounded_liminf_subsequence |
| (fun n => jointEnergy (y n)) E (fun n => jointEnergy_nonneg _) he |
| let B : ℝ≥0 := A+16*E+1 |
| have hBx (n : ℕ) : ‖x n‖ ≤ B := by |
| change ‖x n‖ ≤ (A : ℝ)+16*(E : ℝ)+1 |
| linarith [hx n,E.coe_nonneg] |
| have hBy (n : ℕ) (α : SpinIndex) : ‖y n α‖ ≤ B := by |
| change ‖y n α‖ ≤ (A : ℝ)+16*(E : ℝ)+1 |
| linarith [charge_norm_bound_of_energy (y n) E (he n) α,A.coe_nonneg] |
| have hd := G.normalClosed_energy_extraction hN (h ∘ τ) (fun n => hh (τ n)) |
| (hzero.comp hτ.tendsto_atTop) (x ∘ τ) (y ∘ τ) B (fun n => hg (τ n)) |
| (fun n => hBx (τ n)) (fun n α => hBy (τ n) α) _ ht |
| obtain ⟨D⟩ := hd |
| exact ⟨D.reindex G hτ⟩ |
| end |
| end BFSSQuantum.AlgebraData.GaugeData |
|
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| end OAI |
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