| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.FormCompactness |
|
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| namespace OAI |
|
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| /-! Weighted moment and closed normal-form lower semicontinuity. -/ |
|
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| namespace BFSSNormalAnalysis |
| noncomputable section |
| open MeasureTheory Filter Topology |
| variable {X F : Type*} [MeasurableSpace X] [NormedAddCommGroup F] |
| [NormedSpace ℝ F] {μ : Measure X} |
|
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| theorem weighted_moment_norm_lsc (f : ℕ → Lp F 2 μ) (u : Lp F 2 μ) |
| (w : X → ℝ) (hw : AEStronglyMeasurable w μ) (B : ℝ) |
| (hf : Tendsto f atTop (𝓝 u)) |
| (hm : ∀ n, MemLp (fun x => w x • f n x) 2 μ) |
| (hB : Tendsto (fun n => ‖(hm n).toLp (fun x => w x • f n x)‖) atTop (𝓝 B)) : |
| ∃ hu : MemLp (fun x => w x • u x) 2 μ, |
| ‖hu.toLp (fun x => w x • u x)‖ ≤ B := by |
| have hB0 : 0 ≤ B := ge_of_tendsto' hB (fun n => norm_nonneg _) |
| obtain ⟨σ,hσ,ha⟩ := (tendstoInMeasure_of_tendsto_Lp hf).exists_seq_tendsto_ae |
| have he (n : ℕ) : eLpNorm (fun x => w x • f (σ n) x) 2 μ = |
| ENNReal.ofReal ‖(hm (σ n)).toLp (fun x => w x • f (σ n) x)‖ := by |
| rw [Lp.norm_toLp,ENNReal.ofReal_toReal (hm (σ n)).eLpNorm_ne_top] |
| have ht : Tendsto (fun n => eLpNorm (fun x => w x • f (σ n) x) 2 μ) |
| atTop (𝓝 (ENNReal.ofReal B)) := by |
| simp_rw [he] |
| exact ENNReal.continuous_ofReal.tendsto B |>.comp (hB.comp hσ.tendsto_atTop) |
| have hlim : eLpNorm (fun x => w x • u x) 2 μ ≤ ENNReal.ofReal B := by |
| have hfatou := Lp.eLpNorm_lim_le_liminf_eLpNorm (p := 2) |
| (fun n => (hm (σ n)).aestronglyMeasurable) (fun x => w x • u x) |
| (hw.smul (Lp.memLp u).aestronglyMeasurable) |
| (ha.mono (fun x hx => tendsto_const_nhds.smul hx)) |
| rwa [ht.liminf_eq] at hfatou |
| have hu : MemLp (fun x => w x • u x) 2 μ := |
| hlim.trans_lt ENNReal.ofReal_lt_top |
| refine ⟨hu,?_⟩ |
| rw [Lp.norm_toLp] |
| exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hlim).trans_eq (ENNReal.toReal_ofReal hB0) |
|
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| lemma weighted_moment_sq_lsc (f : ℕ → Lp F 2 μ) (u : Lp F 2 μ) |
| (w : X → ℝ) (hw : AEStronglyMeasurable w μ) (B : ℝ) |
| (hf : Tendsto f atTop (𝓝 u)) |
| (hm : ∀ n, MemLp (fun x => w x • f n x) 2 μ) |
| (hB : Tendsto (fun n => ‖(hm n).toLp (fun x => w x • f n x)‖^2) atTop (𝓝 B)) : |
| ∃ hu : MemLp (fun x => w x • u x) 2 μ, |
| ‖hu.toLp (fun x => w x • u x)‖^2 ≤ B := by |
| have hB0 : 0 ≤ B := ge_of_tendsto' hB (fun n => sq_nonneg _) |
| have hn : Tendsto (fun n => ‖(hm n).toLp (fun x => w x • f n x)‖) atTop (𝓝 (Real.sqrt B)) := by |
| simpa only [Real.sqrt_sq_eq_abs,abs_norm] using hB.sqrt |
| obtain ⟨hu,h⟩ := weighted_moment_norm_lsc f u w hw (Real.sqrt B) hf hm hn |
| refine ⟨hu,?_⟩ |
| exact (pow_le_pow_left₀ (norm_nonneg _) h 2).trans_eq (Real.sq_sqrt hB0) |
| end |
| end BFSSNormalAnalysis |
| namespace BFSSSliceSobolev |
| noncomputable section |
| open MeasureTheory Set Filter Topology |
| variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] |
| [TopologicalSpace.SeparableSpace F] |
| variable (μ : Measure E) [IsFiniteMeasureOnCompacts μ] [IsSeparable μ] |
| variable {ι : Type*} [Fintype ι] (e : ι → E) |
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| omit [FiniteDimensional ℝ E] in |
| lemma closedGraph_extract_sequence (f : ℕ → Lp F 2 μ) |
| (v : ℕ → Gradient (F := F) μ (ι := ι)) (u : Lp F 2 μ) (B : ℝ) |
| (hd : ∀ n, (f n,v n) ∈ closedGraph μ e) |
| (hf : Tendsto f atTop (𝓝 u)) (hB : ∀ n, ‖v n‖ ≤ B) : |
| ∃ (z : Gradient (F := F) μ (ι := ι)) (σ : ℕ → ℕ), StrictMono σ ∧ |
| (u,z) ∈ closedGraph μ e ∧ |
| ∀ y : Gradient (F := F) μ (ι := ι), |
| Tendsto (fun n => inner ℝ (v (σ n)) y) atTop (𝓝 (inner ℝ z y)) := by |
| let : Fact ((2 : ENNReal) ≠ ⊤) := ⟨by norm_num⟩ |
| let : SecondCountableTopology (Lp F 2 μ) := Lp.SecondCountableTopology |
| let : SecondCountableTopology (Gradient (F := F) μ (ι := ι)) := |
| PiLp.secondCountableTopology 2 (fun _ : ι => Lp F 2 μ) |
| obtain ⟨z,σ,hσ,_,hz⟩ := BFSSNormalAnalysis.bounded_weak_subsequence v B hB |
| refine ⟨z,σ,hσ,?_,hz⟩ |
| exact BFSSNormalAnalysis.strong_weak_mem_closed_convex |
| (closedGraph (F := F) μ e : Set _) (closedGraph (F := F) μ e).convex |
| (graph (F := F) μ e).isClosed_topologicalClosure |
| (fun n => f (σ n)) (fun n => v (σ n)) u z |
| (fun n => hd (σ n)) (hf.comp hσ.tendsto_atTop) hz |
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| omit [FiniteDimensional ℝ E] in |
| theorem weighted_closed_form_lsc (w : E → ℝ) (hw : AEStronglyMeasurable w μ) |
| (A : F →L[ℝ] F) (f : ℕ → Lp F 2 μ) |
| (v : ℕ → Gradient (F := F) μ (ι := ι)) (u : Lp F 2 μ) (B C L : ℝ) |
| (hd : ∀ n, (f n,v n) ∈ closedGraph μ e) |
| (hm : ∀ n, MemLp (fun x => w x • f n x) 2 μ) |
| (hf : Tendsto f atTop (𝓝 u)) (hB : ∀ n, ‖v n‖ ≤ B) |
| (hC : ∀ n, ‖(hm n).toLp (fun x => w x • f n x)‖ ≤ C) |
| (hL : Tendsto (fun n => (1/2 : ℝ)*‖v n‖^2+ |
| ‖(hm n).toLp (fun x => w x • f n x)‖^2+inner ℝ (f n) (A.compLp (f n))) atTop (𝓝 L)) : |
| ∃ z : Gradient (F := F) μ (ι := ι), ∃ hu : MemLp (fun x => w x • u x) 2 μ, |
| (u,z) ∈ closedGraph μ e ∧ |
| (1/2 : ℝ)*‖z‖^2+‖hu.toLp (fun x => w x • u x)‖^2+inner ℝ u (A.compLp u) ≤ L := by |
| let : Fact ((2 : ENNReal) ≠ ⊤) := ⟨by norm_num⟩ |
| let : SecondCountableTopology (Lp F 2 μ) := Lp.SecondCountableTopology |
| let : SecondCountableTopology (Gradient (F := F) μ (ι := ι)) := |
| PiLp.secondCountableTopology 2 (fun _ : ι => Lp F 2 μ) |
| let a : ℕ → Fin 2 → ℝ := fun n => ![‖v n‖^2,‖(hm n).toLp (fun x => w x • f n x)‖^2] |
| have ha (n : ℕ) (i : Fin 2) : 0 ≤ a n i := by |
| fin_cases i <;> exact sq_nonneg _ |
| have hb (n : ℕ) : ∑ i, a n i ≤ B^2+C^2 := by |
| simp only [Fin.sum_univ_two,a,Matrix.cons_val_zero,Matrix.cons_val_one] |
| exact add_le_add (pow_le_pow_left₀ (norm_nonneg _) (hB n) 2) |
| (pow_le_pow_left₀ (norm_nonneg _) (hC n) 2) |
| obtain ⟨b,σ,hσ,hb⟩ := BFSSNormalAnalysis.finite_nonnegative_energy_subsequence a _ ha hb |
| have ht0 : Tendsto (fun n => ‖v (σ n)‖^2) atTop (𝓝 (b 0)) := hb 0 |
| have ht1 : Tendsto (fun n => ‖(hm (σ n)).toLp (fun x => w x • f (σ n) x)‖^2) |
| atTop (𝓝 (b 1)) := hb 1 |
| obtain ⟨z,τ,hτ,hz,hwk⟩ := closedGraph_extract_sequence μ e |
| (fun n => f (σ n)) (fun n => v (σ n)) u B (fun n => hd (σ n)) |
| (hf.comp hσ.tendsto_atTop) (fun n => hB (σ n)) |
| have hkin : ‖z‖^2 ≤ b 0 := BFSSNormalAnalysis.weak_norm_sq_le_limit |
| (fun n => v (σ (τ n))) z (b 0) hwk (ht0.comp hτ.tendsto_atTop) |
| obtain ⟨hu,hpot⟩ := BFSSNormalAnalysis.weighted_moment_sq_lsc |
| (fun n => f (σ n)) u w hw (b 1) (hf.comp hσ.tendsto_atTop) |
| (fun n => hm (σ n)) ht1 |
| have hferm : Tendsto (fun n => inner ℝ (f (σ n)) (A.compLp (f (σ n)))) |
| atTop (𝓝 (inner ℝ u (A.compLp u))) := |
| (hf.comp hσ.tendsto_atTop).inner |
| ((A.compLpL 2 μ).continuous.tendsto u |>.comp (hf.comp hσ.tendsto_atTop)) |
| have heq : (1/2 : ℝ)*b 0+b 1+inner ℝ u (A.compLp u) = L := |
| tendsto_nhds_unique ((ht0.const_mul (1/2 : ℝ)).add ht1 |>.add hferm) |
| (hL.comp hσ.tendsto_atTop) |
| refine ⟨z,hu,hz,?_⟩ |
| linarith only [hkin,hpot,heq] |
| end |
| end BFSSSliceSobolev |
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open MeasureTheory Set Filter Topology |
| open scoped NNReal |
| variable {N : ℕ} (M : AlgebraData N) |
| attribute [local irreducible] normalSlice gaugeAction coreToL2 deformedFermionField |
| local instance lowerContinuityFermionNormedAddCommGroup : |
| NormedAddCommGroup (Fermion N →L[ℝ] Fermion N) := inferInstance |
| local instance lowerContinuityFermionNormedSpace : |
| NormedSpace ℝ (Fermion N →L[ℝ] Fermion N) := inferInstance |
|
|
| theorem normalForm_lsc (q : Boson N) |
| (f : ℕ → Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q))) |
| (v : ℕ → M.NormalGradient q) (u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q))) |
| (B E : ℝ≥0) (L : ℝ) |
| (hd : ∀ n, M.NormalFormDomain q (f n) (v n)) |
| (hB : ∀ n, ‖f n‖ ≤ B) (hE : ∀ n, M.normalOscillatorForm q (f n) (v n) ≤ E) |
| (hf : Tendsto f atTop (𝓝 u)) |
| (hL : Tendsto (fun n => M.normalOscillatorForm q (f n) (v n)) atTop (𝓝 L)) : |
| ∃ z : M.NormalGradient q, M.NormalFormDomain q u z ∧ M.normalOscillatorForm q u z ≤ L := by |
| let w := M.normalMomentWeight q |
| let mw (n : ℕ) := (hd n).2.toLp (fun x => w x • f n x) |
| let K : ℝ := E+‖M.deformedFermionField 1 1 q‖*(B : ℝ)^2 |
| have hK : 0 ≤ K := add_nonneg E.coe_nonneg (mul_nonneg (norm_nonneg _) (sq_nonneg _)) |
| have hbound (n : ℕ) : (1/2 : ℝ)*‖v n‖^2+‖mw n‖^2 ≤ K := by |
| exact (M.normalOscillator_coercive q (f n) (v n) (hd n)).trans |
| (add_le_add (hE n) (mul_le_mul_of_nonneg_left |
| (pow_le_pow_left₀ (norm_nonneg _) (hB n) 2) (norm_nonneg _))) |
| have hA (n : ℕ) : ‖v n‖ ≤ Real.sqrt (2*K) := by |
| apply (sq_le_sq₀ (norm_nonneg _) (Real.sqrt_nonneg _)).mp |
| rw [Real.sq_sqrt (by positivity)] |
| nlinarith [hbound n,sq_nonneg ‖mw n‖] |
| have hC (n : ℕ) : ‖mw n‖ ≤ Real.sqrt K := by |
| apply (sq_le_sq₀ (norm_nonneg _) (Real.sqrt_nonneg _)).mp |
| rw [Real.sq_sqrt hK] |
| nlinarith [hbound n,sq_nonneg ‖v n‖] |
| have hform : Tendsto (fun n => (1/2 : ℝ)*‖v n‖^2+‖mw n‖^2+ |
| inner ℝ (f n) ((M.deformedFermionField 1 1 q).compLp (f n))) atTop (𝓝 L) := by |
| convert hL using 1 |
| ext n |
| exact (M.normalOscillatorForm_weighted q (f n) (v n) (hd n)).symm |
| obtain ⟨z,hu,hz,he⟩ := BFSSSliceSobolev.weighted_closed_form_lsc volume |
| (EuclideanSpace.basisFun (M.NormalIndex q) ℝ) w |
| (M.normalMomentWeight_continuous q).aestronglyMeasurable |
| (M.deformedFermionField 1 1 q) f v u (Real.sqrt (2*K)) (Real.sqrt K) L |
| (fun n => (hd n).1) (fun n => (hd n).2) hf hA hC hform |
| have hzu : M.NormalFormDomain q u z := ⟨hz,hu⟩ |
| refine ⟨z,hzu,?_⟩ |
| rw [M.normalOscillatorForm_weighted q u z hzu] |
| exact he |
| end |
| end BFSSQuantum.AlgebraData |
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| end OAI |
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