| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.EnergyExtraction |
| import OAI.MathematicalPhysics.BFSS.LowerContinuity |
|
|
| namespace OAI |
|
|
| /-! Positive normal-form gap and nonzero zero-form extraction. -/ |
|
|
| 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 |
|
|
| def NormalZeroOrthogonal (q : Boson N) |
| (u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q))) : Prop := |
| ∀ z : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q)), |
| ∀ v : M.NormalGradient q, M.NormalFormDomain q z v → |
| M.normalOscillatorForm q z v = 0 → inner ℝ u z = 0 |
|
|
| theorem normal_unit_gap (q : Boson N) (hq : q ∈ M.massiveZeros) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q)), |
| ∀ v : M.NormalGradient q, M.NormalFormDomain q u v → ‖u‖ = 1 → |
| M.NormalZeroOrthogonal q u → δ ≤ M.normalOscillatorForm q u v := by |
| by_contra hgap |
| 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 hbad (n : ℕ) : ∃ u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q)), |
| ∃ v : M.NormalGradient q, M.NormalFormDomain q u v ∧ ‖u‖ = 1 ∧ |
| M.NormalZeroOrthogonal q u ∧ M.normalOscillatorForm q u v < ε n := by |
| by_contra! hn |
| apply hgap |
| exact ⟨ε n,hε n,hn⟩ |
| choose f v hd hn ho he using hbad |
| have hE (n : ℕ) : M.normalOscillatorForm q (f n) (v n) ≤ (1 : ℝ≥0) := |
| (he n).le.trans (hε1 n) |
| have hB (n : ℕ) : ‖f n‖ ≤ (1 : ℝ≥0) := by simp only [hn,NNReal.coe_one,le_refl] |
| have hzero : Tendsto (fun n => M.normalOscillatorForm q (f n) (v n)) atTop (𝓝 0) := |
| squeeze_zero (fun n => M.normalOscillator_nonneg q hq (f n) (v n) (hd n)) |
| (fun n => (he n).le) tendsto_one_div_add_atTop_nhds_zero_nat |
| have hc : IsCompact (closure (range f)) := |
| (M.normalForm_totallyBounded q hq f v 1 1 hd hB hE).closure.isCompact_of_isClosed isClosed_closure |
| obtain ⟨u,_,σ,hσ,hu⟩ := hc.tendsto_subseq (fun n => subset_closure (mem_range_self n)) |
| have hun : ‖u‖ = 1 := tendsto_nhds_unique hu.norm |
| (by simpa only [Function.comp_apply,hn] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (1 : ℝ)) atTop (𝓝 1))) |
| obtain ⟨z,hz,hzE⟩ := M.normalForm_lsc q (fun n => f (σ n)) (fun n => v (σ n)) u |
| 1 1 0 (fun n => hd (σ n)) (fun n => hB (σ n)) (fun n => hE (σ n)) hu |
| (hzero.comp hσ.tendsto_atTop) |
| have hz0 : M.normalOscillatorForm q u z = 0 := le_antisymm hzE (M.normalOscillator_nonneg q hq u z hz) |
| have horth (n : ℕ) : inner ℝ (f (σ n)) u = 0 := ho (σ n) u z hz hz0 |
| have hi : inner ℝ u u = 0 := tendsto_nhds_unique (hu.inner tendsto_const_nhds) |
| (by simpa only [Function.comp_apply,horth] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))) |
| rw [real_inner_self_eq_norm_sq,hun] at hi |
| norm_num at hi |
| end |
| end BFSSQuantum.AlgebraData |
| namespace BFSSNormalAnalysis |
| noncomputable section |
| open MeasureTheory Filter Topology |
| variable {X F : Type*} [MeasurableSpace X] [NormedAddCommGroup F] |
| [NormedSpace ℝ F] {μ : Measure X} |
|
|
| lemma moment_smul_memLp (w : X → ℝ) (u : Lp F 2 μ) |
| (hu : MemLp (fun x => w x • u x) 2 μ) (r : ℝ) : |
| MemLp (fun x => w x • (r • u) x) 2 μ := by |
| apply MemLp.ae_eq _ (hu.const_smul r) |
| filter_upwards [Lp.coeFn_smul r u] with x hx |
| simp only [Pi.smul_apply] at hx |
| rw [hx] |
| exact smul_comm _ _ _ |
|
|
| lemma moment_smul_toLp (w : X → ℝ) (u : Lp F 2 μ) |
| (hu : MemLp (fun x => w x • u x) 2 μ) (r : ℝ) |
| (hr : MemLp (fun x => w x • (r • u) x) 2 μ) : |
| hr.toLp (fun x => w x • (r • u) x) = r • hu.toLp (fun x => w x • u x) := by |
| apply Lp.ext |
| filter_upwards [hr.coeFn_toLp,hu.coeFn_toLp,Lp.coeFn_smul r (hu.toLp _),Lp.coeFn_smul r u] |
| with x h₁ h₂ h₃ h₄ |
| simp only [Pi.smul_apply] at h₃ h₄ |
| rw [h₁,h₃,h₂,h₄,smul_comm] |
|
|
| lemma weighted_energy_smul {F' : Type*} [NormedAddCommGroup F'] |
| [InnerProductSpace ℝ F'] {G : Type*} |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| (A : F' →L[ℝ] F') (u z : Lp F' 2 μ) (v : G) (r : ℝ) : |
| (1/2 : ℝ)*‖r • v‖^2+‖r • z‖^2+inner ℝ (r • u) (A.compLp (r • u)) = |
| r^2*((1/2 : ℝ)*‖v‖^2+‖z‖^2+inner ℝ u (A.compLp u)) := by |
| have he : A.compLp (r • u) = r • A.compLp u := (A.compLpL 2 μ).map_smul r u |
| rw [he,real_inner_smul_left,real_inner_smul_right,norm_smul,norm_smul, |
| Real.norm_eq_abs,mul_pow,mul_pow,sq_abs] |
| ring |
|
|
| end |
| end BFSSNormalAnalysis |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open MeasureTheory Filter Topology Set |
| variable {N : ℕ} (M : AlgebraData N) |
| attribute [local irreducible] normalSlice gaugeAction coreToL2 deformedFermionField |
| local instance normalGapFermionNormedAddCommGroup : |
| NormedAddCommGroup (Fermion N →L[ℝ] Fermion N) := inferInstance |
| local instance normalGapFermionNormedSpace : |
| NormedSpace ℝ (Fermion N →L[ℝ] Fermion N) := inferInstance |
|
|
| lemma NormalFormDomain.smul {q : Boson N} |
| {u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q))} |
| {v : M.NormalGradient q} (hd : M.NormalFormDomain q u v) (r : ℝ) : |
| M.NormalFormDomain q (r • u) (r • v) := |
| ⟨(BFSSSliceSobolev.closedGraph volume (EuclideanSpace.basisFun (M.NormalIndex q) ℝ)).smul_mem r hd.1, |
| BFSSNormalAnalysis.moment_smul_memLp (M.normalMomentWeight q) u hd.2 r⟩ |
|
|
| lemma normalOscillator_smul (q : Boson N) |
| (u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q))) |
| (v : M.NormalGradient q) (hd : M.NormalFormDomain q u v) (r : ℝ) : |
| M.normalOscillatorForm q (r • u) (r • v) = r^2 * M.normalOscillatorForm q u v := by |
| let z := hd.2.toLp (fun x => M.normalMomentWeight q x • u x) |
| let zr := (hd.smul M r).2.toLp (fun x => M.normalMomentWeight q x • (r • u) x) |
| have hm : zr = r • z := BFSSNormalAnalysis.moment_smul_toLp |
| (M.normalMomentWeight q) u hd.2 r (hd.smul M r).2 |
| have hE : (1/2 : ℝ)*‖r • v‖^2+‖r • z‖^2+ |
| inner ℝ (r • u) ((M.deformedFermionField 1 1 q).compLp (r • u)) = |
| r^2*((1/2 : ℝ)*‖v‖^2+‖z‖^2+ |
| inner ℝ u ((M.deformedFermionField 1 1 q).compLp u)) := |
| BFSSNormalAnalysis.weighted_energy_smul (M.deformedFermionField 1 1 q) u z v r |
|
|
| calc |
| M.normalOscillatorForm q (r • u) (r • v) = (1/2 : ℝ)*‖r • v‖^2+‖zr‖^2+ |
| inner ℝ (r • u) ((M.deformedFermionField 1 1 q).compLp (r • u)) := |
| M.normalOscillatorForm_weighted q (r • u) (r • v) (hd.smul M r) |
| _ = (1/2 : ℝ)*‖r • v‖^2+‖r • z‖^2+ |
| inner ℝ (r • u) ((M.deformedFermionField 1 1 q).compLp (r • u)) := |
| congrArg (fun t => (1/2 : ℝ)*‖r • v‖^2+‖t‖^2+ |
| inner ℝ (r • u) ((M.deformedFermionField 1 1 q).compLp (r • u))) hm |
| _ = r^2*((1/2 : ℝ)*‖v‖^2+‖z‖^2+ |
| inner ℝ u ((M.deformedFermionField 1 1 q).compLp u)) := hE |
| _ = r^2*M.normalOscillatorForm q u v := |
| congrArg (r^2 * ·) (M.normalOscillatorForm_weighted q u v hd).symm |
|
|
| theorem normal_gap (q : Boson N) (hq : q ∈ M.massiveZeros) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ u : Lp (Fermion N) 2 (volume : Measure (M.NormalCoordinates q)), |
| ∀ v : M.NormalGradient q, M.NormalFormDomain q u v → |
| M.NormalZeroOrthogonal q u → δ*‖u‖^2 ≤ M.normalOscillatorForm q u v := by |
| obtain ⟨δ,hδ,hgap⟩ := M.normal_unit_gap q hq |
| refine ⟨δ,hδ,?_⟩ |
| intro u v hd ho |
| by_cases hu : u = 0 |
| · simpa only [hu,norm_zero,zero_pow (by decide : 2 ≠ 0),mul_zero] using |
| M.normalOscillator_nonneg q hq u v hd |
| have hn : 0 < ‖u‖ := norm_pos_iff.mpr hu |
| let r : ℝ := ‖u‖⁻¹ |
| have hr : 0 < r := inv_pos.mpr hn |
| have hunit : ‖r • u‖ = 1 := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos hr] |
| exact inv_mul_cancel₀ hn.ne' |
| have hor : M.NormalZeroOrthogonal q (r • u) := by |
| intro z w hz he |
| rw [real_inner_smul_left,ho z w hz he,mul_zero] |
| have hs := hgap (r • u) (r • v) (hd.smul M r) hunit hor |
| rw [M.normalOscillator_smul q u v hd r] at hs |
| have hmul := mul_le_mul_of_nonneg_right hs (sq_nonneg ‖u‖) |
| have he : r^2*‖u‖^2 = 1 := by rw [←mul_pow,inv_mul_cancel₀ hn.ne',one_pow] |
| calc |
| δ*‖u‖^2 ≤ r^2*M.normalOscillatorForm q u v*‖u‖^2 := hmul |
| _ = M.normalOscillatorForm q u v := by rw [mul_right_comm,he,one_mul] |
| end |
| end BFSSQuantum.AlgebraData |
| 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 |
|
|
| theorem NormalClosedEnergyData.zero_forms {h : ℕ → ℝ} {x : ℕ → G.coreNormClosure} |
| (D : G.NormalClosedEnergyData h x 0) (q : D.S) : |
| M.normalOscillatorForm q.val (D.u q) (D.v q) = 0 := by |
| have hp (r : D.S) : 0 ≤ M.normalOscillatorForm r.val (D.u r) (D.v r) := |
| M.normalOscillator_nonneg r.val (D.zero r.val r.property) (D.u r) (D.v r) (D.domain r) |
| apply le_antisymm _ (hp q) |
| exact (Finset.single_le_sum (fun r _ => hp r) (Finset.mem_univ q)).trans D.energy |
|
|
| theorem NormalClosedEnergyData.exists_nonzero_zero_form {h : ℕ → ℝ} |
| {x : ℕ → G.coreNormClosure} (D : G.NormalClosedEnergyData h x 0) |
| (hx : ∀ n, ‖x n‖ = 1) : |
| ∃ q : D.S, D.u q ≠ 0 ∧ M.normalOscillatorForm q.val (D.u q) (D.v q) = 0 := by |
| classical |
| have hn : ∑ q : D.S, ‖D.u q‖^2 = 1 := by |
| apply tendsto_nhds_unique D.norm |
| simpa only [hx,one_pow] using (tendsto_const_nhds : |
| Tendsto (fun _ : ℕ => (1 : ℝ)) atTop (𝓝 1)) |
| have he : ∃ q : D.S, D.u q ≠ 0 := by |
| by_contra! he |
| simp only [he,norm_zero,zero_pow (by decide : 2 ≠ 0),Finset.sum_const_zero] at hn |
| norm_num at hn |
| obtain ⟨q,hq⟩ := he |
| exact ⟨q,hq,D.zero_forms q⟩ |
| end |
| end BFSSQuantum.AlgebraData.GaugeData |
|
|
| end OAI |
|
|
