| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.OrbitDensity |
|
|
| namespace OAI |
|
|
| /-! Gauge-neighborhood cutoffs and local integral bounds. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Matrix MeasureTheory Set TopologicalSpace |
| open scoped Matrix.Norms.L2Operator |
| variable {N : ℕ} (M : AlgebraData N) |
| local instance gaugeMeasurableSpace₅ : MeasurableSpace (unitary (ColorMatrix N)) := borel _ |
| local instance gaugeBorelSpace₅ : BorelSpace (unitary (ColorMatrix N)) := ⟨rfl⟩ |
| attribute [local irreducible] gaugeAction |
|
|
| lemma gauge_neighborhood_cutoff (hN : 2 ≤ N) (q : Boson N) |
| (V : Set (Boson N)) (hV : IsOpen V) (hq : q ∈ V) |
| (hVi : ∀ U : unitary (ColorMatrix N), ∀ x, M.gaugeAction U x ∈ V ↔ x ∈ V) : |
| ∃ φ : Boson N → ℝ, Continuous φ ∧ (∀ x, 0 ≤ φ x ∧ φ x ≤ 1) ∧ |
| 0 < φ q ∧ (∀ U : unitary (ColorMatrix N), ∀ x, φ (M.gaugeAction U x) = φ x) ∧ |
| Function.support φ ⊆ V := by |
| obtain ⟨r,hr,hrsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp (hV.mem_nhds hq) |
| let b : ContDiffBump q := ⟨r/2,r,half_pos hr,half_lt_self hr⟩ |
| let φ := M.gaugeAverage hN b |
| have hbq : (b : Boson N → ℝ) q = 1 := |
| b.one_of_mem_closedBall (Metric.mem_closedBall_self b.rIn_pos.le) |
| have hbs : Function.support (b : Boson N → ℝ) ⊆ V := by |
| apply (subset_tsupport b).trans |
| rw [b.tsupport_eq] |
| exact hrsub |
| let : CompactSpace (unitary (ColorMatrix N)) := |
| isCompact_iff_compactSpace.mp (BFSSOrbit.unitary_compact hN) |
| refine ⟨φ,M.gaugeAverage_continuous hN b b.continuous,?_,?_, |
| M.gaugeAverage_invariant hN b,?_⟩ |
| · intro x |
| have hc : Continuous (fun U : unitary (ColorMatrix N) => b (M.gaugeAction U⁻¹ x)) := |
| b.continuous.comp ((M.gaugeOrbitMap_continuous x).comp continuous_inv) |
| have hi : Integrable (fun U : unitary (ColorMatrix N) => b (M.gaugeAction U⁻¹ x)) (gaugeHaar hN) := |
| hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| refine ⟨integral_nonneg (fun U => b.nonneg),?_⟩ |
| change (∫ U, b (M.gaugeAction U⁻¹ x) ∂gaugeHaar hN) ≤ 1 |
| calc |
| _ ≤ ∫ _U : unitary (ColorMatrix N), (1 : ℝ) ∂gaugeHaar hN := |
| integral_mono hi (integrable_const 1) (fun _ => b.le_one) |
| _ = 1 := by simp |
| · exact M.gaugeAverage_pos hN b b.continuous (fun _ => b.nonneg) q (by rw [hbq]; norm_num) |
| · intro x hx |
| by_contra hnot |
| apply hx |
| apply integral_eq_zero_of_ae |
| filter_upwards with U |
| apply Function.notMem_support.mp |
| intro hb |
| exact hnot ((hVi U⁻¹ x).mp (hbs hb)) |
| |
| end |
| end BFSSQuantum.AlgebraData |
| |
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Matrix MeasureTheory Set TopologicalSpace |
| open scoped Matrix.Norms.L2Operator |
| variable {N : ℕ} (M : AlgebraData N) |
| attribute [local irreducible] gaugeAction |
|
|
| theorem normalOrbit_local_integral_bound (hN : 2 ≤ N) (q : Boson N) : |
| ∃ η B : ℝ, 0 < η ∧ 0 < B ∧ |
| ∀ h R : ℝ, 0 < h → 0 ≤ R → h * R < η → |
| ∀ W : Boson N → ℝ, Continuous W → HasCompactSupport W → |
| (∀ x, 0 ≤ W x) → |
| (∀ U : unitary (ColorMatrix N), ∀ x, W (M.gaugeAction U x) = W x) → |
| (h ^ M.orbitDimension q)⁻¹ * |
| (∫ z : M.normalSlice q in Metric.closedBall 0 R, W (h⁻¹ • q+(z : Boson N))) ≤ |
| B * ∫ x, W x := by |
| obtain ⟨J,V,δ,hJ,hJn,hJ0,hδ,hδJ,_,hV,hqV,hVi,hmeasure⟩ := M.normalOrbit_measure hN q |
| obtain ⟨φ,hφ,hφb,hφq,hφi,hφs⟩ := M.gauge_neighborhood_cutoff hN q V hV |
| (hqV (M.self_mem_gaugeOrbit q)) hVi |
| obtain ⟨s,hs,hssub⟩ := Metric.mem_nhds_iff.mp |
| ((isOpen_lt continuous_const hφ).mem_nhds (half_lt_self hφq)) |
| let C : ℝ := J 0 / 2 * (φ q / 2) |
| have hC : 0 < C := mul_pos (half_pos hJ0) (half_pos hφq) |
| refine ⟨min δ s,C⁻¹,lt_min hδ hs,inv_pos.mpr hC,?_⟩ |
| intro h R hh hR hhR W hW hWK hWn hWi |
| let F : Boson N → ℝ := fun x => φ (h • x) * W x |
| have hsmul : Continuous (fun x : Boson N => h • x) := |
| (continuous_const : Continuous (fun _ : Boson N => h)).smul continuous_id |
| have hFc : Continuous F := (hφ.comp hsmul).mul hW |
| have hFK : HasCompactSupport F := hWK.mul_left |
| have hFi (U : unitary (ColorMatrix N)) (x : Boson N) : F (M.gaugeAction U x) = F x := by |
| dsimp [F] |
| rw [←M.gaugeAction_smul,hφi,hWi] |
| have hFs (x : Boson N) (hx : x ∈ Function.support F) : h • x ∈ V := by |
| apply hφs |
| intro hz |
| exact hx (by simp only [F,hz,zero_mul]) |
| have hm := hmeasure h hh F hFc hFK hFi hFs |
| let a : M.normalSlice q → Boson N := fun z => h⁻¹ • q+(z : Boson N) |
| have ha : Continuous a := continuous_const.add continuous_subtype_val |
| have hWa : Continuous (W ∘ a) := hW.comp ha |
| have haiso : Isometry a := by |
| intro z w |
| simp only [a,edist_add_left] |
| rfl |
| have hWaK : HasCompactSupport (W ∘ a) := hWK.comp_isClosedEmbedding haiso.isClosedEmbedding |
| have hWai : Integrable (W ∘ a) := hWa.integrable_of_hasCompactSupport hWaK |
| let T : M.normalSlice q → ℝ := fun z => J (h • z) * F (a z) |
| have hzsmul : Continuous (fun z : M.normalSlice q => h • z) := |
| (continuous_const : Continuous (fun _ : M.normalSlice q => h)).smul continuous_id |
| have hTc : Continuous T := (hJ.comp hzsmul).mul (hFc.comp ha) |
| have hTK : HasCompactSupport T := |
| (hFK.comp_isClosedEmbedding haiso.isClosedEmbedding).mul_left |
| have hTi : Integrable T := hTc.integrable_of_hasCompactSupport hTK |
| have hTn (z : M.normalSlice q) : 0 ≤ T z := |
| mul_nonneg (hJn _) (mul_nonneg (hφb _).1 (hWn _)) |
| have hscale (z : M.normalSlice q) : h • a z = q+h • (z : Boson N) := by |
| dsimp [a] |
| rw [smul_add,smul_smul,mul_inv_cancel₀ hh.ne',one_smul] |
| have hlower (z : M.normalSlice q) (hz : z ∈ Metric.closedBall 0 R) : |
| C * W (a z) ≤ T z := by |
| have hzn : ‖z‖ ≤ R := by simpa only [Metric.mem_closedBall,dist_zero_right] using hz |
| have hzh : ‖h • z‖ < min δ s := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos hh] |
| exact lt_of_le_of_lt (mul_le_mul_of_nonneg_left hzn hh.le) hhR |
| have hJz := (hδJ (h • z) (lt_of_lt_of_le hzh (min_le_left _ _))).1.le |
| have hφz : φ q / 2 ≤ φ (q+h • (z : Boson N)) := by |
| apply le_of_lt |
| apply hssub |
| simpa only [Metric.mem_ball,dist_self_add_left,norm_neg,dist_eq_norm,add_sub_cancel_left, |
| norm_smul,Submodule.norm_coe] using |
| (lt_of_lt_of_le hzh (min_le_right _ _)) |
| have hp : C ≤ J (h • z) * φ (q+h • (z : Boson N)) := |
| mul_le_mul hJz hφz (half_pos hφq).le (hJn _) |
| dsimp [T,F] |
| rw [hscale] |
| nlinarith [mul_le_mul_of_nonneg_right hp (hWn (a z))] |
| have hloc : C * (∫ z : M.normalSlice q in Metric.closedBall 0 R, W (a z)) ≤ ∫ z, T z := by |
| calc |
| _ = ∫ z : M.normalSlice q in Metric.closedBall 0 R, C * W (a z) := |
| (integral_const_mul _ _).symm |
| _ ≤ ∫ z : M.normalSlice q in Metric.closedBall 0 R, T z := |
| setIntegral_mono_on (hWai.const_mul C).integrableOn hTi.integrableOn |
| Metric.isClosed_closedBall.measurableSet hlower |
| _ ≤ ∫ z, T z := integral_mono_measure Measure.restrict_le_self |
| (Filter.Eventually.of_forall hTn) hTi |
| have hFupper : (∫ x, F x) ≤ ∫ x, W x := by |
| apply integral_mono (hFc.integrable_of_hasCompactSupport hFK) |
| (hW.integrable_of_hasCompactSupport hWK) |
| intro x |
| exact mul_le_of_le_one_left (hWn x) (hφb _).2 |
| have hspos : 0 < (h ^ M.orbitDimension q)⁻¹ := inv_pos.mpr (pow_pos hh _) |
| have hhbound := (mul_le_mul_of_nonneg_left hloc hspos.le).trans |
| (by simpa only [T,a,←hm] using hFupper) |
| have hh' : C * ((h ^ M.orbitDimension q)⁻¹ * ∫ z : M.normalSlice q in Metric.closedBall 0 R, |
| W (h⁻¹ • q+(z : Boson N))) ≤ ∫ x, W x := by |
| simpa only [a,mul_left_comm] using hhbound |
| exact (le_inv_mul_iff₀ hC).mpr hh' |
|
|
| end |
| end BFSSQuantum.AlgebraData |
|
|
| end OAI |
|
|
