| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.NormalPotential |
|
|
| namespace OAI |
|
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| /-! Gauge tangent slices, normal coercivity and Taylor squares. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Finset |
| variable {N : ℕ} (M : BFSSQuantum.AlgebraData N) |
|
|
| def spatialColorMap (i : SpaceIndex) : Boson N →ₗ[ℝ] ColorVector N where |
| toFun x := spatialColor x i |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| def gaugeTangent (q : Boson N) : ColorVector N →ₗ[ℝ] Boson N where |
| toFun w := WithLp.toLp 2 (fun p => M.colorLie w (spatialColor q p.1) p.2) |
| map_add' w v := by |
| ext p |
| exact congrArg (fun a : ColorVector N => a p.2) (M.colorLie_add_left w v (spatialColor q p.1)) |
| map_smul' r w := by |
| ext p |
| exact congrArg (fun a : ColorVector N => a p.2) (M.colorLie_smul_left r w (spatialColor q p.1)) |
|
|
| def normalSlice (q : Boson N) : Submodule ℝ (Boson N) := |
| (LinearMap.range (M.gaugeTangent q))ᗮ |
|
|
| lemma spatialColor_gaugeTangent (q : Boson N) (w : ColorVector N) (i : SpaceIndex) : |
| spatialColor (M.gaugeTangent q w) i = M.colorLie w (spatialColor q i) := rfl |
|
|
| lemma spatialColor_inner (x z : Boson N) : |
| inner ℝ x z = ∑ i : SpaceIndex, inner ℝ (spatialColor x i) (spatialColor z i) := by |
| simp only [PiLp.inner_apply, Fintype.sum_prod_type] |
| rfl |
|
|
| lemma gaugeTangent_pairing (q z : Boson N) (w : ColorVector N) : |
| inner ℝ z (M.gaugeTangent q w) = inner ℝ (M.gaugeDivergence q z) w := by |
| rw [spatialColor_inner] |
| simp only [gaugeDivergence,sum_inner,spatialColor_gaugeTangent] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [M.colorLie_swap w (spatialColor q i),inner_neg_right] |
| exact (M.colorAd_skew _ _ _).symm |
|
|
| lemma mem_normalSlice_iff (q z : Boson N) : |
| z ∈ M.normalSlice q ↔ M.gaugeDivergence q z = 0 := by |
| rw [normalSlice,Submodule.mem_orthogonal'] |
| constructor |
| · intro hz |
| apply (inner_self_eq_zero (𝕜 := ℝ)).mp |
| rw [←M.gaugeTangent_pairing] |
| exact hz _ ⟨M.gaugeDivergence q z,rfl⟩ |
| · intro hz u hu |
| obtain ⟨w,rfl⟩ := hu |
| rw [M.gaugeTangent_pairing,hz,inner_zero_left] |
|
|
| lemma bracketLinearization_eq (q z : Boson N) (i j : SpaceIndex) : |
| M.bracketLinearization q z i j = |
| M.colorAd (spatialColor q i) (spatialColor z j) - |
| M.colorAd (spatialColor q j) (spatialColor z i) := by |
| unfold bracketLinearization |
| rw [M.colorLie_swap (spatialColor z i) (spatialColor q j)] |
| rfl |
|
|
| def bracketLinearizationMap (q : Boson N) (i j : SpaceIndex) : Boson N →ₗ[ℝ] ColorVector N := |
| (M.colorAd (spatialColor q i)).comp (spatialColorMap j) - |
| (M.colorAd (spatialColor q j)).comp (spatialColorMap i) |
|
|
| lemma bracketLinearizationMap_apply (q z : Boson N) (i j : SpaceIndex) : |
| M.bracketLinearizationMap q i j z = M.bracketLinearization q z i j := |
| (M.bracketLinearization_eq q z i j).symm |
|
|
| def cyclicLinearizationMap (q : Boson N) (i j k : SpaceIndex) : Boson N →ₗ[ℝ] ColorVector N := |
| M.bracketLinearizationMap q i j - (2:ℝ) • spatialColorMap k |
|
|
| lemma cyclicLinearizationMap_apply (q z : Boson N) (i j k : SpaceIndex) : |
| M.cyclicLinearizationMap q i j k z = M.cyclicLinearization q z i j k := by |
| change M.bracketLinearizationMap q i j z - (2:ℝ) • spatialColor z k = _ |
| rw [M.bracketLinearizationMap_apply] |
| rfl |
|
|
| lemma normalQuadratic_continuous (q : Boson N) : Continuous (M.normalQuadratic q) := by |
| unfold normalQuadratic |
| simp_rw [←M.cyclicLinearizationMap_apply, ←M.bracketLinearizationMap_apply, |
| show ∀ z i, spatialColor z i = spatialColorMap i z from fun _ _ => rfl] |
| have hc (i j k : SpaceIndex) : Continuous (fun z => ‖M.cyclicLinearizationMap q i j k z‖^2) := |
| (M.cyclicLinearizationMap q i j k).continuous_of_finiteDimensional.norm.pow 2 |
| have hb : Continuous (fun z => ∑ i : SpaceIndex, ∑ j : SpaceIndex, |
| if i < j ∧ 3 ≤ j.val then ‖M.bracketLinearizationMap q i j z‖^2 else 0) := by |
| exact continuous_finsetSum _ fun i _ => continuous_finsetSum _ fun j _ => by |
| split_ifs |
| · exact (M.bracketLinearizationMap q i j).continuous_of_finiteDimensional.norm.pow 2 |
| · exact continuous_const |
| have hs : Continuous (fun z : Boson N => ∑ d : SpaceIndex, |
| if 3 ≤ d.val then ‖spatialColorMap d z‖^2 else 0) := by |
| exact continuous_finsetSum _ fun d _ => by |
| split_ifs |
| · exact (spatialColorMap d).continuous_of_finiteDimensional.norm.pow 2 |
| · exact continuous_const |
| exact (((((hc 0 1 2).add (hc 1 2 0)).add (hc 2 0 1)).add hb).add hs).div_const 2 |
|
|
| lemma normalQuadratic_smul (q z : Boson N) (r : ℝ) : |
| M.normalQuadratic q (r • z) = r^2 * M.normalQuadratic q z := by |
| unfold normalQuadratic |
| simp_rw [←M.cyclicLinearizationMap_apply, ←M.bracketLinearizationMap_apply] |
| simp only [map_smul,spatialColor_smul,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs] |
| have hif (P : Prop) [Decidable P] (a : ℝ) : |
| (if P then r^2*a else 0) = r^2 * (if P then a else 0) := by split_ifs <;> simp |
| simp_rw [hif] |
| simp only [←Finset.mul_sum] |
| ring |
| |
| theorem normalQuadratic_coercive (q : Boson N) (hq : M.deformedBosonicPotential 1 1 q = 0) : |
| ∃ c : ℝ, 0 < c ∧ ∀ z : Boson N, z ∈ M.normalSlice q → |
| c * ‖z‖^2 ≤ M.normalQuadratic q z := by |
| obtain ⟨c,hc,hbound⟩ := BFSSNormal.positive_quadratic_lower |
| (fun z : M.normalSlice q => M.normalQuadratic q z) |
| ((M.normalQuadratic_continuous q).comp continuous_subtype_val) |
| (fun r z => M.normalQuadratic_smul q z r) |
| (by |
| intro z hz |
| have hz' : (z:Boson N) ≠ 0 := by intro he; exact hz (Subtype.ext he) |
| have hne : M.normalQuadratic q z ≠ 0 := by |
| intro he |
| exact hz' (M.normalQuadratic_zero_of_normal q z hq |
| ((M.mem_normalSlice_iff q z).mp z.property) he) |
| exact lt_of_le_of_ne (M.normalQuadratic_nonneg q z) hne.symm) |
| exact ⟨c,hc,fun z hz => hbound ⟨z,hz⟩⟩ |
| end |
| end BFSSQuantum.AlgebraData |
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Finset |
| variable {N : ℕ} (M : BFSSQuantum.AlgebraData N) |
|
|
| lemma colorLie_continuous : Continuous (fun p : ColorVector N × ColorVector N => M.colorLie p.1 p.2) := by |
| unfold colorLie |
| apply (PiLp.continuous_toLp 2 _).comp |
| apply continuous_pi |
| intro A |
| exact continuous_finsetSum _ fun B _ => continuous_finsetSum _ fun C _ => |
| ((continuous_const.mul ((EuclideanSpace.proj (𝕜 := ℝ) B).continuous.comp continuous_fst)).mul |
| ((EuclideanSpace.proj (𝕜 := ℝ) C).continuous.comp continuous_snd)) |
|
|
| lemma spatialBracket_continuous (i j : SpaceIndex) : |
| Continuous (fun x : Boson N => M.colorLie (spatialColor x i) (spatialColor x j)) := |
| M.colorLie_continuous.comp (((spatialColorMap i).continuous_of_finiteDimensional).prodMk |
| ((spatialColorMap j).continuous_of_finiteDimensional)) |
|
|
| lemma potential_color (x : Boson N) : |
| M.deformedBosonicPotential 1 1 x = |
| (‖M.colorLie (spatialColor x 0) (spatialColor x 1) - (2:ℝ) • spatialColor x 2‖^2 + |
| ‖M.colorLie (spatialColor x 1) (spatialColor x 2) - (2:ℝ) • spatialColor x 0‖^2 + |
| ‖M.colorLie (spatialColor x 2) (spatialColor x 0) - (2:ℝ) • spatialColor x 1‖^2 + |
| (∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| ‖M.colorLie (spatialColor x i) (spatialColor x j)‖^2 else 0) + |
| (∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor x d‖^2 else 0))/2 := by |
| have hc (i j k : SpaceIndex) : |
| BFSS.hsNormSq (BFSS.realBracket (M.matrixConfiguration x i).val (M.matrixConfiguration x j).val - |
| (2:ℝ) • (M.matrixConfiguration x k).val) = |
| ‖M.colorLie (spatialColor x i) (spatialColor x j) - (2:ℝ) • spatialColor x k‖^2 := by |
| have he : (2:ℝ) • (M.matrixConfiguration x k).val = (2:ℂ) • (M.matrixConfiguration x k).val := by |
| ext a b |
| simp [Matrix.smul_apply,Complex.real_smul] |
| rw [he,M.residual_normSq] |
| have hb : (∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| BFSS.hsNormSq (BFSS.realBracket (M.matrixConfiguration x i).val (M.matrixConfiguration x j).val) else 0) = |
| ∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| ‖M.colorLie (spatialColor x i) (spatialColor x j)‖^2 else 0 := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro j hj |
| split_ifs |
| · rw [←M.colorEmbed_spatial,←M.colorEmbed_spatial] |
| change BFSS.hsNormSq (AlgebraData.bracket (M.colorEmbed (spatialColor x i)) |
| (M.colorEmbed (spatialColor x j))) = _ |
| rw [←M.colorEmbed_lie,M.colorEmbed_normSq] |
| · rfl |
| have hs : (∑ d : SpaceIndex, if 3 ≤ d.val then BFSS.hsNormSq (M.matrixConfiguration x d).val else 0) = |
| ∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor x d‖^2 else 0 := by |
| apply Finset.sum_congr rfl |
| intro d hd |
| split_ifs |
| · rw [←M.colorEmbed_spatial,M.colorEmbed_normSq] |
| · rfl |
| rw [M.deformedBosonicPotential_matrix, BFSS.massivePotential] |
| simp only [Complex.ofReal_one,one_smul,mul_one,one_pow,one_mul,hc,hb,hs] |
|
|
| lemma cyclicResidual_continuous (i j k : SpaceIndex) : Continuous (fun x : Boson N => |
| ‖M.colorLie (spatialColor x i) (spatialColor x j) - (2:ℝ) • spatialColor x k‖^2) := by |
| have hk : Continuous (fun x : Boson N => spatialColor x k) := |
| (spatialColorMap k).continuous_of_finiteDimensional |
| have hm : Continuous (fun x : Boson N => (2:ℝ) • spatialColor x k) := |
| (continuous_const : Continuous (fun _ : Boson N => (2:ℝ))).smul hk |
| have hr : Continuous (fun x : Boson N => |
| M.colorLie (spatialColor x i) (spatialColor x j) - (2:ℝ) • spatialColor x k) := |
| (M.spatialBracket_continuous i j).sub hm |
| exact hr.norm.pow 2 |
|
|
| lemma potential_continuous : Continuous (M.deformedBosonicPotential 1 1) := by |
| have hc := M.cyclicResidual_continuous |
| have hb : Continuous (fun x : Boson N => ∑ i : SpaceIndex, ∑ j : SpaceIndex, |
| if i < j ∧ 3 ≤ j.val then ‖M.colorLie (spatialColor x i) (spatialColor x j)‖^2 else 0) := by |
| exact continuous_finsetSum _ fun i _ => continuous_finsetSum _ fun j _ => by |
| split_ifs |
| · exact (M.spatialBracket_continuous i j).norm.pow 2 |
| · exact continuous_const |
| have hs : Continuous (fun x : Boson N => ∑ d : SpaceIndex, |
| if 3 ≤ d.val then ‖spatialColor x d‖^2 else 0) := by |
| exact continuous_finsetSum _ fun d _ => by |
| split_ifs |
| · exact (spatialColorMap d).continuous_of_finiteDimensional.norm.pow 2 |
| · exact continuous_const |
| have ht := (((((hc 0 1 2).add (hc 1 2 0)).add (hc 2 0 1)).add hb).add hs).div_const 2 |
| exact ht.congr (fun x => (M.potential_color x).symm) |
|
|
| lemma potential_nonneg (x : Boson N) : 0 ≤ M.deformedBosonicPotential 1 1 x := by |
| rw [M.potential_color] |
| positivity |
|
|
| lemma potential_zero_of_relations (q : Boson N) (hq : M.ZeroRelations q) : |
| M.deformedBosonicPotential 1 1 q = 0 := by |
| rw [M.potential_color,hq.1,hq.2.1,hq.2.2.1] |
| simp only [sub_self,norm_zero,zero_pow (by decide : 2 ≠ 0),zero_add] |
| have hb : (∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| ‖M.colorLie (spatialColor q i) (spatialColor q j)‖^2 else 0) = 0 := by |
| apply Finset.sum_eq_zero |
| intro i hi |
| apply Finset.sum_eq_zero |
| intro j hj |
| split_ifs with hh |
| · rw [hq.2.2.2 j hh.2,M.colorLie_zero_right,norm_zero]; norm_num |
| · rfl |
| have hs : (∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor q d‖^2 else 0) = 0 := by |
| apply Finset.sum_eq_zero |
| intro d hd |
| split_ifs with hh |
| · rw [hq.2.2.2 d hh,norm_zero]; norm_num |
| · rfl |
| rw [hb,hs]; norm_num |
| |
| lemma bracket_taylor (q z : Boson N) (i j : SpaceIndex) : |
| M.colorLie (spatialColor (q+z) i) (spatialColor (q+z) j) = |
| M.colorLie (spatialColor q i) (spatialColor q j) + M.bracketLinearization q z i j + |
| M.colorLie (spatialColor z i) (spatialColor z j) := by |
| simp only [spatialColor_add,M.colorLie_add_left,M.colorLie_add_right,bracketLinearization] |
| abel |
|
|
| lemma cyclic_taylor (q z : Boson N) (i j k : SpaceIndex) |
| (hq : M.colorLie (spatialColor q i) (spatialColor q j) = (2:ℝ) • spatialColor q k) : |
| M.colorLie (spatialColor (q+z) i) (spatialColor (q+z) j) - (2:ℝ) • spatialColor (q+z) k = |
| M.cyclicLinearization q z i j k + M.colorLie (spatialColor z i) (spatialColor z j) := by |
| rw [M.bracket_taylor,hq,spatialColor_add,smul_add] |
| unfold cyclicLinearization |
| abel |
|
|
| theorem potential_taylor_squares (q z : Boson N) (hq : M.deformedBosonicPotential 1 1 q = 0) : |
| M.deformedBosonicPotential 1 1 (q+z) = |
| (‖M.cyclicLinearization q z 0 1 2 + M.colorLie (spatialColor z 0) (spatialColor z 1)‖^2 + |
| ‖M.cyclicLinearization q z 1 2 0 + M.colorLie (spatialColor z 1) (spatialColor z 2)‖^2 + |
| ‖M.cyclicLinearization q z 2 0 1 + M.colorLie (spatialColor z 2) (spatialColor z 0)‖^2 + |
| (∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| ‖M.bracketLinearization q z i j + M.colorLie (spatialColor z i) (spatialColor z j)‖^2 else 0) + |
| (∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor z d‖^2 else 0))/2 := by |
| have hr := M.potential_zero_relations q hq |
| have hb : (∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| ‖M.colorLie (spatialColor (q+z) i) (spatialColor (q+z) j)‖^2 else 0) = |
| ∑ i : SpaceIndex, ∑ j : SpaceIndex, if i < j ∧ 3 ≤ j.val then |
| ‖M.bracketLinearization q z i j + M.colorLie (spatialColor z i) (spatialColor z j)‖^2 else 0 := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro j hj |
| split_ifs with hh |
| · rw [M.bracket_taylor,hr.2.2.2 j hh.2,M.colorLie_zero_right,zero_add] |
| · rfl |
| have hs : (∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor (q+z) d‖^2 else 0) = |
| ∑ d : SpaceIndex, if 3 ≤ d.val then ‖spatialColor z d‖^2 else 0 := by |
| apply Finset.sum_congr rfl |
| intro d hd |
| split_ifs with hh |
| · rw [spatialColor_add,hr.2.2.2 d hh,zero_add] |
| · rfl |
| rw [M.potential_color, M.cyclic_taylor q z 0 1 2 hr.1, |
| M.cyclic_taylor q z 1 2 0 hr.2.1,M.cyclic_taylor q z 2 0 1 hr.2.2.1,hb,hs] |
|
|
| end |
| end BFSSQuantum.AlgebraData |
|
|
| end OAI |
|
|
