| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.FermionQuantization |
|
|
| namespace OAI |
|
|
| /-! Exact field-frequency identification and quantitative frequency lower bounds. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Matrix Finset BFSSSpinHalf |
| open scoped Kronecker |
| variable {N : ℕ} (M : AlgebraData N) |
|
|
| lemma colorEuclidean_matrix_entry (L : Module.End ℝ (ColorVector N)) (A B : ColorIndex N) : |
| (eucAlg.symm L) A B = (L (EuclideanSpace.single B 1)) A := by |
| have h := congrArg (fun K : Module.End ℝ (ColorVector N) => |
| (K (EuclideanSpace.single B 1)) A) (eucAlg.apply_symm_apply L) |
| change ((eucAlg.symm L)*ᵥ (Pi.single B 1)) A = _ at h |
| simpa only [Matrix.mulVec_single_one,Matrix.col_apply] using h |
|
|
| lemma colorJ_matrix_entry (q : Boson N) (a : Fin 3) (A B : ColorIndex N) : |
| (eucAlg.symm (M.colorJ q a)) A B = |
| (1/2:ℝ) * ∑ C : ColorIndex N, |
| M.structureConstant B A C * q (Fin.castLE (by decide) a,C) := by |
| rw [colorEuclidean_matrix_entry] |
| change ((1/2:ℝ) • M.colorLie (spatialColor q (Fin.castLE (by decide) a)) |
| (EuclideanSpace.single B 1)) A = _ |
| simp only [PiLp.smul_apply,smul_eq_mul,colorLie,PiLp.toLp_apply,spatialColor] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro C _ |
| rw [Finset.sum_eq_single B] |
| · simp only [PiLp.single_eq_same,mul_one] |
| rw [←M.structureConstant_cycle B A C] |
| · intro D _ hDB |
| have hz : EuclideanSpace.single B (1:ℝ) D = 0 := by |
| simp [hDB] |
| rw [hz,mul_zero] |
| · simp |
|
|
| lemma normal_frequency_tensor_entry (q : Boson N) (a : Fin 3) |
| (r s : FrequencyIndex N) : |
| (leftLift (κ := ColorIndex N) (M.gamma (Fin.castLE (by decide) a))*M.colorJTensor q a) r s = |
| M.gamma (Fin.castLE (by decide) a) r.1 s.1 * |
| ((1/2:ℝ) * ∑ C : ColorIndex N, |
| M.structureConstant s.2 r.2 C * q (Fin.castLE (by decide) a,C)) := by |
| have hm : leftLift (κ := ColorIndex N) (M.gamma (Fin.castLE (by decide) a))*M.colorJTensor q a = |
| (M.gamma (Fin.castLE (by decide) a)) ⊗ₖ (eucAlg.symm (M.colorJ q a)) := by |
| change ((M.gamma (Fin.castLE (by decide) a)) ⊗ₖ (1 : Matrix (ColorIndex N) (ColorIndex N) ℝ)) * |
| ((1 : GammaMatrix) ⊗ₖ (eucAlg.symm (M.colorJ q a))) = _ |
| rw [←Matrix.mul_kronecker_mul,mul_one,one_mul] |
| rw [hm,Matrix.kronecker_apply,M.colorJ_matrix_entry] |
|
|
| lemma zero_relations_coordinate_tail (q : Boson N) (hq : M.ZeroRelations q) |
| (j : SpaceIndex) (hj : 3 ≤ j.val) (C : ColorIndex N) : q (j,C) = 0 := by |
| have h := congrArg (fun v : ColorVector N => v C) (hq.2.2.2 j hj) |
| exact h |
|
|
| theorem fermionCoefficient_eq_normalFrequency (q : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) : |
| M.fermionCoefficient q = M.normalFermionFrequency q := by |
| have hqr := M.potential_zero_relations q hq |
| ext r s |
| have ht : (∑ j : SpaceIndex, ∑ C : ColorIndex N, |
| M.structureConstant s.2 r.2 C*q (j,C)*M.gamma j r.1 s.1) = |
| ∑ j ∈ BFSSGamma.firstThree, ∑ C : ColorIndex N, |
| M.structureConstant s.2 r.2 C*q (j,C)*M.gamma j r.1 s.1 := by |
| symm |
| apply Finset.sum_subset (Finset.subset_univ _) |
| intro j _ hj |
| apply Finset.sum_eq_zero |
| intro C _ |
| have hge : 3 ≤ j.val := by |
| have h := mt (BFSSGamma.mem_firstThree j).mpr hj |
| exact Nat.le_of_not_gt h |
| rw [M.zero_relations_coordinate_tail q hqr j hge C,mul_zero,zero_mul] |
| have h0 := M.normal_frequency_tensor_entry q 0 r s |
| have h1 := M.normal_frequency_tensor_entry q 1 r s |
| have h2 := M.normal_frequency_tensor_entry q 2 r s |
| change (leftLift (κ := ColorIndex N) (M.gamma 0)*M.colorJTensor q 0) r s = |
| M.gamma 0 r.1 s.1 * ((1/2:ℝ) * ∑ C : ColorIndex N, |
| M.structureConstant s.2 r.2 C*q (0,C)) at h0 |
| change (leftLift (κ := ColorIndex N) (M.gamma 1)*M.colorJTensor q 1) r s = |
| M.gamma 1 r.1 s.1 * ((1/2:ℝ) * ∑ C : ColorIndex N, |
| M.structureConstant s.2 r.2 C*q (1,C)) at h1 |
| change (leftLift (κ := ColorIndex N) (M.gamma 2)*M.colorJTensor q 2) r s = |
| M.gamma 2 r.1 s.1 * ((1/2:ℝ) * ∑ C : ColorIndex N, |
| M.structureConstant s.2 r.2 C*q (2,C)) at h2 |
| simp only [fermionCoefficient,ht,normalFermionFrequency,Matrix.sub_apply, |
| Matrix.smul_apply,smul_eq_mul,Matrix.add_apply,h0,h1,h2] |
| have hT : leftLift (κ := ColorIndex N) (BFSSGamma.massT M.gamma) r s = |
| BFSSGamma.massT M.gamma r.1 s.1 * (if r.2 = s.2 then 1 else 0) := rfl |
| rw [hT] |
| simp only [BFSSGamma.firstThree, |
| Finset.sum_insert (by decide : (0 : Fin 9) ∉ ({1,2} : Finset (Fin 9))), |
| Finset.sum_insert (by decide : (1 : Fin 9) ∉ ({2} : Finset (Fin 9))), |
| Finset.sum_singleton,←Finset.sum_mul] |
| by_cases he : r.2 = s.2 <;> simp only [he,ite_true,ite_false,mul_one,mul_zero] <;> ring |
| |
| end |
| end BFSSQuantum.AlgebraData |
| |
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open BFSSSpinHalf |
| variable {N : ℕ} (M : AlgebraData N) |
|
|
| lemma normalFermionFrequency_skew (q : Boson N) : |
| (M.normalFermionFrequency q).transpose = -M.normalFermionFrequency q := by |
| have hg (a : SpaceIndex) : (M.gamma a).transpose = M.gamma a := (M.gamma_symmetric a).eq |
| have hc (a : Fin 3) : |
| M.colorJTensor q a * leftLift (M.gamma (Fin.castLE (by decide) a)) = |
| leftLift (M.gamma (Fin.castLE (by decide) a)) * M.colorJTensor q a := |
| (lifts_commute _ _).eq.symm |
| have term (a : Fin 3) : |
| (leftLift (κ := ColorIndex N) (M.gamma (Fin.castLE (by decide) a))*M.colorJTensor q a).transpose = |
| -(leftLift (M.gamma (Fin.castLE (by decide) a))*M.colorJTensor q a) := by |
| rw [Matrix.transpose_mul,M.colorJTensor_skew,leftLift_transpose,hg,neg_mul,hc] |
| have h0 := term 0 |
| have h1 := term 1 |
| have h2 := term 2 |
| change (leftLift (M.gamma 0)*M.colorJTensor q 0).transpose = -(leftLift (M.gamma 0)*M.colorJTensor q 0) at h0 |
| change (leftLift (M.gamma 1)*M.colorJTensor q 1).transpose = -(leftLift (M.gamma 1)*M.colorJTensor q 1) at h1 |
| change (leftLift (M.gamma 2)*M.colorJTensor q 2).transpose = -(leftLift (M.gamma 2)*M.colorJTensor q 2) at h2 |
| simp only [normalFermionFrequency,Matrix.transpose_sub,Matrix.transpose_smul, |
| Matrix.transpose_add,h0,h1,h2,leftLift_transpose, |
| BFSSGamma.massT_transpose M.gamma M.gamma_clifford M.gamma_symmetric,map_neg] |
| module |
|
|
| theorem normalFermionFrequency_norm_lower (q : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) |
| (x : EuclideanSpace ℝ (FrequencyIndex N)) : |
| (1/2 : ℝ) * ‖x‖ ≤ ‖eucAlg (M.normalFermionFrequency q) x‖ := by |
| let T : FrequencyMatrix N := leftLift (BFSSGamma.massT M.gamma) |
| let A : FrequencyMatrix N := (3/2:ℝ) • 1-(4:ℝ) • M.spinHalfCoupling q |
| have ht : T*T = -(1 : FrequencyMatrix N) := by |
| change leftLift (BFSSGamma.massT M.gamma)*leftLift (BFSSGamma.massT M.gamma) = _ |
| rw [←map_mul,BFSSGamma.massT_sq M.gamma M.gamma_clifford M.gamma_symmetric,map_neg,map_one] |
| have hts : T.transpose = -T := by |
| simp only [T,leftLift_transpose, |
| BFSSGamma.massT_transpose M.gamma M.gamma_clifford M.gamma_symmetric,map_neg] |
| have hnorm (y : EuclideanSpace ℝ (FrequencyIndex N)) : ‖eucAlg T y‖ = ‖y‖ := by |
| have he := skew_inner T hts y (eucAlg T y) |
| change inner ℝ (eucAlg T y) (eucAlg T y) = -inner ℝ y ((eucAlg T*eucAlg T) y) at he |
| rw [←map_mul,ht,map_neg,map_one,LinearMap.neg_apply,Module.End.one_apply,inner_neg_right, |
| neg_neg,real_inner_self_eq_norm_sq,real_inner_self_eq_norm_sq] at he |
| nlinarith [norm_nonneg y,norm_nonneg (eucAlg T y)] |
| have hp := congrArg (eucAlg (ι := FrequencyIndex N)) |
| (M.spinHalfCoupling_polynomial q (M.potential_zero_relations q hq)) |
| simp only [map_mul,map_add,map_smul] at hp |
| have hb := shift_norm_lower (fun i => eucAlg (M.colorJTensor q i)) (eucAlg (M.spinHalfCoupling q)) |
| (fun i => skew_inner _ (M.colorJTensor_skew q i)) |
| (symmetric_inner _ (M.spinHalfCoupling_symmetric q)) hp x |
| rw [M.normalFermionFrequency_factor] |
| change (1/2:ℝ)*‖x‖ ≤ ‖eucAlg (T*A) x‖ |
| rw [map_mul,Module.End.mul_apply,hnorm] |
| simpa only [A,map_sub,map_smul,map_one,LinearMap.sub_apply,LinearMap.smul_apply, |
| Module.End.one_apply] using hb |
| end |
| end BFSSQuantum.AlgebraData |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| variable {N : ℕ} (M : AlgebraData N) |
|
|
| theorem normal_fermion_field_exact (q : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) : |
| M.deformedFermionField 1 1 q = |
| (M.quantizeFrequency (M.normalFermionFrequency q)).restrictScalars ℝ := by |
| rw [M.deformedFermionField_quantized,M.fermionCoefficient_eq_normalFrequency q hq] |
|
|
| end |
| end BFSSQuantum.AlgebraData |
|
|
| end OAI |
|
|
