| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.PotentialLimits |
|
|
| namespace OAI |
|
|
| /-! Kinetic-mass and color contractions for the literal fermion field. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Matrix Finset |
| open scoped ComplexConjugate |
| variable {N : ℕ} (M : AlgebraData N) |
|
|
| lemma kineticComplexSymbol_sum (α : SpinIndex) (i : SpaceIndex) (A : ColorIndex N) : |
| M.kineticComplexSymbol α (i,A) = |
| ∑ δ : SpinIndex, (M.gamma i α δ : ℂ) • M.theta δ A := by |
| simp only [kineticComplexSymbol,principalSymbol,cliffordLinear,Fintype.sum_prod_type, |
| M.principalCoefficients_coordinate] |
| apply Finset.sum_congr rfl |
| intro β _ |
| rw [Finset.sum_eq_single A] |
| · simp |
| · intro B _ hBA |
| simp only [ite_eq_right (Ne.symm hBA),Complex.ofReal_zero] |
| exact zero_smul ℂ (M.theta β B) |
| · simp |
|
|
| lemma kinetic_matrix_contraction (i : SpaceIndex) (A B : ColorIndex N) |
| (C : GammaMatrix) : |
| (∑ α : SpinIndex, M.kineticComplexSymbol α (i,A) * |
| (∑ β : SpinIndex, (C β α : ℂ) • M.theta β B)) = |
| ∑ δ : SpinIndex, ∑ β : SpinIndex, |
| ((C * M.gamma i) β δ : ℂ) • (M.theta δ A * M.theta β B) := by |
| simp only [M.kineticComplexSymbol_sum,Finset.sum_mul,Finset.mul_sum, |
| smul_mul_assoc,mul_smul_comm,Finset.smul_sum,smul_smul] |
| rw [Finset.sum_comm] |
| conv_lhs => |
| arg 2 |
| intro β |
| rw [Finset.sum_comm] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro δ _ |
| apply Finset.sum_congr rfl |
| intro β _ |
| rw [← Finset.sum_smul] |
| congr 1 |
| simp only [Matrix.mul_apply,Complex.ofReal_sum,Complex.ofReal_mul] |
|
|
| lemma gamma_mass_contraction : |
| (∑ i : SpaceIndex, BFSSGamma.massWeight i • |
| (M.gamma i * BFSSGamma.massT M.gamma * M.gamma i)) = |
| (12 : ℝ) • BFSSGamma.massT M.gamma := by |
| have hs : (∑ i ∈ BFSSGamma.firstThree, BFSSGamma.massWeight i) = (6 : ℝ) := by |
| rw [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] |
| norm_num [BFSSGamma.massWeight,Fin.lt_def] |
| rw [← BFSSGamma.word_firstThree, BFSSGamma.weighted_word_conj M.gamma M.gamma_clifford, |
| BFSSGamma.firstThree_card,hs] |
| norm_num |
|
|
| lemma massLinear_single (α : SpinIndex) (i : SpaceIndex) (A : ColorIndex N) : |
| M.massLinear α (EuclideanSpace.single (i,A) 1) = |
| (∑ β : SpinIndex, |
| ((BFSSGamma.massWeight i * (M.gamma i * BFSSGamma.massT M.gamma) β α : ℝ) : ℂ) • |
| M.theta β A).restrictScalars ℝ := by |
| have hh : M.massLinear α (EuclideanSpace.single (i,A) 1) = |
| ∑ β : SpinIndex, (BFSSGamma.massWeight i * (M.gamma i * BFSSGamma.massT M.gamma) β α) • |
| (M.theta β A).restrictScalars ℝ := by |
| simp only [massLinear,_root_.sum_apply,ContinuousLinearMap.smulRight_apply] |
| apply Finset.sum_congr rfl |
| intro β _ |
| rw [Finset.sum_eq_single A] |
| · rw [Finset.sum_eq_single i] |
| · simp |
| · intro j _ hji |
| simp [Prod.mk.injEq,Ne.symm hji] |
| · simp |
| · intro B _ hBA |
| apply Finset.sum_eq_zero |
| intro j _ |
| simp [Prod.mk.injEq,Ne.symm hBA] |
| · simp |
| rw [hh] |
| apply ContinuousLinearMap.ext |
| intro z |
| simp only [_root_.sum_apply,IsSMulApply.smul_apply, |
| ContinuousLinearMap.coe_restrictScalars'] |
| rfl |
|
|
| lemma kinetic_mass_contraction (A : ColorIndex N) : |
| (∑ i : SpaceIndex, ∑ α : SpinIndex, M.kineticComplexSymbol α (i,A) * |
| (∑ β : SpinIndex, |
| ((BFSSGamma.massWeight i * (M.gamma i * BFSSGamma.massT M.gamma) β α : ℝ) : ℂ) • |
| M.theta β A)) = |
| ∑ δ : SpinIndex, ∑ β : SpinIndex, |
| ((12 * BFSSGamma.massT M.gamma β δ : ℝ) : ℂ) • |
| (M.theta δ A * M.theta β A) := by |
| have hc (i : SpaceIndex) := M.kinetic_matrix_contraction i A A |
| (BFSSGamma.massWeight i • (M.gamma i * BFSSGamma.massT M.gamma)) |
| simp only [Matrix.smul_apply,smul_eq_mul] at hc |
| simp_rw [hc,smul_mul_assoc] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro δ _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro β _ |
| rw [← Finset.sum_smul,← Complex.ofReal_sum] |
| congr 2 |
| have hh := congrArg (fun C : GammaMatrix => C β δ) M.gamma_mass_contraction |
| simpa only [Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul] using hh |
|
|
| lemma deformedFermionField_mass (x : Boson N) : |
| M.deformedFermionField 0 1 x = |
| (((3/4 : ℂ) * Complex.I) • |
| (∑ A : ColorIndex N, ∑ δ : SpinIndex, ∑ β : SpinIndex, |
| (BFSSGamma.massT M.gamma δ β : ℂ) • |
| (M.theta δ A * M.theta β A))).restrictScalars ℝ := by |
| let R := ContinuousLinearMap.restrictScalarsₗ ℂ (Fermion N) (Fermion N) ℝ ℂ |
| let P (i : SpaceIndex) (A : ColorIndex N) (α : SpinIndex) := |
| M.kineticComplexSymbol α (i,A) * |
| (∑ β : SpinIndex, |
| ((BFSSGamma.massWeight i * (M.gamma i * BFSSGamma.massT M.gamma) β α : ℝ) : ℂ) • |
| M.theta β A) |
| have ht (α : SpinIndex) (i : SpaceIndex) (A : ColorIndex N) : |
| M.kineticSkewSymbol α (i,A) * M.massLinear α (EuclideanSpace.single (i,A) 1) = |
| R ((-Complex.I) • P i A α) := by |
| rw [M.massLinear_single] |
| apply ContinuousLinearMap.ext |
| intro z |
| simp only [kineticSkewSymbol,mul_apply_eq_comp, |
| IsSMulApply.smul_apply,ContinuousLinearMap.coe_restrictScalars'] |
| rfl |
| have hs : (∑ α : SpinIndex, ∑ p : SpaceIndex × ColorIndex N, |
| M.kineticSkewSymbol α p * M.massLinear α (EuclideanSpace.single p 1)) = |
| R ((-Complex.I) • (∑ A : ColorIndex N, ∑ i : SpaceIndex, ∑ α : SpinIndex, P i A α)) := by |
| simp only [Fintype.sum_prod_type,ht] |
| have he : (∑ α : SpinIndex, ∑ i : SpaceIndex, ∑ A : ColorIndex N, R ((-Complex.I) • P i A α)) = |
| ∑ A : ColorIndex N, ∑ i : SpaceIndex, ∑ α : SpinIndex, R ((-Complex.I) • P i A α) := by |
| rw [Finset.sum_comm] |
| conv_lhs => arg 2; intro i; rw [Finset.sum_comm] |
| rw [Finset.sum_comm] |
| rw [he] |
| simp only [Finset.smul_sum,map_sum] |
| have hc : (∑ A : ColorIndex N, ∑ i : SpaceIndex, ∑ α : SpinIndex, P i A α) = |
| (-12 : ℂ) • (∑ A : ColorIndex N, ∑ δ : SpinIndex, ∑ β : SpinIndex, |
| (BFSSGamma.massT M.gamma δ β : ℂ) • (M.theta δ A * M.theta β A)) := by |
| have hT := BFSSGamma.massT_transpose M.gamma M.gamma_clifford M.gamma_symmetric |
| have hTe (β δ : SpinIndex) : BFSSGamma.massT M.gamma β δ = -BFSSGamma.massT M.gamma δ β := |
| congrArg (fun C : GammaMatrix => C δ β) hT |
| simp only [P] |
| simp_rw [M.kinetic_mass_contraction] |
| simp only [Complex.ofReal_mul, |
| Complex.ofReal_ofNat,Finset.smul_sum,smul_smul] |
| apply Finset.sum_congr rfl |
| intro A _ |
| apply Finset.sum_congr rfl |
| intro δ _ |
| apply Finset.sum_congr rfl |
| intro β _ |
| rw [hTe β δ,Complex.ofReal_neg] |
| congr 1 |
| ring |
| have hd (α : SpinIndex) (p : SpaceIndex × ColorIndex N) : |
| fderiv ℝ (M.deformedRealField 0 1 α) x (EuclideanSpace.single p 1) = |
| M.massLinear α (EuclideanSpace.single p 1) := by |
| rw [M.deformedRealField_fderiv] |
| module |
| unfold deformedFermionField MixedEnergy.remainderField |
| simp only [hd] |
| rw [hs,hc,smul_smul] |
| rw [RCLike.real_smul_eq_coe_smul (K := ℂ),← map_smul] |
| norm_num only [Complex.ofReal_div,Complex.ofReal_one,Complex.ofReal_ofNat] |
| change R (((1/16 : ℝ) : ℂ) • (((-Complex.I) * (-12)) • _)) = R _ |
| congr 1 |
| rw [smul_smul] |
| congr 1 |
| norm_num |
| ring |
|
|
| end |
| end BFSSQuantum.AlgebraData |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Matrix Finset |
| variable {N : ℕ} (M : AlgebraData N) |
|
|
| def spinColumn (C : GammaMatrix) (α : SpinIndex) (A : ColorIndex N) : |
| Fermion N →L[ℂ] Fermion N := ∑ β : SpinIndex, (C β α : ℂ) • M.theta β A |
|
|
| lemma sum_rotate_five {ι κ ξ μ ν W : Type*} [Fintype ι] [Fintype κ] [Fintype ξ] |
| [Fintype μ] [Fintype ν] [AddCommMonoid W] (f : ι → κ → ξ → μ → ν → W) : |
| (∑ i, ∑ j, ∑ k, ∑ l, ∑ m, f i j k l m) = |
| ∑ j, ∑ k, ∑ l, ∑ m, ∑ i, f i j k l m := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro l _ |
| rw [Finset.sum_comm] |
|
|
| lemma bracketBilinear_restrict (α : SpinIndex) (x y : Boson N) : |
| M.bracketBilinear α x y = |
| (∑ A : ColorIndex N, ∑ p : SpatialPair, ∑ B : ColorIndex N, ∑ C : ColorIndex N, |
| ((-M.structureConstant A B C * x (p.1.1,B) * y (p.1.2,C) : ℝ) : ℂ) • |
| M.spinColumn (BFSSGamma.pairGamma M.gamma p) α A).restrictScalars ℝ := by |
| apply ContinuousLinearMap.ext |
| intro z |
| simp only [bracketBilinear,spinColumn,_root_.sum_apply, |
| ContinuousLinearMap.smulRight_apply,EuclideanSpace.coe_proj, |
| ContinuousLinearMap.coe_restrictScalars',IsSMulApply.smul_apply, |
| Finset.smul_sum,smul_smul] |
| rw [sum_rotate_five] |
| apply Finset.sum_congr rfl |
| intro A _ |
| apply Finset.sum_congr rfl |
| intro p _ |
| apply Finset.sum_congr rfl |
| intro B _ |
| apply Finset.sum_congr rfl |
| intro C _ |
| apply Finset.sum_congr rfl |
| intro β _ |
| rw [RCLike.real_smul_eq_coe_smul (K := ℂ)] |
| simp only [Complex.ofReal_mul,Complex.ofReal_neg] |
| congr 1 |
| change ((x (p.1.1,B) * (y (p.1.2,C) * |
| (-M.structureConstant A B C * BFSSGamma.pairGamma M.gamma p β α)) : ℝ) : ℂ) = |
| -(M.structureConstant A B C : ℂ) * (x (p.1.1,B) : ℂ) * (y (p.1.2,C) : ℂ) * |
| (BFSSGamma.pairGamma M.gamma p β α : ℂ) |
| push_cast |
| ring |
|
|
| lemma structureConstant_cycle (A B C : ColorIndex N) : |
| M.structureConstant A B C = M.structureConstant B C A := by |
| have ht := bracket_trace_skew (M.color C) (M.color A) (M.color B) |
| have hs : AlgebraData.bracket (M.color C) (M.color B) = |
| -AlgebraData.bracket (M.color B) (M.color C) := bracket_neg_swap _ _ |
| rw [hs,mul_neg,Matrix.trace_neg,neg_neg,Matrix.trace_mul_comm] at ht |
| exact (congrArg Complex.re ht).symm |
|
|
| lemma structureConstant_swap (A B C : ColorIndex N) : |
| M.structureConstant A C B = -M.structureConstant A B C := by |
| unfold structureConstant |
| rw [bracket_neg_swap (M.color C) (M.color B),mul_neg,Matrix.trace_neg,Complex.neg_re] |
|
|
| end |
| end BFSSQuantum.AlgebraData |
|
|
| end OAI |
|
|
