| import Mathlib |
| import OAI.MathematicalPhysics.BFSS.FrequencyComparison |
|
|
| namespace OAI |
|
|
| /-! Complex frequency eigenmodes and nonvanishing eigenvalues. -/ |
|
|
| namespace BFSSQuantum.AlgebraData |
| noncomputable section |
| open Matrix Finset |
| open scoped ComplexConjugate |
| variable {N : ℕ} |
|
|
| def complexFrequency (K : FrequencyMatrix N) : |
| Matrix (FrequencyIndex N) (FrequencyIndex N) ℂ := |
| fun r s => Complex.I*(K r s : ℂ) |
|
|
| lemma complexFrequency_hermitian (K : FrequencyMatrix N) (hK : K.transpose = -K) : |
| (complexFrequency K).IsHermitian := by |
| have hn (r s : FrequencyIndex N) : K s r = -K r s := |
| congrArg (fun A : FrequencyMatrix N => A r s) hK |
| ext r s |
| change conj (Complex.I*(K s r : ℂ)) = Complex.I*(K r s : ℂ) |
| rw [map_mul,Complex.conj_I,Complex.conj_ofReal,hn,Complex.ofReal_neg] |
| ring |
|
|
| lemma complexFrequency_eigenmode (K : FrequencyMatrix N) (hK : K.transpose = -K) |
| (j : FrequencyIndex N) (r : FrequencyIndex N) : |
| Complex.I*(∑ s, (K r s : ℂ)* |
| ((complexFrequency_hermitian K hK).eigenvectorBasis j s)) = |
| ((complexFrequency_hermitian K hK).eigenvalues j : ℂ)* |
| ((complexFrequency_hermitian K hK).eigenvectorBasis j r) := by |
| have h := congrFun ((complexFrequency_hermitian K hK).mulVec_eigenvectorBasis j) r |
| simpa only [Matrix.mulVec,dotProduct,complexFrequency,Pi.smul_apply, |
| Complex.real_smul,mul_assoc,Finset.mul_sum] using h |
|
|
| lemma complexFrequency_conjugate_mode (K : FrequencyMatrix N) (v : FrequencyIndex N → ℂ) |
| (ω : ℝ) (hv : ∀ r, Complex.I*(∑ s, (K r s : ℂ)*v s) = (ω : ℂ)*v r) : |
| ∀ r, Complex.I*(∑ s, (K r s : ℂ)*conj (v s)) = ((-ω : ℝ) : ℂ)*conj (v r) := by |
| intro r |
| have h := congrArg conj (hv r) |
| simp only [map_mul,map_sum,Complex.conj_I,Complex.conj_ofReal] at h |
| rw [Complex.ofReal_neg] |
| linear_combination -h |
|
|
| variable (M : AlgebraData N) |
|
|
| theorem normal_frequency_complex_kernel (q : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) (v : FrequencyIndex N → ℂ) |
| (hv : ∀ r, Complex.I*(∑ s, (M.normalFermionFrequency q r s : ℂ)*v s) = 0) : v = 0 := by |
| have hz (r : FrequencyIndex N) : |
| (∑ s, (M.normalFermionFrequency q r s : ℂ)*v s) = 0 := |
| (mul_eq_zero.mp (hv r)).resolve_left Complex.I_ne_zero |
| let vr : EuclideanSpace ℝ (FrequencyIndex N) := WithLp.toLp 2 (fun r => (v r).re) |
| let vi : EuclideanSpace ℝ (FrequencyIndex N) := WithLp.toLp 2 (fun r => (v r).im) |
| have hr : BFSSSpinHalf.eucAlg (M.normalFermionFrequency q) vr = 0 := by |
| apply PiLp.ext |
| intro r |
| have h := congrArg Complex.re (hz r) |
| simpa only [BFSSSpinHalf.eucAlg_eq,Matrix.toLpLin_apply,Matrix.mulVec, |
| dotProduct,vr,PiLp.zero_apply,Complex.re_sum, |
| Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero, |
| Complex.zero_re] using h |
| have hi : BFSSSpinHalf.eucAlg (M.normalFermionFrequency q) vi = 0 := by |
| apply PiLp.ext |
| intro r |
| have h := congrArg Complex.im (hz r) |
| simpa only [BFSSSpinHalf.eucAlg_eq,Matrix.toLpLin_apply,Matrix.mulVec, |
| dotProduct,vi,PiLp.zero_apply,Complex.im_sum, |
| Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero, |
| Complex.zero_im] using h |
| have hvr := M.normalFermionFrequency_trivial_kernel q hq vr hr |
| have hvi := M.normalFermionFrequency_trivial_kernel q hq vi hi |
| funext r |
| apply Complex.ext |
| · exact congrArg (fun w : EuclideanSpace ℝ (FrequencyIndex N) => w r) hvr |
| · exact congrArg (fun w : EuclideanSpace ℝ (FrequencyIndex N) => w r) hvi |
|
|
| theorem normal_frequency_eigenvalues_ne_zero (q : Boson N) |
| (hq : M.deformedBosonicPotential 1 1 q = 0) (j : FrequencyIndex N) : |
| (complexFrequency_hermitian (M.normalFermionFrequency q) |
| (M.normalFermionFrequency_skew q)).eigenvalues j ≠ 0 := by |
| intro hz |
| have hH := complexFrequency_hermitian (M.normalFermionFrequency q) |
| (M.normalFermionFrequency_skew q) |
| have hm := complexFrequency_eigenmode (M.normalFermionFrequency q) |
| (M.normalFermionFrequency_skew q) j |
| have hm0 (r : FrequencyIndex N) : |
| Complex.I*(∑ s, (M.normalFermionFrequency q r s : ℂ)*(hH.eigenvectorBasis j s)) = 0 := by |
| have h := hm r |
| rw [hz] at h |
| simpa only [Complex.ofReal_zero,zero_mul] using h |
| have hb := M.normal_frequency_complex_kernel q hq (hH.eigenvectorBasis j) hm0 |
| have he : hH.eigenvectorBasis j = 0 := by |
| apply PiLp.ext |
| exact congrFun hb |
| exact (hH.eigenvectorBasis.toBasis.ne_zero j) he |
|
|
| end |
| end BFSSQuantum.AlgebraData |
|
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| end OAI |
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