File size: 4,471 Bytes
f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | 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
end OAI
|