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785c727 | 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 | import OAI.Geometry.Manifold.ProjectiveSphere
namespace OAI
/-! Unit representatives and transition probabilities for the actual
Fubini--Study distance on CP2. -/
namespace PositiveEinsteinFour
open EinsteinFour
noncomputable section
abbrev CP2Vector := EuclideanSpace ℂ (Fin 3)
def cp2UnitRep (p : ComplexProjective2) : CP2Vector :=
((‖p.rep‖⁻¹ : ℝ) : ℂ) • p.rep
theorem cp2UnitRep_norm (p : ComplexProjective2) : ‖cp2UnitRep p‖ = 1 := by
have hn : ‖p.rep‖ ≠ 0 := norm_ne_zero_iff.mpr p.rep_nonzero
simp [cp2UnitRep, norm_smul, Complex.norm_real, hn]
theorem cp2UnitRep_ne_zero (p : ComplexProjective2) : cp2UnitRep p ≠ 0 := by
intro h
have hn := cp2UnitRep_norm p
rw [h, norm_zero] at hn
exact zero_ne_one hn
theorem cp2UnitRep_mk (p : ComplexProjective2) :
Projectivization.mk ℂ (cp2UnitRep p) (cp2UnitRep_ne_zero p) = p := by
conv_rhs => rw [← p.mk_rep]
exact (Projectivization.mk_eq_mk_iff' ℂ _ _ _ _).mpr ⟨_, rfl⟩
def cp2Overlap (p q : ComplexProjective2) : ℝ :=
‖inner ℂ (cp2UnitRep p) (cp2UnitRep q)‖
theorem cp2Overlap_nonneg (p q : ComplexProjective2) : 0 ≤ cp2Overlap p q := norm_nonneg _
theorem cp2Overlap_le_one (p q : ComplexProjective2) : cp2Overlap p q ≤ 1 := by
simpa only [cp2Overlap, cp2UnitRep_norm, one_mul] using norm_inner_le_norm (cp2UnitRep p) (cp2UnitRep q)
theorem cp2Overlap_eq_ratio (p q : ComplexProjective2) :
cp2Overlap p q = ‖inner ℂ p.rep q.rep‖ / (‖p.rep‖ * ‖q.rep‖) := by
have hp : (((‖p.rep‖⁻¹ : ℝ) : ℂ)) ≠ 0 := by
exact_mod_cast inv_ne_zero (norm_ne_zero_iff.mpr p.rep_nonzero)
have hq : (((‖q.rep‖⁻¹ : ℝ) : ℂ)) ≠ 0 := by
exact_mod_cast inv_ne_zero (norm_ne_zero_iff.mpr q.rep_nonzero)
have h := normalized_inner_smul p.rep q.rep
(((‖p.rep‖⁻¹ : ℝ) : ℂ)) (((‖q.rep‖⁻¹ : ℝ) : ℂ)) hp hq
change ‖inner ℂ (cp2UnitRep p) (cp2UnitRep q)‖ /
(‖cp2UnitRep p‖ * ‖cp2UnitRep q‖) = _ at h
simpa only [cp2Overlap, cp2UnitRep_norm, one_mul, div_one] using h
theorem fubiniStudyDistance_eq_arccos (p q : ComplexProjective2) :
fubiniStudyDistance p q = Real.arccos (cp2Overlap p q) := by
rw [cp2Overlap_eq_ratio]
rfl
theorem cp2Overlap_eq_iff_distance_eq (p q r s : ComplexProjective2) :
cp2Overlap p q = cp2Overlap r s ↔ fubiniStudyDistance p q = fubiniStudyDistance r s := by
rw [fubiniStudyDistance_eq_arccos, fubiniStudyDistance_eq_arccos]
constructor
· exact congrArg Real.arccos
· exact (Real.strictAntiOn_arccos.injOn
⟨(by linarith [cp2Overlap_nonneg p q]), cp2Overlap_le_one p q⟩
⟨(by linarith [cp2Overlap_nonneg r s]), cp2Overlap_le_one r s⟩)
theorem cp2Overlap_mk (x y : CP2Vector) (hx : x ≠ 0) (hy : y ≠ 0) :
cp2Overlap (Projectivization.mk ℂ x hx) (Projectivization.mk ℂ y hy) =
‖inner ℂ x y‖ / (‖x‖ * ‖y‖) := by
rw [cp2Overlap_eq_ratio, normalized_inner_mk_rep]
theorem cp2Overlap_comm (p q : ComplexProjective2) : cp2Overlap p q = cp2Overlap q p :=
norm_inner_symm _ _
@[simp] theorem cp2Overlap_self (p : ComplexProjective2) : cp2Overlap p p = 1 := by
change ‖inner ℂ (cp2UnitRep p) (cp2UnitRep p)‖ = 1
rw [inner_self_eq_norm_sq_to_K, cp2UnitRep_norm]
norm_num
end
end PositiveEinsteinFour
end OAI
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