| import OAI.Geometry.Manifold.ProjectiveSphere |
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| namespace OAI |
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| /-! Unit representatives and transition probabilities for the actual |
| Fubini |
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| namespace PositiveEinsteinFour |
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| open EinsteinFour |
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| noncomputable section |
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| abbrev CP2Vector := EuclideanSpace ℂ (Fin 3) |
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| def cp2UnitRep (p : ComplexProjective2) : CP2Vector := |
| ((‖p.rep‖⁻¹ : ℝ) : ℂ) • p.rep |
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| 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] |
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| 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 |
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| 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⟩ |
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| def cp2Overlap (p q : ComplexProjective2) : ℝ := |
| ‖inner ℂ (cp2UnitRep p) (cp2UnitRep q)‖ |
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| theorem cp2Overlap_nonneg (p q : ComplexProjective2) : 0 ≤ cp2Overlap p q := norm_nonneg _ |
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| 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) |
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| 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 |
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| theorem fubiniStudyDistance_eq_arccos (p q : ComplexProjective2) : |
| fubiniStudyDistance p q = Real.arccos (cp2Overlap p q) := by |
| rw [cp2Overlap_eq_ratio] |
| rfl |
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| 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⟩) |
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| 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] |
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| theorem cp2Overlap_comm (p q : ComplexProjective2) : cp2Overlap p q = cp2Overlap q p := |
| norm_inner_symm _ _ |
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| @[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 |
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| end |
| end PositiveEinsteinFour |
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| end OAI |
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