184 lines
7.1 KiB
Text
184 lines
7.1 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Lorentz.Group.Basic
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/-!
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# The Proper Lorentz Group
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The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`.
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We define the give a series of lemmas related to the determinant of the Lorentz group.
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-/
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noncomputable section
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open Matrix
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open Complex
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open ComplexConjugate
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namespace LorentzGroup
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open minkowskiMatrix
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variable {d : ℕ}
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/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/
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lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
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refine mul_self_eq_one_iff.mp ?_
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simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
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local notation "ℤ₂" => Multiplicative (ZMod 2)
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instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
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instance : DiscreteTopology ℤ₂ := by
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exact forall_open_iff_discrete.mp fun _ => trivial
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instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
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/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
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@[simps!]
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def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
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toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
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then Additive.toMul 0 else Additive.toMul (1 : ZMod 2)
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continuous_toFun := continuous_of_discreteTopology
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/-- The continuous map taking a Lorentz matrix to its determinant. -/
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def detContinuous : C(𝓛 d, ℤ₂) :=
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ContinuousMap.comp coeForℤ₂ {
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toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
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continuous_toFun := by
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refine Continuous.subtype_mk ?_ _
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exact Continuous.matrix_det $
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Continuous.comp' (continuous_iff_le_induced.mpr fun U a => a) continuous_id'
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}
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lemma detContinuous_eq_one (Λ : LorentzGroup d) :
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detContinuous Λ = Additive.toMul 0 ↔ Λ.1.det = 1 := by
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simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
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Subtype.mk.injEq, ite_eq_left_iff, toMul_eq_one]
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simp only [toMul_zero, ite_eq_left_iff, toMul_eq_one]
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· by_contra hn
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have h' := h hn
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change (1 : Fin 2) = (0 : Fin 2) at h'
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simp only [Fin.isValue, one_ne_zero] at h'
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· intro h'
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exact False.elim (h' h)
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lemma detContinuous_eq_zero (Λ : LorentzGroup d) :
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detContinuous Λ = Additive.toMul (1 : ZMod 2) ↔ Λ.1.det = - 1 := by
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simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk, coeForℤ₂_apply,
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Subtype.mk.injEq, Nat.reduceAdd]
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· by_contra hn
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rw [if_pos] at h
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· change (0 : Fin 2) = (1 : Fin 2) at h
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simp only [Fin.isValue, zero_ne_one] at h
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· cases' det_eq_one_or_neg_one Λ with h2 h2
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· simp_all only [ite_true]
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· simp_all only [not_true_eq_false]
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· rw [if_neg]
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· rfl
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· cases' det_eq_one_or_neg_one Λ with h2 h2
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· rw [h]
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linarith
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· linarith
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lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
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detContinuous Λ = detContinuous Λ' ↔ Λ.1.det = Λ'.1.det := by
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cases' det_eq_one_or_neg_one Λ with h1 h1
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· rw [h1, (detContinuous_eq_one Λ).mpr h1]
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cases' det_eq_one_or_neg_one Λ' with h2 h2
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· rw [h2, (detContinuous_eq_one Λ').mpr h2]
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simp only [toMul_zero]
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· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
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erw [Additive.toMul.apply_eq_iff_eq]
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change (0 : Fin 2) = (1 : Fin 2) ↔ _
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simp only [Fin.isValue, zero_ne_one, false_iff]
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linarith
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· rw [h1, (detContinuous_eq_zero Λ).mpr h1]
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cases' det_eq_one_or_neg_one Λ' with h2 h2
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· rw [h2, (detContinuous_eq_one Λ').mpr h2]
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erw [Additive.toMul.apply_eq_iff_eq]
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change (1 : Fin 2) = (0 : Fin 2) ↔ _
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simp only [Fin.isValue, one_ne_zero, false_iff]
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linarith
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· rw [h2, (detContinuous_eq_zero Λ').mpr h2]
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simp only [Nat.reduceAdd]
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/-- The representation taking a Lorentz matrix to its determinant. -/
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@[simps!]
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def detRep : 𝓛 d →* ℤ₂ where
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toFun Λ := detContinuous Λ
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map_one' := by
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simp only [detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
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lorentzGroupIsGroup_one_coe, det_one, coeForℤ₂_apply, ↓reduceIte]
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map_mul' Λ1 Λ2 := by
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simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
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mul_ite, mul_one, ite_mul, one_mul]
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cases' det_eq_one_or_neg_one Λ1 with h1 h1
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· rw [(detContinuous_eq_one Λ1).mpr h1]
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cases' det_eq_one_or_neg_one Λ2 with h2 h2
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· rw [(detContinuous_eq_one Λ2).mpr h2]
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apply (detContinuous_eq_one _).mpr
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simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
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· rw [(detContinuous_eq_zero Λ2).mpr h2]
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apply (detContinuous_eq_zero _).mpr
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simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one]
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· rw [(detContinuous_eq_zero Λ1).mpr h1]
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cases' det_eq_one_or_neg_one Λ2 with h2 h2
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· rw [(detContinuous_eq_one Λ2).mpr h2]
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apply (detContinuous_eq_zero _).mpr
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simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_one]
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· rw [(detContinuous_eq_zero Λ2).mpr h2]
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apply (detContinuous_eq_one _).mpr
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simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one, neg_neg]
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lemma detRep_continuous : Continuous (@detRep d) := detContinuous.2
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lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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Λ.1.det = Λ'.1.det := by
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obtain ⟨s, hs, hΛ'⟩ := h
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let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
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haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
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simpa [f, detContinuous_eq_iff_det_eq] using
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(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
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(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
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lemma detRep_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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detRep Λ = detRep Λ' := by
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simp only [detRep_apply, detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
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coeForℤ₂_apply, Subtype.mk.injEq]
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rw [det_on_connected_component h]
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lemma det_of_joined {Λ Λ' : LorentzGroup d} (h : Joined Λ Λ') : Λ.1.det = Λ'.1.det :=
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det_on_connected_component $ pathComponent_subset_component _ h
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/-- A Lorentz Matrix is proper if its determinant is 1. -/
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@[simp]
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def IsProper (Λ : LorentzGroup d) : Prop := Λ.1.det = 1
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instance : DecidablePred (@IsProper d) := by
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intro Λ
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apply Real.decidableEq
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lemma IsProper_iff (Λ : LorentzGroup d) : IsProper Λ ↔ detRep Λ = 1 := by
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rw [show 1 = detRep 1 from Eq.symm (MonoidHom.map_one detRep), detRep_apply, detRep_apply,
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detContinuous_eq_iff_det_eq]
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simp only [IsProper, lorentzGroupIsGroup_one_coe, det_one]
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lemma id_IsProper : @IsProper d 1 := by
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simp [IsProper]
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lemma isProper_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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IsProper Λ ↔ IsProper Λ' := by
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simp only [IsProper]
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rw [det_on_connected_component h]
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end LorentzGroup
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end
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