refactor: Major refactor of Lorentz group
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6 changed files with 563 additions and 931 deletions
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@ -3,6 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.FourVelocity
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import HepLean.SpaceTime.LorentzGroup.Proper
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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/-!
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@ -11,6 +12,10 @@ import Mathlib.GroupTheory.SpecificGroups.KleinFour
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We define the give a series of lemmas related to the orthochronous property of lorentz
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matrices.
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## TODO
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- Prove topological properties.
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-/
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@ -24,100 +29,164 @@ open Complex
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open ComplexConjugate
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namespace lorentzGroup
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open PreFourVelocity
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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 : lorentzGroup) : Λ.1.1.det = 1 ∨ Λ.1.1.det = -1 := by
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simpa [← sq, det_one, det_mul, det_mul, det_mul, det_transpose, det_η] using
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(congrArg det ((PreservesηLin.iff_matrix' Λ.1).mp ((mem_iff Λ.1).mp Λ.2)))
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/-- The first column of a lorentz matrix as a `PreFourVelocity`. -/
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@[simp]
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def fstCol (Λ : lorentzGroup) : PreFourVelocity := ⟨Λ.1 *ᵥ stdBasis 0, by
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rw [mem_PreFourVelocity_iff, ηLin_expand]
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simp only [Fin.isValue, stdBasis_mulVec]
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have h00 := congrFun (congrFun ((PreservesηLin.iff_matrix Λ.1).mp ((mem_iff Λ.1).mp Λ.2)) 0) 0
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simp only [Fin.isValue, mul_apply, transpose_apply, Fin.sum_univ_four, ne_eq, zero_ne_one,
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not_false_eq_true, η_off_diagonal, zero_mul, add_zero, Fin.reduceEq, one_ne_zero, mul_zero,
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zero_add, one_apply_eq] at h00
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simp only [η, Fin.isValue, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
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vecCons_const, one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul, cons_val_two,
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Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, cons_val_three, head_fin_const] at h00
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rw [← h00]
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ring⟩
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def IsOrthochronous (Λ : lorentzGroup) : Prop := 0 ≤ Λ.1 0 0
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lemma IsOrthochronous_iff_transpose (Λ : lorentzGroup) :
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IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by
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simp [IsOrthochronous]
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simp [transpose]
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lemma IsOrthochronous_iff_fstCol_IsFourVelocity (Λ : lorentzGroup) :
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IsOrthochronous Λ ↔ IsFourVelocity (fstCol Λ) := by
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simp [IsOrthochronous, IsFourVelocity]
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rw [stdBasis_mulVec]
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def mapZeroZeroComp : C(lorentzGroup, ℝ) := ⟨fun Λ => Λ.1 0 0, by
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refine Continuous.matrix_elem ?_ 0 0
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refine Continuous.comp' Units.continuous_val continuous_subtype_val⟩
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def stepFunction : ℝ → ℝ := fun t =>
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if t ≤ -1 then -1 else
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if 1 ≤ t then 1 else t
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lemma stepFunction_continuous : Continuous stepFunction := by
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apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
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<;> intro a ha
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rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
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rw [ha]
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simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
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have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
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rw [if_neg h1]
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rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
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simp [ha]
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def orthchroMapReal : C(lorentzGroup, ℝ) := ContinuousMap.comp
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⟨stepFunction, stepFunction_continuous⟩ mapZeroZeroComp
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lemma orthchroMapReal_on_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
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orthchroMapReal Λ = 1 := by
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
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simp only [IsFourVelocity] at h
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rw [zero_nonneg_iff] at h
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simp [stdBasis_mulVec] at h
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have h1 : ¬ Λ.1 0 0 ≤ (-1 : ℝ) := by linarith
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change stepFunction (Λ.1 0 0) = 1
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rw [stepFunction, if_neg h1, if_pos h]
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lemma orthchroMapReal_on_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
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orthchroMapReal Λ = - 1 := by
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
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rw [not_IsFourVelocity_iff, zero_nonpos_iff] at h
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simp only [fstCol, Fin.isValue, stdBasis_mulVec] at h
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change stepFunction (Λ.1 0 0) = - 1
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rw [stepFunction, if_pos h]
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lemma orthchroMapReal_minus_one_or_one (Λ : lorentzGroup) :
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orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
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by_cases h : IsOrthochronous Λ
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apply Or.inr $ orthchroMapReal_on_IsOrthochronous h
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apply Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
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local notation "ℤ₂" => Multiplicative (ZMod 2)
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instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
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def orthchroMap : C(lorentzGroup, ℤ₂) :=
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ContinuousMap.comp coeForℤ₂ {
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toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
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continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
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instance : DiscreteTopology ℤ₂ := by
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exact forall_open_iff_discrete.mp fun _ => trivial
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lemma orthchroMap_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
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orthchroMap Λ = 1 := by
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simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
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instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
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lemma orthchroMap_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
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orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
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simp [orthchroMap, orthchroMapReal_on_not_IsOrthochronous h]
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rfl
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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, by simp⟩ then (Additive.toMul 0) else (Additive.toMul (1 : ZMod 2))
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continuous_toFun := by
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haveI : DiscreteTopology ({-1, 1} : Set ℝ) := discrete_of_t1_of_finite
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exact continuous_of_discreteTopology
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lemma zero_zero_mul (Λ Λ' : lorentzGroup) :
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(Λ * Λ').1 0 0 = (fstCol (transpose Λ)).1 0 * (fstCol Λ').1 0 +
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⟪(fstCol (transpose Λ)).1.space, (fstCol Λ').1.space⟫_ℝ := by
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rw [@Subgroup.coe_mul, @GeneralLinearGroup.coe_mul, mul_apply, Fin.sum_univ_four]
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rw [@PiLp.inner_apply, Fin.sum_univ_three]
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simp [transpose, stdBasis_mulVec]
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ring
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/-- The continuous map taking a lorentz matrix to its determinant. -/
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def detContinuous : C(lorentzGroup, ℤ₂) :=
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ContinuousMap.comp coeForℤ₂ {
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toFun := fun Λ => ⟨Λ.1.1.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' Units.continuous_val continuous_subtype_val}
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lemma mul_othchron_of_othchron_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
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(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
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rw [IsOrthochronous_iff_transpose] at h
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
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rw [IsOrthochronous, zero_zero_mul]
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exact euclid_norm_IsFourVelocity_IsFourVelocity h h'
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lemma detContinuous_eq_iff_det_eq (Λ Λ' : lorentzGroup) :
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detContinuous Λ = detContinuous Λ' ↔ Λ.1.1.det = Λ'.1.1.det := by
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apply Iff.intro
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intro h
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simp [detContinuous] at h
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cases' det_eq_one_or_neg_one Λ with h1 h1
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<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
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<;> simp_all [h1, h2, h]
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rw [← toMul_zero, @Equiv.apply_eq_iff_eq] 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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nth_rewrite 2 [← toMul_zero] at h
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rw [@Equiv.apply_eq_iff_eq] at h
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change (1 : Fin 2) = (0 : Fin 2) at h
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simp [Fin.isValue, zero_ne_one] at h
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intro h
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simp [detContinuous, h]
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lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
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(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
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rw [IsOrthochronous_iff_transpose] at h
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
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rw [IsOrthochronous, zero_zero_mul]
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exact euclid_norm_not_IsFourVelocity_not_IsFourVelocity h h'
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lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
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(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
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rw [IsOrthochronous_iff_transpose] at h
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity, not_IsFourVelocity_iff]
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simp [stdBasis_mulVec]
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change (Λ * Λ').1 0 0 ≤ _
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rw [zero_zero_mul]
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exact euclid_norm_IsFourVelocity_not_IsFourVelocity h h'
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/-- The representation taking a lorentz matrix to its determinant. -/
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@[simps!]
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def detRep : lorentzGroup →* ℤ₂ where
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toFun Λ := detContinuous Λ
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lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
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(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
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rw [IsOrthochronous_iff_transpose] at h
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
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rw [IsOrthochronous_iff_fstCol_IsFourVelocity, not_IsFourVelocity_iff]
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simp [stdBasis_mulVec]
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change (Λ * Λ').1 0 0 ≤ _
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rw [zero_zero_mul]
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exact euclid_norm_not_IsFourVelocity_IsFourVelocity h h'
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def orthchroRep : lorentzGroup →* ℤ₂ where
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toFun := orthchroMap
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map_one' := by
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simp [detContinuous]
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map_mul' := by
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intro Λ1 Λ2
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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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<;> cases' (det_eq_one_or_neg_one Λ2) with h2 h2
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<;> simp [h1, h2, detContinuous]
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have h1 : IsOrthochronous 1 := by simp [IsOrthochronous]
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rw [orthchroMap_IsOrthochronous h1]
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map_mul' Λ Λ' := by
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simp
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by_cases h : IsOrthochronous Λ
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<;> by_cases h' : IsOrthochronous Λ'
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rw [orthchroMap_IsOrthochronous h, orthchroMap_IsOrthochronous h',
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orthchroMap_IsOrthochronous (mul_othchron_of_othchron_othchron h h')]
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simp
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rw [orthchroMap_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
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orthchroMap_not_IsOrthochronous (mul_not_othchron_of_othchron_not_othchron h h')]
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simp
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rw [orthchroMap_not_IsOrthochronous h, orthchroMap_IsOrthochronous h',
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orthchroMap_not_IsOrthochronous (mul_not_othchron_of_not_othchron_othchron h h')]
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simp
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rw [orthchroMap_not_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
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orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
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simp only [Nat.reduceAdd]
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rfl
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lemma detRep_continuous : Continuous detRep := detContinuous.2
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lemma det_on_connected_component {Λ Λ' : lorentzGroup} (h : Λ' ∈ connectedComponent Λ) :
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Λ.1.1.det = Λ'.1.1.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 det_of_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') : Λ.1.1.det = Λ'.1.1.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) : Prop := Λ.1.1.det = 1
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instance : DecidablePred IsProper := by
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intro Λ
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apply Real.decidableEq
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lemma IsProper_iff (Λ : lorentzGroup) : IsProper Λ ↔ detRep Λ = 1 := by
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rw [show 1 = detRep 1 by simp]
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rw [detRep_apply, detRep_apply, detContinuous_eq_iff_det_eq]
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simp only [IsProper, OneMemClass.coe_one, Units.val_one, det_one]
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end lorentzGroup
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end spaceTime
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