refactor: Major refactor of lorentz group
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11 changed files with 866 additions and 208 deletions
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@ -3,7 +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.LorentzVector.NormOne
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import HepLean.SpaceTime.LorentzGroup.Proper
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/-!
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# The Orthochronous Lorentz Group
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@ -20,7 +20,6 @@ matrices.
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noncomputable section
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namespace SpaceTime
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open Manifold
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open Matrix
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@ -28,37 +27,50 @@ 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 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 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_explicit, Fin.isValue, of_apply, cons_val', cons_val_zero, empty_val',
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cons_val_fin_one, vecCons_const, one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul,
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cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, cons_val_three,
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head_fin_const] at h00
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exact h00⟩
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variable {d : ℕ}
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variable (Λ : LorentzGroup d)
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open LorentzVector
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open minkowskiMetric
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/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
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def IsOrthochronous (Λ : LorentzGroup) : Prop := 0 ≤ Λ.1 0 0
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def IsOrthochronous : Prop := 0 ≤ timeComp Λ
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lemma IsOrthochronous_iff_transpose (Λ : LorentzGroup) :
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lemma IsOrthochronous_iff_futurePointing :
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IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d := by
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simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]
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rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]
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lemma IsOrthochronous_iff_transpose :
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IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
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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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lemma IsOrthochronous_iff_ge_one :
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IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
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rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
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NormOneLorentzVector.time_pos_iff]
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simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
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erw [Pi.basisFun_apply, mulVec_stdBasis]
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rfl
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lemma not_orthochronous_iff_le_neg_one :
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¬ IsOrthochronous Λ ↔ timeComp Λ ≤ -1 := by
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rw [timeComp, IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.not_mem_iff,
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NormOneLorentzVector.time_nonpos_iff]
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simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
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erw [Pi.basisFun_apply, mulVec_stdBasis]
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lemma not_orthochronous_iff_le_zero :
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¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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rw [not_orthochronous_iff_le_neg_one] at h
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linarith
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rw [IsOrthochronous_iff_ge_one]
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linarith
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/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
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def mapZeroZeroComp : C(LorentzGroup, ℝ) := ⟨fun Λ => Λ.1 0 0,
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Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) 0 0⟩
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def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ ,
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Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
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/-- An auxillary function used in the definition of `orthchroMapReal`. -/
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def stepFunction : ℝ → ℝ := fun t =>
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@ -78,29 +90,23 @@ lemma stepFunction_continuous : Continuous stepFunction := by
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/-- The continuous map from `lorentzGroup` to `ℝ` wh
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taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
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def orthchroMapReal : C(LorentzGroup, ℝ) := ContinuousMap.comp
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⟨stepFunction, stepFunction_continuous⟩ mapZeroZeroComp
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def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
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⟨stepFunction, stepFunction_continuous⟩ timeCompCont
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lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup} (h : IsOrthochronous Λ) :
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lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (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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rw [IsOrthochronous_iff_ge_one, timeComp] at h
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change stepFunction (Λ.1 _ _) = 1
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rw [stepFunction, if_pos h, if_neg]
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linarith
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lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup} (h : ¬ IsOrthochronous Λ) :
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lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (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 [not_orthochronous_iff_le_neg_one] at h
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change stepFunction (timeComp _)= - 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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lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
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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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@ -109,65 +115,50 @@ lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup) :
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local notation "ℤ₂" => Multiplicative (ZMod 2)
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/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
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def orthchroMap : C(LorentzGroup, ℤ₂) :=
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def orthchroMap : C(LorentzGroup d, 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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lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup} (h : IsOrthochronous Λ) :
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lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (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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lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup} (h : ¬ IsOrthochronous Λ) :
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lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (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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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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simp only [Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fin.sum_univ_four, fstCol,
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transpose, stdBasis_mulVec, transpose_apply, space, PiLp.inner_apply, Nat.succ_eq_add_one,
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Nat.reduceAdd, RCLike.inner_apply, conj_trivial, Fin.sum_univ_three, cons_val_zero,
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cons_val_one, head_cons, cons_val_two, tail_cons]
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ring
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lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup} (h : IsOrthochronous Λ)
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lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (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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rw [IsOrthochronous_iff_futurePointing] at h h'
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rw [IsOrthochronous, timeComp_mul]
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exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem h h'
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lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup} (h : ¬ IsOrthochronous Λ)
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lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (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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rw [IsOrthochronous_iff_futurePointing] at h h'
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rw [IsOrthochronous, timeComp_mul]
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exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem h h'
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lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup} (h : IsOrthochronous Λ)
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lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
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(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
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rw [not_orthochronous_iff_le_zero, timeComp_mul]
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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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rw [IsOrthochronous_iff_futurePointing] at h h'
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exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem h h'
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lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup} (h : ¬ IsOrthochronous Λ)
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lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
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(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
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rw [not_orthochronous_iff_le_zero, timeComp_mul]
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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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rw [IsOrthochronous_iff_futurePointing] at h h'
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exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_mem h h'
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/-- The homomorphism from `lorentzGroup` to `ℤ₂` whose kernel are the Orthochronous elements. -/
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def orthchroRep : LorentzGroup →* ℤ₂ where
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def orthchroRep : LorentzGroup d →* ℤ₂ where
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toFun := orthchroMap
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map_one' := orthchroMap_IsOrthochronous (by simp [IsOrthochronous])
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map_mul' Λ Λ' := by
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@ -189,5 +180,4 @@ def orthchroRep : LorentzGroup →* ℤ₂ where
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end LorentzGroup
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end SpaceTime
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end
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