PhysLean/HepLean/Lorentz/Group/Orthochronous.lean

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.Lorentz.RealVector.NormOne
import HepLean.Lorentz.Group.Proper
/-!
# The Orthochronous Lorentz Group

We define the give a series of lemmas related to the orthochronous property of lorentz
matrices.

-/
TODO "Prove topological properties of the Orthochronous Lorentz Group."

noncomputable section

open Matrix
open Complex
open ComplexConjugate

namespace LorentzGroup

variable {d : ℕ}
variable (Λ : LorentzGroup d)
open Lorentz.Contr

/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ Λ.1 (Sum.inl 0) (Sum.inl 0)

/-- A Lorentz transformation is `orthochronous` if and only if its first column is
  future pointing. -/
lemma IsOrthochronous_iff_futurePointing :
    IsOrthochronous Λ ↔ toNormOne Λ ∈ NormOne.FuturePointing d := by
  simp only [IsOrthochronous]
  rw [NormOne.FuturePointing.mem_iff_inl_nonneg, toNormOne_inl]

/-- A Lorentz transformation is orthochronous if and only if its transpose is orthochronous. -/
lemma IsOrthochronous_iff_transpose :
    IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl

/-- A Lorentz transformation is orthochronous if and only if its `0 0` element is greater
  or equal to one. -/
lemma IsOrthochronous_iff_ge_one :
    IsOrthochronous Λ ↔ 1 ≤ Λ.1 (Sum.inl 0) (Sum.inl 0) := by
  rw [IsOrthochronous_iff_futurePointing, NormOne.FuturePointing.mem_iff_inl_one_le_inl,
    toNormOne_inl]

/-- A Lorentz transformation is not orthochronous if and only if its `0 0` element is less then
  or equal to minus one. -/
lemma not_orthochronous_iff_le_neg_one :
    ¬ IsOrthochronous Λ ↔ Λ.1 (Sum.inl 0) (Sum.inl 0) ≤ -1 := by
  rw [IsOrthochronous_iff_futurePointing, NormOne.FuturePointing.not_mem_iff_inl_le_neg_one,
    toNormOne_inl]

/-- A Lorentz transformation is not orthochronous if and only if its `0 0` element is
  non-positive. -/
lemma not_orthochronous_iff_le_zero : ¬ IsOrthochronous Λ ↔ Λ.1 (Sum.inl 0) (Sum.inl 0) ≤ 0 := by
  rw [IsOrthochronous_iff_futurePointing, NormOne.FuturePointing.not_mem_iff_inl_le_zero,
    toNormOne_inl]

/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => Λ.1 (Sum.inl 0) (Sum.inl 0),
    Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩

/-- An auxiliary function used in the definition of `orthchroMapReal`.
  This function takes all elements of `ℝ` less then `-1` to `-1`,
  all elements of `R` greater then `1` to `1` and preserves all other elements. -/
def stepFunction : ℝ → ℝ := fun t =>
  if t ≤ -1 then -1 else
    if 1 ≤ t then 1 else t

/-- The `stepFunction` is continuous. -/
lemma stepFunction_continuous : Continuous stepFunction := by
  apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
    <;> intro a ha
  · rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
    rw [ha]
    simp only [le_neg_self_iff, id_eq]
    have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
    exact Eq.symm (if_neg h1)
  · rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
    exact id (Eq.symm ha)

/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
def orthchroMapReal : C(LorentzGroup d, ℝ) := ContinuousMap.comp
  ⟨stepFunction, stepFunction_continuous⟩ timeCompCont

/-- A Lorentz transformation which is orthochronous maps under `orthchroMapReal` to `1`. -/
lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
    orthchroMapReal Λ = 1 := by
  rw [IsOrthochronous_iff_ge_one] at h
  change stepFunction (Λ.1 _ _) = 1
  rw [stepFunction, if_pos h, if_neg]
  linarith

/-- A Lorentz transformation which is not-orthochronous maps under `orthchroMapReal` to `- 1`. -/
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
    orthchroMapReal Λ = - 1 := by
  rw [not_orthochronous_iff_le_neg_one] at h
  change stepFunction _ = - 1
  rw [stepFunction, if_pos]
  exact h

/-- Every Lorentz transformation maps under `orthchroMapReal` to either `1` or `-1`. -/
lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
    orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
  by_cases h : IsOrthochronous Λ
  · exact Or.inr $ orthchroMapReal_on_IsOrthochronous h
  · exact Or.inl $ orthchroMapReal_on_not_IsOrthochronous h

local notation "ℤ₂" => Multiplicative (ZMod 2)

/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernel are the Orthochronous elements. -/
def orthchroMap : C(LorentzGroup d, ℤ₂) :=
  ContinuousMap.comp coeForℤ₂ {
    toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
    continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}

/-- A Lorentz transformation which is orthochronous maps under `orthchroMap` to `1`
  in `ℤ₂` (the identity element). -/
lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
    orthchroMap Λ = 1 := by
  simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]

/-- A Lorentz transformation which is not-orthochronous maps under `orthchroMap` to
  the non-identity element of `ℤ₂`. -/
lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
    orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
  simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
    orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]
  rw [if_neg]
  · rfl
  · linarith

/-- The product of two orthochronous Lorentz transformations is orthochronous. -/
lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
    (h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
  rw [IsOrthochronous_iff_transpose] at h
  rw [IsOrthochronous_iff_futurePointing] at h h'
  rw [IsOrthochronous, LorentzGroup.inl_inl_mul]
  exact NormOne.FuturePointing.metric_reflect_mem_mem h h'

/-- The product of two non-orthochronous Lorentz transformations is orthochronous. -/
lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
    (h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
  rw [IsOrthochronous_iff_transpose] at h
  rw [IsOrthochronous_iff_futurePointing] at h h'
  rw [IsOrthochronous, LorentzGroup.inl_inl_mul]
  exact NormOne.FuturePointing.metric_reflect_not_mem_not_mem h h'

/-- The product of an orthochronous Lorentz transformations with a
  non-orthochronous Lorentz transformation is not orthochronous. -/
lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
    (h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
  rw [not_orthochronous_iff_le_zero, LorentzGroup.inl_inl_mul]
  rw [IsOrthochronous_iff_transpose] at h
  rw [IsOrthochronous_iff_futurePointing] at h h'
  exact NormOne.FuturePointing.metric_reflect_mem_not_mem h h'

/-- The product of a non-orthochronous Lorentz transformations with an
  orthochronous Lorentz transformation is not orthochronous. -/
lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
    (h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
  rw [not_orthochronous_iff_le_zero, LorentzGroup.inl_inl_mul]
  rw [IsOrthochronous_iff_transpose] at h
  rw [IsOrthochronous_iff_futurePointing] at h h'
  exact NormOne.FuturePointing.metric_reflect_not_mem_mem h h'

/-- The homomorphism from `LorentzGroup` to `ℤ₂` whose kernel are the Orthochronous elements. -/
def orthchroRep : LorentzGroup d →* ℤ₂ where
  toFun := orthchroMap
  map_one' := orthchroMap_IsOrthochronous (by simp [IsOrthochronous])
  map_mul' Λ Λ' := by
    simp only
    by_cases h : IsOrthochronous Λ
      <;> by_cases h' : IsOrthochronous Λ'
    · rw [orthchroMap_IsOrthochronous h, orthchroMap_IsOrthochronous h',
        orthchroMap_IsOrthochronous (mul_othchron_of_othchron_othchron h h')]
      rfl
    · rw [orthchroMap_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
        orthchroMap_not_IsOrthochronous (mul_not_othchron_of_othchron_not_othchron h h')]
      rfl
    · rw [orthchroMap_not_IsOrthochronous h, orthchroMap_IsOrthochronous h',
        orthchroMap_not_IsOrthochronous (mul_not_othchron_of_not_othchron_othchron h h')]
      rfl
    · rw [orthchroMap_not_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
        orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
      rfl

end LorentzGroup

end