refactor: Partial refactor of the lorentz group
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204
HepLean/SpaceTime/LorentzGroup/Basic.lean
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204
HepLean/SpaceTime/LorentzGroup/Basic.lean
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/-
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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.Metric
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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/-!
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# The Lorentz Group
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We define the Lorentz group.
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## TODO
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- Show that the Lorentz is a Lie group.
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- Prove that the restricted Lorentz group is equivalent to the connected component of the
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identity.
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- Define the continuous maps from `ℝ³` to `restrictedLorentzGroup` defining boosts.
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## References
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- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf
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-/
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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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open Complex
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open ComplexConjugate
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/-- We say a matrix `Λ` preserves `ηLin` if for all `x` and `y`,
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`ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y`. -/
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def PreservesηLin (Λ : Matrix (Fin 4) (Fin 4) ℝ) : Prop :=
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∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y
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namespace PreservesηLin
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variable (Λ : Matrix (Fin 4) (Fin 4) ℝ)
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lemma iff_on_right : PreservesηLin Λ ↔
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∀ (x y : spaceTime), ηLin x ((η * Λᵀ * η * Λ) *ᵥ y) = ηLin x y := by
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apply Iff.intro
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intro h
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intro x y
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have h1 := h x y
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rw [ηLin_mulVec_left, mulVec_mulVec] at h1
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exact h1
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intro h
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intro x y
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rw [ηLin_mulVec_left, mulVec_mulVec]
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exact h x y
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lemma iff_matrix : PreservesηLin Λ ↔ η * Λᵀ * η * Λ = 1 := by
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rw [iff_on_right, ηLin_matrix_eq_identity_iff (η * Λᵀ * η * Λ)]
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apply Iff.intro
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· simp_all [ηLin, implies_true, iff_true, one_mulVec]
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· simp_all only [ηLin, LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply,
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mulVec_mulVec, implies_true]
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lemma iff_matrix' : PreservesηLin Λ ↔ Λ * (η * Λᵀ * η) = 1 := by
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rw [iff_matrix]
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apply Iff.intro
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intro h
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exact mul_eq_one_comm.mp h
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intro h
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exact mul_eq_one_comm.mp h
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lemma iff_transpose : PreservesηLin Λ ↔ PreservesηLin Λᵀ := by
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apply Iff.intro
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intro h
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have h1 := congrArg transpose ((iff_matrix Λ).mp h)
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rw [transpose_mul, transpose_mul, transpose_mul, η_transpose,
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← mul_assoc, transpose_one] at h1
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rw [iff_matrix' Λ.transpose, ← h1]
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repeat rw [← mul_assoc]
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intro h
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have h1 := congrArg transpose ((iff_matrix Λ.transpose).mp h)
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rw [transpose_mul, transpose_mul, transpose_mul, η_transpose,
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← mul_assoc, transpose_one, transpose_transpose] at h1
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rw [iff_matrix', ← h1]
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repeat rw [← mul_assoc]
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/-- The lift of a matrix which preserves `ηLin` to an invertible matrix. -/
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def liftGL {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) : GL (Fin 4) ℝ :=
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⟨Λ, η * Λᵀ * η , (iff_matrix' Λ).mp h , (iff_matrix Λ).mp h⟩
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end PreservesηLin
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/-- The Lorentz group as a subgroup of the general linear group over the reals. -/
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def lorentzGroup : Subgroup (GL (Fin 4) ℝ) where
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carrier := {Λ | PreservesηLin Λ}
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mul_mem' {a b} := by
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intros ha hb x y
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simp only [Units.val_mul, mulVec_mulVec]
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rw [← mulVec_mulVec, ← mulVec_mulVec, ha, hb]
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one_mem' := by
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intros x y
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simp
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inv_mem' {a} := by
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intros ha x y
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simp only [coe_units_inv, ← ha ((a.1⁻¹) *ᵥ x) ((a.1⁻¹) *ᵥ y), mulVec_mulVec]
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have hx : (a.1 * (a.1)⁻¹) = 1 := by
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simp only [@Units.mul_eq_one_iff_inv_eq, coe_units_inv]
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simp [hx]
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/-- The Lorentz group is a topological group with the subset topology. -/
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instance : TopologicalGroup lorentzGroup :=
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Subgroup.instTopologicalGroupSubtypeMem lorentzGroup
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def PreservesηLin.liftLor {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) :
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lorentzGroup := ⟨liftGL h, h⟩
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namespace lorentzGroup
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lemma mem_iff (Λ : GL (Fin 4) ℝ): Λ ∈ lorentzGroup ↔ PreservesηLin Λ := by
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rfl
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/-- The transpose of an matrix in the Lorentz group is an element of the Lorentz group. -/
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def transpose (Λ : lorentzGroup) : lorentzGroup :=
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PreservesηLin.liftLor ((PreservesηLin.iff_transpose Λ.1).mp Λ.2)
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def kernalMap : C(GL (Fin 4) ℝ, Matrix (Fin 4) (Fin 4) ℝ) where
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toFun Λ := η * Λ.1ᵀ * η * Λ.1
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continuous_toFun := by
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apply Continuous.mul _ Units.continuous_val
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apply Continuous.mul _ continuous_const
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exact Continuous.mul continuous_const (Continuous.matrix_transpose (Units.continuous_val))
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lemma kernalMap_kernal_eq_lorentzGroup : lorentzGroup = kernalMap ⁻¹' {1} := by
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ext Λ
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erw [mem_iff Λ, PreservesηLin.iff_matrix]
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rfl
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/-- The Lorentz Group is a closed subset of `GL (Fin 4) ℝ`. -/
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theorem isClosed_of_GL4 : IsClosed (lorentzGroup : Set (GL (Fin 4) ℝ)) := by
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rw [kernalMap_kernal_eq_lorentzGroup]
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exact continuous_iff_isClosed.mp kernalMap.2 {1} isClosed_singleton
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section Relations
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/-- The first column of a lorentz matrix. -/
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@[simp]
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def fstCol (Λ : lorentzGroup) : spaceTime := fun i => Λ.1 i 0
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lemma ηLin_fstCol (Λ : lorentzGroup) : ηLin (fstCol Λ) (fstCol Λ) = 1 := by
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rw [ηLin_expand]
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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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simp only [fstCol, Fin.isValue]
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ring
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lemma zero_component (x : { x : spaceTime // ηLin x x = 1}) :
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x.1 0 ^ 2 = 1 + ‖x.1.space‖ ^ 2 := by
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sorry
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/-- The space-like part of the first row of of a Lorentz matrix. -/
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@[simp]
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def fstSpaceRow (Λ : lorentzGroup) : EuclideanSpace ℝ (Fin 3) := fun i => Λ.1 0 i.succ
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/-- The space-like part of the first column of of a Lorentz matrix. -/
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@[simp]
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def fstSpaceCol (Λ : lorentzGroup) : EuclideanSpace ℝ (Fin 3) := fun i => Λ.1 i.succ 0
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lemma fstSpaceRow_transpose (Λ : lorentzGroup) : fstSpaceRow (transpose Λ) = fstSpaceCol Λ := by
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rfl
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lemma fstSpaceCol_transpose (Λ : lorentzGroup) : fstSpaceCol (transpose Λ) = fstSpaceRow Λ := by
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rfl
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lemma fst_col_normalized (Λ : lorentzGroup) :
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(Λ.1 0 0) ^ 2 - ‖fstSpaceCol Λ‖ ^ 2 = 1 := by
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rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
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simp
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rw [show Fin.succ 2 = 3 by rfl]
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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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lemma fst_row_normalized
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end Relations
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end lorentzGroup
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end spaceTime
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124
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
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124
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
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@ -0,0 +1,124 @@
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/-
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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.LorentzGroup.Proper
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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/-!
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# The Orthochronous Lorentz Group
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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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-/
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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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open Complex
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open ComplexConjugate
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namespace lorentzGroup
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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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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, 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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/-- 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 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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/-- 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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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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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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end
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123
HepLean/SpaceTime/LorentzGroup/Proper.lean
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123
HepLean/SpaceTime/LorentzGroup/Proper.lean
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@ -0,0 +1,123 @@
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/-
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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.LorentzGroup.Basic
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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/-!
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# The Proper Lorentz Group
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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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namespace spaceTime
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open Manifold
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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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/-- 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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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, 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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/-- 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 _)⟩,
|
||||
continuous_toFun := by
|
||||
refine Continuous.subtype_mk ?_ _
|
||||
exact Continuous.matrix_det $
|
||||
Continuous.comp' Units.continuous_val continuous_subtype_val}
|
||||
|
||||
lemma detContinuous_eq_iff_det_eq (Λ Λ' : lorentzGroup) :
|
||||
detContinuous Λ = detContinuous Λ' ↔ Λ.1.1.det = Λ'.1.1.det := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
simp [detContinuous] at h
|
||||
cases' det_eq_one_or_neg_one Λ with h1 h1
|
||||
<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
|
||||
<;> simp_all [h1, h2, h]
|
||||
rw [← toMul_zero, @Equiv.apply_eq_iff_eq] at h
|
||||
change (0 : Fin 2) = (1 : Fin 2) at h
|
||||
simp only [Fin.isValue, zero_ne_one] at h
|
||||
nth_rewrite 2 [← toMul_zero] at h
|
||||
rw [@Equiv.apply_eq_iff_eq] at h
|
||||
change (1 : Fin 2) = (0 : Fin 2) at h
|
||||
simp [Fin.isValue, zero_ne_one] at h
|
||||
intro h
|
||||
simp [detContinuous, h]
|
||||
|
||||
|
||||
/-- The representation taking a lorentz matrix to its determinant. -/
|
||||
@[simps!]
|
||||
def detRep : lorentzGroup →* ℤ₂ where
|
||||
toFun Λ := detContinuous Λ
|
||||
map_one' := by
|
||||
simp [detContinuous]
|
||||
map_mul' := by
|
||||
intro Λ1 Λ2
|
||||
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
|
||||
mul_ite, mul_one, ite_mul, one_mul]
|
||||
cases' (det_eq_one_or_neg_one Λ1) with h1 h1
|
||||
<;> cases' (det_eq_one_or_neg_one Λ2) with h2 h2
|
||||
<;> simp [h1, h2, detContinuous]
|
||||
rfl
|
||||
|
||||
lemma detRep_continuous : Continuous detRep := detContinuous.2
|
||||
|
||||
lemma det_on_connected_component {Λ Λ' : lorentzGroup} (h : Λ' ∈ connectedComponent Λ) :
|
||||
Λ.1.1.det = Λ'.1.1.det := by
|
||||
obtain ⟨s, hs, hΛ'⟩ := h
|
||||
let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
|
||||
haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
|
||||
simpa [f, detContinuous_eq_iff_det_eq] using
|
||||
(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
|
||||
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
|
||||
|
||||
lemma det_of_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') : Λ.1.1.det = Λ'.1.1.det :=
|
||||
det_on_connected_component $ pathComponent_subset_component _ h
|
||||
|
||||
/-- A Lorentz Matrix is proper if its determinant is 1. -/
|
||||
@[simp]
|
||||
def IsProper (Λ : lorentzGroup) : Prop := Λ.1.1.det = 1
|
||||
|
||||
instance : DecidablePred IsProper := by
|
||||
intro Λ
|
||||
apply Real.decidableEq
|
||||
|
||||
lemma IsProper_iff (Λ : lorentzGroup) : IsProper Λ ↔ detRep Λ = 1 := by
|
||||
rw [show 1 = detRep 1 by simp]
|
||||
rw [detRep_apply, detRep_apply, detContinuous_eq_iff_det_eq]
|
||||
simp only [IsProper, OneMemClass.coe_one, Units.val_one, det_one]
|
||||
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
end spaceTime
|
||||
end
|
||||
|
|
@ -118,6 +118,20 @@ lemma ηLin_expand (x y : spaceTime) : ηLin x y = x 0 * y 0 - x 1 * y 1 - x 2 *
|
|||
cons_val_zero, cons_val_one, head_cons, mul_neg, cons_val_two, tail_cons, cons_val_three]
|
||||
ring
|
||||
|
||||
lemma ηLin_expand_self (x : spaceTime) : ηLin x x = x 0 ^ 2 - ‖x.space‖ ^ 2 := by
|
||||
rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three, ηLin_expand]
|
||||
simp only [Fin.isValue, space, cons_val_zero, RCLike.inner_apply, conj_trivial, cons_val_one,
|
||||
head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons]
|
||||
ring
|
||||
|
||||
lemma time_elm_sq_of_ηLin (x : spaceTime) : x 0 ^ 2 = ηLin x x + ‖x.space‖ ^ 2 := by
|
||||
rw [ηLin_expand_self]
|
||||
ring
|
||||
|
||||
lemma ηLin_leq_time_sq (x : spaceTime) : ηLin x x ≤ x 0 ^ 2 := by
|
||||
rw [time_elm_sq_of_ηLin]
|
||||
apply (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖x.space‖
|
||||
|
||||
lemma ηLin_space_inner_product (x y : spaceTime) :
|
||||
ηLin x y = x 0 * y 0 - ⟪x.space, y.space⟫_ℝ := by
|
||||
rw [ηLin_expand, @PiLp.inner_apply, Fin.sum_univ_three]
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue