refactor: Major refactor of lorentz group
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11 changed files with 866 additions and 208 deletions
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@ -3,8 +3,9 @@ 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 HepLean.SpaceTime.MinkowskiMetric
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import HepLean.SpaceTime.AsSelfAdjointMatrix
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import HepLean.SpaceTime.LorentzVector.NormOne
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/-!
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# The Lorentz Group
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@ -26,7 +27,6 @@ identity.
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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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@ -34,146 +34,139 @@ open Complex
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open ComplexConjugate
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/-!
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## Matrices which preserve `ηLin`
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## Matrices which preserves the Minkowski metric
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We start studying the properties of matrices which preserve `ηLin`.
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These matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.
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-/
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variable {d : ℕ}
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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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open minkowskiMetric in
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/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/
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def LorentzGroup (d : ℕ) : Set (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :=
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{Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ |
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∀ (x y : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ y⟫ₘ = ⟪x, y⟫ₘ}
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namespace PreservesηLin
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variable (Λ : Matrix (Fin 4) (Fin 4) ℝ)
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namespace LorentzGroup
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/-- Notation for the Lorentz group. -/
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scoped[LorentzGroup] notation (name := lorentzGroup_notation) "𝓛" => LorentzGroup
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lemma iff_norm : PreservesηLin Λ ↔
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∀ (x : SpaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ x) = ηLin x x := by
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open minkowskiMetric
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variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
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/-!
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# Membership conditions
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-/
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lemma mem_iff_norm : Λ ∈ LorentzGroup d ↔
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∀ (x : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ x⟫ₘ = ⟪x, x⟫ₘ := by
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refine Iff.intro (fun h x => h x x) (fun h x y => ?_)
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have hp := h (x + y)
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have hn := h (x - y)
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rw [mulVec_add] at hp
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rw [mulVec_sub] at hn
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simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn
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rw [ηLin_symm (Λ *ᵥ y) (Λ *ᵥ x), ηLin_symm y x] at hp hn
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rw [symm (Λ *ᵥ y) (Λ *ᵥ x), symm y x] at hp hn
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linear_combination hp / 4 + -1 * hn / 4
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lemma iff_det_selfAdjoint : PreservesηLin Λ ↔
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∀ (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)),
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det ((toSelfAdjointMatrix ∘ toLin stdBasis stdBasis Λ ∘ toSelfAdjointMatrix.symm) x).1
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= det x.1 := by
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rw [iff_norm]
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apply Iff.intro
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intro h x
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have h1 := congrArg ofReal $ h (toSelfAdjointMatrix.symm x)
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simpa [← det_eq_ηLin] using h1
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intro h x
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have h1 := h (toSelfAdjointMatrix x)
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simpa [det_eq_ηLin] using h1
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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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lemma mem_iff_on_right : Λ ∈ LorentzGroup d ↔
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∀ (x y : LorentzVector d), ⟪x, (dual Λ * Λ) *ᵥ y⟫ₘ = ⟪x, y⟫ₘ := by
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apply Iff.intro
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intro h 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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rw [← dual_mulVec_right, mulVec_mulVec] at h1
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exact h1
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intro h x y
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rw [ηLin_mulVec_left, mulVec_mulVec]
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rw [← dual_mulVec_right, 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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· exact fun a x y => Eq.symm (Real.ext_cauchy (congrArg Real.cauchy (a x y)))
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lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
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rw [mem_iff_on_right, matrix_eq_id_iff]
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exact forall_comm
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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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lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := by
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rw [mem_iff_dual_mul_self]
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exact mul_eq_one_comm
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lemma iff_transpose : PreservesηLin Λ ↔ PreservesηLin Λᵀ := by
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lemma mem_iff_transpose : Λ ∈ LorentzGroup d ↔ Λᵀ ∈ LorentzGroup d := 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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noncomm_ring
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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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· intro h
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have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)
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rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,
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← mul_assoc, transpose_one] at h1
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rw [mem_iff_self_mul_dual, ← h1, dual]
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noncomm_ring
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· intro h
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have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)
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rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,
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← mul_assoc, transpose_one, transpose_transpose] at h1
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rw [mem_iff_self_mul_dual, ← h1, dual]
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noncomm_ring
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lemma mem_mul (hΛ : Λ ∈ LorentzGroup d) (hΛ' : Λ' ∈ LorentzGroup d) : Λ * Λ' ∈ LorentzGroup d := by
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rw [mem_iff_dual_mul_self, dual_mul]
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trans dual Λ' * (dual Λ * Λ) * Λ'
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noncomm_ring
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rw [(mem_iff_dual_mul_self).mp hΛ]
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simp [(mem_iff_dual_mul_self).mp hΛ']
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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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lemma mul {Λ Λ' : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) (h' : PreservesηLin Λ') :
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PreservesηLin (Λ * Λ') := by
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intro x y
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rw [← mulVec_mulVec, ← mulVec_mulVec, h, h']
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lemma one : PreservesηLin 1 := by
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intro x y
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lemma one_mem : 1 ∈ LorentzGroup d := by
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rw [mem_iff_dual_mul_self]
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simp
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lemma η : PreservesηLin η := by
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simp [iff_matrix, η_transpose, η_sq]
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lemma dual_mem (h : Λ ∈ LorentzGroup d) : dual Λ ∈ LorentzGroup d := by
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rw [mem_iff_dual_mul_self, dual_dual]
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exact mem_iff_self_mul_dual.mp h
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end PreservesηLin
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end LorentzGroup
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/-!
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## The Lorentz group
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We define the Lorentz group as the set of matrices which preserve `ηLin`.
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We show that the Lorentz group is indeed a group.
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# The Lorentz group as a group
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-/
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/-- The Lorentz group is the subset of matrices which preserve ηLin. -/
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def LorentzGroup : Type := {Λ : Matrix (Fin 4) (Fin 4) ℝ // PreservesηLin Λ}
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@[simps mul_coe one_coe inv div]
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instance lorentzGroupIsGroup : Group LorentzGroup where
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mul A B := ⟨A.1 * B.1, PreservesηLin.mul A.2 B.2⟩
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instance lorentzGroupIsGroup : Group (LorentzGroup d) where
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mul A B := ⟨A.1 * B.1, LorentzGroup.mem_mul A.2 B.2⟩
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mul_assoc A B C := by
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apply Subtype.eq
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exact Matrix.mul_assoc A.1 B.1 C.1
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one := ⟨1, PreservesηLin.one⟩
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one := ⟨1, LorentzGroup.one_mem⟩
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one_mul A := by
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apply Subtype.eq
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exact Matrix.one_mul A.1
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mul_one A := by
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apply Subtype.eq
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exact Matrix.mul_one A.1
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inv A := ⟨η * A.1ᵀ * η , PreservesηLin.mul (PreservesηLin.mul PreservesηLin.η
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((PreservesηLin.iff_transpose A.1).mp A.2)) PreservesηLin.η⟩
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inv A := ⟨minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2⟩
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mul_left_inv A := by
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apply Subtype.eq
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exact (PreservesηLin.iff_matrix A.1).mp A.2
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exact LorentzGroup.mem_iff_dual_mul_self.mp A.2
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/-- Notation for the Lorentz group. -/
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scoped[SpaceTime] notation (name := lorentzGroup_notation) "𝓛" => LorentzGroup
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/-- `lorentzGroup` has the subtype topology. -/
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instance : TopologicalSpace LorentzGroup := instTopologicalSpaceSubtype
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/-- `LorentzGroup` has the subtype topology. -/
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instance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype
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namespace LorentzGroup
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lemma coe_inv (A : LorentzGroup) : (A⁻¹).1 = A.1⁻¹:= by
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open minkowskiMetric
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variable {Λ Λ' : LorentzGroup d}
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@[simp]
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lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= by
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refine (inv_eq_left_inv ?h).symm
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exact (PreservesηLin.iff_matrix A.1).mp A.2
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exact mem_iff_dual_mul_self.mp Λ.2
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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 := ⟨Λ.1ᵀ, (PreservesηLin.iff_transpose Λ.1).mp Λ.2⟩
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def transpose (Λ : LorentzGroup d) : LorentzGroup d :=
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⟨Λ.1ᵀ, mem_iff_transpose.mp Λ.2⟩
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/-!
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@ -186,9 +179,9 @@ embedding.
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-/
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/-- The homomorphism of the Lorentz group into `GL (Fin 4) ℝ`. -/
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def toGL : LorentzGroup →* GL (Fin 4) ℝ where
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toFun A := ⟨A.1, (A⁻¹).1, mul_eq_one_comm.mpr $ (PreservesηLin.iff_matrix A.1).mp A.2,
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(PreservesηLin.iff_matrix A.1).mp A.2⟩
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def toGL : LorentzGroup d →* GL (Fin 1 ⊕ Fin d) ℝ where
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toFun A := ⟨A.1, (A⁻¹).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,
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mem_iff_dual_mul_self.mp A.2⟩
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map_one' := by
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simp
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rfl
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@ -197,7 +190,7 @@ def toGL : LorentzGroup →* GL (Fin 4) ℝ where
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ext
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rfl
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lemma toGL_injective : Function.Injective toGL := by
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lemma toGL_injective : Function.Injective (@toGL d) := by
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intro A B h
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apply Subtype.eq
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rw [@Units.ext_iff] at h
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@ -206,20 +199,21 @@ lemma toGL_injective : Function.Injective toGL := by
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/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of
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the monoid of matrices. -/
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@[simps!]
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def toProd : LorentzGroup →* (Matrix (Fin 4) (Fin 4) ℝ) × (Matrix (Fin 4) (Fin 4) ℝ)ᵐᵒᵖ :=
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def toProd : LorentzGroup d →* (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) ×
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(Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)ᵐᵒᵖ :=
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MonoidHom.comp (Units.embedProduct _) toGL
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lemma toProd_eq_transpose_η : toProd A = (A.1, ⟨η * A.1ᵀ * η⟩) := rfl
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lemma toProd_eq_transpose_η : toProd Λ = (Λ.1, MulOpposite.op $ minkowskiMetric.dual Λ.1) := rfl
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lemma toProd_injective : Function.Injective toProd := by
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lemma toProd_injective : Function.Injective (@toProd d) := by
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intro A B h
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rw [toProd_eq_transpose_η, toProd_eq_transpose_η] at h
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rw [@Prod.mk.inj_iff] at h
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apply Subtype.eq
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exact h.1
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lemma toProd_continuous : Continuous toProd := by
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change Continuous (fun A => (A.1, ⟨η * A.1ᵀ * η⟩))
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lemma toProd_continuous : Continuous (@toProd d) := by
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change Continuous (fun A => (A.1, ⟨dual A.1⟩))
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refine continuous_prod_mk.mpr (And.intro ?_ ?_)
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exact continuous_iff_le_induced.mpr fun U a => a
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refine Continuous.comp' ?_ ?_
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@ -230,7 +224,7 @@ lemma toProd_continuous : Continuous toProd := by
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/-- The embedding from the Lorentz Group into the monoid of matrices times the opposite of
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the monoid of matrices. -/
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lemma toProd_embedding : Embedding toProd where
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lemma toProd_embedding : Embedding (@toProd d) where
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inj := toProd_injective
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induced := by
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refine (inducing_iff ⇑toProd).mp ?_
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@ -238,7 +232,7 @@ lemma toProd_embedding : Embedding toProd where
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exact (inducing_iff (Prod.fst ∘ ⇑toProd)).mpr rfl
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/-- The embedding from the Lorentz Group into `GL (Fin 4) ℝ`. -/
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lemma toGL_embedding : Embedding toGL.toFun where
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lemma toGL_embedding : Embedding (@toGL d).toFun where
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inj := toGL_injective
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induced := by
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refine ((fun {X} {t t'} => TopologicalSpace.ext_iff.mpr) ?_).symm
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@ -248,11 +242,48 @@ lemma toGL_embedding : Embedding toGL.toFun where
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exact exists_exists_and_eq_and
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instance : TopologicalGroup LorentzGroup := Inducing.topologicalGroup toGL toGL_embedding.toInducing
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instance : TopologicalGroup (LorentzGroup d) :=
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Inducing.topologicalGroup toGL toGL_embedding.toInducing
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section
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open LorentzVector
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/-!
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# To a norm one Lorentz vector
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-/
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/-- The first column of a lorentz matrix as a `NormOneLorentzVector`. -/
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@[simps!]
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def toNormOneLorentzVector (Λ : LorentzGroup d) : NormOneLorentzVector d :=
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⟨Λ.1 *ᵥ timeVec, by rw [NormOneLorentzVector.mem_iff, Λ.2, minkowskiMetric.on_timeVec]⟩
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/-!
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# The time like element
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-/
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/-- The time like element of a Lorentz matrix. -/
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@[simp]
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def timeComp (Λ : LorentzGroup d) : ℝ := Λ.1 (Sum.inl 0) (Sum.inl 0)
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lemma timeComp_eq_toNormOneLorentzVector : timeComp Λ = (toNormOneLorentzVector Λ).1.time := by
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simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue, timeComp]
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erw [Pi.basisFun_apply, mulVec_stdBasis]
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lemma timeComp_mul (Λ Λ' : LorentzGroup d) : timeComp (Λ * Λ') =
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⟪toNormOneLorentzVector (transpose Λ), (toNormOneLorentzVector Λ').1.spaceReflection⟫ₘ := by
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simp only [timeComp, Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fintype.sum_sum_type,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton, toNormOneLorentzVector,
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transpose, timeVec, right_spaceReflection, time, space, PiLp.inner_apply, Function.comp_apply,
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RCLike.inner_apply, conj_trivial]
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erw [Pi.basisFun_apply, mulVec_stdBasis]
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simp
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end
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end LorentzGroup
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end SpaceTime
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