100 lines
3.3 KiB
Text
100 lines
3.3 KiB
Text
/-
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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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/-!
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# The Lorentz Group
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We define the Lorentz group, and show it is a closed subgroup General Linear Group.
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## TODO
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- Show that the Lorentz is a Lie group.
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- Define the proper Lorentz group.
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- Define the restricted Lorentz group, and prove it is connected.
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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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/-- The Lorentz group as a subgroup of the general linear group over the reals. -/
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def lorentzGroup : Subgroup (GeneralLinearGroup (Fin 4) ℝ) where
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carrier := {Λ | ∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y}
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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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lemma mem_lorentzGroup_iff (Λ : GeneralLinearGroup (Fin 4) ℝ) :
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Λ ∈ lorentzGroup ↔ ∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y := by
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rfl
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lemma mem_lorentzGroup_iff' (Λ : GeneralLinearGroup (Fin 4) ℝ) :
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Λ ∈ lorentzGroup ↔ ∀ (x y : spaceTime), ηLin (x) ((η * Λ.1ᵀ * η * Λ.1) *ᵥ y) = ηLin x y := by
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rw [mem_lorentzGroup_iff]
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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 mem_lorentzGroup_iff'' (Λ : GL (Fin 4) ℝ) :
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Λ ∈ lorentzGroup ↔ η * Λ.1ᵀ * η * Λ.1 = 1 := by
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rw [mem_lorentzGroup_iff', ηLin_matrix_eq_identity_iff (η * Λ.1ᵀ * η * Λ.1)]
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apply Iff.intro
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· simp_all only [ηLin_apply_apply, implies_true, iff_true, one_mulVec]
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· simp_all only [ηLin_apply_apply, mulVec_mulVec, implies_true]
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lemma det_lorentzGroup (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 ((mem_lorentzGroup_iff'' Λ.1).mp Λ.2))
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/-- A continous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` for which the Lorentz
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group is the kernal. -/
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def lorentzMap (Λ : GL (Fin 4) ℝ) : Matrix (Fin 4) (Fin 4) ℝ := η * Λ.1ᵀ * η * Λ.1
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lemma lorentzMap_continuous : Continuous lorentzMap := 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 lorentzGroup_kernal : lorentzGroup = lorentzMap ⁻¹' {1} := by
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ext Λ
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erw [mem_lorentzGroup_iff'' Λ]
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rfl
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theorem lorentzGroup_isClosed : IsClosed (lorentzGroup : Set (GeneralLinearGroup (Fin 4) ℝ)) := by
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rw [lorentzGroup_kernal]
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exact continuous_iff_isClosed.mp lorentzMap_continuous {1} isClosed_singleton
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end spaceTime
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end
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