109 lines
3.7 KiB
Text
109 lines
3.7 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.Basic
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import HepLean.SpaceTime.MinkowskiMetric
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import Mathlib.Algebra.Lie.Classical
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/-!
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# The Lorentz Algebra
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We define
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- Define `lorentzAlgebra` via `LieAlgebra.Orthogonal.so'` as a subalgebra of
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`Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`.
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- In `mem_iff` prove that a matrix is in the Lorentz algebra if and only if it satisfies the
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condition `Aᵀ * η = - η * A`.
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-/
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namespace SpaceTime
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open Matrix
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open TensorProduct
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/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
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def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :=
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(LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
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namespace lorentzAlgebra
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open minkowskiMatrix
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lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
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have h1 := A.2
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erw [mem_skewAdjointMatricesLieSubalgebra] at h1
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simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
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lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
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(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
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erw [mem_skewAdjointMatricesLieSubalgebra]
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simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h
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lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
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A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
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Iff.intro (fun h => transpose_eta ⟨A, h⟩) (fun h => mem_of_transpose_eta_eq_eta_mul_self h)
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lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
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A ∈ lorentzAlgebra ↔ A = - η * Aᵀ * η := by
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rw [mem_iff]
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· trans -η * (Aᵀ * η)
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rw [h]
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trans (η * η) * A
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rw [minkowskiMatrix.sq]
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all_goals noncomm_ring
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· nth_rewrite 2 [h]
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trans (η * η) * Aᵀ * η
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rw [minkowskiMatrix.sq]
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all_goals noncomm_ring
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lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 := by
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have h := congrArg (fun M ↦ M μ μ) $ mem_iff.mp Λ.2
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simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
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transpose_apply, diagonal_neg, diagonal_mul, neg_mul] at h
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rcases μ with μ | μ
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simpa using h
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simpa using h
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lemma time_comps (Λ : lorentzAlgebra) (i : Fin 3) :
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Λ.1 (Sum.inr i) (Sum.inl 0) = Λ.1 (Sum.inl 0) (Sum.inr i) := by
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simpa only [Fin.isValue, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
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transpose_apply, Sum.elim_inr, mul_neg, mul_one, diagonal_neg, diagonal_mul, Sum.elim_inl,
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neg_mul, one_mul, neg_inj] using congrArg (fun M ↦ M (Sum.inl 0) (Sum.inr i)) $ mem_iff.mp Λ.2
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lemma space_comps (Λ : lorentzAlgebra) (i j : Fin 3) :
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Λ.1 (Sum.inr i) (Sum.inr j) = - Λ.1 (Sum.inr j) (Sum.inr i) := by
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simpa only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_neg, diagonal_mul,
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Sum.elim_inr, neg_neg, one_mul, mul_diagonal, transpose_apply, mul_neg, mul_one] using
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(congrArg (fun M ↦ M (Sum.inr i) (Sum.inr j)) $ mem_iff.mp Λ.2).symm
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end lorentzAlgebra
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@[simps!]
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instance lorentzVectorAsLieRingModule : LieRingModule lorentzAlgebra (LorentzVector 3) where
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bracket Λ x := Λ.1.mulVec x
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add_lie Λ1 Λ2 x := by
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simp [add_mulVec]
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lie_add Λ x1 x2 := by
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simp only
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exact mulVec_add _ _ _
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leibniz_lie Λ1 Λ2 x := by
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simp [mulVec_add, Bracket.bracket, sub_mulVec]
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@[simps!]
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instance spaceTimeAsLieModule : LieModule ℝ lorentzAlgebra (LorentzVector 3) where
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smul_lie r Λ x := by
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simp [Bracket.bracket, smul_mulVec_assoc]
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lie_smul r Λ x := by
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simp [Bracket.bracket]
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rw [mulVec_smul]
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end SpaceTime
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