2024-06-11 11:16:31 -04:00
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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.LorentzAlgebra.Basic
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/-!
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# Basis of the Lorentz Algebra
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We define the standard basis of the Lorentz group.
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-/
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namespace spaceTime
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namespace lorentzAlgebra
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open Matrix
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2024-06-11 11:33:50 -04:00
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/-- The matrices which form the basis of the Lorentz algebra. -/
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2024-06-11 11:16:31 -04:00
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@[simp]
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def σMat (μ ν : Fin 4) : Matrix (Fin 4) (Fin 4) ℝ := fun ρ δ ↦
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η^[ρ]_[μ] * η_[ν]_[δ] - η_[μ]_[δ] * η^[ρ]_[ν]
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2024-06-12 11:47:38 -04:00
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lemma σMat_in_lorentzAlgebra (μ ν : Fin 4) : σMat μ ν ∈ lorentzAlgebra := by
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rw [mem_iff]
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funext ρ δ
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rw [Matrix.neg_mul, Matrix.neg_apply, η_mul, mul_η, transpose_apply]
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apply Eq.trans ?_ (by ring :
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- ((η^[ρ]_[μ] * η_[ρ]_[ρ]) * η_[ν]_[δ] - η_[μ]_[δ] * (η^[ρ]_[ν] * η_[ρ]_[ρ])) =
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-(η_[ρ]_[ρ] * (η^[ρ]_[μ] * η_[ν]_[δ] - η_[μ]_[δ] * η^[ρ]_[ν])))
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apply Eq.trans (by ring : (η^[δ]_[μ] * η_[ν]_[ρ] - η_[μ]_[ρ] * η^[δ]_[ν]) * η_[δ]_[δ]
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= ((η^[δ]_[μ] * η_[δ]_[δ]) * η_[ν]_[ρ] - η_[μ]_[ρ] * (η^[δ]_[ν] * η_[δ]_[δ])))
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rw [η_mul_self, η_mul_self, η_mul_self, η_mul_self]
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ring
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/-- Elements of the Lorentz algebra which form a basis thereof. -/
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@[simps!]
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def σ (μ ν : Fin 4) : lorentzAlgebra := ⟨σMat μ ν, σMat_in_lorentzAlgebra μ ν⟩
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lemma σ_anti_symm (μ ν : Fin 4) : σ μ ν = - σ ν μ := by
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refine SetCoe.ext ?_
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funext ρ δ
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simp only [σ_coe, σMat, NegMemClass.coe_neg, neg_apply, neg_sub]
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ring
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lemma σMat_mul (α β γ δ a b: Fin 4) :
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(σMat α β * σMat γ δ) a b =
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η^[a]_[α] * (η_[δ]_[b] * η_[β]_[γ] - η_[γ]_[b] * η_[β]_[δ])
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- η^[a]_[β] * (η_[δ]_[b] * η_[α]_[γ]- η_[γ]_[b] * η_[α]_[δ]) := by
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rw [Matrix.mul_apply]
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simp only [σMat]
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trans (η^[a]_[α] * η_[δ]_[b]) * ∑ x, η^[x]_[γ] * η_[β]_[x]
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- (η^[a]_[α] * η_[γ]_[b]) * ∑ x, η^[x]_[δ] * η_[β]_[x]
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- (η^[a]_[β] * η_[δ]_[b]) * ∑ x, η^[x]_[γ] * η_[α]_[x]
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+ (η^[a]_[β] * η_[γ]_[b]) * ∑ x, η^[x]_[δ] * η_[α]_[x]
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repeat rw [Fin.sum_univ_four]
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ring
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rw [η_contract_self', η_contract_self', η_contract_self', η_contract_self']
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ring
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lemma σ_comm (α β γ δ : Fin 4) :
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⁅σ α β , σ γ δ⁆ =
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η_[α]_[δ] • σ β γ + η_[α]_[γ] • σ δ β + η_[β]_[δ] • σ γ α + η_[β]_[γ] • σ α δ := by
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refine SetCoe.ext ?_
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change σMat α β * σ γ δ - σ γ δ * σ α β = _
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funext a b
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simp only [σ_coe, sub_apply, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid,
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Submodule.coe_smul_of_tower, add_apply, smul_apply, σMat, smul_eq_mul]
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rw [σMat_mul, σMat_mul, η_symmetric α γ, η_symmetric α δ, η_symmetric β γ, η_symmetric β δ]
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ring
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2024-06-11 11:16:31 -04:00
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end lorentzAlgebra
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end spaceTime
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