/- Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license. Authors: Joseph Tooby-Smith -/ import HepLean.SpaceTime.LorentzAlgebra.Basic /-! # Basis of the Lorentz Algebra We define the standard basis of the Lorentz group. -/ namespace spaceTime namespace lorentzAlgebra open Matrix /-- The matrices which form the basis of the Lorentz algebra. -/ @[simp] def σMat (μ ν : Fin 4) : Matrix (Fin 4) (Fin 4) ℝ := fun ρ δ ↦ η_[μ]_[δ] * η^[ρ]_[ν] - η^[ρ]_[μ] * η_[ν]_[δ] lemma σMat_in_lorentzAlgebra (μ ν : Fin 4) : σMat μ ν ∈ lorentzAlgebra := by rw [mem_iff] funext ρ δ rw [Matrix.neg_mul, Matrix.neg_apply, η_mul, mul_η, transpose_apply] apply Eq.trans ?_ (by ring : ((η^[ρ]_[μ] * η_[ρ]_[ρ]) * η_[ν]_[δ] - η_[μ]_[δ] * (η^[ρ]_[ν] * η_[ρ]_[ρ])) = -(η_[ρ]_[ρ] * (η_[μ]_[δ] * η^[ρ]_[ν] - η^[ρ]_[μ] * η_[ν]_[δ] ))) apply Eq.trans (by ring : (η_[μ]_[ρ] * η^[δ]_[ν] - η^[δ]_[μ] * η_[ν]_[ρ]) * η_[δ]_[δ] = (- (η^[δ]_[μ] * η_[δ]_[δ]) * η_[ν]_[ρ] + η_[μ]_[ρ] * (η^[δ]_[ν] * η_[δ]_[δ]))) rw [η_mul_self, η_mul_self, η_mul_self, η_mul_self] ring /-- Elements of the Lorentz algebra which form a basis thereof. -/ @[simps!] def σ (μ ν : Fin 4) : lorentzAlgebra := ⟨σMat μ ν, σMat_in_lorentzAlgebra μ ν⟩ lemma σ_anti_symm (μ ν : Fin 4) : σ μ ν = - σ ν μ := by refine SetCoe.ext ?_ funext ρ δ simp only [σ_coe, σMat, NegMemClass.coe_neg, neg_apply, neg_sub] ring lemma σMat_mul (α β γ δ a b: Fin 4) : (σMat α β * σMat γ δ) a b = η^[a]_[α] * (η_[δ]_[b] * η_[β]_[γ] - η_[γ]_[b] * η_[β]_[δ]) - η^[a]_[β] * (η_[δ]_[b] * η_[α]_[γ]- η_[γ]_[b] * η_[α]_[δ]) := by rw [Matrix.mul_apply] simp only [σMat] trans (η^[a]_[α] * η_[δ]_[b]) * ∑ x, η^[x]_[γ] * η_[β]_[x] - (η^[a]_[α] * η_[γ]_[b]) * ∑ x, η^[x]_[δ] * η_[β]_[x] - (η^[a]_[β] * η_[δ]_[b]) * ∑ x, η^[x]_[γ] * η_[α]_[x] + (η^[a]_[β] * η_[γ]_[b]) * ∑ x, η^[x]_[δ] * η_[α]_[x] repeat rw [Fin.sum_univ_four] ring rw [η_contract_self', η_contract_self', η_contract_self', η_contract_self'] ring lemma σ_comm (α β γ δ : Fin 4) : ⁅σ α β , σ γ δ⁆ = η_[α]_[δ] • σ γ β + η_[α]_[γ] • σ β δ + η_[β]_[δ] • σ α γ + η_[β]_[γ] • σ δ α := by refine SetCoe.ext ?_ change σMat α β * σ γ δ - σ γ δ * σ α β = _ funext a b simp only [σ_coe, sub_apply, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, Submodule.coe_smul_of_tower, Matrix.add_apply, Matrix.smul_apply, σMat, smul_eq_mul] rw [σMat_mul, σMat_mul, η_symmetric α γ, η_symmetric α δ, η_symmetric β γ, η_symmetric β δ] ring lemma eq_span_σ (Λ : lorentzAlgebra) : Λ = Λ.1 0 1 • σ 0 1 + Λ.1 0 2 • σ 0 2 + Λ.1 0 3 • σ 0 3 + Λ.1 1 2 • σ 1 2 + Λ.1 1 3 • σ 1 3 + Λ.1 2 3 • σ 2 3 := by apply SetCoe.ext ?_ funext a b fin_cases a <;> fin_cases b <;> simp only [Fin.zero_eta, Fin.isValue, Fin.mk_one, Fin.reduceFinMk, AddSubmonoid.coe_add, Submodule.coe_smul_of_tower, σ_coe, Matrix.add_apply, Matrix.smul_apply, σMat, ηUpDown, ne_eq, zero_ne_one, not_false_eq_true, one_apply_ne, η_explicit, of_apply, cons_val_zero, mul_zero, one_apply_eq, mul_one, sub_neg_eq_add, zero_add, smul_eq_mul, Fin.reduceEq, cons_val_one, vecHead, vecTail, Nat.reduceAdd, Function.comp_apply, Fin.succ_zero_eq_one, sub_self, add_zero, cons_val_two, cons_val_three, Fin.succ_one_eq_two, mul_neg, neg_zero, sub_zero] · exact diag_comp Λ 0 · exact time_comps Λ 0 · exact diag_comp Λ 1 · exact time_comps Λ 1 · exact space_comps Λ 1 0 · exact diag_comp Λ 2 · exact time_comps Λ 2 · exact space_comps Λ 2 0 · exact space_comps Λ 2 1 · exact diag_comp Λ 3 /-- The coordinate map for the basis formed by the matrices `σ`. -/ @[simps!] noncomputable def σCoordinateMap : lorentzAlgebra ≃ₗ[ℝ] Fin 6 →₀ ℝ where toFun Λ := Finsupp.equivFunOnFinite.invFun fun i => match i with | 0 => Λ.1 0 1 | 1 => Λ.1 0 2 | 2 => Λ.1 0 3 | 3 => Λ.1 1 2 | 4 => Λ.1 1 3 | 5 => Λ.1 2 3 map_add' S T := by ext i fin_cases i <;> rfl map_smul' c S := by ext i fin_cases i <;> rfl invFun c := c 0 • σ 0 1 + c 1 • σ 0 2 + c 2 • σ 0 3 + c 3 • σ 1 2 + c 4 • σ 1 3 + c 5 • σ 2 3 left_inv Λ := by simp only [Fin.isValue, Equiv.invFun_as_coe, Finsupp.equivFunOnFinite_symm_apply_toFun] exact (eq_span_σ Λ).symm right_inv c := by ext i fin_cases i <;> simp only [Fin.isValue, Set.Finite.toFinset_setOf, ne_eq, Finsupp.coe_mk, Fin.zero_eta, Fin.isValue, Fin.mk_one, Fin.reduceFinMk, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, Submodule.coe_smul_of_tower, σ_coe, Matrix.add_apply, Matrix.smul_apply, σMat, ηUpDown, ne_eq, zero_ne_one, not_false_eq_true, one_apply_ne, η_explicit, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const, mul_zero, one_apply_eq, mul_one, sub_neg_eq_add, zero_add, smul_eq_mul, Fin.reduceEq, cons_val_one, vecHead, vecTail, Nat.succ_eq_add_one, Nat.reduceAdd, Function.comp_apply, Fin.succ_zero_eq_one, sub_self, add_zero, cons_val_two, cons_val_three, Fin.succ_one_eq_two, mul_neg, neg_zero, sub_zero, Finsupp.equivFunOnFinite] /-- The basis formed by the matrices `σ`. -/ @[simps! repr_apply_support_val repr_apply_toFun] noncomputable def σBasis : Basis (Fin 6) ℝ lorentzAlgebra where repr := σCoordinateMap instance : Module.Finite ℝ lorentzAlgebra := Module.Finite.of_basis σBasis /-- The Lorentz algebra is 6-dimensional. -/ theorem finrank_eq_six : FiniteDimensional.finrank ℝ lorentzAlgebra = 6 := by have h := Module.mk_finrank_eq_card_basis σBasis simp_all simp [FiniteDimensional.finrank] rw [h] simp only [Cardinal.toNat_ofNat] end lorentzAlgebra end spaceTime