refactor: Major refactor of Lorentz vecs
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jstoobysmith
parent
3eb5da875f
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5cc188146f
20 changed files with 494 additions and 1005 deletions
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@ -35,21 +35,19 @@ variable {d : ℕ}
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/-- An auxillary linear map used in the definition of a generalised boost. -/
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def genBoostAux₁ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d where
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toFun x := (2 * toField d ⟪x, u.val.val⟫ₘ) • v.1.1
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toFun x := (2 * ⟪x, u.val.val⟫ₘ) • v.1.1
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map_add' x y := by
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simp [map_add, LinearMap.add_apply, tmul_add, add_tmul, mul_add, add_smul]
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map_smul' c x := by
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simp only [toField, Action.instMonoidalCategory_tensorObj_V,
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simp only [ Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_tensorUnit_V, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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smul_tmul, tmul_smul, map_smul, smul_eq_mul, RingHom.id_apply]
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erw [LinearMap.id_apply]
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rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
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rfl
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/-- An auxillary linear map used in the definition of a genearlised boost. -/
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def genBoostAux₂ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d where
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toFun x := - (toField d ⟪x, u.1.1 + v.1.1⟫ₘ / (1 + toField d ⟪u.1.1, v.1.1⟫ₘ)) • (u.1.1 + v.1.1)
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toFun x := - (⟪x, u.1.1 + v.1.1⟫ₘ / (1 + ⟪u.1.1, v.1.1⟫ₘ)) • (u.1.1 + v.1.1)
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map_add' x y := by
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simp only
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rw [← add_smul]
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@ -62,7 +60,7 @@ def genBoostAux₂ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d
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rw [map_smul]
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conv =>
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lhs; lhs; rhs; lhs
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change (c * toField d (contrContrContract.hom (x ⊗ₜ[ℝ] (u.val.val + v.val.val))))
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change (c * (contrContrContractField (x ⊗ₜ[ℝ] (u.val.val + v.val.val))))
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rw [mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
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rfl
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@ -75,21 +73,47 @@ lemma genBoostAux₂_self (u : FuturePointing d) : genBoostAux₂ u u = - genBoo
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rw [smul_smul]
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congr 1
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rw [u.1.2]
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simp [toField]
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conv => lhs; lhs; rhs; rhs; change 1
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rw [show 1 + (1 : ℝ) = (2 : ℝ ) by ring]
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simp only [isUnit_iff_ne_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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IsUnit.div_mul_cancel]
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rw [← (two_smul ℝ u.val.val)]
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simp only [tmul_smul, map_smul, smul_eq_mul]
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rfl
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/-- An generalised boost. This is a Lorentz transformation which takes the four velocity `u`
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to `v`. -/
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def genBoost (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d :=
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LinearMap.id + genBoostAux₁ u v + genBoostAux₂ u v
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lemma genBoost_mul_one_plus_contr (u v : FuturePointing d) (x : Contr d) :
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(1 + ⟪u.1.1, v.1.1⟫ₘ) • genBoost u v x =
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(1 + ⟪u.1.1, v.1.1⟫ₘ) • x +
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(2 * ⟪x, u.val.val⟫ₘ * (1 + ⟪u.1.1, v.1.1⟫ₘ)) • v.1.1
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- (⟪x, u.1.1 + v.1.1⟫ₘ) • (u.1.1 + v.1.1) := by
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simp only [genBoost, LinearMap.add_apply, LinearMap.id_apply, id_eq]
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rw [smul_add, smul_add]
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trans (1 + ⟪u.1.1, v.1.1⟫ₘ) • x +
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(2 * ⟪x, u.val.val⟫ₘ * (1 + ⟪u.1.1, v.1.1⟫ₘ)) • v.1.1
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+ (- ⟪x, u.1.1 + v.1.1⟫ₘ) • (u.1.1 + v.1.1)
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· congr 1
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· congr 1
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rw [genBoostAux₁]
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simp
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rw [smul_smul]
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congr 1
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ring
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· rw [genBoostAux₂]
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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CategoryTheory.Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, neg_smul, LinearMap.coe_mk,
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AddHom.coe_mk, smul_neg]
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rw [smul_smul]
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congr
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have h1 := FuturePointing.one_add_metric_non_zero u v
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field_simp
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· rw [neg_smul]
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rfl
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namespace genBoost
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/--
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@ -110,7 +134,7 @@ lemma toMatrix_mulVec (u v : FuturePointing d) (x : Contr d) :
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(toMatrix u v) *ᵥ x = genBoost u v x :=
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ContrMod.ext (LinearMap.toMatrix_mulVec_repr ContrMod.stdBasis ContrMod.stdBasis (genBoost u v) x)
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open minkowskiMatrix LorentzVector in
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open minkowskiMatrix in
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@[simp]
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lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
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(toMatrix u v) μ ν = η μ μ * (toField d ⟪ContrMod.stdBasis μ, ContrMod.stdBasis ν⟫ₘ + 2 *
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@ -118,31 +142,39 @@ lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
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- toField d ⟪ContrMod.stdBasis μ, u.val.val + v.val.val⟫ₘ *
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toField d ⟪ContrMod.stdBasis ν, u.val.val + v.val.val⟫ₘ /
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(1 + toField d ⟪u.val.val, v.val.val⟫ₘ)) := by
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sorry
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rw [matrix_apply_stdBasis (toMatrix u v) μ ν, toMatrix_mulVec]
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rw [contrContrContractField.matrix_apply_stdBasis (Λ := toMatrix u v) μ ν, toMatrix_mulVec]
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simp only [genBoost, genBoostAux₁, genBoostAux₂, map_add, smul_add, neg_smul, LinearMap.add_apply,
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LinearMap.id_apply, LinearMap.coe_mk, AddHom.coe_mk, basis_left, map_smul, smul_eq_mul, map_neg,
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mul_eq_mul_left_iff]
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LinearMap.id_apply, LinearMap.coe_mk, AddHom.coe_mk, contrContrContractField.basis_left, map_smul, smul_eq_mul, map_neg,
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mul_eq_mul_left_iff, toField_apply]
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ring_nf
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exact (true_or (η μ μ = 0)).mpr trivial
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simp only [Pi.add_apply, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_tensorUnit_V, CategoryTheory.Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Pi.smul_apply, smul_eq_mul, Pi.sub_apply, Pi.neg_apply]
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apply Or.inl
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ring
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open minkowskiMatrix LorentzVector in
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open minkowskiMatrix in
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lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) := by
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refine continuous_matrix ?_
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intro i j
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simp only [toMatrix_apply]
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refine (continuous_mul_left (η i i)).comp' ?_
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refine Continuous.sub ?_ ?_
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· refine .comp' (continuous_add_left (minkowskiMetric (e i) (e j))) ?_
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refine .comp' (continuous_mul_left (2 * ⟪e j, u⟫ₘ)) ?_
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· refine .comp' (continuous_add_left _) ?_
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refine .comp' (continuous_mul_left _) ?_
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exact FuturePointing.metric_continuous _
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refine .mul ?_ ?_
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· refine .mul ?_ ?_
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· simp only [(minkowskiMetric _).map_add]
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· simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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CategoryTheory.Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, tmul_add, map_add]
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refine .comp' ?_ ?_
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· exact continuous_add_left ((minkowskiMetric (stdBasis i)) ↑u)
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· exact continuous_add_left _
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· exact FuturePointing.metric_continuous _
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· simp only [(minkowskiMetric _).map_add]
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· simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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CategoryTheory.Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, tmul_add, map_add]
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refine .comp' ?_ ?_
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· exact continuous_add_left _
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· exact FuturePointing.metric_continuous _
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@ -150,17 +182,28 @@ lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) := by
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· refine .comp' (continuous_add_left 1) (FuturePointing.metric_continuous _)
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exact fun x => FuturePointing.one_add_metric_non_zero u x
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lemma toMatrix_in_lorentzGroup (u v : FuturePointing d) : (toMatrix u v) ∈ LorentzGroup d:= by
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lemma toMatrix_in_lorentzGroup (u v : FuturePointing d) : (toMatrix u v) ∈ LorentzGroup d := by
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rw [LorentzGroup.mem_iff_invariant]
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intro x y
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rw [toMatrix_mulVec, toMatrix_mulVec, genBoost, genBoostAux₁, genBoostAux₂]
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have h1 : (1 + (minkowskiMetric ↑u) ↑v.1.1) ≠ 0 := FuturePointing.one_add_metric_non_zero u v
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simp only [map_add, smul_add, neg_smul, LinearMap.add_apply, LinearMap.id_coe,
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id_eq, LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, map_neg, LinearMap.smul_apply,
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smul_eq_mul, LinearMap.neg_apply]
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field_simp
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rw [u.1.2, v.1.2, minkowskiMetric.symm v.1.1 u, minkowskiMetric.symm u y,
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minkowskiMetric.symm v y]
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ring
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rw [toMatrix_mulVec, toMatrix_mulVec]
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have h1 : (((1 + ⟪u.1.1, v.1.1⟫ₘ)) * (1 + ⟪u.1.1, v.1.1⟫ₘ)) •
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(contrContrContractField ((genBoost u v) y ⊗ₜ[ℝ] (genBoost u v) x))
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= (((1 + ⟪u.1.1, v.1.1⟫ₘ)) * (1 + ⟪u.1.1, v.1.1⟫ₘ)) • ⟪y, x⟫ₘ := by
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conv_lhs =>
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erw [← map_smul]
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rw [← smul_smul]
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rw [← tmul_smul, ← smul_tmul, ← tmul_smul, genBoost_mul_one_plus_contr,
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genBoost_mul_one_plus_contr]
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simp only [add_smul, one_smul, tmul_add, map_add, smul_add, tmul_sub, sub_tmul, add_tmul,
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smul_tmul, tmul_smul, map_sub, map_smul, smul_eq_mul, contr_self, mul_one]
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rw [contrContrContractField.symm v.1.1 u, contrContrContractField.symm v.1.1 x,
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contrContrContractField.symm u.1.1 x]
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simp
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ring
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have hn (a : ℝ ) {b c : ℝ} (h : a ≠ 0) (hab : a * b = a * c ) : b = c := by
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simp_all only [smul_eq_mul, ne_eq, mul_eq_mul_left_iff, or_false]
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refine hn _ ?_ h1
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simpa using (FuturePointing.one_add_metric_non_zero u v)
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/-- A generalised boost as an element of the Lorentz Group. -/
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def toLorentz (u v : FuturePointing d) : LorentzGroup d :=
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