feat: Add metric invariance for Real Lorentz Tensors
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jstoobysmith
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7b0b979d51
7 changed files with 175 additions and 24 deletions
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@ -175,18 +175,41 @@ open scoped minkowskiMatrix
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variable {d : ℕ}
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/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
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def asProdLorentzVector : LorentzVector d ⊗[ℝ] LorentzVector d :=
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∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ)
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def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
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∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
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lemma asTenProd_diag :
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@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) := by
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simp only [asTenProd]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [Finset.sum_eq_single μ]
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intro ν _ hμν
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rw [minkowskiMatrix.off_diag_zero hμν.symm]
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simp only [zero_smul]
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intro a
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simp_all only
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/-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/
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def asProdCovariantLorentzVector : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
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∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ)
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def asCoTenProd : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
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∑ μ, ∑ ν, η μ ν • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν)
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lemma asCoTenProd_diag :
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@asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
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CovariantLorentzVector.stdBasis μ) := by
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simp only [asCoTenProd]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [Finset.sum_eq_single μ]
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intro ν _ hμν
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rw [minkowskiMatrix.off_diag_zero hμν.symm]
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simp only [zero_smul]
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intro a
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simp_all only
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@[simp]
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lemma contrLeft_asProdLorentzVector (x : CovariantLorentzVector d) :
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contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asProdLorentzVector) =
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lemma contrLeft_asTenProd (x : CovariantLorentzVector d) :
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contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asTenProd) =
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∑ μ, ((η μ μ * x μ) • LorentzVector.stdBasis μ) := by
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simp only [asProdLorentzVector]
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simp only [asTenProd_diag]
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rw [tmul_sum]
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rw [map_sum]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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@ -196,10 +219,10 @@ lemma contrLeft_asProdLorentzVector (x : CovariantLorentzVector d) :
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exact smul_smul (η μ μ) (x μ) (e μ)
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@[simp]
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lemma contrLeft_asProdCovariantLorentzVector (x : LorentzVector d) :
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contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asProdCovariantLorentzVector) =
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lemma contrLeft_asCoTenProd (x : LorentzVector d) :
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contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asCoTenProd) =
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∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) := by
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simp only [asProdCovariantLorentzVector]
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simp only [asCoTenProd_diag]
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rw [tmul_sum]
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rw [map_sum]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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@ -209,17 +232,17 @@ lemma contrLeft_asProdCovariantLorentzVector (x : LorentzVector d) :
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exact smul_smul (η μ μ) (x μ) (CovariantLorentzVector.stdBasis μ)
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@[simp]
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lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
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(contrMidAux (contrUpDown) (asProdLorentzVector ⊗ₜ[ℝ] asProdCovariantLorentzVector))
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lemma asTenProd_contr_asCoTenProd :
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(contrMidAux (contrUpDown) (asTenProd ⊗ₜ[ℝ] asCoTenProd))
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= TensorProduct.comm ℝ _ _ (@unitUp d) := by
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simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asProdLorentzVector, LinearMap.coe_comp,
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simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asTenProd_diag, LinearMap.coe_comp,
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LinearEquiv.coe_coe, Function.comp_apply]
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rw [sum_tmul, map_sum, map_sum, unitUp]
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simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, map_add, comm_tmul, map_sum]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [← tmul_smul, TensorProduct.assoc_tmul]
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simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asProdCovariantLorentzVector]
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simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asCoTenProd]
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rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul]
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change (η μ μ * (η μ μ * e μ μ)) • e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ = _
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rw [LorentzVector.stdBasis]
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@ -236,10 +259,10 @@ lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
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exact fun a => False.elim (hμν (id (Eq.symm a)))
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@[simp]
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lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
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(contrMidAux (contrDownUp) (asProdCovariantLorentzVector ⊗ₜ[ℝ] asProdLorentzVector))
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lemma asCoTenProd_contr_asTenProd :
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(contrMidAux (contrDownUp) (asCoTenProd ⊗ₜ[ℝ] asTenProd))
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= TensorProduct.comm ℝ _ _ (@unitDown d) := by
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simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asProdCovariantLorentzVector,
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simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asCoTenProd_diag,
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LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
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rw [sum_tmul, map_sum, map_sum, unitDown]
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simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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@ -247,7 +270,7 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [smul_tmul, TensorProduct.assoc_tmul]
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simp only [tmul_smul, LinearMapClass.map_smul, map_tmul, LinearMap.id_coe, id_eq,
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contrLeft_asProdLorentzVector]
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contrLeft_asTenProd]
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rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul, smul_smul, LorentzVector.stdBasis]
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erw [Pi.basisFun_apply]
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simp only [LinearMap.stdBasis_same, mul_one, η_apply_mul_η_apply_diag, one_smul]
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@ -260,6 +283,86 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
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ite_eq_right_iff, smul_eq_zero, mul_eq_zero]
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exact fun a => False.elim (hμν (id (Eq.symm a)))
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lemma asTenProd_invariant (g : LorentzGroup d) :
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TensorProduct.map (LorentzVector.rep g) (LorentzVector.rep g) asTenProd = asTenProd := by
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simp only [asTenProd, map_sum, LinearMapClass.map_smul, map_tmul, rep_apply_stdBasis,
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Matrix.transpose_apply]
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trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d,
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η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] ∑ ρ : Fin 1 ⊕ Fin d, g.1 ρ ν • e ρ
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refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))
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rw [sum_tmul, Finset.smul_sum]
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trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
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(η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] (g.1 ρ ν • e ρ))
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· refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ =>
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Finset.sum_congr rfl (fun φ _ => ?_)))
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rw [tmul_sum, Finset.smul_sum]
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rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_comm)]
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rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => Finset.sum_comm))]
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rw [Finset.sum_comm]
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rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_comm)]
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trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d, ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d,
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((g.1 φ μ * η μ ν * g.1 ρ ν) • (e φ) ⊗ₜ[ℝ] (e ρ))
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· refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
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Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))))
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rw [smul_tmul, tmul_smul, tmul_smul, smul_smul, smul_smul]
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ring_nf
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rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
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Finset.sum_congr rfl (fun μ _ => Finset.sum_smul.symm)))]
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rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => Finset.sum_smul.symm))]
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trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
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(((g.1 * minkowskiMatrix * g.1.transpose : Matrix _ _ _) φ ρ) • (e φ) ⊗ₜ[ℝ] (e ρ))
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· refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => ?_))
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apply congrFun
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apply congrArg
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simp only [Matrix.mul_apply, Matrix.transpose_apply]
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rw [Finset.sum_comm]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [Finset.sum_mul]
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simp
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open CovariantLorentzVector in
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lemma asCoTenProd_invariant (g : LorentzGroup d) :
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TensorProduct.map (CovariantLorentzVector.rep g) (CovariantLorentzVector.rep g)
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asCoTenProd = asCoTenProd := by
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simp only [asCoTenProd, map_sum, LinearMapClass.map_smul, map_tmul,
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CovariantLorentzVector.rep_apply_stdBasis, Matrix.transpose_apply]
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trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d,
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η μ ν • (g⁻¹.1 μ φ • CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ]
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∑ ρ : Fin 1 ⊕ Fin d, g⁻¹.1 ν ρ • CovariantLorentzVector.stdBasis ρ
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· refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))
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rw [sum_tmul, Finset.smul_sum]
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trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
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(η μ ν • (g⁻¹.1 μ φ • CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ]
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(g⁻¹.1 ν ρ • CovariantLorentzVector.stdBasis ρ))
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· refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ =>
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Finset.sum_congr rfl (fun φ _ => ?_)))
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rw [tmul_sum, Finset.smul_sum]
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rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_comm)]
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rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => Finset.sum_comm))]
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rw [Finset.sum_comm]
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rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_comm)]
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trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d, ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d,
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((g⁻¹.1 μ φ * η μ ν * g⁻¹.1 ν ρ) • (CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ]
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(CovariantLorentzVector.stdBasis ρ))
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· refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
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Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))))
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rw [smul_tmul, tmul_smul, tmul_smul, smul_smul, smul_smul]
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ring_nf
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rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
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Finset.sum_congr rfl (fun μ _ => Finset.sum_smul.symm)))]
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rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => Finset.sum_smul.symm))]
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trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
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(((g⁻¹.1.transpose * minkowskiMatrix * g⁻¹.1 : Matrix _ _ _) φ ρ) •
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(CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis ρ))
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· refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => ?_))
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apply congrFun
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apply congrArg
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simp only [Matrix.mul_apply, Matrix.transpose_apply]
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rw [Finset.sum_comm]
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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [Finset.sum_mul]
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rw [LorentzGroup.transpose_mul_minkowskiMatrix_mul_self]
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end minkowskiMatrix
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end
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