feat: Make MulActionTensor
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jstoobysmith
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11 changed files with 177 additions and 55 deletions
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@ -3,7 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzVector.Basic
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import HepLean.SpaceTime.LorentzVector.LorentzAction
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import HepLean.SpaceTime.LorentzVector.Covariant
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import HepLean.SpaceTime.LorentzTensor.Basic
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/-!
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@ -24,9 +24,9 @@ open TensorProduct
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namespace LorentzVector
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open Matrix
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variable {d : ℕ} (v : LorentzVector d)
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def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
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toFun v := {
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toFun := fun w => ∑ i, v i * w i,
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@ -60,6 +60,10 @@ def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[
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def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
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TensorProduct.lift contrUpDownBi
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lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
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contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
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rfl
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@[simp]
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lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
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contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
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@ -92,16 +96,26 @@ lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ F
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def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
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contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
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lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
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contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
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rw [dotProduct_comm x y]
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rfl
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/-!
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## The unit of the contraction
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-/
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def unitUp : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
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∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
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lemma unitUp_lid (x : LorentzVector d) :
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TensorProduct.rid ℝ _
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(TensorProduct.map (LinearEquiv.refl ℝ (_)).toLinearMap
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(contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap )
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((TensorProduct.assoc ℝ _ _ _) (unitUp ⊗ₜ[ℝ] x ))) = x := by
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simp only [LinearEquiv.refl_toLinearMap, unitUp]
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(TensorProduct.map (LinearEquiv.refl ℝ _).toLinearMap
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(contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap)
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((TensorProduct.assoc ℝ _ _ _) (unitUp ⊗ₜ[ℝ] x))) = x := by
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simp only [LinearEquiv.refl_toLinearMap, unitUp]
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rw [sum_tmul]
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simp only [Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, map_add, assoc_tmul, map_sum, map_tmul, LinearMap.id_coe, id_eq,
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@ -113,8 +127,8 @@ def unitDown : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
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lemma unitDown_lid (x : CovariantLorentzVector d) :
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TensorProduct.rid ℝ _
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(TensorProduct.map (LinearEquiv.refl ℝ (_)).toLinearMap
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(contrDownUp ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap )
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(TensorProduct.map (LinearEquiv.refl ℝ _).toLinearMap
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(contrDownUp ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap)
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((TensorProduct.assoc ℝ _ _ _) (unitDown ⊗ₜ[ℝ] x))) = x := by
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simp only [LinearEquiv.refl_toLinearMap, unitDown]
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rw [sum_tmul]
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@ -123,6 +137,29 @@ lemma unitDown_lid (x : CovariantLorentzVector d) :
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id_eq, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, comm_tmul,
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contrUpDown_stdBasis_right, rid_tmul, CovariantLorentzVector.decomp_stdBasis']
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/-!
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# Contractions and the Lorentz actions
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-/
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open Matrix
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@[simp]
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lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
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(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
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rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
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simp only [rep_apply, CovariantLorentzVector.rep_apply]
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rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
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simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
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@[simp]
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lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
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(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
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rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
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rw [dotProduct_comm, dotProduct_comm x y]
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simp only [rep_apply, CovariantLorentzVector.rep_apply]
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rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
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simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
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end LorentzVector
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@ -132,7 +169,7 @@ open scoped minkowskiMatrix
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variable {d : ℕ}
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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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∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ)
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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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@ -173,16 +210,15 @@ lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
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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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rw [tmul_sum, Finset.sum_eq_single_of_mem μ]
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rw [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 [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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erw [Pi.basisFun_apply]
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simp
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simp only [LinearMap.stdBasis_same, mul_one, η_apply_mul_η_apply_diag, one_smul]
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exact Finset.mem_univ μ
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intro ν _ hμν
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rw [tmul_smul]
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change (η ν ν * (η μ μ * e μ ν)) • (e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν) = _
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change (η ν ν * (η μ μ * e μ ν)) • (e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν) = _
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rw [LorentzVector.stdBasis]
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erw [Pi.basisFun_apply]
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simp only [LinearMap.stdBasis_apply', mul_ite, mul_one, mul_zero, ite_smul, zero_smul,
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@ -197,15 +233,12 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
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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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refine Finset.sum_congr rfl (fun μ _ => ?_)
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rw [smul_tmul,]
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rw [TensorProduct.assoc_tmul]
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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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rw [tmul_sum, Finset.sum_eq_single_of_mem μ]
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rw [tmul_smul, smul_smul]
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rw [LorentzVector.stdBasis]
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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
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simp only [LinearMap.stdBasis_same, mul_one, η_apply_mul_η_apply_diag, one_smul]
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exact Finset.mem_univ μ
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intro ν _ hμν
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rw [tmul_smul]
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@ -217,5 +250,4 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
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end minkowskiMatrix
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end
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