272 lines
12 KiB
Text
272 lines
12 KiB
Text
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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 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.Complex.Basic
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import Mathlib.LinearAlgebra.TensorProduct.Matrix
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/-!
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# Tensor products of two complex Lorentz vectors
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-/
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noncomputable section
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open Matrix
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open MatrixGroups
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open Complex
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open TensorProduct
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open SpaceTime
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open CategoryTheory.MonoidalCategory
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namespace Lorentz
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/-- Equivalence of `complexContr ⊗ complexContr` to `4 x 4` complex matrices. -/
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def contrContrToMatrix : (complexContr ⊗ complexContr).V ≃ₗ[ℂ]
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Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ :=
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(Basis.tensorProduct complexContrBasis complexContrBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)
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/-- Equivalence of `complexCo ⊗ complexCo` to `4 x 4` complex matrices. -/
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def coCoToMatrix : (complexCo ⊗ complexCo).V ≃ₗ[ℂ]
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Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ :=
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(Basis.tensorProduct complexCoBasis complexCoBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)
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/-- Equivalence of `complexContr ⊗ complexCo` to `4 x 4` complex matrices. -/
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def contrCoToMatrix : (complexContr ⊗ complexCo).V ≃ₗ[ℂ]
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Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ :=
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(Basis.tensorProduct complexContrBasis complexCoBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)
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/-- Equivalence of `complexCo ⊗ complexContr` to `4 x 4` complex matrices. -/
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def coContrToMatrix : (complexCo ⊗ complexContr).V ≃ₗ[ℂ]
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Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ :=
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(Basis.tensorProduct complexCoBasis complexContrBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)
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/-!
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## Group actions
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-/
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lemma contrContrToMatrix_ρ (v : (complexContr ⊗ complexContr).V) (M : SL(2,ℂ)) :
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contrContrToMatrix (TensorProduct.map (complexContr.ρ M) (complexContr.ρ M) v) =
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M)) * contrContrToMatrix v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))ᵀ := by
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nth_rewrite 1 [contrContrToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)) ((LinearMap.toMatrix
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(complexContrBasis.tensorProduct complexContrBasis)
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(complexContrBasis.tensorProduct complexContrBasis)
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(TensorProduct.map (complexContr.ρ M) (complexContr.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)))
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((complexContrBasis.tensorProduct complexContrBasis).repr v)))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (complexContrBasis.tensorProduct complexContrBasis)
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(complexContrBasis.tensorProduct complexContrBasis)
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(TensorProduct.map (complexContr.ρ M) (complexContr.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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funext i j
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change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
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((LinearMap.toMatrix complexContrBasis complexContrBasis) (complexContr.ρ M))
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((LinearMap.toMatrix complexContrBasis complexContrBasis) (complexContr.ρ M)) (i, j) k)
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* contrContrToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x, (∑ x1 , LorentzGroup.toComplex (SL2C.toLorentzGroup M) i x1 *
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contrContrToMatrix v x1 x) * LorentzGroup.toComplex (SL2C.toLorentzGroup M) j x
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= ∑ x , ∑ x1 , (LorentzGroup.toComplex (SL2C.toLorentzGroup M) i x1
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* contrContrToMatrix v x1 x) * LorentzGroup.toComplex (SL2C.toLorentzGroup M) j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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erw [h1]
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rw [Finset.sum_comm]
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congr
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funext x
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congr
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funext x1
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simp only [complexContrBasis_ρ_apply, transpose_apply, Action.instMonoidalCategory_tensorObj_V]
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ring
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lemma coCoToMatrix_ρ (v : (complexCo ⊗ complexCo).V) (M : SL(2,ℂ)) :
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coCoToMatrix (TensorProduct.map (complexCo.ρ M) (complexCo.ρ M) v) =
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ * coCoToMatrix v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ := by
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nth_rewrite 1 [coCoToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)) ((LinearMap.toMatrix
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(complexCoBasis.tensorProduct complexCoBasis)
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(complexCoBasis.tensorProduct complexCoBasis)
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(TensorProduct.map (complexCo.ρ M) (complexCo.ρ M))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)))
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((complexCoBasis.tensorProduct complexCoBasis).repr v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (complexCoBasis.tensorProduct complexCoBasis)
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(complexCoBasis.tensorProduct complexCoBasis)
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(TensorProduct.map (complexCo.ρ M) (complexCo.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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funext i j
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change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
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((LinearMap.toMatrix complexCoBasis complexCoBasis) (complexCo.ρ M))
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((LinearMap.toMatrix complexCoBasis complexCoBasis) (complexCo.ρ M)) (i, j) k)
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* coCoToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x, (∑ x1 , (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x1 i *
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coCoToMatrix v x1 x) * (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x j
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= ∑ x , ∑ x1 , ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x1 i
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* coCoToMatrix v x1 x) * (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x j := by
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congr
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funext x
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rw [Finset.sum_mul]
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erw [h1]
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rw [Finset.sum_comm]
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congr
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funext x
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congr
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funext x1
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simp only [complexCoBasis_ρ_apply, transpose_apply, Action.instMonoidalCategory_tensorObj_V]
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ring
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lemma contrCoToMatrix_ρ (v : (complexContr ⊗ complexCo).V) (M : SL(2,ℂ)) :
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contrCoToMatrix (TensorProduct.map (complexContr.ρ M) (complexCo.ρ M) v) =
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M)) * contrCoToMatrix v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ := by
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nth_rewrite 1 [contrCoToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)) ((LinearMap.toMatrix
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(complexContrBasis.tensorProduct complexCoBasis)
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(complexContrBasis.tensorProduct complexCoBasis)
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(TensorProduct.map (complexContr.ρ M) (complexCo.ρ M))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)))
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((complexContrBasis.tensorProduct complexCoBasis).repr v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (complexContrBasis.tensorProduct complexCoBasis)
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(complexContrBasis.tensorProduct complexCoBasis)
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(TensorProduct.map (complexContr.ρ M) (complexCo.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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funext i j
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change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
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((LinearMap.toMatrix complexContrBasis complexContrBasis) (complexContr.ρ M))
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((LinearMap.toMatrix complexCoBasis complexCoBasis) (complexCo.ρ M)) (i, j) k)
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* contrCoToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply]
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have h1 : ∑ x, (∑ x1 , LorentzGroup.toComplex (SL2C.toLorentzGroup M) i x1 *
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contrCoToMatrix v x1 x) * (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x j
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= ∑ x , ∑ x1 , (LorentzGroup.toComplex (SL2C.toLorentzGroup M) i x1
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* contrCoToMatrix v x1 x) * (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x j := by
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congr
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funext x
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rw [Finset.sum_mul]
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erw [h1]
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rw [Finset.sum_comm]
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congr
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funext x
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congr
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funext x1
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simp only [complexContrBasis_ρ_apply, complexCoBasis_ρ_apply, transpose_apply,
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Action.instMonoidalCategory_tensorObj_V]
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ring
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lemma coContrToMatrix_ρ (v : (complexCo ⊗ complexContr).V) (M : SL(2,ℂ)) :
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coContrToMatrix (TensorProduct.map (complexCo.ρ M) (complexContr.ρ M) v) =
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ * coContrToMatrix v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))ᵀ := by
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nth_rewrite 1 [coContrToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3)) ((LinearMap.toMatrix
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(complexCoBasis.tensorProduct complexContrBasis)
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(complexCoBasis.tensorProduct complexContrBasis)
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(TensorProduct.map (complexCo.ρ M) (complexContr.ρ M))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ ((Fin 1 ⊕ Fin 3) × (Fin 1 ⊕ Fin 3)))
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((complexCoBasis.tensorProduct complexContrBasis).repr v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (complexCoBasis.tensorProduct complexContrBasis)
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(complexCoBasis.tensorProduct complexContrBasis)
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(TensorProduct.map (complexCo.ρ M) (complexContr.ρ M)) v)
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erw [h1]
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rfl
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rw [TensorProduct.toMatrix_map]
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funext i j
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change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
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((LinearMap.toMatrix complexCoBasis complexCoBasis) (complexCo.ρ M))
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((LinearMap.toMatrix complexContrBasis complexContrBasis) (complexContr.ρ M)) (i, j) k)
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* coContrToMatrix v k.1 k.2) = _
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erw [Finset.sum_product]
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simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
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have h1 : ∑ x, (∑ x1 , (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x1 i *
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coContrToMatrix v x1 x) * (LorentzGroup.toComplex (SL2C.toLorentzGroup M)) j x
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= ∑ x , ∑ x1 , ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ x1 i
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* coContrToMatrix v x1 x) * (LorentzGroup.toComplex (SL2C.toLorentzGroup M)) j x := by
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congr
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funext x
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rw [Finset.sum_mul]
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erw [h1]
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rw [Finset.sum_comm]
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congr
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funext x
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congr
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funext x1
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simp only [complexCoBasis_ρ_apply, complexContrBasis_ρ_apply, transpose_apply,
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Action.instMonoidalCategory_tensorObj_V]
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ring
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/-!
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## The symm version of the group actions.
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-/
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lemma contrContrToMatrix_ρ_symm (v : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ) (M : SL(2,ℂ)) :
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TensorProduct.map (complexContr.ρ M) (complexContr.ρ M) (contrContrToMatrix.symm v) =
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contrContrToMatrix.symm ((LorentzGroup.toComplex (SL2C.toLorentzGroup M)) * v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))ᵀ) := by
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have h1 := contrContrToMatrix_ρ (contrContrToMatrix.symm v) M
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
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rw [← h1]
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simp
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lemma coCoToMatrix_ρ_symm (v : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ) (M : SL(2,ℂ)) :
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TensorProduct.map (complexCo.ρ M) (complexCo.ρ M) (coCoToMatrix.symm v) =
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coCoToMatrix.symm ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ * v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹) := by
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have h1 := coCoToMatrix_ρ (coCoToMatrix.symm v) M
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
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rw [← h1]
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simp
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lemma contrCoToMatrix_ρ_symm (v : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ) (M : SL(2,ℂ)) :
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TensorProduct.map (complexContr.ρ M) (complexCo.ρ M) (contrCoToMatrix.symm v) =
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contrCoToMatrix.symm ((LorentzGroup.toComplex (SL2C.toLorentzGroup M)) * v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹) := by
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have h1 := contrCoToMatrix_ρ (contrCoToMatrix.symm v) M
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
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rw [← h1]
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simp
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lemma coContrToMatrix_ρ_symm (v : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℂ) (M : SL(2,ℂ)) :
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TensorProduct.map (complexCo.ρ M) (complexContr.ρ M) (coContrToMatrix.symm v) =
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coContrToMatrix.symm ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ * v *
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(LorentzGroup.toComplex (SL2C.toLorentzGroup M))ᵀ) := by
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have h1 := coContrToMatrix_ρ (coContrToMatrix.symm v) M
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
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rw [← h1]
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simp
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end Lorentz
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end
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