feat: Weyl fermion contraction, unit, metric
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HepLean/SpaceTime/WeylFermion/Two.lean
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HepLean/SpaceTime/WeylFermion/Two.lean
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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.WeylFermion.Basic
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import HepLean.SpaceTime.WeylFermion.Contraction
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import Mathlib.LinearAlgebra.TensorProduct.Matrix
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/-!
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# Tensor product of two Weyl fermion
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-/
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namespace Fermion
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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 CategoryTheory.MonoidalCategory
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/-!
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## Equivalences to matrices.
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-/
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/-- Equivalence of `leftHanded ⊗ leftHanded` to `2 x 2` complex matrices. -/
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def leftLeftToMatrix : (leftHanded ⊗ leftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct leftBasis leftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `altLeftHanded ⊗ altLeftHanded` to `2 x 2` complex matrices. -/
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def altLeftaltLeftToMatrix : (altLeftHanded ⊗ altLeftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altLeftBasis altLeftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `leftHanded ⊗ altLeftHanded` to `2 x 2` complex matrices. -/
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def leftAltLeftToMatrix : (leftHanded ⊗ altLeftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct leftBasis altLeftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `altLeftHanded ⊗ leftHanded` to `2 x 2` complex matrices. -/
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def altLeftLeftToMatrix : (altLeftHanded ⊗ leftHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altLeftBasis leftBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `rightHanded ⊗ rightHanded` to `2 x 2` complex matrices. -/
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def rightRightToMatrix : (rightHanded ⊗ rightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct rightBasis rightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `altRightHanded ⊗ altRightHanded` to `2 x 2` complex matrices. -/
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def altRightAltRightToMatrix : (altRightHanded ⊗ altRightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altRightBasis altRightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `rightHanded ⊗ altRightHanded` to `2 x 2` complex matrices. -/
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def rightAltRightToMatrix : (rightHanded ⊗ altRightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct rightBasis altRightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-- Equivalence of `altRightHanded ⊗ rightHanded` to `2 x 2` complex matrices. -/
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def altRightRightToMatrix : (altRightHanded ⊗ rightHanded).V ≃ₗ[ℂ] Matrix (Fin 2) (Fin 2) ℂ :=
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(Basis.tensorProduct altRightBasis rightBasis).repr ≪≫ₗ
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Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) ≪≫ₗ
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LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)
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/-!
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## Group actions
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-/
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/-- The group action of `SL(2,ℂ)` on `leftHanded ⊗ leftHanded` is equivalent to
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`M.1 * leftLeftToMatrix v * (M.1)ᵀ`. -/
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lemma leftLeftToMatrix_ρ (v : (leftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
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leftLeftToMatrix (TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M) v) =
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M.1 * leftLeftToMatrix v * (M.1)ᵀ := by
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nth_rewrite 1 [leftLeftToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
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(leftBasis.tensorProduct leftBasis) (leftBasis.tensorProduct leftBasis)
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(TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
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((leftBasis.tensorProduct leftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct leftBasis)
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(leftBasis.tensorProduct leftBasis) (TensorProduct.map (leftHanded.ρ M) (leftHanded.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 leftBasis leftBasis) (leftHanded.ρ M))
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((LinearMap.toMatrix leftBasis leftBasis) (leftHanded.ρ M)) (i, j) k)
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* leftLeftToMatrix 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 : Fin 2, (∑ j : Fin 2, M.1 i j * leftLeftToMatrix v j x) * M.1 j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftLeftToMatrix v x1 x) * M.1 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 [leftBasis_ρ_apply, Finsupp.linearEquivFunOnFinite_apply,
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Action.instMonoidalCategory_tensorObj_V]
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rw [mul_assoc]
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nth_rewrite 2 [mul_comm]
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rw [← mul_assoc]
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/-- The group action of `SL(2,ℂ)` on `altLeftHanded ⊗ altLeftHanded` is equivalent to
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`(M.1⁻¹)ᵀ * leftLeftToMatrix v * (M.1⁻¹)`. -/
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lemma altLeftaltLeftToMatrix_ρ (v : (altLeftHanded ⊗ altLeftHanded).V) (M : SL(2,ℂ)) :
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altLeftaltLeftToMatrix (TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M) v) =
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(M.1⁻¹)ᵀ * altLeftaltLeftToMatrix v * (M.1⁻¹) := by
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nth_rewrite 1 [altLeftaltLeftToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
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(altLeftBasis.tensorProduct altLeftBasis) (altLeftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
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((altLeftBasis.tensorProduct altLeftBasis).repr v)))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct altLeftBasis)
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(altLeftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.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 altLeftBasis altLeftBasis) (altLeftHanded.ρ M))
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((LinearMap.toMatrix altLeftBasis altLeftBasis) (altLeftHanded.ρ M)) (i, j) k)
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* altLeftaltLeftToMatrix 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 : Fin 2, (∑ x1 : Fin 2, (M.1)⁻¹ x1 i * altLeftaltLeftToMatrix v x1 x) * (M.1)⁻¹ x j
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1)⁻¹ x1 i * altLeftaltLeftToMatrix v x1 x) * (M.1)⁻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 [altLeftBasis_ρ_apply, transpose_apply, Action.instMonoidalCategory_tensorObj_V]
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ring
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/-- The group action of `SL(2,ℂ)` on `leftHanded ⊗ altLeftHanded` is equivalent to
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`M.1 * leftAltLeftToMatrix v * (M.1⁻¹)`. -/
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lemma leftAltLeftToMatrix_ρ (v : (leftHanded ⊗ altLeftHanded).V) (M : SL(2,ℂ)) :
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leftAltLeftToMatrix (TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M) v) =
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M.1 * leftAltLeftToMatrix v * (M.1⁻¹) := by
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nth_rewrite 1 [leftAltLeftToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
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(leftBasis.tensorProduct altLeftBasis) (leftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
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((leftBasis.tensorProduct altLeftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (leftBasis.tensorProduct altLeftBasis)
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(leftBasis.tensorProduct altLeftBasis)
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(TensorProduct.map (leftHanded.ρ M) (altLeftHanded.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 leftBasis leftBasis) (leftHanded.ρ M))
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((LinearMap.toMatrix altLeftBasis altLeftBasis) (altLeftHanded.ρ M)) (i, j) k)
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* leftAltLeftToMatrix 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 : Fin 2, (∑ x1 : Fin 2, M.1 i x1 * leftAltLeftToMatrix v x1 x) * (M.1⁻¹) x j
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= ∑ x : Fin 2, ∑ x1 : Fin 2, (M.1 i x1 * leftAltLeftToMatrix v x1 x) * (M.1⁻¹) 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 [leftBasis_ρ_apply, altLeftBasis_ρ_apply, transpose_apply,
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Action.instMonoidalCategory_tensorObj_V]
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ring
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/-- The group action of `SL(2,ℂ)` on `altLeftHanded ⊗ leftHanded` is equivalent to
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`(M.1⁻¹)ᵀ * leftAltLeftToMatrix v * (M.1)ᵀ`. -/
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lemma altLeftLeftToMatrix_ρ (v : (altLeftHanded ⊗ leftHanded).V) (M : SL(2,ℂ)) :
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altLeftLeftToMatrix (TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M) v) =
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(M.1⁻¹)ᵀ * altLeftLeftToMatrix v * (M.1)ᵀ := by
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nth_rewrite 1 [altLeftLeftToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
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(altLeftBasis.tensorProduct leftBasis) (altLeftBasis.tensorProduct leftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
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((altLeftBasis.tensorProduct leftBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altLeftBasis.tensorProduct leftBasis)
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(altLeftBasis.tensorProduct leftBasis)
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(TensorProduct.map (altLeftHanded.ρ M) (leftHanded.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 altLeftBasis altLeftBasis) (altLeftHanded.ρ M))
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((LinearMap.toMatrix leftBasis leftBasis) (leftHanded.ρ M)) (i, j) k)
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* altLeftLeftToMatrix 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 : Fin 2, (∑ x1 : Fin 2, (M.1)⁻¹ x1 i * altLeftLeftToMatrix v x1 x) * M.1 j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1)⁻¹ x1 i * altLeftLeftToMatrix v x1 x) * M.1 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 [altLeftBasis_ρ_apply, leftBasis_ρ_apply, transpose_apply,
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Action.instMonoidalCategory_tensorObj_V]
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ring
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/-- The group action of `SL(2,ℂ)` on `rightHanded ⊗ rightHanded` is equivalent to
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`(M.1.map star) * rightRightToMatrix v * ((M.1.map star))ᵀ`. -/
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lemma rightRightToMatrix_ρ (v : (rightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)) :
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rightRightToMatrix (TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M) v) =
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(M.1.map star) * rightRightToMatrix v * ((M.1.map star))ᵀ := by
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nth_rewrite 1 [rightRightToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
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(rightBasis.tensorProduct rightBasis) (rightBasis.tensorProduct rightBasis)
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(TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
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((rightBasis.tensorProduct rightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct rightBasis)
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(rightBasis.tensorProduct rightBasis) (TensorProduct.map (rightHanded.ρ M) (rightHanded.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 rightBasis rightBasis) (rightHanded.ρ M))
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((LinearMap.toMatrix rightBasis rightBasis) (rightHanded.ρ M)) (i, j) k)
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* rightRightToMatrix 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 : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) j x
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= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightRightToMatrix v x1 x) * (M.1.map star) 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 [rightBasis_ρ_apply, Finsupp.linearEquivFunOnFinite_apply,
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Action.instMonoidalCategory_tensorObj_V]
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ring
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/-- The group action of `SL(2,ℂ)` on `altRightHanded ⊗ altRightHanded` is equivalent to
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`((M.1⁻¹).conjTranspose * rightRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ`. -/
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lemma altRightAltRightToMatrix_ρ (v : (altRightHanded ⊗ altRightHanded).V) (M : SL(2,ℂ)) :
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altRightAltRightToMatrix (TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M) v) =
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((M.1⁻¹).conjTranspose) * altRightAltRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ) := by
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nth_rewrite 1 [altRightAltRightToMatrix]
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simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
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trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
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(altRightBasis.tensorProduct altRightBasis) (altRightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M)))
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*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
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((altRightBasis.tensorProduct altRightBasis).repr (v))))
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· apply congrArg
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have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct altRightBasis)
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(altRightBasis.tensorProduct altRightBasis)
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(TensorProduct.map (altRightHanded.ρ M) (altRightHanded.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 altRightBasis altRightBasis) (altRightHanded.ρ M))
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((LinearMap.toMatrix altRightBasis altRightBasis) (altRightHanded.ρ M)) (i, j) k)
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* altRightAltRightToMatrix v k.1 k.2) = _
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||||
erw [Finset.sum_product]
|
||||
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
|
||||
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
|
||||
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
|
||||
congr
|
||||
funext x
|
||||
rw [Finset.sum_mul]
|
||||
erw [h1]
|
||||
rw [Finset.sum_comm]
|
||||
congr
|
||||
funext x
|
||||
congr
|
||||
funext x1
|
||||
simp only [altRightBasis_ρ_apply, transpose_apply, Action.instMonoidalCategory_tensorObj_V]
|
||||
ring
|
||||
|
||||
/-- The group action of `SL(2,ℂ)` on `rightHanded ⊗ altRightHanded` is equivalent to
|
||||
`(M.1.map star) * rightAltRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ`. -/
|
||||
lemma rightAltRightToMatrix_ρ (v : (rightHanded ⊗ altRightHanded).V) (M : SL(2,ℂ)) :
|
||||
rightAltRightToMatrix (TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M) v) =
|
||||
(M.1.map star) * rightAltRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ) := by
|
||||
nth_rewrite 1 [rightAltRightToMatrix]
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
|
||||
trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
|
||||
(rightBasis.tensorProduct altRightBasis) (rightBasis.tensorProduct altRightBasis)
|
||||
(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)))
|
||||
*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
|
||||
((rightBasis.tensorProduct altRightBasis).repr (v))))
|
||||
· apply congrArg
|
||||
have h1 := (LinearMap.toMatrix_mulVec_repr (rightBasis.tensorProduct altRightBasis)
|
||||
(rightBasis.tensorProduct altRightBasis)
|
||||
(TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M)) v)
|
||||
erw [h1]
|
||||
rfl
|
||||
rw [TensorProduct.toMatrix_map]
|
||||
funext i j
|
||||
change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
|
||||
((LinearMap.toMatrix rightBasis rightBasis) (rightHanded.ρ M))
|
||||
((LinearMap.toMatrix altRightBasis altRightBasis) (altRightHanded.ρ M)) (i, j) k)
|
||||
* rightAltRightToMatrix v k.1 k.2) = _
|
||||
erw [Finset.sum_product]
|
||||
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
|
||||
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x
|
||||
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((M.1.map star) i x1 * rightAltRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by
|
||||
congr
|
||||
funext x
|
||||
rw [Finset.sum_mul]
|
||||
erw [h1]
|
||||
rw [Finset.sum_comm]
|
||||
congr
|
||||
funext x
|
||||
congr
|
||||
funext x1
|
||||
simp only [rightBasis_ρ_apply, altRightBasis_ρ_apply, transpose_apply,
|
||||
Action.instMonoidalCategory_tensorObj_V]
|
||||
ring
|
||||
|
||||
/-- The group action of `SL(2,ℂ)` on `altRightHanded ⊗ rightHanded` is equivalent to
|
||||
`((M.1⁻¹).conjTranspose * rightAltRightToMatrix v * ((M.1.map star)).ᵀ`. -/
|
||||
lemma altRightRightToMatrix_ρ (v : (altRightHanded ⊗ rightHanded).V) (M : SL(2,ℂ)) :
|
||||
altRightRightToMatrix (TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M) v) =
|
||||
((M.1⁻¹).conjTranspose) * altRightRightToMatrix v * (M.1.map star)ᵀ := by
|
||||
nth_rewrite 1 [altRightRightToMatrix]
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.trans_apply]
|
||||
trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix
|
||||
(altRightBasis.tensorProduct rightBasis) (altRightBasis.tensorProduct rightBasis)
|
||||
(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)))
|
||||
*ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2))
|
||||
((altRightBasis.tensorProduct rightBasis).repr (v))))
|
||||
· apply congrArg
|
||||
have h1 := (LinearMap.toMatrix_mulVec_repr (altRightBasis.tensorProduct rightBasis)
|
||||
(altRightBasis.tensorProduct rightBasis)
|
||||
(TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M)) v)
|
||||
erw [h1]
|
||||
rfl
|
||||
rw [TensorProduct.toMatrix_map]
|
||||
funext i j
|
||||
change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
|
||||
((LinearMap.toMatrix altRightBasis altRightBasis) (altRightHanded.ρ M))
|
||||
((LinearMap.toMatrix rightBasis rightBasis) (rightHanded.ρ M)) (i, j) k)
|
||||
* altRightRightToMatrix v k.1 k.2) = _
|
||||
erw [Finset.sum_product]
|
||||
simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply]
|
||||
have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x
|
||||
= ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * altRightRightToMatrix v x1 x) * (M.1.map star) j x := by
|
||||
congr
|
||||
funext x
|
||||
rw [Finset.sum_mul]
|
||||
erw [h1]
|
||||
rw [Finset.sum_comm]
|
||||
congr
|
||||
funext x
|
||||
congr
|
||||
funext x1
|
||||
simp only [altRightBasis_ρ_apply, rightBasis_ρ_apply, transpose_apply,
|
||||
Action.instMonoidalCategory_tensorObj_V]
|
||||
ring
|
||||
|
||||
/-!
|
||||
|
||||
## The symm version of the group actions.
|
||||
|
||||
-/
|
||||
|
||||
lemma leftLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (leftHanded.ρ M) (leftHanded.ρ M) (leftLeftToMatrix.symm v) =
|
||||
leftLeftToMatrix.symm (M.1 * v * (M.1)ᵀ) := by
|
||||
have h1 := leftLeftToMatrix_ρ (leftLeftToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma altLeftaltLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (altLeftHanded.ρ M) (altLeftHanded.ρ M) (altLeftaltLeftToMatrix.symm v) =
|
||||
altLeftaltLeftToMatrix.symm ((M.1⁻¹)ᵀ * v * (M.1⁻¹)) := by
|
||||
have h1 := altLeftaltLeftToMatrix_ρ (altLeftaltLeftToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma leftAltLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (leftHanded.ρ M) (altLeftHanded.ρ M) (leftAltLeftToMatrix.symm v) =
|
||||
leftAltLeftToMatrix.symm (M.1 * v * (M.1⁻¹)) := by
|
||||
have h1 := leftAltLeftToMatrix_ρ (leftAltLeftToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma altLeftLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (altLeftHanded.ρ M) (leftHanded.ρ M) (altLeftLeftToMatrix.symm v) =
|
||||
altLeftLeftToMatrix.symm ((M.1⁻¹)ᵀ * v * (M.1)ᵀ) := by
|
||||
have h1 := altLeftLeftToMatrix_ρ (altLeftLeftToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma rightRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (rightHanded.ρ M) (rightHanded.ρ M) (rightRightToMatrix.symm v) =
|
||||
rightRightToMatrix.symm ((M.1.map star) * v * ((M.1.map star))ᵀ) := by
|
||||
have h1 := rightRightToMatrix_ρ (rightRightToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma altRightAltRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (altRightHanded.ρ M) (altRightHanded.ρ M) (altRightAltRightToMatrix.symm v) =
|
||||
altRightAltRightToMatrix.symm (((M.1⁻¹).conjTranspose) * v * ((M.1⁻¹).conjTranspose)ᵀ) := by
|
||||
have h1 := altRightAltRightToMatrix_ρ (altRightAltRightToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma rightAltRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (rightHanded.ρ M) (altRightHanded.ρ M) (rightAltRightToMatrix.symm v) =
|
||||
rightAltRightToMatrix.symm ((M.1.map star) * v * (((M.1⁻¹).conjTranspose)ᵀ) ) := by
|
||||
have h1 := rightAltRightToMatrix_ρ (rightAltRightToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
lemma altRightRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :
|
||||
TensorProduct.map (altRightHanded.ρ M) (rightHanded.ρ M) (altRightRightToMatrix.symm v) =
|
||||
altRightRightToMatrix.symm (((M.1⁻¹).conjTranspose) * v * (M.1.map star)ᵀ) := by
|
||||
have h1 := altRightRightToMatrix_ρ (altRightRightToMatrix.symm v) M
|
||||
simp only [Action.instMonoidalCategory_tensorObj_V, LinearEquiv.apply_symm_apply] at h1
|
||||
rw [← h1]
|
||||
simp
|
||||
|
||||
|
||||
|
||||
end
|
||||
end Fermion
|
||||
Loading…
Add table
Add a link
Reference in a new issue