Dual Right Dual Right To Matrix ρ
Fermion.dualRightDualRightToMatrix_ρ
Plain-language statement
The group action of SL(2,ℂ) on dualRightHanded ⊗ dualRightHanded is equivalent to ((M.1⁻¹).conjTranspose * rightRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ.
Exact Lean statement
lemma dualRightDualRightToMatrix_ρ (v : (DualRightHandedWeyl ⊗[ℂ] DualRightHandedWeyl))
(M : SL(2,ℂ)) :
dualRightDualRightToMatrix (TensorProduct.map (DualRightHandedWeyl.rep M)
(DualRightHandedWeyl.rep M) v) =
((M.1⁻¹).conjTranspose) * dualRightDualRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ)Formal artifact
Lean source
lemma dualRightDualRightToMatrix_ρ (v : (DualRightHandedWeyl ⊗[ℂ] DualRightHandedWeyl)) (M : SL(2,ℂ)) : dualRightDualRightToMatrix (TensorProduct.map (DualRightHandedWeyl.rep M) (DualRightHandedWeyl.rep M) v) = ((M.1⁻¹).conjTranspose) * dualRightDualRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ) := by nth_rewrite 1 [dualRightDualRightToMatrix] simp only [LinearEquiv.trans_apply] trans (LinearEquiv.curry ℂ ℂ (Fin 2) (Fin 2)) ((LinearMap.toMatrix (DualRightHandedWeyl.basis.tensorProduct DualRightHandedWeyl.basis) (DualRightHandedWeyl.basis.tensorProduct DualRightHandedWeyl.basis) (TensorProduct.map (DualRightHandedWeyl.rep M) (DualRightHandedWeyl.rep M))) *ᵥ ((Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2)) ((DualRightHandedWeyl.basis.tensorProduct DualRightHandedWeyl.basis).repr (v)))) · apply congrArg have h1 := (LinearMap.toMatrix_mulVec_repr (DualRightHandedWeyl.basis.tensorProduct DualRightHandedWeyl.basis) (DualRightHandedWeyl.basis.tensorProduct DualRightHandedWeyl.basis) (TensorProduct.map (DualRightHandedWeyl.rep M) (DualRightHandedWeyl.rep M)) v) simp only [coe_linearEquivFunOnFinite] rw [h1] rw [TensorProduct.toMatrix_map] funext i j change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2) ((LinearMap.toMatrix DualRightHandedWeyl.basis DualRightHandedWeyl.basis) (DualRightHandedWeyl.rep M)) ((LinearMap.toMatrix DualRightHandedWeyl.basis DualRightHandedWeyl.basis) (DualRightHandedWeyl.rep M)) (i, j) k) * dualRightDualRightToMatrix v k.1 k.2) = _ rw [Fintype.sum_prod_type] simp_rw [kroneckerMap_apply, Matrix.mul_apply, Matrix.transpose_apply] have h1 : ∑ x : Fin 2, (∑ x1 : Fin 2, (↑M)⁻¹ᴴ i x1 * dualRightDualRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x = ∑ x : Fin 2, ∑ x1 : Fin 2, ((↑M)⁻¹ᴴ i x1 * dualRightDualRightToMatrix v x1 x) * (↑M)⁻¹ᴴ j x := by congr funext x rw [Finset.sum_mul] rw [h1] rw [Finset.sum_comm] congr funext x congr funext x1 simp only [DualRightHandedWeyl.rep_toMatrix] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Fermions/Weyl/Two.lean:468-511
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