Rep Gauge Group I tmul basis eq sum
StandardModel.QuarkDoublet.repGaugeGroupI_tmul_basis_eq_sum
Plain-language statement
The action of the full gauge group on a tensor product of basis elements, expanded as a sum over the columns of the SU(3) and SU(2) matrices.
Exact Lean statement
lemma repGaugeGroupI_tmul_basis_eq_sum (g : GaugeGroupI) (k : Fin 2) (i : Fin 3) (j : Fin 2) :
repGaugeGroupI g ⟨LeftHandedWeyl.basis k ⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 3) ℂ i
⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 2) ℂ j⟩ =
∑ i' : Fin 3, ∑ j' : Fin 2, (g.toU1.1 * g.toSU3.1 i' i * g.toSU2.1 j' j)
• (⟨LeftHandedWeyl.basis k ⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 3) ℂ i'
⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 2) ℂ j'⟩ : QuarkDoublet)Formal artifact
Lean source
lemma repGaugeGroupI_tmul_basis_eq_sum (g : GaugeGroupI) (k : Fin 2) (i : Fin 3) (j : Fin 2) : repGaugeGroupI g ⟨LeftHandedWeyl.basis k ⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 3) ℂ i ⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 2) ℂ j⟩ = ∑ i' : Fin 3, ∑ j' : Fin 2, (g.toU1.1 * g.toSU3.1 i' i * g.toSU2.1 j' j) • (⟨LeftHandedWeyl.basis k ⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 3) ℂ i' ⊗ₜ[ℂ] EuclideanSpace.basisFun (Fin 2) ℂ j'⟩ : QuarkDoublet) := by apply valLinEquiv.injective apply (((LeftHandedWeyl.basis).tensorProduct (EuclideanSpace.basisFun (Fin 3) ℂ).toBasis).tensorProduct (EuclideanSpace.basisFun (Fin 2) ℂ).toBasis).repr.injective ext ⟨⟨k, l⟩, m⟩ simp only [EuclideanSpace.basisFun_apply, repGaugeGroupI_tmul, Submonoid.smul_def, valLinEquiv_apply, map_smul, Finsupp.coe_smul, Pi.smul_apply, Module.Basis.tensorProduct_repr_tmul_apply, OrthonormalBasis.coe_toBasis_repr_apply, EuclideanSpace.basisFun_repr, ofLp_toLpLin, PiLp.ofLp_single, toLin'_apply, mulVec_single, MulOpposite.op_one, col_apply, one_smul, Module.Basis.repr_self, smul_eq_mul, map_sum, Finsupp.coe_finsetSum, Finset.sum_apply, PiLp.single_apply, ite_mul, one_mul, zero_mul, mul_ite, mul_zero, Finset.sum_ite_irrel, Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, Finset.sum_const_zero] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Particles/StandardModel/Fermions/QuarkDoublet.lean:143-162
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