feat: Weyl fermion contraction, unit, metric
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jstoobysmith
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5 changed files with 1176 additions and 168 deletions
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@ -47,6 +47,18 @@ def leftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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simp only [LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply, LinearEquiv.apply_symm_apply,
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mulVec_mulVec]}
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/-- The standard basis on left-handed Weyl fermions. -/
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def leftBasis : Basis (Fin 2) ℂ leftHanded := Basis.ofEquivFun
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(Equiv.linearEquiv ℂ LeftHandedModule.toFin2ℂFun)
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@[simp]
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lemma leftBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
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(LinearMap.toMatrix leftBasis leftBasis) (leftHanded.ρ M) i j = M.1 i j := by
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rw [LinearMap.toMatrix_apply]
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simp [leftBasis]
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change (M.1 *ᵥ (Pi.single j 1)) i = _
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simp only [mulVec_single, mul_one]
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/-- The vector space ℂ^2 carrying the representation of SL(2,C) given by
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M → (M⁻¹)ᵀ. In index notation corresponds to a Weyl fermion with indices ψ^a. -/
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def altLeftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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@ -70,6 +82,18 @@ def altLeftHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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rw [Matrix.mul_inv_rev]
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exact transpose_mul _ _}
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/-- The standard basis on alt-left-handed Weyl fermions. -/
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def altLeftBasis : Basis (Fin 2) ℂ altLeftHanded := Basis.ofEquivFun
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(Equiv.linearEquiv ℂ AltLeftHandedModule.toFin2ℂFun)
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@[simp]
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lemma altLeftBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
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(LinearMap.toMatrix altLeftBasis altLeftBasis) (altLeftHanded.ρ M) i j = (M.1⁻¹)ᵀ i j := by
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rw [LinearMap.toMatrix_apply]
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simp only [altLeftBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
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change ((M.1⁻¹)ᵀ *ᵥ (Pi.single j 1)) i = _
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simp only [mulVec_single, transpose_apply, mul_one]
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/-- The vector space ℂ^2 carrying the conjugate representation of SL(2,C).
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In index notation corresponds to a Weyl fermion with indices ψ_{dot a}. -/
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def rightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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@ -90,6 +114,18 @@ def rightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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simp only [SpecialLinearGroup.coe_mul, RCLike.star_def, Matrix.map_mul, LinearMap.coe_mk,
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AddHom.coe_mk, LinearMap.mul_apply, LinearEquiv.apply_symm_apply, mulVec_mulVec]}
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/-- The standard basis on right-handed Weyl fermions. -/
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def rightBasis : Basis (Fin 2) ℂ rightHanded := Basis.ofEquivFun
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(Equiv.linearEquiv ℂ RightHandedModule.toFin2ℂFun)
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@[simp]
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lemma rightBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
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(LinearMap.toMatrix rightBasis rightBasis) (rightHanded.ρ M) i j = (M.1.map star) i j := by
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rw [LinearMap.toMatrix_apply]
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simp only [rightBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
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change (M.1.map star *ᵥ (Pi.single j 1)) i = _
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simp only [mulVec_single, transpose_apply, mul_one]
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/-- The vector space ℂ^2 carrying the representation of SL(2,C) given by
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M → (M⁻¹)^†.
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In index notation this corresponds to a Weyl fermion with index `ψ^{dot a}`. -/
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@ -114,6 +150,19 @@ def altRightHanded : Rep ℂ SL(2,ℂ) := Rep.of {
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rw [Matrix.mul_inv_rev]
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exact conjTranspose_mul _ _}
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/-- The standard basis on alt-right-handed Weyl fermions. -/
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def altRightBasis : Basis (Fin 2) ℂ altRightHanded := Basis.ofEquivFun
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(Equiv.linearEquiv ℂ AltRightHandedModule.toFin2ℂFun)
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@[simp]
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lemma altRightBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :
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(LinearMap.toMatrix altRightBasis altRightBasis) (altRightHanded.ρ M) i j =
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((M.1⁻¹).conjTranspose) i j := by
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rw [LinearMap.toMatrix_apply]
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simp only [altRightBasis, Basis.coe_ofEquivFun, Basis.ofEquivFun_repr_apply, transpose_apply]
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change ((M.1⁻¹).conjTranspose *ᵥ (Pi.single j 1)) i = _
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simp only [mulVec_single, transpose_apply, mul_one]
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/-!
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## Equivalences between Weyl fermion vector spaces.
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@ -224,174 +273,6 @@ informal_lemma rightHandedWeylAltEquiv_equivariant where
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action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
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deps :≈ [``rightHandedWeylAltEquiv]
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/-!
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## Contraction of Weyl fermions.
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-/
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open CategoryTheory.MonoidalCategory
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/-- The bi-linear map corresponding to contraction of a left-handed Weyl fermion with a
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alt-left-handed Weyl fermion. -/
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def leftAltBi : leftHanded →ₗ[ℂ] altLeftHanded →ₗ[ℂ] ℂ where
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toFun ψ := {
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toFun := fun φ => ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ,
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map_add' := by
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intro φ φ'
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simp only [map_add]
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rw [dotProduct_add]
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map_smul' := by
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intro r φ
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simp only [LinearEquiv.map_smul]
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rw [dotProduct_smul]
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rfl}
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map_add' ψ ψ':= by
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refine LinearMap.ext (fun φ => ?_)
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simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
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rw [add_dotProduct]
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map_smul' r ψ := by
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refine LinearMap.ext (fun φ => ?_)
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simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
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rw [smul_dotProduct]
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rfl
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/-- The bi-linear map corresponding to contraction of a alt-left-handed Weyl fermion with a
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left-handed Weyl fermion. -/
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def altLeftBi : altLeftHanded →ₗ[ℂ] leftHanded →ₗ[ℂ] ℂ where
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toFun ψ := {
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toFun := fun φ => ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ,
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map_add' := by
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intro φ φ'
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simp only [map_add]
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rw [dotProduct_add]
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map_smul' := by
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intro r φ
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simp only [LinearEquiv.map_smul]
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rw [dotProduct_smul]
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rfl}
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map_add' ψ ψ':= by
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refine LinearMap.ext (fun φ => ?_)
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simp only [map_add, add_dotProduct, vec2_dotProduct, Fin.isValue, LinearMap.coe_mk,
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AddHom.coe_mk, LinearMap.add_apply]
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map_smul' ψ ψ' := by
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refine LinearMap.ext (fun φ => ?_)
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simp only [_root_.map_smul, smul_dotProduct, vec2_dotProduct, Fin.isValue, smul_eq_mul,
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LinearMap.coe_mk, AddHom.coe_mk, RingHom.id_apply, LinearMap.smul_apply]
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/-- The linear map from leftHandedWeyl ⊗ altLeftHandedWeyl to ℂ given by
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summing over components of leftHandedWeyl and altLeftHandedWeyl in the
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standard basis (i.e. the dot product).
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Physically, the contraction of a left-handed Weyl fermion with a alt-left-handed Weyl fermion.
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In index notation this is ψ_a φ^a. -/
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def leftAltContraction : leftHanded ⊗ altLeftHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
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hom := TensorProduct.lift leftAltBi
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comm M := by
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apply TensorProduct.ext'
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intro ψ φ
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change (M.1 *ᵥ ψ.toFin2ℂ) ⬝ᵥ (M.1⁻¹ᵀ *ᵥ φ.toFin2ℂ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ
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rw [dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
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simp
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lemma leftAltContraction_hom_tmul (ψ : leftHanded) (φ : altLeftHanded) :
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leftAltContraction.hom (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ := by
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rw [leftAltContraction]
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erw [TensorProduct.lift.tmul]
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rfl
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/-- The linear map from altLeftHandedWeyl ⊗ leftHandedWeyl to ℂ given by
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summing over components of altLeftHandedWeyl and leftHandedWeyl in the
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standard basis (i.e. the dot product).
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Physically, the contraction of a alt-left-handed Weyl fermion with a left-handed Weyl fermion.
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In index notation this is φ^a ψ_a. -/
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def altLeftContraction : altLeftHanded ⊗ leftHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
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hom := TensorProduct.lift altLeftBi
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comm M := by
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apply TensorProduct.ext'
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intro φ ψ
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change (M.1⁻¹ᵀ *ᵥ φ.toFin2ℂ) ⬝ᵥ (M.1 *ᵥ ψ.toFin2ℂ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ
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rw [dotProduct_mulVec, mulVec_transpose, vecMul_vecMul]
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simp
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lemma altLeftContraction_hom_tmul (φ : altLeftHanded) (ψ : leftHanded) :
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altLeftContraction.hom (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ := by
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rw [altLeftContraction]
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erw [TensorProduct.lift.tmul]
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rfl
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lemma leftAltContraction_apply_symm (ψ : leftHanded) (φ : altLeftHanded) :
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leftAltContraction.hom (ψ ⊗ₜ φ) = altLeftContraction.hom (φ ⊗ₜ ψ) := by
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rw [altLeftContraction_hom_tmul, leftAltContraction_hom_tmul]
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exact dotProduct_comm ψ.toFin2ℂ φ.toFin2ℂ
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/-- A manifestation of the statement that `ψ ψ' = - ψ' ψ` where `ψ` and `ψ'`
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are `leftHandedWeyl`. -/
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lemma leftAltContraction_apply_leftHandedAltEquiv (ψ ψ' : leftHanded) :
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leftAltContraction.hom (ψ ⊗ₜ leftHandedAltEquiv.hom.hom ψ') =
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- leftAltContraction.hom (ψ' ⊗ₜ leftHandedAltEquiv.hom.hom ψ) := by
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rw [leftAltContraction_hom_tmul, leftAltContraction_hom_tmul,
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leftHandedAltEquiv_hom_hom_apply, leftHandedAltEquiv_hom_hom_apply]
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simp only [CategoryTheory.Monoidal.transportStruct_tensorUnit,
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CategoryTheory.Equivalence.symm_functor, Action.functorCategoryEquivalence_inverse,
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Action.FunctorCategoryEquivalence.inverse_obj_V, CategoryTheory.Monoidal.tensorUnit_obj,
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cons_mulVec, cons_dotProduct, zero_mul, one_mul, dotProduct_empty, add_zero, zero_add, neg_mul,
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empty_mulVec, LinearEquiv.apply_symm_apply, dotProduct_cons, mul_neg, neg_add_rev, neg_neg]
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ring
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/-- A manifestation of the statement that `φ φ' = - φ' φ` where `φ` and `φ'` are
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`altLeftHandedWeyl`. -/
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lemma leftAltContraction_apply_leftHandedAltEquiv_inv (φ φ' : altLeftHanded) :
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leftAltContraction.hom (leftHandedAltEquiv.inv.hom φ ⊗ₜ φ') =
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- leftAltContraction.hom (leftHandedAltEquiv.inv.hom φ' ⊗ₜ φ) := by
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rw [leftAltContraction_hom_tmul, leftAltContraction_hom_tmul,
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leftHandedAltEquiv_inv_hom_apply, leftHandedAltEquiv_inv_hom_apply]
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simp only [CategoryTheory.Monoidal.transportStruct_tensorUnit,
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CategoryTheory.Equivalence.symm_functor, Action.functorCategoryEquivalence_inverse,
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Action.FunctorCategoryEquivalence.inverse_obj_V, CategoryTheory.Monoidal.tensorUnit_obj,
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cons_mulVec, cons_dotProduct, zero_mul, neg_mul, one_mul, dotProduct_empty, add_zero, zero_add,
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empty_mulVec, LinearEquiv.apply_symm_apply, neg_add_rev, neg_neg]
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ring
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informal_lemma leftAltWeylContraction_symm_altLeftWeylContraction where
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math :≈ "The linear map altLeftWeylContraction is leftAltWeylContraction composed
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with the braiding of the tensor product."
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deps :≈ [``leftAltContraction, ``altLeftContraction]
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informal_lemma altLeftWeylContraction_invariant where
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math :≈ "The contraction altLeftWeylContraction is invariant with respect to
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the action of SL(2,C) on leftHandedWeyl and altLeftHandedWeyl."
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deps :≈ [``altLeftContraction]
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informal_definition rightAltWeylContraction where
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math :≈ "The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
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summing over components of rightHandedWeyl and altRightHandedWeyl in the
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standard basis (i.e. the dot product)."
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physics :≈ "The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
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In index notation this is ψ_{dot a} φ^{dot a}."
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deps :≈ [``rightHanded, ``altRightHanded]
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informal_lemma rightAltWeylContraction_invariant where
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math :≈ "The contraction rightAltWeylContraction is invariant with respect to
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the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
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deps :≈ [``rightAltWeylContraction]
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informal_definition altRightWeylContraction where
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math :≈ "The linear map from altRightHandedWeyl ⊗ rightHandedWeyl to ℂ given by
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summing over components of altRightHandedWeyl and rightHandedWeyl in the
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standard basis (i.e. the dot product)."
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physics :≈ "The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
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In index notation this is φ^{dot a} ψ_{dot a}."
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deps :≈ [``rightHanded, ``altRightHanded]
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informal_lemma rightAltWeylContraction_symm_altRightWeylContraction where
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math :≈ "The linear map altRightWeylContraction is rightAltWeylContraction composed
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with the braiding of the tensor product."
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deps :≈ [``rightAltWeylContraction, ``altRightWeylContraction]
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informal_lemma altRightWeylContraction_invariant where
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math :≈ "The contraction altRightWeylContraction is invariant with respect to
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the action of SL(2,C) on rightHandedWeyl and altRightHandedWeyl."
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deps :≈ [``altRightWeylContraction]
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end
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end Fermion
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