feat: Expansion of metrics in terms of basis
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jstoobysmith
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b70e9bd005
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ca55da6a34
3 changed files with 154 additions and 6 deletions
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@ -70,6 +70,15 @@ lemma metricRaw_comm_star (M : SL(2,ℂ)) : metricRaw * M.1.map star = ((M.1)⁻
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def leftMetricVal : (leftHanded ⊗ leftHanded).V :=
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leftLeftToMatrix.symm (- metricRaw)
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/-- Expansion of `leftMetricVal` into the left basis. -/
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lemma leftMetricVal_expand_tmul : leftMetricVal =
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- leftBasis 0 ⊗ₜ[ℂ] leftBasis 1 + leftBasis 1 ⊗ₜ[ℂ] leftBasis 0 := by
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simp only [Action.instMonoidalCategory_tensorObj_V, leftMetricVal, Fin.isValue]
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erw [leftLeftToMatrix_symm_expand_tmul]
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simp only [metricRaw, neg_apply, of_apply, cons_val', empty_val', cons_val_fin_one, neg_smul,
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Finset.sum_neg_distrib, Fin.sum_univ_two, Fin.isValue, cons_val_zero, cons_val_one, head_cons,
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neg_add_rev, one_smul, zero_smul, neg_zero, add_zero, head_fin_const, neg_neg, zero_add]
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/-- The metric `εₐₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ leftHanded`,
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making manifest its invariance under the action of `SL(2,ℂ)`. -/
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def leftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ leftHanded where
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@ -99,10 +108,25 @@ def leftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ leftHanded where
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simp only [SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq, one_ne_zero,
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not_false_eq_true, mul_nonsing_inv, transpose_one, mul_one]
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lemma leftMetric_apply_one : leftMetric.hom (1 : ℂ) = leftMetricVal := by
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change leftMetric.hom.toFun (1 : ℂ) = leftMetricVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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leftMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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/-- The metric `εᵃᵃ` as an element of `(altLeftHanded ⊗ altLeftHanded).V`. -/
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def altLeftMetricVal : (altLeftHanded ⊗ altLeftHanded).V :=
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altLeftaltLeftToMatrix.symm metricRaw
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/-- Expansion of `altLeftMetricVal` into the left basis. -/
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lemma altLeftMetricVal_expand_tmul : altLeftMetricVal =
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altLeftBasis 0 ⊗ₜ[ℂ] altLeftBasis 1 - altLeftBasis 1 ⊗ₜ[ℂ] altLeftBasis 0 := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altLeftMetricVal, Fin.isValue]
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erw [altLeftaltLeftToMatrix_symm_expand_tmul]
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simp only [metricRaw, neg_apply, of_apply, cons_val', empty_val', cons_val_fin_one, neg_smul,
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Finset.sum_neg_distrib, Fin.sum_univ_two, Fin.isValue, cons_val_zero, cons_val_one, head_cons,
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neg_add_rev, one_smul, zero_smul, neg_zero, add_zero, head_fin_const, neg_neg, zero_add]
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rfl
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/-- The metric `εᵃᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHanded`,
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making manifest its invariance under the action of `SL(2,ℂ)`. -/
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def altLeftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHanded where
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@ -132,10 +156,24 @@ def altLeftMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altLeftHanded ⊗ altLeftHande
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simp only [SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq, one_ne_zero,
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not_false_eq_true, mul_nonsing_inv, mul_one]
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lemma altLeftMetric_apply_one : altLeftMetric.hom (1 : ℂ) = altLeftMetricVal := by
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change altLeftMetric.hom.toFun (1 : ℂ) = altLeftMetricVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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altLeftMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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/-- The metric `ε_{dot a}_{dot a}` as an element of `(rightHanded ⊗ rightHanded).V`. -/
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def rightMetricVal : (rightHanded ⊗ rightHanded).V :=
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rightRightToMatrix.symm (- metricRaw)
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/-- Expansion of `rightMetricVal` into the left basis. -/
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lemma rightMetricVal_expand_tmul : rightMetricVal =
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- rightBasis 0 ⊗ₜ[ℂ] rightBasis 1 + rightBasis 1 ⊗ₜ[ℂ] rightBasis 0 := by
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simp only [Action.instMonoidalCategory_tensorObj_V, rightMetricVal, Fin.isValue]
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erw [rightRightToMatrix_symm_expand_tmul]
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simp only [metricRaw, neg_apply, of_apply, cons_val', empty_val', cons_val_fin_one, neg_smul,
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Finset.sum_neg_distrib, Fin.sum_univ_two, Fin.isValue, cons_val_zero, cons_val_one, head_cons,
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neg_add_rev, one_smul, zero_smul, neg_zero, add_zero, head_fin_const, neg_neg, zero_add]
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/-- The metric `ε_{dot a}_{dot a}` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ rightHanded`,
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making manifest its invariance under the action of `SL(2,ℂ)`. -/
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def rightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ rightHanded where
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@ -173,10 +211,25 @@ def rightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ rightHanded wher
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· rw [← rightRightToMatrix_ρ_symm metricRaw M]
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rfl
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lemma rightMetric_apply_one : rightMetric.hom (1 : ℂ) = rightMetricVal := by
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change rightMetric.hom.toFun (1 : ℂ) = rightMetricVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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rightMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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/-- The metric `ε^{dot a}^{dot a}` as an element of `(altRightHanded ⊗ altRightHanded).V`. -/
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def altRightMetricVal : (altRightHanded ⊗ altRightHanded).V :=
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altRightAltRightToMatrix.symm (metricRaw)
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/-- Expansion of `rightMetricVal` into the left basis. -/
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lemma altRightMetricVal_expand_tmul : altRightMetricVal =
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altRightBasis 0 ⊗ₜ[ℂ] altRightBasis 1 - altRightBasis 1 ⊗ₜ[ℂ] altRightBasis 0 := by
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simp only [Action.instMonoidalCategory_tensorObj_V, altRightMetricVal, Fin.isValue]
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erw [altRightAltRightToMatrix_symm_expand_tmul]
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simp only [metricRaw, neg_apply, of_apply, cons_val', empty_val', cons_val_fin_one, neg_smul,
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Finset.sum_neg_distrib, Fin.sum_univ_two, Fin.isValue, cons_val_zero, cons_val_one, head_cons,
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neg_add_rev, one_smul, zero_smul, neg_zero, add_zero, head_fin_const, neg_neg, zero_add]
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rfl
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/-- The metric `ε^{dot a}^{dot a}` as a morphism
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`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ altRightHanded`,
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making manifest its invariance under the action of `SL(2,ℂ)`. -/
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@ -217,5 +270,9 @@ def altRightMetric : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ altRightHanded ⊗ altRightHa
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· rw [← altRightAltRightToMatrix_ρ_symm metricRaw M]
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rfl
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lemma altRightMetric_apply_one : altRightMetric.hom (1 : ℂ) = altRightMetricVal := by
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change altRightMetric.hom.toFun (1 : ℂ) = altRightMetricVal
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simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
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altRightMetric, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
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end
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end Fermion
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