DibyashanuPati/PhysLean
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

feat: Add expansion of Pauli matrices as basis vect

Browse source
This commit is contained in:
jstoobysmith 2024-10-23 06:50:55 +00:00
parent 1148234929
commit 74d4b2c2c0
4 changed files with 162 additions and 4 deletions
Download patch file Download diff file Expand all files Collapse all files

75
HepLean/SpaceTime/PauliMatrices/AsTensor.lean
View file

@ -35,6 +35,76 @@ open SpaceTime
def asTensor : (complexContr ⊗ leftHanded ⊗ rightHanded).V :=
∑ i, complexContrBasis i ⊗ₜ leftRightToMatrix.symm (σSA i)
/-- The expansion of `asTensor` into complexContrBasis basis vectors . -/
lemma asTensor_expand_complexContrBasis : asTensor =
complexContrBasis (Sum.inl 0) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inl 0))
+ complexContrBasis (Sum.inr 0) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inr 0))
+ complexContrBasis (Sum.inr 1) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inr 1))
+ complexContrBasis (Sum.inr 2) ⊗ₜ leftRightToMatrix.symm (σSA (Sum.inr 2)) := by
simp only [Action.instMonoidalCategory_tensorObj_V, asTensor,
CategoryTheory.Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, Fintype.sum_sum_type, Finset.univ_unique,
Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, Fin.sum_univ_three]
rfl
/-- The expansion of the pauli matrix `σ₀` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inl_0_expand : leftRightToMatrix.symm (σSA (Sum.inl 0)) =
leftBasis 0 ⊗ₜ rightBasis 0 + leftBasis 1 ⊗ₜ rightBasis 1 := by
simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
erw [leftRightToMatrix_symm_expand_tmul]
simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ0, of_apply, cons_val', empty_val',
cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, one_smul, zero_smul,
add_zero, head_fin_const, zero_add, CategoryTheory.Equivalence.symm_inverse,
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]
/-- The expansion of the pauli matrix `σ₁` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inr_0_expand : leftRightToMatrix.symm (σSA (Sum.inr 0)) =
leftBasis 0 ⊗ₜ rightBasis 1 + leftBasis 1 ⊗ₜ rightBasis 0:= by
simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
erw [leftRightToMatrix_symm_expand_tmul]
simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ1, of_apply, cons_val', empty_val',
cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, zero_smul, one_smul,
zero_add, head_fin_const, add_zero, CategoryTheory.Equivalence.symm_inverse,
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]
/-- The expansion of the pauli matrix `σ₂` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inr_1_expand : leftRightToMatrix.symm (σSA (Sum.inr 1)) =
-(I • leftBasis 0 ⊗ₜ[ℂ] rightBasis 1) + I • leftBasis 1 ⊗ₜ[ℂ] rightBasis 0 := by
simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
erw [leftRightToMatrix_symm_expand_tmul]
simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ2, of_apply, cons_val', empty_val',
cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, zero_smul, neg_smul,
zero_add, head_fin_const, add_zero]
/-- The expansion of the pauli matrix `σ₃` in terms of a basis of tensor product vectors. -/
lemma leftRightToMatrix_σSA_inr_2_expand : leftRightToMatrix.symm (σSA (Sum.inr 2)) =
leftBasis 0 ⊗ₜ rightBasis 0 - leftBasis 1 ⊗ₜ rightBasis 1 := by
simp only [Action.instMonoidalCategory_tensorObj_V, Fin.isValue]
erw [leftRightToMatrix_symm_expand_tmul]
simp only [σSA, Fin.isValue, Basis.coe_mk, σSA', σ3, of_apply, cons_val', empty_val',
cons_val_fin_one, Fin.sum_univ_two, cons_val_zero, cons_val_one, head_cons, one_smul, zero_smul,
add_zero, head_fin_const, neg_smul, zero_add, CategoryTheory.Equivalence.symm_inverse,
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]
rfl
/-- The expansion of `asTensor` into complexContrBasis basis of tensor product vectors. -/
lemma asTensor_expand : asTensor =
complexContrBasis (Sum.inl 0) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 0)
+ complexContrBasis (Sum.inl 0) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 1)
+ complexContrBasis (Sum.inr 0) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 1)
+ complexContrBasis (Sum.inr 0) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 0)
- I • complexContrBasis (Sum.inr 1) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 1)
+ I • complexContrBasis (Sum.inr 1) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 0)
+ complexContrBasis (Sum.inr 2) ⊗ₜ (leftBasis 0 ⊗ₜ rightBasis 0)
- complexContrBasis (Sum.inr 2) ⊗ₜ (leftBasis 1 ⊗ₜ rightBasis 1) := by
rw [asTensor_expand_complexContrBasis]
rw [leftRightToMatrix_σSA_inl_0_expand, leftRightToMatrix_σSA_inr_0_expand,
leftRightToMatrix_σSA_inr_1_expand, leftRightToMatrix_σSA_inr_2_expand]
simp only [Action.instMonoidalCategory_tensorObj_V, CategoryTheory.Equivalence.symm_inverse,
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
Fin.isValue, tmul_add, tmul_neg, tmul_smul, tmul_sub]
rfl
/-- The tensor `σ^μ^a^{dot a}` based on the Pauli-matrices as a morphism,
`𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ leftHanded ⊗ rightHanded` manifesting
the invariance under the `SL(2,ℂ)` action. -/
@ -124,5 +194,10 @@ def asConsTensor : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ leftHanded ⊗
CategoryTheory.Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj]
lemma asConsTensor_apply_one : asConsTensor.hom (1 : ℂ) = asTensor := by
change asConsTensor.hom.toFun (1 : ℂ) = asTensor
simp only [Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
asConsTensor, AddHom.toFun_eq_coe, AddHom.coe_mk, one_smul]
end
end PauliMatrix