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

42
HepLean/Tensors/ComplexLorentz/Basis.lean
View file

@ -81,8 +81,7 @@ lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
- basisVector ![Color.up, Color.up] (fun _ => 2)
- basisVector ![Color.up, Color.up] (fun _ => 3) := by
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Functor.id_obj, Fin.isValue]
Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V]
erw [Lorentz.contrMetric_apply_one, Lorentz.contrMetricVal_expand_tmul]
simp only [Fin.isValue, map_sub]
congr 1
@ -100,6 +99,45 @@ lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
rfl
/-- The expansion of the Pauli matrices `σ^μ^a^{dot a}` in terms of basis vectors. -/
lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 0 | 2 => 0)
+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 1 | 2 => 1)
+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 0 | 2 => 1)
+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 1 | 2 => 0)
- I • basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 0 | 2 => 1)
+ I • basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 1 | 2 => 0)
+ basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 0 | 2 => 0)
- basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constThreeNode_tensor,
Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue]
erw [PauliMatrix.asConsTensor_apply_one, PauliMatrix.asTensor_expand]
simp only [Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, Action.instMonoidalCategory_tensorObj_V,
Fin.isValue, map_sub, map_add, _root_.map_smul]
congr 1
congr 1
congr 1
congr 1
congr 1
congr 1
congr 1
all_goals
erw [tripleIsoSep_tmul, basisVector]
apply congrArg
try apply congrArg
funext i
match i with
| (0 : Fin 3) =>
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue, OverColor.mk_hom,
cons_val_zero, Fin.cases_zero]
change _ = Lorentz.complexContrBasisFin4 _
simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
rfl
| (1 : Fin 3) => rfl
| (2 : Fin 3) => rfl
end complexLorentzTensor
end Fermion
end