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feat: Add Pauli-matrices as tensor.

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jstoobysmith 2024-10-16 10:39:11 +00:00
parent a60ade65f0
commit 691b7e112e
13 changed files with 989 additions and 154 deletions
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59
HepLean/SpaceTime/SL2C/Basic.lean
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@ -43,6 +43,7 @@ lemma inverse_coe (M : SL(2, ℂ)) : M.1⁻¹ = (M⁻¹).1 := by
· simp
· simp
lemma transpose_coe (M : SL(2, ℂ)) : M.1ᵀ = (M.transpose).1 := rfl
/-!
## Representation of SL(2, ℂ) on spacetime
@ -133,6 +134,64 @@ def toLorentzGroup : SL(2, ℂ) →* LorentzGroup 3 where
simp only [toLorentzGroupElem, _root_.map_mul, LinearMap.toMatrix_mul,
lorentzGroupIsGroup_mul_coe]
lemma toLorentzGroup_eq_σSAL (M : SL(2, ℂ)) :
toLorentzGroup M = LinearMap.toMatrix
PauliMatrix.σSAL PauliMatrix.σSAL (repSelfAdjointMatrix M) := by
rfl
lemma toLorentzGroup_eq_stdBasis (M : SL(2, ℂ)) :
toLorentzGroup M = LinearMap.toMatrix LorentzVector.stdBasis LorentzVector.stdBasis
(repLorentzVector M) := by rfl
lemma repLorentzVector_apply_eq_mulVec (v : LorentzVector 3) :
SL2C.repLorentzVector M v = (SL2C.toLorentzGroup M).1 *ᵥ v := by
simp [toLorentzGroup]
have hv : v = (Finsupp.linearEquivFunOnFinite ℝ ℝ (Fin 1 ⊕ Fin 3)) (LorentzVector.stdBasis.repr v) := by rfl
nth_rewrite 2 [hv]
change _ = toLorentzGroup M *ᵥ (LorentzVector.stdBasis.repr v)
rw [toLorentzGroup_eq_stdBasis, LinearMap.toMatrix_mulVec_repr]
rfl
lemma repSelfAdjointMatrix_basis (i : Fin 1 ⊕ Fin 3) :
SL2C.repSelfAdjointMatrix M (PauliMatrix.σSAL i) =
∑ j, (toLorentzGroup M).1 j i •
PauliMatrix.σSAL j := by
rw [toLorentzGroup_eq_σSAL]
simp only [LinearMap.toMatrix_apply, Finset.univ_unique,
Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton]
nth_rewrite 1 [← (Basis.sum_repr PauliMatrix.σSAL ((repSelfAdjointMatrix M) (PauliMatrix.σSAL i)))]
congr
lemma repSelfAdjointMatrix_σSA (i : Fin 1 ⊕ Fin 3) :
SL2C.repSelfAdjointMatrix M (PauliMatrix.σSA i) =
∑ j, (toLorentzGroup M⁻¹).1 i j • PauliMatrix.σSA j := by
have h1 : (toLorentzGroup M⁻¹).1 = minkowskiMetric.dual (toLorentzGroup M).1 := by
simp
simp only [h1]
rw [PauliMatrix.σSA_minkowskiMetric_σSAL, _root_.map_smul]
rw [repSelfAdjointMatrix_basis]
rw [Finset.smul_sum]
apply congrArg
funext j
rw [smul_smul, PauliMatrix.σSA_minkowskiMetric_σSAL, smul_smul]
apply congrFun
apply congrArg
exact Eq.symm (minkowskiMetric.dual_apply_minkowskiMatrix ((toLorentzGroup M).1) i j)
lemma repLorentzVector_stdBasis (i : Fin 1 ⊕ Fin 3) :
SL2C.repLorentzVector M (LorentzVector.stdBasis i) =
∑ j, (toLorentzGroup M).1 j i • LorentzVector.stdBasis j := by
simp only [repLorentzVector, MonoidHom.coe_mk, OneHom.coe_mk, LinearMap.coe_comp,
LinearEquiv.coe_coe, Function.comp_apply]
rw [toSelfAdjointMatrix_stdBasis]
rw [repSelfAdjointMatrix_basis]
rw [map_sum]
apply congrArg
funext j
simp
/-!
## Homomorphism to the restricted Lorentz group