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feat: Add Pauli-matrices as tensor.

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This commit is contained in:
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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8
HepLean/SpaceTime/WeylFermion/Contraction.lean
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@ -125,7 +125,7 @@ def altRightBi : altRightHanded →ₗ[ℂ] rightHanded →ₗ[ℂ] ℂ where
summing over components of leftHandedWeyl and altLeftHandedWeyl in the
standard basis (i.e. the dot product).
Physically, the contraction of a left-handed Weyl fermion with a alt-left-handed Weyl fermion.
In index notation this is ψ_a φ^a. -/
In index notation this is ψ^a φ_a. -/
def leftAltContraction : leftHanded ⊗ altLeftHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift leftAltBi
comm M := TensorProduct.ext' fun ψ φ => by
@ -143,7 +143,7 @@ lemma leftAltContraction_hom_tmul (ψ : leftHanded) (φ : altLeftHanded) :
summing over components of altLeftHandedWeyl and leftHandedWeyl in the
standard basis (i.e. the dot product).
Physically, the contraction of a alt-left-handed Weyl fermion with a left-handed Weyl fermion.
In index notation this is φ^a ψ_a. -/
In index notation this is φ_a ψ^a. -/
def altLeftContraction : altLeftHanded ⊗ leftHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift altLeftBi
comm M := TensorProduct.ext' fun φ ψ => by
@ -162,7 +162,7 @@ The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
summing over components of rightHandedWeyl and altRightHandedWeyl in the
standard basis (i.e. the dot product).
The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
In index notation this is ψ_{dot a} φ^{dot a}.
In index notation this is ψ^{dot a} φ_{dot a}.
-/
def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift rightAltBi
@ -187,7 +187,7 @@ def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2
summing over components of altRightHandedWeyl and rightHandedWeyl in the
standard basis (i.e. the dot product).
The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
In index notation this is φ^{dot a} ψ_{dot a}.
In index notation this is φ_{dot a} ψ^{dot a}.
-/
def altRightContraction : altRightHanded ⊗ rightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
hom := TensorProduct.lift altRightBi