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

feat: Some lemmas about Bispinors

Browse source
This commit is contained in:
jstoobysmith 2024-10-25 15:02:40 +00:00
parent f2eaa2ee43
commit 7ea91f459c
2 changed files with 30 additions and 11 deletions
Download patch file Download diff file Expand all files Collapse all files

23
HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean
View file

@ -29,10 +29,31 @@ open Lorentz
def contrBispinorUp (p : complexContr) :=
{p | μ ⊗ pauliCo | μ α β}ᵀ.tensor
lemma tensorNode_contrBispinorUp (p : complexContr) :
(tensorNode (contrBispinorUp p)).tensor = {p | μ ⊗ pauliCo | μ α β}ᵀ.tensor := by
rw [contrBispinorUp, tensorNode_tensor]
/-- A bispinor `pₐₐ` created from a lorentz vector `p^μ`. -/
def contrBispinorDown (p : complexContr) :=
{Fermion.altRightMetric | β β' ⊗ Fermion.altLeftMetric | α α' ⊗
{Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
(contrBispinorUp p) | α β}ᵀ.tensor
/-- Expands the tensor node of `contrBispinorDown` into a tensor tree based on
`contrBispinorUp`. -/
lemma tensorNode_contrBispinorDown (p : complexContr) :
{contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β' ⊗ (contrBispinorUp p) | α β}ᵀ.tensor := by
rw [contrBispinorDown, tensorNode_tensor]
/-- Expansion of a `contrBispinorDown` into the original contravariant tensor nested
between pauli matrices and metrics. -/
lemma contrBispinorDown_full_nested (p : complexContr) :
{contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β' ⊗ (p | μ ⊗ pauliCo | μ α β)}ᵀ.tensor := by
conv =>
lhs
rw [tensorNode_contrBispinorDown]
rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_contrBispinorUp p]
end complexLorentzTensor
end