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Refactor: Metrics as complex lorentz tensors

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jstoobysmith 2024-10-29 13:46:18 +00:00
parent 324f448ec7
commit 34c3643bac
8 changed files with 448 additions and 240 deletions
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28
HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean
View file

@ -10,6 +10,7 @@ import HepLean.Tensors.Tree.NodeIdentities.ContrContr
import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
import HepLean.Tensors.Tree.NodeIdentities.PermContr
import HepLean.Tensors.Tree.NodeIdentities.Congr
import HepLean.Tensors.ComplexLorentz.Metrics.Lemmas
/-!
## Pauli matrices as complex Lorentz tensors
@ -41,15 +42,13 @@ open Fermion
def pauliContr := {PauliMatrix.asConsTensor | ν α β}ᵀ.tensor
/-- The Pauli matrices as the complex Lorentz tensor `σ_μ^α^{dot β}`. -/
def pauliCo := {Lorentz.coMetric | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor
def pauliCo := {η' | μ ν ⊗ pauliContr | ν α β}ᵀ.tensor
/-- The Pauli matrices as the complex Lorentz tensor `σ_μ_α_{dot β}`. -/
def pauliCoDown := {pauliCo | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β'}ᵀ.tensor
def pauliCoDown := {pauliCo | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor
/-- The Pauli matrices as the complex Lorentz tensor `σ^μ_α_{dot β}`. -/
def pauliContrDown := {pauliContr | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β'}ᵀ.tensor
def pauliContrDown := {pauliContr | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor
/-!
@ -64,19 +63,17 @@ lemma tensorNode_pauliContr : {pauliContr | μ α β}ᵀ.tensor =
/-- The definitional tensor node relation for `pauliCo`. -/
lemma tensorNode_pauliCo : {pauliCo | μ α β}ᵀ.tensor =
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
{η' | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
rfl
/-- The definitional tensor node relation for `pauliCoDown`. -/
lemma tensorNode_pauliCoDown : {pauliCoDown | μ α β}ᵀ.tensor =
{pauliCo | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β'}ᵀ.tensor := by
{pauliCo | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor := by
rfl
/-- The definitional tensor node relation for `pauliContrDown`. -/
lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
{pauliContr | μ α β ⊗ Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β'}ᵀ.tensor := by
{pauliContr | μ α β ⊗ εL' | α α' ⊗ εR' | β β'}ᵀ.tensor := by
rfl
/-!
@ -88,8 +85,7 @@ lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
set_option maxRecDepth 5000 in
/-- A rearanging of `pauliCoDown` to place the pauli matrices on the right. -/
lemma pauliCoDown_eq_metric_mul_pauliCo :
{pauliCoDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β' ⊗ pauliCo | μ α β}ᵀ := by
{pauliCoDown | μ α' β' = εL' | α α' ⊗ εR' | β β' ⊗ pauliCo | μ α β}ᵀ := by
conv =>
lhs
rw [tensorNode_pauliCoDown]
@ -138,8 +134,8 @@ lemma pauliCoDown_eq_metric_mul_pauliCo :
set_option maxRecDepth 5000 in
/-- A rearanging of `pauliContrDown` to place the pauli matrices on the right. -/
lemma pauliContrDown_eq_metric_mul_pauliContr :
{pauliContrDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β' ⊗ pauliContr | μ α β}ᵀ := by
{pauliContrDown | μ α' β' = εL' | α α' ⊗
εR' | β β' ⊗ pauliContr | μ α β}ᵀ := by
conv =>
lhs
rw [tensorNode_pauliContrDown]
@ -224,7 +220,7 @@ lemma action_pauliCoDown (g : SL(2,ℂ)) : {g •ₐ pauliCoDown | μ α β}ᵀ.
action_pauliCo _]
rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_snd <|
action_constTwoNode _ _]
rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
erw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
rfl
/-- The tensor `pauliContrDown` is invariant under the action of `SL(2,ℂ)`. -/
@ -241,7 +237,7 @@ lemma action_pauliContrDown (g : SL(2,ℂ)) : {g •ₐ pauliContrDown | μ α
action_pauliContr _]
rw [contr_tensor_eq <| prod_tensor_eq_fst <| contr_tensor_eq <| prod_tensor_eq_snd <|
action_constTwoNode _ _]
rw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
erw [contr_tensor_eq <| prod_tensor_eq_snd <| action_constTwoNode _ _]
rfl
end complexLorentzTensor

3
HepLean/Tensors/ComplexLorentz/PauliMatrices/CoContractContr.lean
View file

@ -517,8 +517,7 @@ lemma pauliCo_contr_pauliContr_expand :
/-- The statement that `η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}`. -/
theorem pauliCo_contr_pauliContr :
{pauliCo | ν α β ⊗ pauliContr | ν α' β' =
2 •ₜ Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ := by
{pauliCo | ν α β ⊗ pauliContr | ν α' β' = 2 •ₜ εL | α α' ⊗ εR | β β'}ᵀ := by
rw [pauliCo_contr_pauliContr_expand]
rw [perm_tensor_eq <| smul_tensor_eq <| leftMetric_mul_rightMetric_tree]
rw [perm_smul]