105 lines
3.6 KiB
Text
105 lines
3.6 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.Tensors.Tree.Elab
|
||
import HepLean.Tensors.ComplexLorentz.Basic
|
||
import Mathlib.LinearAlgebra.TensorProduct.Basis
|
||
import HepLean.Tensors.Tree.NodeIdentities.Basic
|
||
import HepLean.Tensors.Tree.NodeIdentities.PermProd
|
||
import HepLean.Tensors.Tree.NodeIdentities.PermContr
|
||
import HepLean.Tensors.Tree.NodeIdentities.ProdComm
|
||
import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
|
||
import HepLean.Tensors.Tree.NodeIdentities.ContrContr
|
||
/-!
|
||
|
||
## Basis vectors associated with complex Lorentz tensors
|
||
|
||
Note that this file will be much improved once:
|
||
https://github.com/leanprover-community/mathlib4/pull/11156
|
||
is merged.
|
||
|
||
-/
|
||
open IndexNotation
|
||
open CategoryTheory
|
||
open MonoidalCategory
|
||
open Matrix
|
||
open MatrixGroups
|
||
open Complex
|
||
open TensorProduct
|
||
open IndexNotation
|
||
open CategoryTheory
|
||
open TensorTree
|
||
open OverColor.Discrete
|
||
noncomputable section
|
||
namespace Fermion
|
||
namespace complexLorentzTensor
|
||
|
||
/-- Basis vectors for complex Lorentz tensors. -/
|
||
def basisVector {n : ℕ} (c : Fin n → complexLorentzTensor.C)
|
||
(b : Π j, Fin (complexLorentzTensor.repDim (c j))) :
|
||
complexLorentzTensor.F.obj (OverColor.mk c) :=
|
||
PiTensorProduct.tprod ℂ (fun i => complexLorentzTensor.basis (c i) (b i))
|
||
|
||
/-!
|
||
|
||
## Useful expansions.
|
||
|
||
-/
|
||
|
||
/-- The expansion of the Lorentz covariant metric in terms of basis vectors. -/
|
||
lemma coMetric_basis_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
|
||
basisVector ![Color.down, Color.down] (fun _ => 0)
|
||
- basisVector ![Color.down, Color.down] (fun _ => 1)
|
||
- basisVector ![Color.down, Color.down] (fun _ => 2)
|
||
- basisVector ![Color.down, Color.down] (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]
|
||
erw [Lorentz.coMetric_apply_one, Lorentz.coMetricVal_expand_tmul]
|
||
simp only [Fin.isValue, map_sub]
|
||
congr 1
|
||
congr 1
|
||
congr 1
|
||
all_goals
|
||
erw [pairIsoSep_tmul, basisVector]
|
||
apply congrArg
|
||
funext i
|
||
fin_cases i
|
||
all_goals
|
||
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.complexCoBasisFin4 _
|
||
simp only [Fin.isValue, Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
|
||
rfl
|
||
|
||
/-- The expansion of the Lorentz contrvariant metric in terms of basis vectors. -/
|
||
lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
|
||
basisVector ![Color.up, Color.up] (fun _ => 0)
|
||
- basisVector ![Color.up, Color.up] (fun _ => 1)
|
||
- 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]
|
||
erw [Lorentz.contrMetric_apply_one, Lorentz.contrMetricVal_expand_tmul]
|
||
simp only [Fin.isValue, map_sub]
|
||
congr 1
|
||
congr 1
|
||
congr 1
|
||
all_goals
|
||
erw [pairIsoSep_tmul, basisVector]
|
||
apply congrArg
|
||
funext i
|
||
fin_cases i
|
||
all_goals
|
||
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
|
||
|
||
end complexLorentzTensor
|
||
end Fermion
|
||
end
|